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5Inductive Reasoning
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Learning Objectives After reading this chapter, you should be able to:
1. Define key terms and concepts in inductive logic, including strength and cogency.
2. Differentiate between strong inductive arguments and weak inductive arguments.
3. Identify general methods for strengthening inductive arguments.
4. Identify statistical syllogisms and describe how they can be strong or weak.
5. Evaluate the strength of inductive generalizations.
6. Differentiate between causal and correlational relationships and describe various types of causes.
7. Use Mill’s methods to evaluate causal arguments.
8. Recognize arguments from authority and evaluate their quality.
9. Identify key features of arguments from analogy and use them to evaluate the strength of such arguments.
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Section 5.1 Basic Concepts in Inductive Reasoning
When talking about logic, people often think about formal deductive reasoning. However, most of the arguments we encounter in life are not deductive at all. They do not intend to establish the truth of the conclusion beyond any possible doubt; they simply try to provide good evidence for the truth of their conclusions. Arguments that intend to reason in this way are called inductive arguments. Inductive arguments are not any worse than deductive ones. Often the best evidence available is not final or conclusive but can still be very good.
For example, to infer that the sun will rise tomorrow because it has every day in the past is inductive reasoning. The inference, however, is very strongly supported. Not all inductive arguments are as strong as that one. This chapter will explore different types of inductive arguments and some principles we can use to determine whether they are strong or weak. The chapter will also discuss some specific methods that we can use to try to make good inferences about causation. The goal of this chapter is to enable you to identify inductive arguments, evaluate their strength, and create strong inductive arguments about impor- tant issues.
5.1 Basic Concepts in Inductive Reasoning Inductive is a technical term in logic: It has a precise definition, and that definition may be different from the definition used in other fields or in everyday conversation. An inductive argument is one in which the premises provide support for the conclusions but fall short of establishing complete certainty. If you stop to think about arguments you have encountered recently, you will probably find that most of them are inductive. We are seldom in a position to prove something absolutely, even when we have very good reasons for believing it.
Take, for example, the following argument:
The odds of a given lottery ticket being the winning ticket are extremely low. You just bought a lottery ticket. Therefore, your lottery ticket is probably not the winning ticket.
If the odds of each ticket winning are 1 in millions, then this argument gives very good evi- dence for the truth of its conclusion. However, the argument is not deductively valid. Even if its premises are true, its conclusion is still not absolutely certain. This means that there is still a remote possibility that you bought the winning ticket.
Chapter 3 discussed how an argument is valid if our premises guarantee the truth of the con- clusion. In the case of the lottery, even our best evidence cannot be used to make a valid argu- ment for the conclusion. The given reasons do not guarantee that you will not win; they just make it very likely that you will not win.
This argument, however, helps us establish the likelihood of its conclusion. If it were not for this type of reasoning, we might spend all our money on lottery tickets. We would also not be able to know whether we should do such things as drive our car because we would not be able
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Section 5.1 Basic Concepts in Inductive Reasoning
to reason about the likelihood of getting into a crash on the way to the store. Therefore, this and other types of inductive reasoning are essential in daily life. Consequently, it is important that we learn how to evaluate their strength.
Inductive Strength Some inductive arguments can be bet- ter or worse than others, depending on how well their premises increase the likelihood of the truth of their con- clusion. Some arguments make their conclusions only a little more likely; other arguments make their conclu- sions a lot more likely. Arguments that greatly increase the likelihood of their conclusions are called strong argu- ments; those that do not substantially increase the likelihood are called weak arguments.
Here is an example of an argument that could be considered very strong:
A random fan from the crowd is going to race (in a 100 meter dash) against Usain Bolt. Usain Bolt is the fastest sprinter of all time. Therefore, the fan is going to lose.
It is certainly possible that the fan could win—say, for example, if Usain Bolt breaks an ankle— but it seems highly unlikely. This next argument, however, could be considered weak:
I just scratched off two lottery tickets and won $2 each time. Therefore, I will win $2 on the next ticket, too.
The previous lottery tickets would have no bearing on the likelihood of winning on the next one. Now this next argument’s strength might be somewhere in between:
The Bears have beaten the Lions the last four times they have played. The Bears have a much better record than the Lions this season. Therefore, the Bears will beat the Lions again tomorrow.
This sounds like good evidence, but upsets happen all the time in sports, so its strength is only moderate.
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Weather forecasters use inductive reasoning when giving their predictions. They have tools at their disposal that provide support for their arguments, but some arguments are weaker than others.
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Section 5.1 Basic Concepts in Inductive Reasoning
Considering the Context It is important to realize that inductive strength and weakness are relative terms. As such, they are like the terms tall and short. A person who is short in one context may be tall in another. At 6’0”, professional basketball player Allen Iverson was considered short in the National Bas- ketball Association. But outside of basketball, someone of his height might be considered tall. Similarly, an argument that is strong in one context may be considered weak in another. You
would probably be reasonably happy if you could reliably predict sports (or lottery) results at an accuracy rate of 70%, but researchers in the social sciences typically aim for certainty upward of 90%. In high-energy physics, the goal is a result that is supported at the level of 5 sigma—a probability of more than 99.99997%!
The same is true when it comes to legal arguments. A case tried in a civil court needs to be shown to be true with a preponderance of evidence, which is much less stringent than in a criminal case, in which the defendant must be proved guilty beyond rea- sonable doubt. Therefore, whether the argument is strong or weak is a matter of context.
Moreover, some subjects have the sort of evi- dence that allows for extremely strong arguments, whereas others do not. A psychologist trying to pre- dict human behavior is unlikely to have the same strength of argument as an astronomer trying to predict the path of a comet. These are important things to keep in mind when it comes to evaluating inductive strength.
Strengthening Inductive Arguments Regardless of the subject matter of an argument, we generally want to create the strongest arguments we can. In general, there are two ways of strengthening inductive arguments. We can either claim more in the premises or claim less in the conclusion.
Claiming more in the premises is straightforward in theory, though it can be difficult in prac- tice. The idea is simply to increase the amount of evidence for the conclusion. Suppose you are trying to convince a friend that she will enjoy a particular movie. You have shown her that she has liked other movies by the same director and that the movie is of the general kind that she likes. How could you strengthen your argument? You might show her that her favorite actors are cast in the lead roles, or you might appeal to the reviews of critics with which she often agrees. By adding these additional pieces of evidence, you have increased the strength of your argument that your friend will enjoy the movie.
Oksana Kostyushko/iStock/Thinkstock
Context plays an important role in inductive arguments. What makes an argument strong in one context might not be strong enough in another. Would you be more likely to play the lottery if your chances of winning were supported at 99%?
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The strength of an inductive argument can change when new premises are added. When evaluating or presenting an inductive argument, gather as many details as possible to have a more complete understanding of the strength of the argument.
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Section 5.1 Basic Concepts in Inductive Reasoning
However, if your friend looks at all the evidence and still is not sure, you might take the approach of weakening your conclusion. You might say some- thing like, “Please go with me; you may not actually like the movie, but at least you can be pretty sure you won’t hate it.” The very same evidence you pre- sented earlier—about the director, the genre, the actors, and so on—actually makes a stronger argument for your new, less ambitious claim: that your friend won’t hate the movie.
It might help to have another example of how each of the two approaches can help strengthen an inductive argu- ment. Take the following argument:
Every crow I have ever seen has been black. Therefore, all crows are black.
This seems to provide decent evidence, provided that you have seen a lot of crows. Here is one way to make the argument stronger:
Studies by ornithologists have examined thousands of crows in every conti- nent in which they live, and they have all been black. Therefore, all crows are black.
This argument is much stronger because there is much more evidence for the truth of the conclusion within the premise. Another way to strengthen the argument—if you do not have access to lots of ornithological studies—would simply be to weaken the stated conclusion:
Every crow I have ever seen has been black. Therefore, most crows are probably black.
This argument makes a weaker claim in the conclusion, but the argument is actually much stronger than the original because the premises make this (weaker) conclusion much more likely to be true than the original (stronger) conclusion.
By the same token, an inductive argument can also be made weaker either by subtracting evidence from the premises or by making a stronger claim in the conclusion. (For another way to weaken or strengthen inductive arguments, see A Closer Look: Using Premises to Affect Inductive Strength.)
Considering the Context It is important to realize that inductive strength and weakness are relative terms. As such, they are like the terms tall and short. A person who is short in one context may be tall in another. At 6’0”, professional basketball player Allen Iverson was considered short in the National Bas- ketball Association. But outside of basketball, someone of his height might be considered tall. Similarly, an argument that is strong in one context may be considered weak in another. You
would probably be reasonably happy if you could reliably predict sports (or lottery) results at an accuracy rate of 70%, but researchers in the social sciences typically aim for certainty upward of 90%. In high-energy physics, the goal is a result that is supported at the level of 5 sigma—a probability of more than 99.99997%!
The same is true when it comes to legal arguments. A case tried in a civil court needs to be shown to be true with a preponderance of evidence, which is much less stringent than in a criminal case, in which the defendant must be proved guilty beyond rea- sonable doubt. Therefore, whether the argument is strong or weak is a matter of context.
Moreover, some subjects have the sort of evi- dence that allows for extremely strong arguments, whereas others do not. A psychologist trying to pre- dict human behavior is unlikely to have the same strength of argument as an astronomer trying to predict the path of a comet. These are important things to keep in mind when it comes to evaluating inductive strength.
Strengthening Inductive Arguments Regardless of the subject matter of an argument, we generally want to create the strongest arguments we can. In general, there are two ways of strengthening inductive arguments. We can either claim more in the premises or claim less in the conclusion.
Claiming more in the premises is straightforward in theory, though it can be difficult in prac- tice. The idea is simply to increase the amount of evidence for the conclusion. Suppose you are trying to convince a friend that she will enjoy a particular movie. You have shown her that she has liked other movies by the same director and that the movie is of the general kind that she likes. How could you strengthen your argument? You might show her that her favorite actors are cast in the lead roles, or you might appeal to the reviews of critics with which she often agrees. By adding these additional pieces of evidence, you have increased the strength of your argument that your friend will enjoy the movie.
Oksana Kostyushko/iStock/Thinkstock
Context plays an important role in inductive arguments. What makes an argument strong in one context might not be strong enough in another. Would you be more likely to play the lottery if your chances of winning were supported at 99%?
Fuse/Thinkstock
The strength of an inductive argument can change when new premises are added. When evaluating or presenting an inductive argument, gather as many details as possible to have a more complete understanding of the strength of the argument.
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Section 5.1 Basic Concepts in Inductive Reasoning
A Closer Look: Using Premises to Affect Inductive Strength Suppose we have a valid deductive argument. That means that, if its premises are all true, then its conclusion must be true as well. Suppose we add a new premise. Is there any way that the argument could become invalid? The answer is no, because if the premises of the new argu- ment are all true, then so are all the premises of the old argument. Therefore, the conclusion still must be true.
This is a principle with a fancy name; it is called monotonicity: Adding a new premise can never make a deductive argument go from valid to invalid. However, this principle does not hold for inductive strength: It is possible to weaken an inductive argument by adding new premises.
The following argument, for example, might be strong:
99% of birds can fly. Jonah is a bird. Therefore, Jonah can fly.
This argument may be strong as it is, but what happens if we add a new premise, “Jonah is an ostrich”? The addition of this new premise just made the argument’s strength plummet. We now have a fairly weak argument! To use our new big word, this means that inductive reasoning is nonmonotonic. The addition of new premises can either enhance or diminish an argument’s inductive strength.
An interesting “game” is to see if you can continue to add premises that continue to flip the inductive argument’s degree of strength back and forth. For example, we could make the argu- ment strong again by adding “Jonah is living in the museum of amazing flying ostriches.” Then we could weaken it again with “Jonah is now retired.” It could be strengthened again with “Jonah is still sometimes seen flying to the roof of the museum,” but it could be weakened again with “He was seen flying by the neighbor child who has been known to lie.” The game demonstrates the sensitivity of inductive arguments to new information.
Thus, when using inductive reasoning, we should always be open to learning more details that could further serve to strengthen or weaken the case for the truth of the conclusion. Inductive strength is a never-ending process of gathering and evaluating new and relevant information. For scientists and logicians, that is partly what makes induction so exciting!
Inductive Cogency Notice that, like deductive validity, inductive strength has to do with the strength of the con- nection between the premises and the conclusion, not with the truth of the premises. There- fore, an inductive argument can be strong even with false premises. Here is an example of an inductively strong argument:
Every lizard ever discovered is purple. Therefore, most lizards are probably purple.
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Section 5.2 Statistical Arguments: Statistical Syllogisms
Of course, as with deductive reasoning, for an argument to give good evidence for the truth of the conclusion, we also want the premises to actually be true. An inductive argument is called cogent if it is strong and all of its premises are true. Whereas inductive strength is the counterpart of deductive validity, cogency is the inductive counterpart of deductive soundness.
5.2 Statistical Arguments: Statistical Syllogisms The remainder of this chapter will go over some examples of the different types of inductive arguments: statistical arguments, causal arguments, arguments from authority, and argu- ments from analogy. You will likely find that you have already encountered many of these various types in your daily life. Statistical arguments, for example, should be quite familiar. From politics, to sports, to science and health, many of the arguments we encounter are based on statistics, drawing conclusions from percentages and other data.
In early 2013 American actress Angelina Jolie elected to have a preventive double mastec- tomy. This surgery is painful and costly, and the removal of both breasts is deeply disturbing for many women. We might have expected Jolie to avoid the surgery until it was absolutely necessary. Instead, she had the surgery before there was any evidence of the cancer that nor- mally prompts a mastectomy. Why did she do this?
Jolie explained some of her reasoning in an opinion piece in the New York Times.
I carry a “faulty” gene, BRCA1, which sharply increases my risk of developing breast cancer and ovarian cancer.
My doctors estimated that I had an 87 percent risk of breast cancer and a 50 percent risk of ovarian cancer, although the risk is different in the case of each woman. (Jolie, 2013, para. 2–3)
As you can see, Jolie’s decision was based on probabilities and statistics. If these types of rea- soning can have such profound effects in our lives, it is essential that we have a good grasp on how they work and how they might fail. In this section, we will be looking at the basic struc- ture of some simple statistical arguments and some of the things to pay attention to as we use these arguments in our lives.
One of the main types of statistical arguments we will discuss is the statistical syllogism. Let us start with a basic example. If you are not a cat fancier, you may not know that almost all calico cats are female—to be more precise, about 99.97% of calico cats are female (Becker, 2013). Suppose you are introduced to a calico cat named Puzzle. If you had to guess, would you say that Puzzle is female or male? How confident are you in your guess?
Since you do not have any other information except that 99.97% of calico cats are female and Puzzle is a calico cat, it should seem far more likely to you that Puzzle is female. This is a
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Section 5.2 Statistical Arguments: Statistical Syllogisms
statistical syllogism: You are using a general statistic about calico cats to make an argument for a specific case. In its simplest form, the argument would look like this:
99.97% of calico cats are female. Puzzle is a calico cat. Therefore, Puzzle is female.
Clearly, this argument is not deductively valid, but inductively it seems quite strong. Given that male calico cats are extremely rare, you can be reasonably confident that Puzzle is female. In this case we can actually put a number to how confident you can be: 99.97% confident.
Of course, you might be mistaken. After all, male calico cats do exist; this is what makes the argument inductive rather than deductive. However, statistical syllogisms like this one can establish a high degree of certainty about the truth of the conclusion.
Form If we consider the calico cat example, we can see that the general form for a statistical syl- logism looks like this:
X% of S are P. i is an S. Therefore, i is (probably) a P.
There are also statistical syllogisms that conclude that the individual i does not have the prop- erty P. Take the following example:
Only 1% of college males are on the football team. Mike is a college male. Therefore, Mike is probably not on the football team.
This type of statistical syllogism has the following form:
X% of S are P. i is an S. Therefore, i is (probably) not a P.
In this case, for the argument to be strong, we want X to be a low percentage.
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Section 5.2 Statistical Arguments: Statistical Syllogisms
Note that statistical syllogisms are similar to two kinds of categorical syllogisms presented in Chapter 3 (see Table 5.1). We see from the table that statistical syllogisms become valid categorical syllogisms when the percentage, X, becomes 100% or 0%.
Table 5.1: Statistical syllogism versus categorical syllogism
Statistical syllogism Similar valid categorical syllogism
Example 99.97% of calico cats are female. Puzzle is calico. Therefore, Puzzle is female.
All calico cats are female. Puzzle is calico. Therefore, Puzzle is female.
Form X% of S are P. i is an S. Therefore, i is (probably) P.
All M are P. S is M. Therefore, S is P.
Example 1% of college males are on the football team. Mike is a college male. Therefore, Mike is not on the football team.
No college males are on the football team. Mike is a college male. Therefore, Mike is not on the football team.
Form X% of S are P. i is an S. Therefore, i is P.
X% of S are P. i is an S. Therefore, i is not P.
When identifying a statistical syllogism, it is important to keep the specific form in mind, since there are other kinds of statistical arguments that are not statistical syllogisms. Con- sider the following example:
85% of community college students are younger than 40. John is teaching a community college course. Therefore, about 85% of the students in John’s class are under 40.
This argument is not a statistical syllogism because it does not fit the form. If we make i “John” then the conclusion states that John, the teacher, is probably under 40, but that is not the con- clusion of the original argument. If we make i “the students in John’s class,” then we get the conclusion that it is 85% likely that the students in John’s class are under 40. Does this mean that all of them or that some of them are? Either way, it does not seem to be the same as the original conclusion, since that conclusion has to do with the percentage of students under 40 in his class. Though this argument has the same “feel” as a statistical syllogism, it is not one because it does not have the same form as a statistical syllogism.
Weak Statistical Syllogisms There are at least two ways in which a statistical syllogism might not be strong. One way is if the percentage is not high enough (or low enough in the second type). If an argument simply includes the premise that most of S are P, that means only that more than half of S are P. A probability of only 51% does not make for a strong inductive argument.
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Section 5.3 Statistical Arguments: Inductive Generalizations
Another way that statistical syllogisms can be weak is if the individual in question is more (or less) likely to have the relevant characteristic P than the average S. For example, take the reasoning:
99% of birds do not talk. My pet parrot is a bird. Therefore, my pet parrot cannot talk.
The premises of this argument may well be true, and the percentage is high, but the argument may be weak. Do you see why? The reason is that a pet parrot has a much higher likelihood of being able to talk than the average bird. We have to be very careful when coming to final conclusions about inductive reasoning until we consider all of the relevant information.
5.3 Statistical Arguments: Inductive Generalizations In the example about Puzzle, the calico cat, the first premise said that 99.97% of calico cats are female. How did someone come up with that figure? Clearly, she or he did not go out and look at every calico cat. Instead, he or she likely looked at a bunch of calicos, figured out what percentage of those cats were female, and then reasoned that the percentage of females would have been the same if they had looked at all calico cats. In this sort of reasoning, the group of calico cats that were actually examined is called the sample, and all the calico cats taken as a group are called the population. An inductive generalization is an argument in which we reason from data about a sample population to a claim about a large population that includes the sample. Its general form looks like this:
X% of observed Fs are Gs. Therefore, X% of all Fs are Gs.
In the case of the calico cats, the argument looks like this:
99.97% of calico cats in the sample were female. Therefore, 99.97% of all calico cats are female.
Whether the argument is strong or weak depends crucially on whether the sample popula- tion is representative of the whole population. We say that a sample is representative of a population when the sample and the population both have the same distribution of the trait we are interested in—when the sample “looks like” the population for our purposes. In the case of the cats, the strength of the argument depends on whether our sample group of calico cats had about the same proportion of females as the entire population of all calico cats.
There is a lot of math and research design—which you might learn about if you take a course in applied statistics or in quantitative research design—that goes into determining the likeli- hood that a sample is representative. However, even with the best math and design, all we can infer is that a sample is extremely likely to be representative; we can never be absolutely
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Section 5.3 Statistical Arguments: Inductive Generalizations
certain it is without checking the entire population. However, if we are careful enough, our arguments can still be very strong, even if they do not produce absolute certainty. This sec- tion will examine how researchers try to ensure the sample population is representative of the whole population and how researchers assess how confident they can be in their results.
Representativeness The main way that researchers try to ensure that the sample population is representative of the whole population is to make sure that the sample population is random and sufficiently large. Researchers also consider a measure called the margin of error to determine how simi- lar the sample population is to the whole population.
Randomness Suppose you want to know how many marshmallow treats are in a box of your favorite break- fast cereal. You do not have time to count the whole box, so you pour out one cup. You can count the number of marshmallows in your cup and then reason that the box should have the same proportion of marshmallows as the cup. You found 15 marshmallows in the cup, and the box holds eight cups of cereal, so you figure that there should be about 120 marshmallows in the box. Your argument looks something like this:
A one-cup sample of cereal contains 15 marshmallows. The box holds eight cups of cereal. Therefore, the box contains 120 marshmallows.
What entitles you to claim that the sample is representative? Is there any way that the sample may not represent the percentage of marshmallows in the whole box? One potential problem is that marshmallows tend to be lighter than the cereal pieces. As a result, they tend to rise to the top of the box as the cereal pieces settle toward the bottom of the box over time. If you just scoop out a cup of cereal from the top, then, your sample may not be representa- tive of the whole box and may have too many marshmallows.
One way to solve this problem might be to shake the box. Vigorously shak- ing the box would probably distribute the marshmallows fairly evenly. After a good shake, a particular piece of
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To ensure a sample is representative, participants should be randomly selected from the larger population. Careful consideration is required to ensure selections truly represent the larger population.
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Section 5.3 Statistical Arguments: Inductive Generalizations
marshmallow or cereal might equally end up anywhere in the box, so the ones that make it into your sample will be largely random. In this case the argument may be fairly strong.
In a random sample, every member of the population has an equal chance of being included. Understanding how randomness works to ensure representativeness is a bit tricky, but another example should help clear it up.
Almost all students at my high school have laptops. Therefore, almost all high school students in the United States have laptops.
This reasoning might seem pretty strong, especially if you go to a large high school. However, is there a way that the sample population (the students at the high school) may not be truly random? Perhaps if the high school is in a relatively wealthy area, then the students will be more likely to have laptops than random American high schoolers. If the sample population is not truly random but has a greater or lesser tendency to have the relevant characteristic than a random member of the whole population, this is known as a biased sample. Biased samples will be discussed further in Chapter 7, but note that they often help reinforce people’s biased viewpoints (see Everyday Logic: Why You Might Be Wrong).
The principle of randomness applies to other types of statistical arguments as well. Consider the argument about John’s community college class. The argument, again, goes as follows:
85% of community college students are younger than 40. John is teaching a community college course. Therefore, about 85% of the students in John’s class are under 40.
Since 85% of community college students are younger than 40, we would expect a sufficiently large random sample of community college students to have about the same percentage. There are several ways, however, that John’s class may not be a random sample. Before going on to the next paragraph, stop and see how many ways you can think of on your own.
So how is John’s class not a random sample? Notice first that the argument references a course at a single community college. The average student age likely varies from college to college, depending on the average age of the nearby population. Even within this one community college, John’s class is not random. What time is John’s class? Night classes tend to attract a higher percentage of older students than daytime classes. Some subjects also attract different age groups. Finally, we should think about John himself. His age and reputation may affect the kind of students who enroll in his classes.
In all these ways, and maybe others, John’s class is not a random sample: There is not an equal chance that every community college student might be included. As a result, we do not really have good reason to think that John’s class will be representative of the general popu- lation of community college students. So we have little reason to expect it to be representa- tive of the larger population. As a result, we cannot use his class to reliably predict what the population will look like, nor can we use the population to reliably predict what John’s class will look like.
Everyday Logic: Why You Might Be Wrong
People are often very confident about their views, even when it comes to very controversial issues that may have just as many people on the other side. There are prob- ably several reasons for this, but one of them is due to the use of biased sampling. Consider whether you think your views about the world are shared by many people or by only a few. It is not uncommon for people to think that their views are more widespread than they actually are. Why is that?
Think about how you form your opinion about how much of the nation or world agrees with your view. You prob- ably spend time talking with your friends about these views and notice how many of your friends agree or dis- agree with you. You may watch television shows or read news articles that agree or disagree with you. If most of the sources you interact with agree with your view, you might conclude that most people agree with you.
However, this would be a mistake. Most of us tend to interact more with people and infor- mation sources with which we agree, rather than those with which we disagree. Our circle of friends tends to be concentrated near us both geographically and ideologically. We share similar concerns, interests, and views; that is part of what makes us friends. As with choosing friends, we also tend to select information sources that confirm our beliefs. This is a well- known psychological tendency known as confirmation bias (this will be discussed further in Chapter 8).
We seem to reason as follows:
A large percentage of my friends and news sources agree with my view. Therefore, a large percentage of all people and sources agree with my view.
We have seen that this reasoning is based on a biased sample. If you take your friends and information sources as a sample, they are not likely to be representative of the larger popu- lation of the nation or world. This is because rather than being a random sample, they have been selected, in part, because they hold views similar to yours. A good critical thinker takes sampling bias into account when thinking about controversial issues.
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Confirmation bias, or the tendency to seek out support for our beliefs, can be seen in the friends we choose, books we read, and news sources we select.
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Section 5.3 Statistical Arguments: Inductive Generalizations
Sample Size Even a perfectly random sample may not be representative, due to bad luck. If you flip a coin 10 times, for example, there is a decent chance that it will come up heads 8 of the 10 times. However, the more times you flip the coin, the more likely it is that the percentage of heads will approach 50%.
The smaller the sample, the more likely it is to be nonrepresentative. This variable is known as the sample size. Suppose a teacher wants to know the average height of students in his school. He randomly picks one student and measures her height. You should see that this is
marshmallow or cereal might equally end up anywhere in the box, so the ones that make it into your sample will be largely random. In this case the argument may be fairly strong.
In a random sample, every member of the population has an equal chance of being included. Understanding how randomness works to ensure representativeness is a bit tricky, but another example should help clear it up.
Almost all students at my high school have laptops. Therefore, almost all high school students in the United States have laptops.
This reasoning might seem pretty strong, especially if you go to a large high school. However, is there a way that the sample population (the students at the high school) may not be truly random? Perhaps if the high school is in a relatively wealthy area, then the students will be more likely to have laptops than random American high schoolers. If the sample population is not truly random but has a greater or lesser tendency to have the relevant characteristic than a random member of the whole population, this is known as a biased sample. Biased samples will be discussed further in Chapter 7, but note that they often help reinforce people’s biased viewpoints (see Everyday Logic: Why You Might Be Wrong).
The principle of randomness applies to other types of statistical arguments as well. Consider the argument about John’s community college class. The argument, again, goes as follows:
85% of community college students are younger than 40. John is teaching a community college course. Therefore, about 85% of the students in John’s class are under 40.
Since 85% of community college students are younger than 40, we would expect a sufficiently large random sample of community college students to have about the same percentage. There are several ways, however, that John’s class may not be a random sample. Before going on to the next paragraph, stop and see how many ways you can think of on your own.
So how is John’s class not a random sample? Notice first that the argument references a course at a single community college. The average student age likely varies from college to college, depending on the average age of the nearby population. Even within this one community college, John’s class is not random. What time is John’s class? Night classes tend to attract a higher percentage of older students than daytime classes. Some subjects also attract different age groups. Finally, we should think about John himself. His age and reputation may affect the kind of students who enroll in his classes.
In all these ways, and maybe others, John’s class is not a random sample: There is not an equal chance that every community college student might be included. As a result, we do not really have good reason to think that John’s class will be representative of the general popu- lation of community college students. So we have little reason to expect it to be representa- tive of the larger population. As a result, we cannot use his class to reliably predict what the population will look like, nor can we use the population to reliably predict what John’s class will look like.
Everyday Logic: Why You Might Be Wrong
People are often very confident about their views, even when it comes to very controversial issues that may have just as many people on the other side. There are prob- ably several reasons for this, but one of them is due to the use of biased sampling. Consider whether you think your views about the world are shared by many people or by only a few. It is not uncommon for people to think that their views are more widespread than they actually are. Why is that?
Think about how you form your opinion about how much of the nation or world agrees with your view. You prob- ably spend time talking with your friends about these views and notice how many of your friends agree or dis- agree with you. You may watch television shows or read news articles that agree or disagree with you. If most of the sources you interact with agree with your view, you might conclude that most people agree with you.
However, this would be a mistake. Most of us tend to interact more with people and infor- mation sources with which we agree, rather than those with which we disagree. Our circle of friends tends to be concentrated near us both geographically and ideologically. We share similar concerns, interests, and views; that is part of what makes us friends. As with choosing friends, we also tend to select information sources that confirm our beliefs. This is a well- known psychological tendency known as confirmation bias (this will be discussed further in Chapter 8).
We seem to reason as follows:
A large percentage of my friends and news sources agree with my view. Therefore, a large percentage of all people and sources agree with my view.
We have seen that this reasoning is based on a biased sample. If you take your friends and information sources as a sample, they are not likely to be representative of the larger popu- lation of the nation or world. This is because rather than being a random sample, they have been selected, in part, because they hold views similar to yours. A good critical thinker takes sampling bias into account when thinking about controversial issues.
Jakubzak/iStock/Thinkstock
Confirmation bias, or the tendency to seek out support for our beliefs, can be seen in the friends we choose, books we read, and news sources we select.
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Section 5.3 Statistical Arguments: Inductive Generalizations
not a big enough sample. By measuring only one student, there is a decent chance that the teacher may have randomly picked someone extremely tall or extremely short. Generalizing on an overly small sample would be making a hasty generalization, an error in reasoning that will be discussed in greater detail in Chapter 7. If the teacher chooses a sample of two stu- dents, it is less likely that they will both be tall or both be short. The more students the teacher chooses for his sample, the less likely it is that the average height of the sample will be much different than the average height of all students. Assuming that the selection process is unbiased, therefore, the larger the sample population is, the more likely it is that the sam- ple will be representative of the whole population (see A Closer Look: How Large Must a Sample Be?).
Margin of Error It is always possible that a sample will be wildly different than the population. But equally important is the fact that it is quite likely that any sample will be slightly different than the population. Statisticians know how to calculate just how big this difference is likely to be. You will see this reported in some studies or polls as the margin of error. The margin of error can be used to determine the range of values that are likely for the population.
A Closer Look: How Large Must a Sample Be? In general, the larger a sample is, the more likely it is to be representative of the popula- tion from which it is drawn. However, even relatively small samples can lead to powerful conclusions if they have been carefully drawn to be random and to be representative of the population. As of this writing, the population of the United States is in the neighborhood of 317 million, yet Gallup, one of the most respected polling organizations in the country, often publishes results based on a sample of fewer than 3,000 people. Indeed, its typical sample size is around 1,000 (Gallup, 2010). That is a sample size of less than 1 in every 300,000 people!
Gallup can do this because it goes to great lengths to make sure that its samples are randomly drawn in a way that matches the makeup of the country’s population. If you want to know about people’s political views, you have to be very careful because these views can vary based on a person’s locale, income, race or ethnicity, gender, age, religion, and a host of other factors.
There is no single, simple rule for how large a sample should be. When samples are small or incautiously collected, you should be suspicious of the claims made on their basis. Profes- sional research will generally provide clear descriptions of the samples used and a justifica- tion of why they are adequate to support their conclusions. That is not a guarantee that the results are correct, but they are bound to be much more reliable than conclusions reached on the basis of small and poorly collected samples.
For example, sometimes politicians tour a state with the stated aim of finding out what the people think. However, given that people who attend political rallies are usually those with similar opinions as the speaker, it is unlikely that the set of people sampled will be both large enough and random enough to provide a solid basis for a reliable conclusion. If politicians really want to find out what people think, there are better ways of doing so.
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Section 5.3 Statistical Arguments: Inductive Generalizations
For example, suppose that a poll finds that 52% of a sample prefers Ms. Frazier in an elec- tion. When you read about the result of this poll, you will probably read that 52% of peo- ple prefer Ms. Frazier with a margin of error of ±3% (plus or minus 3%). This means that although the real number probably is not 52%, it is very likely to be somewhere between 49% (3% lower than 52%) and 55% (3% higher than 52%). Since the real percentage may be as low as 49%, Ms. Frazier should not start picking out curtains for her office just yet: She may actually be losing!
Confidence Level We want large, random samples because we want to be confident that our sample is repre- sentative of the population. The more confident we are that are sample is representative, the more confident we can be in conclusions we draw from it. Nonetheless, even a small, poorly drawn sample can yield informative results if we are cautious about our reasoning.
If you notice that many of your friends and acquaintances are out of work, you may conclude that unemployment levels are up. Clearly, you have some evidence for your conclusion, but is it enough? The answer to this question depends on how strong you take your argument to be. Remember that inductive arguments vary from extremely weak to extremely strong. The strength of an argument is essentially the level of confidence we should have in the con- clusion based on the reasons presented. Consider the following ways you might state your confidence that unemployment levels were up, based on noting unemployment among your friends and acquaintances.
a. “I’m certain that unemployment is up.” b. “I’m reasonably sure that unemployment is up.” c. “It’s more likely than not that unemployment is up.” d. “Unemployment might be up.”
Clearly, A is too strong. Your acquaintances just are not likely to represent the population enough for you to be certain that unemployment is up. On the other hand, D is weak enough that it really does not need much evidence to support it. B and C will depend on how wide and varied your circle of acquaintances is and on how much unemployment you see among them. If you know a lot of people and your acquaintances are quite varied in terms of profession, income, age, race, gender, and so on, then you can have more confidence in your conclusion than if you had only a small circle of acquaintances and they tended to all be like each other in these ways. B also depends on just what you mean by “reasonably sure.” Does that mean 60% sure? 75%? 85%?
Most reputable studies will include a “confidence level” that indicates how confident one can be that their conclusions are supported by the reasons they give. The degree of confidence can vary quite a bit, so it is worth paying attention to. In most social sciences, researchers aim to reach a 95% or 99% confidence level. A confidence level of 95% means that if we did the same study 100 times, then in 95 of those tests the results would fall within the margin of error. As noted earlier, the field of physics requires a confidence level of about 99.99997%, much higher than is typically required or attained in the social sciences. On the other end, sometimes a confidence level of just over 50% is enough if you are only interested in knowing whether something is more likely than not.
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Section 5.3 Statistical Arguments: Inductive Generalizations
Applying This Knowledge Now that we have learned something about statistical arguments, what can we say about Angelina Jolie’s argument, presented at the beginning of the prior section? First, notice that it has the form of a statistical syllogism. We can put it this way, written as if from her perspective:
87% of women with certain genetic and other factors develop breast cancer. I am a woman with those genetic and other factors. Therefore, I have an 87% risk of getting breast cancer.
We can see that the argument fits the form correctly. While not deductive, the argument is inductively strong. Unless we have reason to believe that she is more or less likely than the average person with those factors to develop breast cancer, if these premises are true then they give strong evidence for the truth of the conclusion. However, what about the first prem- ise? Should we believe it?
In evaluating the first premise, we need to consider the evidence for it. Were the samples of women studied sufficiently random and large that we can be confident they were representa- tive of the population of all women? With what level of confidence are the results established? If the samples were small or not randomized, then we may have less confidence in them. Jolie’s doctors said that Jolie had an 87% chance of developing breast cancer, but there’s a big difference between being 60% confident that she has this level of risk and being 99% certain that she does. To know how confident we should be, we would need to look at the back- ground studies that establish that 87% of women with those factors develop breast cancer. Anyone making such an important decision would be well advised to look at these issues in the research before acting.
Practice Problems 5.1
Which of the following attributes might negatively influence the data drawn from the following samples?
1. A teacher surveys the gifted students in the district about the curriculum that should be adopted at the high school. a. sample size b. representativeness of the sample c. a and b d. There is no negative influence in this case.
2. A researcher for Apple analyzes a large group of tribal people in the Amazon to determine which new apps she should create in 2014. a. sample size b. representativeness of the sample c. a and b d. There is no negative influence in this case.
(continued)
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Section 5.4 Causal Relationships: The Meaning of Cause
5.4 Causal Relationships: The Meaning of Cause It is difficult to say exactly what we mean when we say that one thing causes another. Think about turning on the lights in your room. What is the cause of the lights turning on? Is it the flipping of the switch? The electricity in the wires? The fact that the bulb is not broken? Your initial desire for the lights to be on? There are many things we could identify as a plausible cause of the lights turning on. However, for practical purposes, we generally look for the set of conditions without which the event in question would not have occurred and with which it will occur. In other words, logicians aim to be more specific about causal relationships by dis- cussing them in terms of sufficient and necessary conditions. Recall that we used these terms in Chapter 4 when discussing propositional logic. Here we will discuss how these terms can help us understand causal relationships.
Sufficient Conditions According to British philosopher David Hume, the notion of cause is based on nothing more than a “con- stant conjunction” that holds between events—the two events always occur together (Morris & Brown, 2014). We notice that events of kind A are always followed by events of kind B, and we say “A causes B.” Thus, to claim a causal relationship between events of type A and B might be to say: Whenever A occurs, B will occur.
Logicians have a fancy phrase for this relationship: We say that A is a sufficient condition for B. A factor is a sufficient condition for the occurrence of an event if whenever the factor occurs, the event also occurs: Whenever A occurs, B occurs as well. Or in other words:
If A occurs, then B occurs.
iStock/Thinkstock
Sufficient conditions are present in classroom grading systems. If you need a total of 850 points to receive an A, the sufficient condition to receive an A is earning 850 points.
3. A researcher on a college campus interviews 10 students after a yoga class about their drug use habits and determines that 80% of the student population probably smokes marijuana. a. sample size b. representativeness of the sample c. a and b d. There is no negative influence in this case.
Practice Problems 5.1 (continued)
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Section 5.4 Causal Relationships: The Meaning of Cause
For example, having a billion dollars is a sufficient condition for being rich; being hospitalized is a sufficient condition for being excused from jury duty; having a ticket is a sufficient condi- tion for being able to be admitted to the concert.
Often several factors are jointly required to create sufficient conditions. For example, each state has a set of jointly sufficient conditions for being able to vote, including being over 18, being registered to vote, and not having been convicted of a felony, among other possible qualifications.
Here is an example of how to think about sufficient conditions when thinking about real-life causation.
We know room lights do not go on just because you flip the switch. The points of the switch must come into contact with a power source, electricity must be present, a working lightbulb has to be properly secured in the socket, the socket has to be properly connected, and so forth. If any one of the conditions is not satisfied, the light will not come on. Strictly speaking, then, the whole set of conditions constitutes the sufficient condition for the event.
We often choose one factor from a set of factors and call it the cause of an event. The one we call the cause is the one with which we are most concerned for some reason or other; often it is the one that represents a change from the normal state of things. A working car is the nor- mal state of affairs; a hole in the radiator tube is the change to that state of affairs that results in the overheated engine. Similarly, the electricity and lightbulb are part of the normal state of things; what changed most recently to make the light turn on was the flipping of the switch.
Necessary Conditions A factor is a necessary condition for an event if the event would not occur in the absence of the factor. Without the necessary condition, the effect will not occur. A is a necessary condi- tion for B if the following statement is always true:
If A is not present, then neither is B.
This statement happens to be equivalent to the statement that if B is present, then A is pres- ent. Thus, a handy way to understand the difference between necessary and sufficient condi- tions is as follows:
“A is sufficient for B” means that if A occurs, then B occurs.
“A is necessary for B” means that if B occurs, then A occurs.
Let us take a look at a real example. Poliomyelitis, or polio, is a disease caused by a specific virus. In only a small minority of those with poliovirus does the virus infect the central ner- vous system and lead to the terrible condition known as paralytic polio. In the large majority of cases, however, the virus goes undetected and does not result in paralysis. Thus, infec- tion with poliovirus is not a sufficient condition for getting paralytic polio. However, because one must have the virus to have that condition, being infected with poliovirus is a necessary
Stockbyte/Thinkstock
Although water is a necessary condition for life, it is not a sufficient condition for life because humans also need oxygen and food.
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Section 5.4 Causal Relationships: The Meaning of Cause
condition for getting paralytic polio (Mayo Clinic, 2014).
On the other hand, being squashed by a steamroller is a sufficient condition for death, but it is not a necessary con- dition. Whenever someone has been squashed by a steamroller, that person is quite dead. However, it is not the case that anyone who is dead has been run over by a steamroller.
If our purpose in looking for causes is to be able to produce an effect, it is reasonable to look for sufficient conditions for that effect. If we can manipulate circumstances so that the sufficient condition is present, the effect will also be present. If we are looking for causes in order to prevent
an effect, it is reasonable to look for necessary conditions for that effect. If we prevent a neces- sary condition from materializing, we can prevent the effect.
The eradication of yellow fever is a striking example. Research showed that being bitten by a certain type of mosquito was a necessary condition for contracting yellow fever (though it was not a sufficient condition, for some people who were bitten by these mosquitoes did not contract yellow fever). Consequently, a campaign to destroy that particular species of mos- quito through the widespread use of insecticides virtually eliminated yellow fever in many parts of the world (World Health Organization, 2014).
Necessary and Sufficient Conditions The most restrictive interpretation of a causal relationship consists of construing “cause” as a condition both necessary and sufficient for the occurrence of an event. If factor A is necessary and sufficient for the occurrence of event B, then whenever A occurs, B occurs, and whenever A does not occur, B does not occur. In other words:
If A, then B, and if not-A, then not-B.
For example, to produce diamonds, certain very specific conditions must exist. Diamonds are produced if and only if carbon is subjected to immense pressure and heat for a certain period of time. Diamonds do not occur through any other process. If all of the conditions exist, then diamonds will result; diamonds exist only when all of those conditions have been met. There- fore, carbon subjected to the right combination of pressure, heat, and time constitutes both a necessary and sufficient condition for diamond production.
This construction of cause is so restrictive that very few actual relationships in ordinary expe- rience can satisfy it. However, some scientists think that this is the kind of invariant relation- ship that scientific laws must express. For instance, according to Newton’s law of gravitation,
For example, having a billion dollars is a sufficient condition for being rich; being hospitalized is a sufficient condition for being excused from jury duty; having a ticket is a sufficient condi- tion for being able to be admitted to the concert.
Often several factors are jointly required to create sufficient conditions. For example, each state has a set of jointly sufficient conditions for being able to vote, including being over 18, being registered to vote, and not having been convicted of a felony, among other possible qualifications.
Here is an example of how to think about sufficient conditions when thinking about real-life causation.
We know room lights do not go on just because you flip the switch. The points of the switch must come into contact with a power source, electricity must be present, a working lightbulb has to be properly secured in the socket, the socket has to be properly connected, and so forth. If any one of the conditions is not satisfied, the light will not come on. Strictly speaking, then, the whole set of conditions constitutes the sufficient condition for the event.
We often choose one factor from a set of factors and call it the cause of an event. The one we call the cause is the one with which we are most concerned for some reason or other; often it is the one that represents a change from the normal state of things. A working car is the nor- mal state of affairs; a hole in the radiator tube is the change to that state of affairs that results in the overheated engine. Similarly, the electricity and lightbulb are part of the normal state of things; what changed most recently to make the light turn on was the flipping of the switch.
Necessary Conditions A factor is a necessary condition for an event if the event would not occur in the absence of the factor. Without the necessary condition, the effect will not occur. A is a necessary condi- tion for B if the following statement is always true:
If A is not present, then neither is B.
This statement happens to be equivalent to the statement that if B is present, then A is pres- ent. Thus, a handy way to understand the difference between necessary and sufficient condi- tions is as follows:
“A is sufficient for B” means that if A occurs, then B occurs.
“A is necessary for B” means that if B occurs, then A occurs.
Let us take a look at a real example. Poliomyelitis, or polio, is a disease caused by a specific virus. In only a small minority of those with poliovirus does the virus infect the central ner- vous system and lead to the terrible condition known as paralytic polio. In the large majority of cases, however, the virus goes undetected and does not result in paralysis. Thus, infec- tion with poliovirus is not a sufficient condition for getting paralytic polio. However, because one must have the virus to have that condition, being infected with poliovirus is a necessary
Stockbyte/Thinkstock
Although water is a necessary condition for life, it is not a sufficient condition for life because humans also need oxygen and food.
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Section 5.4 Causal Relationships: The Meaning of Cause
objects attract each other with a force proportional to the inverse of the square of their dis- tance. Therefore, if we know the force of attraction between two bodies, we can calculate the distance between them (assuming we know their masses). Conversely, if we know the distance between them, we can calculate the force of attraction. Thus, having a certain degree of attraction between two bodies constitutes both a necessary and sufficient condition for the distance between them. It happens frequently in math and science that the values assigned to one factor determine the values assigned to another, and this relationship can be understood in terms of necessary and sufficient conditions.
Other Types of Causes The terms necessary condition and sufficient condition give us concrete and technical ways to describe types of causes. However, in everyday life, the factor we mention as the cause of an event is rarely one we consider sufficient or even necessary. We frequently select one factor from a set and say it is the cause of the event. Our aims and interests, as well as our knowledge, affect that choice. Thus, practical, moral, or legal considerations may influence our selection. There are three principal considerations that may lead us to choose a single factor as “the cause,” although this is not an exhaustive listing.
Trigger cause. The trigger cause, or the factor that initiates an event, is often designated the cause of the event. Usually, this is the factor that occurs last and completes a causal chain— the set of sufficient conditions—producing the effect. Flipping the switch triggers the lights. All the other factors may be present and as such constitute the standing conditions that allow the event to be triggered. The trigger factor is sometimes referred to as the proximate cause since it is the factor nearest the final event (or effect).
Unusual factor. Let us suppose that someone turns on a light and an explosion follows. Turn- ing on the light caused an explosion because the room was full of methane gas. Now being in a room is fairly normal, turning on lights is fairly normal, having oxygen in a room is fairly normal, and having an unsealed light switch is fairly normal. The only condition outside the norm is the presence of a large quantity of explosive gas. Therefore, the presence of methane is referred to as the cause of the explosion. What is unusual, what is outside the norm, is the cause. If we are concerned with fixing moral or legal responsibility for an effect, we are likely to focus on the person who left the gas on, not the person who turned on the lights.
Controllable factor. Sometimes we call attention to a controllable factor instrumental in pro- ducing the event and point out that since the factor could have been controlled, so could the event. Thus, although smoking is neither a sufficient nor a necessary condition for lung cancer, it is a controllable factor. Therefore, over and above uncontrollable factors like hered- ity and chance, we are likely to single out smoking as the cause. Similarly, drunk driving is neither a sufficient nor a necessary condition for getting into a car accident, but it is a control- lable factor, so we are likely to point to it as a cause.
Correlational Relationships In both the case of smoking and drunk driving, neither were necessary nor sufficient condi- tions for the subsequent event in question (lung cancer and car accidents). Instead, we would say that both are highly correlated with the respective events. Two things can be said to be
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Section 5.4 Causal Relationships: The Meaning of Cause
correlated, or in correlation, when they occur together frequently. In other words, A is cor- related with B, so B is more likely to occur if A occurs, and vice versa. For example, having gray hair is correlated with age. The older someone is, the more likely he or she is to have gray hair, and vice versa. Of course, not all people with gray hair are old, and not all old people have gray hair, so age is neither a necessary nor a sufficient condition for gray hair. However, the two are highly correlated because they have a strong tendency to go together.
Two things that vary in the same direction are said to be directly correlated or to vary directly; the higher one’s age, the more gray hair. Things that are correlated may also vary in opposite directions; these are said to vary inversely. For example, there is an inverse correlation between the size of a car and its fuel economy. In general, the bigger a car is, the lower its fuel economy is. If you want a car that gets high miles per gallon, you should focus on cars that are smaller. There are other factors to consider too, of course. A small sports car may get lower fuel economy than a larger car with less power. Correla- tion does not mean that the relationship is perfect, only that variables tend to vary in a certain way.
You may have heard the phrase “correlation does not imply causation,” or something similar. Just because two things happen together, it does not necessarily follow that one causes the other. For example, there is a well-known correlation between shoe size and reading ability in elementary children. Children with larger feet have a strong tendency to read bet- ter than children with smaller feet. Of course, no one supposes that a child’s shoe size has a direct effect on his or her reading ability, or vice versa. Instead, both of these things are related to a child’s age. Older children tend to have bigger feet than younger children; they also tend to read better. Sometimes the connection between correlated things is simple, as in the case of shoe size and reading, and sometimes it is more complicated.
Whenever you read that two things have been shown to be linked, you should pay attention to the possibility that the correlation is spurious or possibly has another explanation. Consider, for example, a study showing a strong correlation between the amount of fat in a country’s diet and the amount of certain types of cancer in that country (such as K. K. Carroll’s 1975 study, as cited in Paulos, 1997). Such a correlation may lead you to think that eating fat causes cancer, but this could potentially be a mistake. Instead, we should consider whether there might be some other connection between the two.
It turns out that countries with high fat consumption also have high sugar consumption— perhaps sugar is the culprit. Also, countries with high fat and sugar consumption tend to be wealthier; fat and sugar are expensive compared to grain. Perhaps the correlation is the result of some other aspect of a wealthier lifestyle, such as lower rates of physical exercise. (Note that wealth is a particularly common confounding factor, or a factor that correlates with the
Hagen/Cartoonstock
Variables, such as buffalo and White men, can be correlated in two ways— directly and inversely. Which type of correlation is being discussed in this cartoon?
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Divorce rate in Maine Per capita consumption of margarine (U.S.)
2000
5.0
4.8
4.6
4.4
4.2
4.0
9
8
7
6
5
4
3
D iv
o rc
e s
p e r
10 0 0 p
e o p le
P o u n d s
2001 2002 2003 2004 2005 2006 2007 2008 2009
Section 5.5 Causal Arguments: Mill’s Methods
dependent and independent variables being studied, as it bestows a wide range of advantages and difficulties on those who have it.) Perhaps it is a combination of factors, and perhaps it is the fat after all; however, we cannot simply conclude with certainty from a correlation that one causes the other, not without further research.
Sometimes correlation between two things is simply random. If you search through enough data, you may find two factors that are strongly correlated but that have nothing at all to do with each other. For example, consider Figure 5.1. At first glance, you might think the two fac- tors must be closely connected. But then you notice that one of them is the divorce rate in Maine and the other is the per capita consumption of margarine in the United States. Could it be that by eating less margarine you could help save the marriages of people in Maine?
On the other hand, although correlation does not imply causation, it does point to it. That is, when we see a strong correlation, there is at least some reason to suspect a causal connection of some sort between the two correlates. It may be that one of the correlates causes the other, a third thing causes them both, there is some more complicated causal relation between them, or there is no connection at all.
However, the possibility that the correlation is merely accidental becomes increasingly unlikely if the sample size is large and the correlation is strong. In such cases we may have to be very thoughtful in seeking and testing possible explanations of the correlation. The next section discusses ways that we might find and narrow down potential factors involved in a causal relationship.
5.5 Causal Arguments: Mill’s Methods Reasoning about causes is extremely important. If we can correctly identify what causes a particular effect, then we have a much better chance of controlling or preventing the effect. Consider the search for a cure for a disease. If we do not understand what causes a particular disease, then our chances of being able to cure it are small. If we can identify the cause of the
Figure 5.1: Are these two factors correlated?
Although it may seem like two factors are correlated, we sometimes have to look harder to understand the relationship.
Source: www.tylervigen.com.
Divorce rate in Maine Per capita consumption of margarine (U.S.)
2000
5.0
4.8
4.6
4.4
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Section 5.5 Causal Arguments: Mill’s Methods
disease, we can be much more precise in searching for a way to prevent the disease. On the other hand, if we think we know the cause when we do not, then we are likely to look in the wrong direction for a cure.
A causal argument—an argument about causes and effects—is almost always an inductive argument. This is because, although we can gather evidence about these relationships, we are almost never in a position to prove them absolutely.
The following four experimental methods were formally stated in the 19th century by John Stuart Mill in his book A System of Logic and so are often referred to as Mill’s methods. Mill’s methods express the most basic underlying logic of many current methods for investigat- ing causality. They provide a great introduction to some of the basic concepts involved—but know that modern methods are much more rigorous.
Used with caution, Mill’s methods can provide a guide for exploring causal connections, espe- cially when one is looking at specific cases against the background of established theory. It is important to remember that although they can be useful, Mill’s methods are only the begin- ning of the study of causation. By themselves, they are probably most useful as methods for identifying potential subjects for further study using more robust methods that are beyond the scope of this book.
Method of Agreement In 1976 an unknown illness affected numerous people in Philadelphia. Although it took some time to fully identify the cause of the disease, a bacterium now called Legionella pneumophila, the first step in the investigation was to find common features of those who became ill. Research- ers were quick to note that sufferers had all attended an American Legion convention at the Bellevue-Stratford Hotel. As you can guess, the focus of the investigation quickly narrowed to conditions at the hotel. Of course, the convention and the hotel were not the actual cause of get- ting sick, but neither was it mere coincidence that all of the ill had attended the convention. By finding the common elements shared by those who became ill, investigators were able to quickly narrow their search for the cause. Ultimately, the bacterium was located in a fountain in the hotel.
The method of agreement involves comparing situations in which the same kind of event occurs. If the presence of a certain factor is the only respect in which the situations are the same (that is, they agree), then this factor may be related to the cause of the event. We can represent this with something like Table 5.2. The table indicates whether each of four factors was present in a specific case (A, B, or C) and, in the last column, whether the effect mani- fested itself (in the earlier case of what is now known as Legionnaires’ disease, the effect we would be interested in is whether infection occurred).
Table 5.2: Example of method of agreement
Case Factor 1 Factor 2 Factor 3 Factor 4 Effect
A No Yes Yes No Yes
B No No Yes Yes Yes
C Yes Yes Yes Yes Yes
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Section 5.5 Causal Arguments: Mill’s Methods
The three cases all resulted in the same effect but differed in which factors were present— with the exception of Factor 3, which was present in all three cases. We may then suspect that Factor 3 may be causally related to the effect. Our notion of cause here is that of sufficient condition. The common factor is sufficient to account for the effect.
In general, the method of agreement works best when we have a large group of cases that is as varied as possible. A large group is much more likely to vary across many different factors than a small group. Unfortunately, the world almost never presents us with two situations wholly unlike except for one factor. We may have three or more situations that are greatly similar. For example, all of the afflicted in the 1976 outbreak were members of the American Legion, all were adults, all were men, all lived in Pennsylvania. Here is where we have to use common sense and what we already know. It is unlikely that merely being a member of an organization is the cause of a disease. We expect diseases to be caused by environmental fac- tors: bacteria, viruses, contaminants, and so on. As a result, we can focus our search on those similarities that seem most likely to be relevant to the cause. Of course, we may be wrong; that is a hallmark of inductive reasoning generally, but by being as careful and as reasonable as we can, we can often make great progress.
Method of Difference The method of difference involves comparing a situation in which an event occurs with sim- ilar situations in which it does not. If the presence of a certain factor is the only difference between the two kinds of situations, it is likely to be causally related to the effect.
Suppose your mother comes to visit you and makes your favorite cake. Unfortunately, it just does not turn out. You know she made it in the same way she always does. What could the problem be? Start by looking at differences between how she made the cake at your house and how she makes it at hers. Ultimately, the only difference you can find is that your mom lives in Tampa and you live in Denver. Since that is the only difference, that difference is likely to be causally related to the effect. In fact, Denver is both much higher and much drier than Tampa. Both of these factors make a difference in baking cakes.
Let us suppose we are interested in two cases, A and B, in which A has the effect we are inter- ested in (the cake not turning out right) and B does not. This is outlined in Table 5.3. If we can find only one factor that is different between the two cases—in this case, Factor 1—then that factor is likely to be causally related to the effect. This does not tell us whether the factor directly causes the effect, but it does suggest a causal link. Further investigation might reveal just exactly what the connection is.
Table 5.3: Example of method of difference
Case Factor 1 Factor 2 Factor 3 Factor 4 Effect
A Yes No No Yes Yes
B No No No Yes No
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Section 5.5 Causal Arguments: Mill’s Methods
In this example, Factor 1 is the one factor that is different between the two cases. Perhaps the presence of Factor 1 is related to why Case A had the effect but Case B did not. Here we are seeing Factor 1 as a necessary condition for the effect.
The method of difference is employed frequently in clinical trials of experimental drugs. Researchers carefully choose or construct two situations that resemble each other in as many respects as possible. If a drug is employed in one but not the other, then they can ascribe to the drug any change in one situation not matched by a change in the other. Note that the two sets must be as similar as possible, since variation could introduce other possible causal links. The group in which change is expected is often referred to as the experimental group, and the group in which change is not expected is often referred to as the control group.
The method of difference may seem obvious and its results reliable. Yet even in a relatively simple experimental setup like this one, we may easily find grounds for doubting that the causal claim has been adequately established.
One important factor is that the two cases, A and B, have to be as similar as possible in all other respects for the method of difference to be used effectively. If your 8-year-old son made the cake without supervision, there are likely to be a whole host of differences that could explain the failure. The same principle applies to scientific studies. One thing that can subtly skew experimental results is experimental bias. For example, if the experimenters know which people are receiving the experimental drug, they might unintentionally treat them differently.
To prevent such possibilities, so-called blind experiments are often used. Those conducting the experiment are kept in ignorance about which subjects are in the control group and which are in the experimental group so that they do not even unintentionally treat the subjects dif- ferently. Experimenters therefore, do not know whether they are injecting distilled water or the actual drug. In this way the possibility of a systematic error is minimized.
We also have to keep in mind that our inquiry is guided by background beliefs that may be incorrect. No two cases will ever be completely the same except for a single factor. Your mother made the cake on a different day than she did at home, she used a different spoon, different people were present in the house, and so on. We naturally focus on similarities and differences that we expect to be relevant. However, we should always realize that reality may disagree with our expectations.
Causal inquiry is usually not a matter of conducting a single experiment. Often we cannot even control for all relevant factors at the same time, and once an experiment is concluded, doubts about other factors may arise. A series of experiments in which different factors are kept constant while others are varied one by one is always preferable.
Joint Method of Agreement and Difference The joint method of agreement and difference is, as the name suggests, a combination of the methods of agreement and difference. It is the most powerful of Mill’s methods. The basic idea is to have two groups of cases: One group shows the effect, and the other does not. The method of agreement is used within each group, by seeing what they have in common, and the method of difference is used between the two groups, by looking for the differences between the two. Table 5.4 shows how such a chart would look, if we were comparing three different cases (1, 2, and 3) among two groups (A and B).
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Section 5.5 Causal Arguments: Mill’s Methods
Table 5.4: Example of joint method of agreement and difference
Case/group Factor 1 Factor 2 Factor 3 Factor 4 Effect
1/A Yes No No Yes Yes
2/A No No Yes Yes Yes
3/A No Yes No Yes Yes
1/B No Yes Yes No No
2/B Yes Yes No No No
3/B Yes No Yes No No
As you can see, within each group the cases agree only on Factor 4 and the effect. But when you compare the two groups, the only consistent differences between them are in Factor 4 and the effect. This result suggests the possibility that Factor 4 may be causally related to the effect in question. In this method, we are using the notion of a necessary and sufficient condi- tion. The effect happens whenever Factor 4 is present and never when it is absent.
The joint method is the basis for modern randomized controlled experiments. Suppose you want to see if a new medicine is effective. You begin by recruiting a large group of volun- teers. You then randomly assign them to either receive the medicine or a placebo. The random assignment ensures that each group is as varied as possible and that you are not unknowingly deciding whether to give someone the medicine based on some common factor. If it turns out that everyone who gets the medicine improves and everyone who gets the placebo stays the same or gets worse, then you can infer that the medicine is probably effective.
In fact, advanced statistics allow us to make inferences from such studies even when there is not perfect agreement on the presence or absence of the effect. So, in reading studies, you may note that the discussion talks about the percentage of each group that shows or does not show the effect. Yet we may still make good inferences about causation by using the method of concomitant variation.
Method of Concomitant Variation The method of concomitant variation is simply the method of looking for correlation between two things. As we noted in our discussion of correlation, this cannot be used to conclude con- clusively that one thing causes the other, but it is suggestive that there is perhaps some causal connection between the two. Stronger evidence can be found by further scientific study.
You may have noticed that, in discussing causes, we are trying to explain a phenomenon. We observe something that is interesting or important to us, and we seek to know why it happened. Therefore, the study of Mill’s methods, as well as correlation and concomitant variation, can be seen as part of a broader type of reasoning known as inference to the best explanation, the effort to find the best or most accurate explanation of our observations. Because this type of reasoning is sometimes classified as a separate type of reasoning (sometimes called abduc- tive reasoning), it will be covered in Chapter 6.
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Section 5.5 Causal Arguments: Mill’s Methods
In summary, Mill’s methods provide a framework for exploring causal relationships. It is impor- tant to remember that although they can be useful, they are only the beginning of this important field. By themselves, they are probably most useful as methods for identifying potential subjects for further study using more robust methods that are beyond the scope of this book.
Practice Problems 5.2
Identify which of Mill’s methods discussed in the chapter relates to the following examples.
1. After going to dinner, all the members of a family came down with vomiting. They all had different entrées but shared a salad as an appetizer. The mother of the family determines that it must have been the salad that caused the sickness. a. method of agreement b. method of difference c. joint method d. method of concomitant variation
2. A couple goes to dinner and shares an appetizer, entrée, and dessert. Only one of the two gets sick. She drank a glass of wine, and her husband drank a beer. She believes that the wine was the cause of her sickness. a. method of agreement b. method of difference c. joint method d. method of concomitant variation
3. In a specific city, the number of people going to emergency rooms for asthma attacks increases as the level of pollution increases in the summer. When the winter comes and pollution goes down, the number of people with asthma attacks decreases. a. method of agreement b. method of difference c. joint method d. method of concomitant variation
4. In the past 15 years there has never been a safety accident in the warehouse. Each day for the past 15 years Lorena has been conducting the morning safety inspec- tions. However, today Lorena missed work, and there was an accident. a. method of agreement b. method of difference c. joint method d. method of concomitant variation
5. Since we have hired Earl, productivity in the office has decreased by 20%. a. method of agreement b. method of difference c. joint method d. method of concomitant variation
6. In the past, lead was put into many paints. It was found that the number of infant fatalities increased in relation to the amount of exposure these infants had to lead- based paints that were used on their cribs. a. method of agreement b. method of difference
(continued)
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Section 5.6 Arguments From Authority
5.6 Arguments From Authority An argument from authority, also known as an appeal to authority, is an inductive argu- ment in which one infers that a claim is true because someone said so. The general reasoning looks like this:
Person A said that X is true. Person A is an authority on the subject. Therefore, X is true.
Whether this type of reasoning is strong depends on the issue discussed and the authority cited. If it is the kind of issue that can be settled by an argument from authority and if the per- son is actually an authority on the subject, then it can actually be a strong inductive argument.
Some people think that arguments from authority in general are fallacious. However, that is not generally the case. To see why, try to imagine life without any appeals to authority. You could not believe anyone’s statements, no matter how credible. You could not believe books; you could not believe published journals, and so on. How would you do in college if you did not listen to your textbooks, teachers, or any other sources of information?
c. joint method d. method of concomitant variation
7. It appears that the likelihood of catching the Zombie virus increases the more one is around people who have already been turned into zombies. a. method of agreement b. method of difference c. joint method d. method of concomitant variation
8. In order to determine how a disease was spread in humans, researchers placed two groups of people into two rooms. Both rooms were exactly alike. However, in one room they placed someone who was infected with the disease. The researchers found that those who were in the room with the infected person got sick, whereas those who were not with an infected person remained well. a. method of agreement b. method of difference c. joint method d. method of concomitant variation
9. In a certain IQ test, students in a specific group performed at a much higher level than those of the other groups. After analyzing the group, the researchers found that the high-performing students all smoked marijuana before the exam. a. method of agreement b. method of difference c. joint method d. method of concomitant variation
Practice Problems 5.2 (continued)
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Section 5.7 Arguments From Analogy
Even in science class, you would have to do every experiment on your own because you could not believe pub- lished reports. In math, you could not trust the book or teacher, so you would have to prove every theorem by your- self. History class would be a complete waste of time because, unless you had a time machine, there would be no way to verify any claims about what happened in the past without appeal to historical records, newspapers, journals, and so forth. You would also have a hard time following medical advice, so you might end up with serious health problems. Finally, why would you go to school or work if you could not trust the claim that you were going to get a degree or a paycheck after all of your efforts?
Therefore, in order to learn from oth- ers and to succeed in life, it is essen-
tial that we listen to appropriate authorities. However, since many sources are unreliable, misleading, or even downright deceptive, it is essential that we learn to distinguish reli- able sources of authority from unreliable ones. Chapter 7 will discuss how to distinguish between legitimate and fallacious appeals to authority.
Here are some examples of legitimate arguments from authority:
“The theory of relativity is true. I know because my physics professor and my physics textbook teach that it is true.”
“Pine trees are not deciduous; it says so right here in this tree book.”
“The Giants won the pennant! I read it on ESPN.com.”
“Mike hates radishes. He told me so yesterday.”
All of these inferences seem pretty strong. For examples of arguments to authority that are not as strong, or even downright fallacious, visit Chapter 7.
5.7 Arguments From Analogy An argument from analogy is an inductive argument that draws conclusions based on the use of analogy. An analogy is a comparison of two items. For example, many object to deficit spending (when the country spends more money than it takes in) based on the reasoning that debt is bad for household budgets. The person’s argument depends on an analogy that compares the national budget to a household budget. The two items being compared may be referred to as analogs (or analogues, depending on where you live) but are referred to
badahos/iStock/Thinkstock
The ability to think critically about an authority’s argument will allow you to determine reliable sources from unreliable ones, which can be quite helpful when writing research papers, reading news articles, or taking advice from someone.
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Section 5.7 Arguments From Analogy
technically as cases. Of the two analogs, one should be well known, with a body of knowledge behind it, and so is referred to as the familiar case; the second analog, about which much less is known, is called the unfamiliar case.
The basic structure of an argument from analogy is as follows:
B is similar to A. A has feature F. Therefore, B probably also has feature F.
Here, A is the familiar case and B is the unfamiliar case. We made an inference about thing B based on its similarity to the more familiar A.
Analogical reasoning proceeds from this premise: Since the analogs are similar either in many ways or in some very important ways, they are likely to be similar in other ways as well. If there are many similarities, or if the similarities are significant, then the analogy can be strong. If the analogs are different in many ways, or if the differences are important, then it is a weak analogy. Conclusions arrived at through strong analogies are fairly reliable; con- clusions reached through weak analogies are less reliable and often fallacious (the fallacy is called false analogy). Therefore, when confronted with an analogy (“A is like B”), the first question to be asked is this: Are the two analogs very similar in ways that are relevant to the current discussion, or are they different in relevant ways?
Analogies occur in both arguments and explanations. As we saw in Chapter 2, arguments and explanations are not the same thing. The key difference is whether the analogy is being used to give evidence that a certain claim is true—an argument—or to give a better understanding of how or why a claim is true—an explanation. In explanations, the analogy aims to provide deeper understanding of the issue. In arguments, the analogy aims to provide reasons for believing a conclusion. The next section provides some tips for evaluating the strength of such arguments.
Evaluating Arguments From Analogy Again, the strength of the argument depends on just how much A is like B, and the degree to which the similarities between A and B are relevant to F. Let us consider an example. Suppose that you are in the market for a new car, and your primary concern is that the car be reliable. You have the opportunity to buy a Nissan. One of your friends owns a Nissan. Since you want to buy a reliable car, you ask a friend how reliable her car is. In this case you are depending on an analogy between your friend’s car and the car you are looking to buy. Suppose your friend says that her car is reliable. You can now make the following argument:
The car I’m looking at is like my friend’s car. My friend’s car is reliable. Therefore, the car I’m looking at will be reliable.
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Section 5.7 Arguments From Analogy
How strong is this argument? That depends on how similar the two cases are. If the only thing the cars have in common is the brand, then the argument is fairly weak. On the other hand, if the cars are the same model and year, with all the same options and a similar driving history, then the argument is stronger. We can list the similarities in a chart (see Table 5.5). Initially, the analogy is based only on the make of the car. We will call the car you are looking at A and your friend’s car B.
Table 5.5: Comparing cars by make
Car Make Reliable?
B Nissan Yes
A Nissan ?
The make of a car is relevant to its reliability, but the argument is weak because that is the only similarity we know about. To strengthen the argument, we can note further relevant similarities. For example, if you find out that your friend’s car is the same model and year, then the argument is strengthened (see Table 5.6).
Table 5.6: Comparing cars by make, model, and year
Car Make Model Year Reliable?
B Nissan Sentra 2000 Yes
A Nissan Sentra 2000 ?
The more relevant similarities there are between the two cars, the stronger the argument. However, the word relevant is critical here. Finding out that the two cars have the same engine and similar driving histories is relevant and will strengthen the argument. Finding out that both cars are the same color and have license plates beginning with the same let- ter will not strengthen the argument. Thus, arguments from analogy typically require that we already have some idea of which features are relevant to the feature we are interested in. If you really had no idea at all what made some cars reliable and others not reliable, then you would have no way to evaluate the strength of an argument from analogy about reliability.
Another way we can strengthen an argument from analogy is by increasing the number of analogs. If you have two more friends who also own a car of the same make, model, and year, and if those cars are reliable, then you can be more confident that your new car will be reli- able. Table 5.7 shows what the chart would look like. The more analogs you have that match the car you are looking at, the more confidence you can have that the car you’re looking at will be reliable.
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Section 5.7 Arguments From Analogy
Table 5.7: Comparing multiple analogs
Car Make Model Year Reliable?
B Nissan Sentra 2000 Yes
C Nissan Sentra 2000 Yes
D Nissan Sentra 2000 Yes
A Nissan Sentra 2000 ?
In general, then, analogical arguments are stronger when they have more analogous cases with more relevant similarities. They are weaker when there are significant differences between the familiar cases and the unfamiliar case. If you discover a significant difference between the car you are looking at and the analogs, that reduces the strength of the argument. If, for example, you find that all your friends’ cars have manual transmission, whereas the one you are looking at has an automatic transmission, this counts against the strength of the anal- ogy and hence against the strength of the argument.
Another way that an argument from analogy can be weakened is if there are cases that are similar but do not have the feature in question. Suppose you find a fourth friend who has the same model and year of car but whose car has been unreliable. As a result, you should have less confidence that the car you are looking at is reliable.
Here are a couple more examples, with questions about how to gauge the strength.
“Except for size, chickens and turkeys are very similar birds. Therefore, if a food is good for chickens, it is probably good for turkeys.”
Relevant questions include how similar chickens and turkeys are, whether there are signifi- cant differences, and whether the difference in size is enough to allow turkeys to eat things that would be too big for chickens.
“Seattle’s climate is similar, in many ways to the United Kingdom’s. Therefore, this plant is likely to grow well in Seattle, because it grows well in the United Kingdom.”
Just how similar is the climate between the two places? Is the total about of rain about the same? How about the total amount of sun? Are the low and high temperatures comparable? Are there soil differences that would matter?
“I am sure that my favorite team will win the bowl game next week; they have won every game so far this season.”
This example might seem strong at first, but it hides a very relevant difference: In a bowl game, college football teams are usually matched up with an opponent of approximately equal strength. It is therefore likely that the team being played will be much better than the other teams played so far this season. This difference weakens the analogy in a relevant way, so the argument is much weaker than it may at first appear. It is essential when studying the strength of analogical arguments to be thorough in our search for relevant similarities and differences.
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Section 5.7 Arguments From Analogy
Analogies in Moral Reasoning Analogical reasoning is often used in moral reasoning and moral arguments. Examples of ana- logical reasoning are found in ethical or legal debates over contentious or controversial issues such as abortion, gun control, and medical practices of all sorts (including vaccinations and transplants). Legal arguments are often based on finding precedents—analogous cases that have already been decided. Recent arguments presented in the debate over gun control have drawn conclusions based on analogies that compare the United States with other countries, including Switzerland and Japan. Whether these and similar arguments are strong enough to establish their conclusions depends on just how similar the cases are and the degree and number of dissimilarities and contrary cases. Being aware of similar cases that have already occurred or that are occurring in other areas can vastly improve one’s wisdom about how best to address the topic at hand.
The importance of analogies in moral reasoning is sometimes captured in the principle of equal treatment—that if two things are analogous in all morally relevant respects, then what is right (or wrong) to do in one case will be right (or wrong) to do in the other case as well. For example, if it is right for a teacher to fail a student for missing the final exam, then another student who does the same thing should also be failed. Whether the teacher happens to like one student more than the other should not make a difference, because that is not a morally relevant difference when it comes to grading.
The reasoning could look as follows:
Things that are similar in all morally relevant respects should be treated the same. Student A was failed for missing the final exam. Student B also missed the final exam. Therefore, student B should be failed as well.
It follows from the principle of equal treatment that if two things should be treated differ- ently, then there must be a morally relevant difference between them to justify this different treatment. An example of the application of this principle might be in the interrogation of prisoners of war. If one country wants to subject prisoners of war to certain kinds of harsh treatments but objects to its own prisoners being treated the same way by other countries, then there need to be relevant differences between the situations that justify the different treatment. Otherwise, the country is open to the charge of moral inconsistency.
This principle, or something like it, comes up in many other types of moral debates, such as about abortion and animal ethics. Animal rights advocates, for example, say that if we object to people harming cats and dogs, then we are morally inconsistent to accept to the same treat- ment of cows, pigs, and chickens. One then has to address the question of whether there are differences in the beings or in their use for food that justify the differences in moral consid- eration we give to each.
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Section 5.7 Arguments From Analogy
Other Uses of Analogies Analogies are the basis for parables, allegories, and forms of writing that try to give a moral. The phrase “The moral of the story is . . .” may be featured at the end of such stories, or the author may simply imply that there is a lesson to be learned from the story. Aesop’s Fables are one well-known example of analogy used in writing. Consider the fable of the ant and the grasshop- per, which compares the hardwork- ing, industrious ant with the footloose and fancy-free grasshopper. The ant gathers and stores food all summer to prepare for winter; the grasshopper fiddles around and plays all summer, giving no thought for tomorrow. When winter comes, the ant lives warm and comfortable while the grasshopper starves, freezes, and dies. The fable argues that we should be like the ant if we want to survive harsh times. The ant and grasshopper are analogs for industrious people and lazy people. How strong is the argument? Clearly, ants and grasshop- pers are quite different from people. Are the differences relevant to the conclusion? What are the relevant similarities? These are the questions that must be addressed to get an idea of whether the argument is strong or weak.
Jupiterimages/BananaStock/Thinkstock
Retailers such as bookstores commonly use arguments from analogy when they suggest purchases based on their similarity to other items.
Practice Problems 5.3
Determine whether the following arguments are inductive or deductive.
1. All voters are residents of California. But some residents of California are Republican. Therefore, some voters are Republican. a. deductive b. inductive
2. All doctors are people who are committed to enhancing the health of their patients. No people who purposely harm others can consider themselves to be doctors. Therefore, some people who harm others do not enhance the health of their patients. a. deductive b. inductive
3. Guns are necessary. Guns protect people. They give people confidence that they can defend themselves. Guns also ensure that the government will not be able to take over its citizenry. a. deductive b. inductive
(continued)
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Section 5.7 Arguments From Analogy
4. Every time I turn on the radio, all I hear is vulgar language about sex, violence, and drugs. Whether it’s rock and roll or rap, it’s all the same. The trend toward vulgarity has to change. If it doesn’t, younger children will begin speaking in these ways and this will spoil their innocence. a. deductive b. inductive
5. Letting your kids play around on the Internet all day is like dropping them off in downtown Chicago to spend the day by themselves. They will find something that gets them into trouble. a. deductive b. inductive
6. Many people today claim that men and women are basically the same. Although I believe that men and women are equally capable of completing the same tasks physically as well as mentally, to say that they are intrinsically the same detracts from the differences between men and women that are displayed every day in their social interactions, the way they use their resources, and the way in which they find themselves in the world. a. deductive b. inductive
7. Too many intravenous drug users continue to risk their lives by sharing dirty nee- dles. This situation could be changed if we were to supply drug addicts with a way to get clean needles. This would lower the rate of AIDS in this high-risk population as well as allow for the opportunity to educate and attempt to aid those who are addicted to heroin and other intravenous drugs. a. deductive b. inductive
8. I know that Stephen has a lot of money. His parents drive a Mercedes. His dogs wear cashmere sweaters, and he paid cash for his Hummer. a. deductive b. inductive
9. Dogs are better than cats, since they always listen to what their masters say. They also are more fun and energetic. a. deductive b. inductive
10. All dogs are warm-blooded. All warm-blooded creatures are mammals. Hence, all dogs are mammals. a. deductive b. inductive
11. Chances are that I will not be able to get in to see Slipknot since it is an over 21 show, and Jeffrey, James, and Sloan were all carded when they tried to get in to the club. a. deductive b. inductive
12. This is not the best of all possible worlds, because the best of all possible worlds would not contain suffering, and this world contains much suffering. a. deductive b. inductive
Practice Problems 5.3 (continued)
(continued)
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Section 5.7 Arguments From Analogy
13. Some apples are not bananas. Some bananas are things that are yellow. Therefore, some things that are yellow are not apples. a. deductive b. inductive
14. Since all philosophers are seekers of truth, it follows that no evil human is a seeker after truth, since no philosophers are evil humans. a. deductive b. inductive
15. All squares are triangles, and all triangles are rectangles. Therefore, all squares are rectangles. a. deductive b. inductive
16. Deciduous trees are trees that shed their leaves. Maple trees are deciduous trees. Therefore, maple trees will shed their leaves at some point during the growing season. a. deductive b. inductive
17. Joe must make a lot of money teaching philosophy, since most philosophy professors are rich. a. deductive b. inductive
18. Since all mammals are cold-blooded, and all cold-blooded creatures are aquatic, all mammals must be aquatic. a. deductive b. inductive
19. I felt fine until I missed lunch. I must be feeling tired because I don’t have anything in my stomach. a. deductive b. inductive
20. If you drive too fast, you will get into an accident. If you get into an accident, your insurance premiums will increase. Therefore, if you drive too fast, your insurance premiums will increase. a. deductive b. inductive
21. The economy continues to descend into chaos. The stock market still moves down after it makes progress forward, and unemployment still hovers around 10%. It is going to be a while before things get better in the United States. a. deductive b. inductive
22. Football is the best sport. The athletes are amazing, and it is extremely complex. a. deductive b. inductive
(continued)
Practice Problems 5.3 (continued)
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Section 5.7 Arguments From Analogy
23. We should go to see Avatar tonight. I hear that it has amazing special effects. a. deductive b. inductive
24. Pigs are smarter than dogs. It’s easier to train them. a. deductive b. inductive
25. Seventy percent of the students at this university come from upper-class families. The school budget has taken a hit since the economic downturn. We need funding for the three new buildings on campus. I think it’s time for us to start a phone cam- paign to raise funds so that we don’t plunge into bankruptcy. a. deductive b. inductive
26. Justin was working at IBM. The last person we got from IBM was a horrible worker. I don’t think that it’s a good idea for us to go with Justin for this job. a. deductive b. inductive
27. If she wanted me to buy her a drink, she would’ve looked over at me. But she never looked over at me. So that means that she doesn’t want me to buy her a drink. a. deductive b. inductive
28. Almost all the people I know who are translators have their translator’s license from the ATA. Carla is a translator. Therefore, she must have a license from the ATA. a. deductive b. inductive
29. The economy will not recover anytime soon. Big businesses are struggling to keep their profits high. This is due to the fact that consumers no longer have enough money to purchase things that are luxuries. Most of them buy only those things that they need and don’t have much left over. Those same businesses have been firing employees left and right. If America’s largest businesses are losing employees, then there won’t be any jobs for the people who are already unemployed. That means that these people will not have money to pump back into the system, and the circle will continue to descend into recession. a. deductive b. inductive
Determine which of the following forms of inductive reasoning are taking place.
30. The purpose of ancient towers that were discovered in Italy are unknown. However, similar towers were discovered in Albania, and historical accounts in that country indicate that the towers were used to store grain. Therefore, the towers in Italy were probably used for the same purpose. a. argument from analogy b. statistical syllogism c. inductive generalization d. causal argument
Practice Problems 5.3 (continued)
(continued)
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Section 5.7 Arguments From Analogy
31. After the current presidential administration passes a bill that increases the amount of time people can be on unemployment, the unemployment rate in the country increases. Economists studying the bill claim that there is a direct relation between the bill and the unemployment rate. a. argument from analogy b. statistical syllogism c. inductive generalization d. causal argument
32. When studying a group of electricians, it was found that 60% of them did not have knowledge of the new safety laws governing working on power lines. Therefore, 60% of the electricians in the United States probably do not have knowledge of the new laws. a. argument from analogy b. statistical syllogism c. inductive generalization d. causal argument
33. In the state of California, studies found that violent criminals who were released on parole had a 68% chance of committing another violent crime. Therefore, a majority of violent criminals in society are likely to commit more violent crimes if they are released from prison. a. argument from analogy b. statistical syllogism c. inductive generalization d. causal argument
34. Psilocybin mushrooms cause hallucinations in humans who ingest them. A new spe- cies of mushroom shares similar visual characteristics to many forms of psilocybin mushrooms. Therefore, it is likely that this form of mushroom has compounds that have neurological effects. a. argument from analogy b. statistical syllogism c. inductive generalization d. causal argument
35. A recent survey at work indicates that 60% of the employees believe that they do not make enough money for the work that they do. It is likely that a majority of the people that work for this company are unhappy in their jobs. a. argument from analogy b. statistical syllogism c. inductive generalization d. causal argument
36. A family is committed to buying Hondas because every Honda they have owned has had few problems and been very reliable. They believe that all Hondas must be reliable. a. argument from analogy b. statistical syllogism c. inductive generalization d. causal argument
Practice Problems 5.3 (continued)
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Summary and Resources
Summary and Resources
Chapter Summary The key feature of inductive arguments is that the support they provide for a conclusion is always less than perfect. Even if all the premises of an inductive argument are true, there is at least some possibility that the conclusion may be false. Of course, when an inductive argu- ment is very strong, the evidence for the conclusion may still be overwhelming. Even our best scientific theories are supported by inductive arguments.
This chapter has looked at four broad types of inductive arguments: statistical arguments, causal arguments, arguments from authority, and arguments from analogy. We have seen that each type can be quite strong, very weak, or anywhere in between. The key to success in evalu- ating their strength is to be able to (a) identify the type of argument being used, (b) know the criteria by which to evaluate its strength, and (c) notice the strengths and weaknesses of the specific argument in question within the context that it is given. If we can perform all of these tasks well, then we should be good evaluators of inductive reasoning.
Critical Thinking Questions
1. What are some ways that you can now protect yourself from making hasty general- izations through inductive reasoning?
2. Can you think of an example that relates to each one of Mill’s methods of determining causation? What are they, and how did you determine that it fit with Mill’s methods?
3. Think of a time where you reasoned improperly about correlation and causation. Have you seen anyone in the news or in your place of employment fall into improper analysis of causation? What did they do, and what errors did they make?
4. Learning how to evaluate arguments is a great way to empower the mind. What are three forms of empowerment that result when people understand how to identify and evaluate arguments?
5. Why do you believe that superstitions are so prevalent in many societies? What forms of illogical reasoning lead to belief in superstitions? Are there any superstitions that you believe are true? What evidence do you have that supports your claims?
6. Think of an example of a strong inductive argument, then think of a premise that you can add that significantly weakens the argument. Now think of a new premise that you can add that strengthens it again. Now find one that makes it weaker, and so on. Repeat this process several times to notice how the strength of inductive arguments can change with new premises.
Web Resources http://austhink.com/critical/pages/stats_prob.html This website offers a number of resources and essays designed to help you learn more about statistics and probability.
http://www.nss.gov.au/nss/home.nsf/pages/Sample+size+calculator The Australian government hosts a sample size calculator that allows users to approximate how large a sample they need.
http://www.gutenberg.org/ebooks/27942 Read John Stuart Mill’s A System of Logic, which is where Mill first introduces his methods for identifying causality.
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Summary and Resources
appeal to authority See argument from authority.
argument from analogy Reasoning in which we draw a conclusion about something based on characteristics of other similar things.
argument from authority An argument in which we infer that something is true because someone (a purported authority) said that it was true.
causal argument An argument about causes and effects.
cogent An inductive argument that is strong and has all true premises.
confidence level In an inductive general- ization, the likelihood that a random sample from a population will have results that fall within the estimated margin of error.
correlation An association between two factors that occur together frequently or that vary in relation to each other.
inductive arguments Arguments in which the premises increase the likelihood of the conclusion being true but do not guarantee that it is.
inductive generalization An argument in which one draws a conclusion about a whole population based on results from a sample population.
joint method of agreement and differ- ence A way of selecting causal candidates by looking for a factor that is present in all cases in which the effect occurs and absent in all cases in which it does not.
margin of error A range of values above and below the estimated value in which it is predicted that the actual result will fall.
method of agreement A way of selecting causal candidates by looking for a factor that is present in all cases in which the effect occurs.
method of concomitant variation A way of selecting causal candidates by looking for a factor that is highly correlated with the effect in question.
method of difference A way of selecting causal candidates by looking for a factor that is present when effect occurs and absent when it does not.
necessary condition A condition for an event without which the event will not occur; A is a necessary condition of B if A occurs whenever B does.
population In an inductive generalization, the whole group about which the generaliza- tion is made; it is the group discussed in the conclusion.
proximate cause See trigger cause.
random sample A group selected from within the whole population using a selec- tion method such that every member of the population has an equal chance of being included.
sample A smaller group selected from among the population.
sample size The number of individuals within the sample.
statistical arguments Arguments involv- ing statistics, either in the premises or in the conclusion.
statistical syllogism An argument of the form X% of S are P; i is an S; Therefore, i is (probably) a P.
Key Terms
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Summary and Resources
strong arguments Inductive arguments in which the premises greatly increase the likelihood that the conclusion is true.
sufficient condition A condition for an event that guarantees that the event will occur; A is a sufficient condition of B if B occurs whenever A does.
trigger cause The factor that completes the cause chain resulting in the effect. Also known as proximate cause.
weak arguments Inductive arguments in which the premises only minimally increase the likelihood that the conclusion is true.
Answers to Practice Problems Practice Problems 5.1: Statistical Arguments: Inductive Generalization
1. b 2. b 3. c
Practice Problems 5.2: Mill’s Methods: Ways of Exploring Causality
1. a 2. a 3. b 4. b 5. b 6. b 7. b 8. b 9. b
10. a 11. b 12. a 13. a 14. a 15. a 16. a 17. b 18. a
19. b 20. a 21. b 22. b 23. b 24. b 25. b 26. b 27. a 28. b 29. b 30. a 31. d 32. c 33. c 34. a 35. c 36. c
1. a 2. b 3. d 4. b 5. b
6. d 7. d 8. c 9. a
Practice Problems 5.3: Arguments From Analogy
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"The Sovereign People Are in a Beastly State": The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness Author(s): Nicholas Mason Reviewed work(s): Source: Victorian Literature and Culture, Vol. 29, No. 1 (2001), pp. 109-127 Published by: Cambridge University Press Stable URL: http://www.jstor.org/stable/25058542 . Accessed: 05/04/2012 15:16
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Copyright ? 2001 Cambridge University Press. 1060-1503/01 $9.50
"THE SOVEREIGN PEOPLE ARE IN A BEASTLY STATE": THE BEER ACT OF 1830 AND VICTORIAN DISCOURSE ON
WORKING-CLASS DRUNKENNESS
By Nicholas Mason
i
On July 23, 1830, Parliament passed "An Act to permit the general Sale of Beer and
Cyder by Retail in England." Commonly known as the Beer Act of 1830, this law called for a major overhaul of the way beer was taxed and distributed in England and Wales. In
place of a sixteenth-century statute that had given local magistrates complete control over the licensing of brewers and publicans, the Beer Act stipulated that a new type of drinking establishment, the beer shop, or beer house, could now be opened by any rate-paying householder in England or Wales (Scotland and Ireland had their own drink laws). For the modest annual licensing fee of two guineas, rate-payers in England could now pur chase a license to brew and vend from their own residence.1
In addition to dramatically deregulating the licensing of drink establishments, the Beer Act also repealed all duties on strong beer and cider. By conservative estimates,
eliminating this tax immediately reduced the cost of a pot of beer by approximately twenty percent (Harrison, Drink 80). The only major restriction in the new law came in an amendment added in the House of Lords requiring all beer shops to close by 10 P.M.
Eventually beer-sellers would complain vociferously about the competitive advantage this
early closing time gave to publicans, who could remain open at all times except during Sunday morning church services. But in the months following the Beer Act's passage, beer-sellers had few complaints, as the law granted liberties and conveniences never
imagined under the old system. So attractive was the idea of the beer house to both retailers and consumers, in fact, that within six months of the Beer Act's taking effect, over
24,000 beer houses had sprung up throughout England and Wales (Gourvish and Wilson
16). As might be expected, the laboring poor, for whom beer had traditionally been a
dietary staple, were the chief beneficiaries of the Beer Act. Several decades after the Act's
passage, beer houses in England still rang with the chorus,
109
110 VICTORIAN LITERATURE AND CULTURE
Come, neighbours all, both great and small,
Perform your duties here, And loudly sing Live Billy our King,
For bating the tax upon beer. (Hughes 116; see also Hackwood 102)
However historically inaccurate this song may be ?
"Billy," or William IV, in reality had
little to do with the new law ? it effectively captures the general popularity the Beer Act
enjoyed among the nation's laborers.
At the opposite end of the social spectrum, however, the Beer Act quickly proved a
cause for concern. From the moment the new law took effect on October 10,1830, many members of England's privileged classes complained about the widespread debauchery the
law had supposedly incited. In a steady stream of sermons, poems, crime reports, and stump
speeches, the beer house came to represent intemperance, idleness, and a lack of discipline ?in short, all the self-destructive vices of the working class. The Reverend Robert Ousby, a
Lincolnshire curate, spoke for many when in 1834 he insisted that the only solution to Eng land's drunkenness problem was "'to close every beer-shop as soon as possible; to cut them
up root and branch.'" He continued, "The public-houses, I thought, were bad before, and I
endeavoured to counteract them; but it is absolutely impossible to do anything with these
beer-houses'" (Report from the Select Committee on Inquiry into Drunkenness 288). It is claims such as Ousby's that I would like to explore in this paper. As I will show
below, most evidence, both statistical and anecdotal, suggests that the Beer Act at least
initially increased levels of drunkenness among England's working class. But the most
significant long-term effect of the Beer Act of 1830,1 would argue, was not so much the
real levels of drunkenness it produced as the perceptions that were formed in its after
math. From 1830 until the 1870s, it was counted as something of a truism in middle- and
upper-class society that the Beer Act had touched off an irreversible course of working class drunkenness. Social commentators of all political persuasions, ranging from the conservative Henry Mayhew to the communist Friedrich Engels, viewed the Beer Act as
a defining moment in the fortunes of England's working class, and few flinched when in
1884 the historian Richard Valpy French argued that the Beer Act was "prominent among the legislative beacons of the present century" (349). In this essay, then, I will analyze the
discourse surrounding the Act of 1830, showing how writers and speakers in a wide variety of genres depicted this law as a major turning-point in working-class history. In the end, I
hope to have shown that while the year 1830 lacks the associative power of an 1832 or an
1848, it warrants recognition by cultural historians as a crucial moment in working-class
history for the simple reason that it is the year in which this landmark piece of legislation
passed into law.
//
Eight years after the passage of the Beer Act, James Bishop, Secretary of the Metropoli tan Protection Society, attempted to explain Parliament's motivations for passing this law:
The year 1830, it will be recollected, was ushered in by a period of unexampled privation and want of employment among the working classes, agricultural and commercial; insubordina
tion was spreading; and breaking of machinery and tumultuous meetings in the factory and
The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness 111
commercial districts, and riots and incendiary burnings in the agricultural parts of the king
dom, were unhappily too general. In this state of affairs, Government was
appealed to for
relief .... [T]he greatest measure of relief it was in their power to bestow was, the repeal of
the beer duty, and the opening of the Beer-trade. (6)
In his own terse manner, Cobbett shared this interpretation, dubbing the Beer Act "a sop to pot-house politicians" (398). Without question, the law offered much to like for English laborers, as it provided them with beer in greater abundance and at a lower price. In an
era when many laborers questioned whether any Act of Parliament had been designed with the nation's workers in mind, the Beer Act seemed strong proof that England's leaders were indeed concerned about the plight of the masses.
Much more contributed to the passage of the Beer Act, however, than compassion for
the laboring poor or paranoia over the prospects of insurrection. Most of the Parliamentary debates on the subject, in fact, centered not so much on the plight of workers as on theories
of free trade and the dangers of monopolies.2 Prior to 1830, the only way to obtain public house licenses was through local magistrates, whose stinginess is evident in the steep decline in the number of public houses per capita during the eighteenth and early nine
teenth centuries (Clark 333). By the 1820s, the common perception in England was that
magistrates were in the pockets of the dominant brewers, issuing licenses only to those who
agreed to sell a certain brand of beer and purposely keeping the number of pubs low so the
brewers could maintain their watchful eye over the industry. The actual extent of the
conspiracy between brewers and magistrates is, of course, difficult to measure, but statistics
bear out the degree to which the nation's leading brewers flourished in the late eighteenth and early nineteenth centuries. In 1748, most brewing still took place in the home or the
small shop, with the twelve major brewers of England only producing 42% of the country's beer. In contrast, by 1830 new technology and the now-standard contracts between large brewers and publicans had pushed the major brewers' market share up to 85%, essentially twice what it had been eighty years earlier (Mathias 26; see also Park 64). Although in some
regions small brewers still prospered, the trend clearly pointed to a future where the major brewers would have absolute control over the distribution of beer in England.
Not surprisingly, in an era when the public was increasingly enamored of free trade and
hostile towards anything resembling a monopoly, the supposed collusion between the big brewers and the country's magistrates became a frequent subject of complaint. During the
first quarter of the century, several official and unofficial expos?s attempted to unearth the misdeeds of the brewing industry. Perhaps the most damning of these was an 1818 survey
by the Committee on Public Breweries which documented not only the extent to which the
major brewers had dominated the manufacturing and retailing of beer, but also how they had colluded to fix prices and adulterate their product with cheap stimulants (Gourvish and
Wilson 6). Enflamed by reports such as this, fourteen thousand Londoners signed an 1818
petition protesting the high prices and poor quality of the city's liquor (Clark 334).
By the early 1820s, free-traders and monopoly busters had persuaded Parliament
to begin debating brewing reform. It took another drink-related crisis, however, to pro duce the final momentum for the Beer Act. Independent of the beer debates, in 1825
Parliament passed an act calling for a 40% reduction of the duties on spirits, a measure
theoretically designed to reduce the temptation toward illicit trading and tax-dodging
(Gourvish and Wilson 10). As would be the case five years later with the Beer Act,
112 VICTORIAN LITERATURE AND CULTURE
this reduction of the spirits tax at least initially prompted a significant increase in con
sumption. Whereas between 1821 and 1825 the annual quantity of spirits consumed in
England fluctuated between 3.7 and 4.7 million gallons, the average annual consumption between 1826 and 1830 exceeded 7.4 million gallons (Harrison, Drink 66-67).
Much of the blame for this upsurge in consumption fell on the working class, who were
said to have abandoned their pot of beer for the much more intoxicating dram of gin. Once
again, statistics are useful here in demonstrating the declining popularity of beer. In the
early nineteenth century, when the population of England and Wales was increasing by slightly more than 15% per decade, beer sales were essentially stagnant. Between 1818 and
1830, for instance, English brewers never produced more than 4.1 million nor less than 3.6 million barrels of beer in a given year. All told, in the decade prior to the Beer Act, roughly the same amount of beer was being consumed in England and Wales as had been consumed
thirty years earlier, when the population was approximately one-third smaller (Mathias 543).3
In the eyes of many, beer's declining popularity signified much more than a simple shift in taste. Rather, it represented the abandonment of a key component of Englishness.
As George Evans Light has recently shown, since the sixteenth century, when the English began to cultivate hops on a large scale, beer had factored prominently into the English national identity. From a practical standpoint, beer was not only safer than the cholera-in fested water of many communities, but it was also widely seen as a major source of nutrition for the poor. Into the late Victorian age, laborers made claims such as, "Beer's
made of corn as well as bread, and so it's stand to reason it's nourishing." Others argued that beer helped them "keep up their strength" during the long workday (Humpherys 59-60). Over time, however, beer became much more than a practical drink, but a means
by which the English distinguished themselves from the wine-drinking French. In 1909, Frederick Hackwood recollected how "at a recent Conference of Brewers, Lord Burton claimed that this country owed its high and proud position among the nations of the earth
simply on account of its characteristic dietary, 'Beef and Beer.' Whereupon some one
made the waggish comment, 'Why drag in the Beef?'" (94). That the Beer Act of 1830 was at least in part designed to turn English workers from gin
back to the national drink is evidenced in a commentary appearing in the October 21,1830 issue of the Times. Written less than two weeks after the Act went into effect, this brief arti cle defends the new law primarily on the grounds that beer was the lesser of two alcoholic evils. The anonymous writer of the article contends, "Now if, as is assumed (not proved) by the cavillers, a greater number of people do indulge themselves inordinately with the Eng lish beverage of beer than formerly, it is plain to us, at least, that the consumption of beer has
been increased at the expense of ardent spirits, and to their positive diminution." After a
paragraph championing the free trade principles demonstrated by the Act, the writer goes on to point out that, compared to spirits, beer is "a far more salubrious, or rather a far less
destructive liquor" and that "to commit excess in beer costs considerably more money, time, and trouble, than a similar performance with British gin, or whisky."
///
With its promises of breaking up a monopoly, promoting free trade, and converting the poor away from the false religion of gin and back to the orthodoxy of beer, the
The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness 113
Beer Bill received unusually bipartisan support in the House of Commons, where it
passed on its second reading by a margin of 245 to 29 votes. The only serious opponents of the bill were publicans and brewers, who anticipated that the new beer houses would
vastly diminish their market share, and High Tories, who traditionally had allied them selves with the brewing industry. Both of these groups, however, lacked the numbers in the House of Commons to seriously challenge the popular measure, and thus it easily
passed to the House of Lords, where after further discussion and a decision to add the
closing-time amendment, the bill became law on July 12, 1830 (Harrison, Drink 75-79). Even evangelists and moral reformers, who were laying the foundations of the Tem
perance Movement in 1830, voiced little objection to the new law, holding out hope that the spread of beer shops would put an end to the gin craze of the late 1820s (Clark 336).
Over the years, historians have presented a wide range of views on what the actual
legacy of the Beer Act turned out to be. Prior to the 1970s, the standard interpretation was
what Brian Harrison has labeled the "debauchery theory." Most famously promoted in
Sidney and Beatrice Webb's 1903 study, The History of Liquor Licensing in England, this
theory holds that the Beer Act led to widespread degeneracy and should thus be remem
bered as one of the great legislative blunders of British history. For nearly three-quarters of a century, the Webbs' reading of the Beer Act remained largely unquestioned. Since
then, however, several of the most prominent historians of England's drink industry have raised doubts about just how large of an effect the Beer Act actually had on English culture. In his landmark study, Drink and the Victorians (1971), Harrison concedes that for some "working people the beerhouse helped to emancipate their leisure from super vision" (83). This said, however, he calls for restraint when interpreting the long-term effects of the Beer Act, arguing that the Webbs' strong feelings about this law were less the product of rigorous analysis than the couple's "taste for discipline, their puritanism, and their distance from popular culture" (84). Similarly, T. R. Gourvish and R. G. Wilson have recently argued that while the Beer Act certainly had a short-term impact on English society, by the 1840s the law's novelty had all but died out and it ceased to have a
significant impact on daily life in England (16). Although picking sides in this debate is not my goal here, it is important to examine
at least briefly the evidence that has produced such varying interpretations. As I alluded to in the beginning of this paper, the statistics measuring the Beer Act's influence can be
staggering. In the first six months after the law took effect, the existing 51,000 licensed
public houses in England and Wales were joined by over 24,000 beer shops (Gourvish and Wilson 3,16). According to some counts, Liverpool alone supported eight hundred beer
shops within three weeks of the Act's implementation, a number which only grew over the next several months, when fifty new beer shops reportedly opened in the city every week
(Gourvish and Wilson 16; Webb and Webb 116). As might be expected, the steepest rise in the number of beer shops occurred in late 1830 and early 1831, when the idea of vending from one's own home remained something of a novelty. Nevertheless, it wasn't until
nearly a decade later that the rate at which beer licenses were issued began to decrease
significantly. By 1838 at least 40,000 ?
and, according to one count, closer to 46,000 ?
beer shops were operating in England and Wales, a number which neared the sum total of other public houses, which had grown in number from 51,000 to 56,000 during the decade (Inhabitant 8; Hamer 3; Gourvish and Wilson 16).
114 VICTORIAN LITERATURE AND CULTURE
Not surprisingly, this exponential increase in the number of retailers reflected a
corresponding rise in the amount of beer that was being consumed during the 1830s. For
the years following the repeal of the beer tax, the most reliable figures on beer con
sumption come from records on malt duties. These figures indicate that between 1829
and 1831 there was a 40% increase in the amount of malt taxed in England and Wales.
Tracking the shifts in malt consumption over a longer period, one sees an increase from
an average of 26.5 million imperial bushels being taxed annually in the 1820s to 33.5
million in the 1830s, a growth of 26% (Mitchell 402). Even factoring in the 16% popu lation increase that occurred in England and Wales between 1821 and 1831, the malt
duty figures still strongly suggest that the Beer Act led to immediate increases in con
sumption. For the first time in the nineteenth century, beer drinking was on the rise.
As Harrison and Gourvish and Wilson point out, however, it is debatable whether the
beer house continued to have so significant an impact on the English cultural landscape
beyond the 1830s. Technically, the number of beer houses in England and Wales actually increased between 1840 and 1869, the year in which the system of magisterial licensing was
reinstituted. But, all told, the increase in the number of beer shops was relatively small in
light of the population boom of the mid-nineteenth century. Moreover, malt duty figures
suggest that after the significant upswing in beer drinking during the 1830s, beer produc tion fluctuated between 30 and 40 million imperial bushels per year throughout the 1840s
and 1850s (Mitchell 402). Statistics on drunkenness arrests in London show a similar trend of peak years
coming on the heels of the Act of 1830 followed by a gradual leveling out or decline.
During each of the first three years following the passage of the Beer Act, the Met
ropolitan Police arrested approximately 20 people per 1,000 residents of London on
drunkenness charges. Between 1834 and 1839, drunkenness arrests dipped slightly, but
continued at a rate of roughly 13 per 1,000 residents. In 1840, however, the number of
drunkenness arrests dropped to 8 per 1,000 residents, beginning a 35-year trend in which
arrests for intoxication never again exceeded 1% of the population (Report from the
Select Committee of the House of Lords 1:342). At least part of the decline in arrests
per capita, of course, may have resulted from changes in police department policies or
the simple fact that the police force was overwhelmed by the swelling population. It
seems at least as likely, however, that these statistics point to a general trend also seen
in the figures for the licensing of beer houses and the taxing of malt ?
namely, that
beer consumption surged in the aftermath of the Beer Act but began to decline roughly a decade later.
IV
EVEN USING THE BEST available statistics, tracing trends in alcohol consumption or drunk
enness from a distance of more than a century and a half is at best an inexact science.4 This
is particularly the case when dealing with England, given the sheer number of recorded
binges in the country's past. Nearly every period of modern history is replete with ac
counts of a soused English populace. During the sixteenth century, Rabelais coined the
simile "as drunk as an Englishman" (Hackwood 154). A century later, Shakespeare's lago observed that the English were "most potent in potting" and that "your Dane, your
German, and your swag-bellied Hollander ... are nothing to your English" in their ability
The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness 115
to imbibe (II.iii.77?79). During the first half of the eighteenth century, several authors and
painters ? most notably Gay in The Beggar's Opera and Hogarth in Gin Lane
? depicted
what they saw as the general drunkenness of England's underclass. Even during the
Romantic period, an era historians have often treated as a moment of sobriety between
the alcohol sprees of the mid-eighteenth and mid-nineteenth centuries, many Britons
described their age as one marked by excessive drinking. As Anna Taylor has recently shown, during the late eighteenth and early nineteenth centuries several physicians began studying England's "crisis" of heavy drinking, and Coleridge
? far from a teetotaler himself ? lamented, "'No Country in God's Earth labours under the tremendous curse of
Drunkenness equally with England'" (qtd. in Taylor 13). In light of commentaries such as these, it can be difficult when reviewing English
history to tell when one binge ends and another begins. In the case of the increased levels
of drunkenness in the 1830s, for example, it might be asked whether this phenomenon was
merely a continuation of the gin craze of the late 1820s or a distinctive consequence of the
Beer Act. Pushing the issue even further, in light of Taylor's recent findings on drunken ness in the Romantic era, it might even be argued that England experienced one unbroken
spell of intoxication from the 1720s through the end of the Victorian Age.
Considering the frequency of drunkenness in England's cultural past, I would argue that what ultimately makes the Beer Act of 1830 distinctive is not the actual debauchery it produced, but the degree to which for many Victorians it came to symbolize a clear
turning point in the behavior of the laboring people. Whether in Temperance tracts, satirical poems, or Parliamentary reports, much of the Victorian discourse on working class drunkenness repeats a narrative in which the Act of 1830 almost instantly placed a
beer house in every neighborhood, thus exposing the poor to temptations they were too
weak to resist. In the perception of many middle- and upper-class Britons, after 1830 the
nation's poor were never the same. A working-class culture previously centered around
the home, the church, and the work-site now quite clearly found its focal point in the
neighborhood beer house.
Literally within hours of the Beer Act's passage, several members of the privileged classes believed they were witnessing unprecedented levels of drunkenness among the nation's laborers. In perhaps the most famous observation on the Beer Act's immediate
effects, the Reverend Sydney Smith reported to John Murray, "Everybody is drunk. Those who are not singing are sprawling. The sovereign people are in a beastly state"
(Holland 481). In another account, an observer at King's Lynn recorded how October 10, 1830 "was kept as a jubilee by all the devotees of Sir John Barleycorn," with drunken
workers spilling out of the town's forty-plus new beer houses into the streets (Clark 336). Three years later, William Holmes, a former mayor of Arundel, recalled before a Select Committee on the Sale of Beer, "T was obliged to get out of my gig three times from
people coming along, waggoners drunk, when I was returning from shooting on the very
day of the operation of this Bill'" {Report from the Select Committee on the Sale of Beer
45-46). Even an ardent supporter of the Beer Act had to admit in the October 21,1830 issue of the Times that "people have now and then, since the Act was passed, been seen
'summot fresh' with beer."
Within a few years, a number of voices were calling for a repeal of the new law. An
anonymous pamphlet from 1833, for instance, labels the statute "one of the most mischie vous measures, which a mistaking policy ever devised" and claims that in a few short years
116 VICTORIAN LITERATURE AND CULTURE
the law had "materially increased the domestic distress" of the laboring poor (A Few
Remarks 4). To illustrate how the Beer Act had proven particularly disastrous in the
provinces, the pamphleteer includes a "General Report of the disturbed district of East
Sussex." After recounting how beer shops had immediately become havens for prosti tutes, thieves, and radicals, this report quotes an East Kent magistrate who believes "'no
single measure ever caused so much mischief in so short a time, in demoralizing the
labourers'" (23). One of the best indicators of how widely discussed the Beer Act was during the 1830s
is its appearance in satire. In the mid-1830s, for instance, the parodist James Smith penned an eight-line poem simply entitled "Beer Shops":
"These beer shops," quoth Barnabas, speaking in alt,
"Are ruinous ?
down with the growers of malt!"
"Too true," answers Ben, with a shake of the head,
"Wherever they congregate, honesty's dead.
That beer breeds dishonesty causes no wonder,
'Tis nurtured in crime ?
'tis concocted in plunder; In Kent, while surrounded by flourishing crops, I saw a rogue picking a pocket of hops."
While the agricultural wordplay of the punch-line ? a "pocket" as used here is literally a
large bag used for harvesting hops ?
may be lost on most modern readers, the poem's
general idea remains fairly clear. Smith's target is not the beer shop itself, but the debates
that have followed in its wake. Specifically, he lampoons the frenzied morality of Temper ance advocates, suggesting that detractors of the Beer Act have seized upon the slightest
misdeeds of the working class to make sweeping statements about the law's "ruinous"
effects.
By 1833 the types of complaints Smith parodies had become so numerous that Parlia
ment saw fit to organize a Select Committee on the Sale of Beer. This would be the first
in a long series of such bodies that summoned magistrates, physicians, temperance work
ers, and other members of the privileged classes to testify concerning the social problems excessive drinking was causing among England's poor. Typical of the testimonies given before the 1833 committee is that of the chaplain of Reading gaol, who estimated that
"'four-fifths of the offences committed by the agricultural population are traceable to beer
houses'" (Report by the Select Committee on the Sale of Beer 86). Similarly, the chief
constable of Leeds reported that there had been three times as many arrests for drunken
ness in his city in the thirty months since the Beer Act as in the preceding three years.
Following up on the work of the 1833 committee, in 1834 the Select Committee on
Inquiry into Drunkenness deposed witnesses from around the country, who gave further
testimony about the general dissipation that had followed the Beer Act. Edwin Chadwick, a member of Parliamentary committees on factory labor and the poor laws, reported how
"'the workman when he comes home from work, in passing through the village where
there was formerly only one public-house, has now to run the gauntlet of three or four
beer-shops, in each of which are fellow-labourers carousing, who urge him to stay and
drink with them'" (Report from the Select Committee on Inquiry into Drunkenness 31). At
the same session, the Temperance worker Joseph Livesey related how he had long been
The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness 111
in the practice of visiting the poor on Sunday mornings, which had given him occasion to note the significant increase in levels of drunkenness since the appearance of beer houses
(89-92).
Perhaps the most interesting trend in the 1834 testimonies, however, is the number of witnesses who suggested that the Beer Act had not only created beer houses, but had also
precipitated the conversion of traditional pubs into "gin palaces." Witness after witness detailed a pattern in which competition from beer houses drove publicans to remodel their taverns or inns into extravagant gin palaces. The glamour of these new establishments,
according to most accounts, lured in curious laborers, which led to another wave of
gin-drinking. In the end, then, rather than turning workers away from gin, the Beer Act had only increased the amount of spirits being consumed by the working class. The most succinct testimony concerning this trend is that of Robert Edwards Broughton, a London barrister and magistrate, who explained,
"In the course of things, [beer houses] very much interfered with the business of the regular publicans, and the capital laid out by the original houses was materially wasted and damaged, and therefore the persons are driven, many of them as a matter of necessity, to try those
schemes which should retrieve them, or prevent them from failing, and that is the cause of a great number of what are called in the newspapers gin-palaces. The old public-houses,
where a man could have his steak dressed, and sit down and take his ale, are extinct; they are obliged to convert them into splendid houses, and sell gin at the bar." {Report from the Select Committee on Inquiry into Drunkenness 14-15)
As Broughton implied, whereas the only enticements of the beer house tended to be the company and the drink, the gin palace offered workers an escape from reality, complete with hired musicians, comfortable surroundings, and strong drink. Observing the rapid proliferation of gin palaces following the Beer Act, in 1835 Dickens wrote "Gin Shops," an essay he eventually included in Sketches by Boz. In this piece, Dickens describes how the fashion for ostentatiously decorating one's shop had begun with the haberdashers and drapers of London but had recently spread "with tenfold violence" to the city's publicans. Trying to build the most extravagant gin palace yet, publicans had taken to "knocking down all the old public houses, and depositing splendid man
sions, stone balustrades, rosewood fittings, immense lamps, and illuminated clocks, at the corner of every street" (182). The allure of such edifices for workers is perhaps
best captured in another commentary from 1835, Cruikshank's The Gin Juggarnath, Or, the Worship of the Great Spirit of the Age (Figure 1). With dark clouds gathering ominously in the background, Cruikshank's gaudy gin palace stands as a beacon for the drunken masses, who throng forward to partake of its splendors. As if the scene
weren't already self-explanatory, Cruikshank includes a caption, warning that the Gin
Juggarnath's "Devotees destroy themselves ? It's progress is marked with desolation,
misery and crime."
Concluding that the Beer Act was at least partially responsible for the rise of the gin palace and a number of other social ills, the Select Committee on Drunkenness issued a
report in 1834 calling for a major crackdown on beer houses. This report demonstrates the extent to which the new beer law was already being recognized as a turning point in the behavior of English laborers. Rather than condemning drunkenness as a vice prevalent
118 VICTORIAN LITERATURE AND CULTURE
Figure 1. George Cruikshank, "The Gin Juggarnath," 1835. Etching, from William Bates, George Cruikshank: The Artist, the Humorist, and the Man (London: Houlston and Sons, 1879). Between
56-57.
among all ranks of society, the Committee's report focuses explicitly on the working class,
going so far as to maintain that "the vice of Intoxication has been for some years past on
the decline in the higher and middle ranks of society; but has increased within the same
period among the labouring classes" (Report from the Select Committee on Inquiry into
Drunkenness iii). In the Committee's assessment, the crisis of working-class drunkenness
had reached such a point that it now constituted a distinct threat to the nation's economic
well-being. Across the country, one work day in six was reportedly being lost to drunken ness (v), and, all told, the Committee concluded that "the retardation of improvement caused by the excessive use of Intoxicating Drinks, may be fairly estimated at little short
of fifty millions sterling per annum" (vi). Based on findings such as these, in 1834 Parliament conceded that "much Evil" had
arisen from the creation of beer houses and determined to make amends by revising the
Act of 1830 ("An Act to amend an Act"). The new law increased the annual beer-shop
licensing fee from two to three guineas, granted police unlimited rights to inspect beer
shops for fugitives, and required beer-sellers to obtain a "Certificate of Good Character"
The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness 119
signed by six rate-payers in their district.5 The most radical aspect of the 1834 Act, however, came in its division of beer shops into two types: those that could serve drinks on-site and those that could only sell "take-out" beer.
According to the dictates of the new law, beer-sellers with an on-site license were to
post a sign above their door declaring "To be drunk on the Premises." The potential ambiguity in this wording provoked Richard Polwhele to write "Beershops," another satirical poem from the era that suggests the extent to which the Beer Act had become a
favorite subject of popular discourse:
Whether Beershops encourage or not inebriety, Of opinions, it seems, there has been a variety. But, unless he would fly in the face of an Act,
(The product, too true, of the crazy or crackt) The Lord or the villein, will hail kidliwinks, An honester subject the deeper he drinks; And a sot tho' he be, who can fancy the blame is his,
Required by the law "to be drunk on the premises?"
Unlike James Smith, who in the previously quoted poem mocks the doomsday rhetoric of the Temperance crowd, Polwhele sees the nation's lawmakers as being the most deserving target of ridicule. Suggesting that the Beer Act of 1834 could only have proceeded from the "crazy or crackt," Polwhele questions the competence of a legislative body that
attempts to curb drunkenness by posting signs in beer shops advising laborers "to be drunk on the premises." Cruikshank also noticed the potential humor in the new signs, drawing a caricature of workers who had become "'Drunk' ? according to Act of Parliament"
(Figure 2).
Considering Polwhele's and Cruikshank's mockery of the Act of 1834, it comes as
little surprise that the new law ultimately did little to calm the storm over beer shops. That the original Beer Act of 1830 continued to be viewed as a pivotal event in English cultural life even after the Act of 1834 is manifest in an 1838 Manchester pamphlet debate. The
participants in this dispute were two local citizens: "An Inhabitant of Manchester," who
not-so-artfully attempts to disguise his actual identity as a publican, and John Hamer, one
of the new class of beer-sellers. The first shot in the battle came from the Inhabitant, who
begins his pamphlet with the sweeping claim that "if ever public opinion was unanimous
upon a parliamentary measure, it is in its condemnation" of the Beer Act of 1830 (3). He
proceeds to suggest that publicans and beer-sellers alike were on the verge of financial
collapse as a result of the competition the Beer Act had introduced. Furthermore, he
argues that beer shops had quickly established themselves as hubs for England's under world. In the pamphlet's most agitated passage, the Inhabitant maintains, "The Beer Act has planted the source of vice at every man's door. As if drunkenness was not before
sufficiently prevalent, that Act has sent for 40,000 missionaries to inculcate it. Under its
influence, the most odious exhibitions of indecency have acquired a locomotive power, by which they have penetrated every lane, alley, and street" (9).
Hamer's relatively polished reply to the Inhabitant's "malignant and scurrilous at tack" addresses his antagonist's arguments point by point. From Hamer's perspective, by ending the tyranny of magistrates and brewers, lowering prices, and improving the quality
120 VICTORIAN LITERATURE AND CULTURE
Figure 2. George Cruikshank, "To Be Drunk on the Premises," c. 1834. Illustration, from George
Cruikshank, Four Hundred Humorous Illustrations (London: Simpkin, Marshall, Hamilton, 1895),
of England's beer, the Act of 1830 had accomplished all of its original purposes (5).
Significantly, though, the only point from the Inhabitant's pamphlet that Hamer sidesteps is the link between the rise of beer shops and the increase in levels of working-class drunkenness. Perhaps perceiving that public opinion on this matter was firmly set, Hamer
chooses first to express how strongly he "deplores" the "great national evil" of drunken
ness and then to deflect a portion of the blame away from beer houses and toward "those
public-houses which are regularly the scenes of midnight revelry and dissipation" (16). In
the end, Hamer's inability to dissociate the beer house from working-class drunkenness
suggests that in the late 1830s even a beer-seller had little hope of disproving what most
middle- and upper-class Britons had come to accept as an established fact ? that the
Beer Act of 1830 had had a disastrous impact on the behavior of England's laborers.
V
While the conversations from the 1830s about the Beer Act's effects are certainly
important, the Act's real legacy comes through the continued prominence it held in the
The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness 121
discourse on working-class drunkenness for decades to come. Even after the growth rate of beer shops began to decline in the early 1840s, the beer shop remained a prominent symbol of working-class degeneracy in the discourse of the privileged classes. For several
generations to come, middle- and upper-class Britons would remember how seemingly overnight the beer house went from being non-existent to being conspicuously present on
nearly every block of the average English town. Given this memorable explosion of
drinking establishments, 1830 seemed an obvious (and conveniently round) date for the
beginning of the most recent phase in working-class history. In surveying the plight of England's poor during the 1840s, both Friedrich Engels and
Henry Mayhew devoted a significant amount of space to what they saw as the trail of drunkenness that followed the Beer Act. In The Condition of the Working Class in
England (1844), Engels notes how beer shops had become the hubs of Manchester's slums. This was not due to the elegance of these houses or the comforts they provided, but because the poor "are deprived of all enjoyments except sexual indulgence and drunken
ness, are worked every day to the point of complete exhaustion of their mental and
physical energies, and are thus constantly spurred on to the maddest excess in the only two
enjoyments at their command" (129). While Engels assures his readers that he advocates neither promiscuity nor intoxication, he also makes it clear that the blame for working class drunkenness lay not with the workers themselves, but with the nation's leaders and the system they had created. In ratifying the Beer Act, he argues, Parliament "facilitated the spread of intemperance by bringing a beerhouse, so to say, to everybody's door" (152).
Although he stops short of arguing, as some Temperance workers were wont to do, that the Beer Act was little more than a conspiracy of the rich to subjugate the poor (Harrison, "Pubs" 183), Engels insists that the poor should not be held accountable for conditions over which they have no control.7
In this era's other monumental survey of working-class life, London Labour and the London Poor (1849-52), Mayhew records numerous tales of families on the brink of starvation because of the ever-present appeal of the local beer shop. The typical pattern is for a husband to receive his wages on Saturday night and to have spent them all on
drink by Sunday morning (423-24). One working-class woman observes that in post-Beer Act London, "a shilling goes further with a poor couple that's sober than two shillings does with a drunkard" (127). Overall, Mayhew describes a potentially catastrophic trend
among the poor of beer shops leading to drunkenness, drunkenness leading to poverty, poverty leading to children on the streets, and children on the streets leading to robbery, prostitution, and the general decay of English society (162). Whereas Engels sees a
proletarian revolution as the only solution to the cycles of poverty, Mayhew suggests that the most effectual means of reducing the poor rates would be providing "wholesome amusements" as alternatives to drink (41-42). In Mayhew's thinking, if working people have alternatives to "the conversation, warmth, and merriment of the beer-shop" (17), they will be able to save their wages and rise from the squalor into which they have sunk.8
To the extent that Mayhew's comments on the evils of drunkenness derive from his
participation in the Temperance Movement, London Labour and the London Poor be
longs to a large body of Temperance literature designed to counteract or overturn the Beer Act. Perhaps the figure most responsible for the dissemination of the Temperance message during the mid-nineteenth century was Cruikshank, whose Gin Juggernath was
122 VICTORIAN LITERATURE AND CULTURE
just one of many images he created to convince the nation of the dangers of drink.
Cruikshank's extremely popular series The Bottle (1847), which tells of a father's addiction
to drink and how it reduces his family from relative comfort to misery, murder, and
insanity, was priced at a shilling so that, as Cruikshank explained, "'it might be within the
reach of the working classes'" (qtd. in Evans and Evans 127). A year after the remarkable success of The Bottle, Cruikshank produced a sequel
entitled The Drunkard's Children. As had been the case with The Bottle, The Drunkard's
Children was accompanied by a Charles Mackay poem that added details to the story found in Cruikshank's drawings. One scene from The Drunkard's Children that Mackay
develops at some length is how the drunkard's son, Edward, receives lessons in debauch
ery amid "the 'Beer-shop's' wild uproarious throng." Mackay's poem invokes all of the
increasingly stereotypical images of beer-shop culture, as is evidenced in the fourth stanza
of Part III:
There Ben, half-drunken, bounets drunken Hal,
There Jack, that swept the crossing all the day, Calls for his pipe and pot: there joyous Sal Takes from her prostrate Joe his "yard of clay"; Places her bonnet, decked in ribbons gay, Upon his head, and sports his fantail hat; And Costermonger Dick attempts a lay From the "Flash Songster," dull, obscene, and flat
All noises mixed in one, songs, laughter, shrieks, and chat.
For most middle- and upper-class Victorians, for whom beer shops were socially
off-limits, images such as these provided the only access they had to beer-shop society. Not
surprisingly, then, beer shops increasingly became favorite symbols of the moral depths to
which the English working class had sunk. Typical of the mid-nineteenth-century rhetoric
surrounding beer shops and the Beer Act is the Reverend J. M. Calvert's 1852 sermon The
Impropriety of Religious Characters Keeping Beer-Houses. As his title suggests, Calvert's
general message is that, no matter how religiously inclined, Christians who sell beer from
their homes inevitably fall under condemnation for bringing drunkenness to their commu
nities and, more deplorably, vice into their homes. When addressing the specific effects of
the Beer Act, Calvert falls back on what had by 1852 become the conventional ways of
assessing the law's effects. Near the beginning of his sermon, he reflects, "I am just old
enough to remember the law being passed which gave leave to open houses of an inferior
character for the sale of beer and porter; and well do I recollect the change for the worse
which took place in the village where I then resided, after the opening of two or three of
these Beerhouses" (4).
By 1864 public distress over the Act of 1830's ruinous effects had become so wide
spread that a group of concerned citizens organized the "Special Committee of Temper ance Reformers for Procuring the Repeal of the Beer Act of 1830." Five years later, the
Convocation of Canterbury published a Report by the Committee on Intemperance, which
demonstrates the extent to which the discourse surrounding the Beer Act remained
virtually unchanged since the Select Committee hearings of the early 1830s. For example, one clergyman quoted in this report maintains,
The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness 123
"If I am asked to point out the great cause and encouragement of Intemperance, I have no
hesitation in ascribing it in a great measure to that most disastrous Act of Parliament which set Beer-Shops on foot. It has inflicted
a terrible curse on this country. I would sooner see a
dozen Public-Houses in a parish than one Beer-Shop. I believe no greater boon could be
conferred on the working classes than to repeal that Act." (23)
Another veteran minister recollected, '"Many families in which the wives and children were formerly well clad and apparently well fed have since the introduction of the
Beer-Shops been in rags and poverty-stricken'" (25). Records from the numerous Parliamentary hearings on drunkenness held during the
1870s are filled with similar claims about the Beer Act of 1830. In 1872, for instance, several magistrates, constables, and physicians told the Select Committee on Habitual
Drunkards that the simple solution to England's drunkenness problem would be reducing or altogether eliminating beer houses (12-13,136,176). Similar testimonies occur through out the three reports issued in 1877 by the Select Committee of the House of Lords on the
Prevalence of Habits of Intemperance. Speaking before this committee, Henry William
Schneider, the Mayor of Burrow-in-Furness, called the beer shop "the very worst style of
house that is licensed in England; it is impossible to have anything so bad" (2: 333). Another witness, the Reverend Canon Ellison, a former chairman of the Church of
England Temperance Society, conceded that the original intentions of the Beer Act were
"essentially and thoroughly good" (3: 84). The Act's results, however, had been nothing short of disastrous in his estimation. According to a statistical report Ellison prepared for
the Committee, between 1824 and 1874 there was an 88% increase in England's popula tion but a 92% increase in the consumption of beer, a 237% increase in the consumption of British spirits, a 152% increase in the consumption of foreign spirits, and a 250%
increase in the consumption of wine {Reports from the Select Committee of the House of Lords on the Prevalence of Habits of Intemperance 3:85). From Ellison's perspective, the
Beer Act of 1830 had sparked a general revolution in the way all alcoholic beverages were
marketed and distributed in England. Simply put, the single most significant cause of
England's drunkenness problem in 1877, as Ellison saw it, was the Beer Act that had been
passed forty-seven years earlier.
Although Ellison doesn't appear to have noticed, by 1877 what he had long been
campaigning for ? the demise of the English beer house
? was already underway. The first major blow to the common beer house came in 1869, when Parliament placed the licensing of beer-sellers under the control of magistrates, effectively reversing one
of the most radical elements of the Act of 1830. At approximately the same time, the
country's major brewers began aggressively purchasing beer shops and pubs and ex
panding their distribution networks. Under the watchful eye of both the magistrate and
the large brewer, the distinctively working-class character of the beer house gradually
disappeared (Clark 338, Gourvish and Wilson 19-20). Because of this, historians of
British brewing have traditionally defined the "beer shop era" as stretching from 1830
until roughly 1870.
During this forty-year span, the beer shop undeniably had a profound effect on the
English cultural landscape. For the poor, the rise of the beer house impacted where they
congregated, what they drank, and how much beer they consumed. At the same time, the
Beer Act also influenced the drinking customs of the middle and upper classes. In the
124 VICTORIAN LITERATURE AND CULTURE
years following 1830, the social codes for what respectable Englishmen could drink and
where they could drink it changed dramatically. Whereas in the early nineteenth century beer was a common form of refreshment for the wealthy, from roughly 1830 forward, tea,
spirits, and wine increasingly replaced beer in the homes of the well-to-do (Davidoff and
Hall 385, Pool 212). Moreover, during this same era London gentlemen began to do their
drinking at home, establishing a trend that would eventually spread to the provinces. As
Harrison documents, "By the late 1830s the village inn, where all classes drank together, had become a nostalgic memory
? even if it had never been as widespread as its
admirers imagined. By the 1850s no respectable urban Englishman entered an ordinary
public-house" (Drink 46).9 In the end, the Beer Act of 1830, a law initially designed to ease class tensions, only
exacerbated the rifts between the various ranks of society. On one level, the Beer Act
literally isolated the rich from the poor, replacing the public house of the eighteenth
century ? which in many ways had been the embodiment of the Habermasian public
sphere ? with the exclusively working-class beer house. Although no laws prohibited the
rich from frequenting a beer house or the poor from drinking at an inn, the Beer Act
intentionally made the beer house the most convenient and inexpensive place for the poor to do their drinking. In addition to increasing the actual space between the rich and the
poor, the Beer Act also widened the imaginary gap separating the classes. Cumulatively, the myriad attacks on the Beer Act and the beer shops it produced only reinforced
stereotypes of the poor. During an era when wealthy Britons increasingly prided them
selves on their domestic virtues, the discourse surrounding the Beer Act suggested that
the working poor were moving in the opposite direction, abandoning the simple comforts
of the home for the revelry of the beer shop. The ultimate legacy of this law, then, was
much more than the beer binge of the 1830s. In significant ways, the Beer Act of 1830
helped shape how the rich viewed the poor and, undoubtedly, how the poor came to view
themselves.
Brigham Young University
NOTES
1. At this early point in my argument, I should clarify some of the terms I will be using
frequently. Technically speaking, a
"publican" is anyone who operates a
"public house." A
"public house," in the term's broadest sense, could be any site where alcohol is legally consumed on the premises. In the debates surrounding the Beer Act, however, the term
"public house" was narrowed to refer primarily to the old-style establishments, such
as
taverns and inns. In contrast to beer houses, which were generally quite spartan and were
only licensed to sell beer, public houses tended to be more comfortably furnished and could
sell all types of alcoholic beverages. Another distinction which grew out of the Beer Act was
between the publican, who operated a public house, and the beer-seller, who operated
a beer
shop. Occasionally the generic meanings of public house and publican were still used after
1830, but for the sake of clarity, in this essay I follow the tradition of speaking of public houses and beer houses, and publicans and beer-sellers,
as mutually exclusive categories.
Two terms I will use interchangeably are "beer shop" and "beer house," since in the
nineteenth century there was generally no distinction between the two. In various regions of
The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness 125
the country, unique nicknames developed for beer-only establishments. In the Midlands, for
instance, beer shops were often called "Tom and Jerries," while in the West they were
commonly known as "kiddle-a-winks" or "kidliwinks" (Clark 336). 2. While nearly every historical account of the Beer Act of 1830 discusses the role theories of
free trade played in the law's passage, Gourvish and Wilson and Harrison provide particu larly detailed narratives of the principal figures involved in the debates and the history behind the issues at hand. See Gourvish and Wilson 3-22 and Harrison, Drink 64-86.
3. According to census figures, 8,893,000 people lived in England and Wales in 1801. By 1831, the total population had risen to 13,897,000, a 64% increase (Mitchell 9).
4. In an 1834 pamphlet on the Improvement of the Working People, Francis Place implies that
many crusaders against working-class drunkenness manipulate statistics to support their
arguments. According to Place, if statisticians were to study patterns of consumption among the different classes, they might discover that the incidences of drunkenness per capita were
just as high among the rich as the poor. "When a man in easy circumstances gets drunk," he
argues,
it is either at his own house or at the house of a friend, whence he goes home in a coach and is not exposed to the public gaze. A working man gets drunk at a
public-house and staggers along the streets; here he is seen by every body, and is
inconsiderately taken as a fair example of his class; and thus, through the occa
sional drunkard, or the drunken vagabond, the whole body are stigmatized and condemned as drunkards, when in fact the number of those who are really drunkards is, when compared with the whole body, a very small number. (21)
5. Supposedly to compensate for these new restrictions, the Beer Act of 1834 allowed beer-sell ers to remain open until 11 P.M. upon approval from the local magistrate. As beer-sellers
later complained, however, the magistrates, still stinging from the limits the original Beer Act had placed on their authority, often abused this new power, forcing some beer shops to close as early as 9 P.M., an hour before the closing time mandated in the Act of 1830. Of the 1834 Act, James Bishop complained in 1838,
These circumstances left the Beerseller in a position much worse than that in which he was previously. He has to pay an additional price for his license, and to
give additional guarantee for good behaviour ....
[T]he very hour, for the
attainment of which he had agreed to pay an increase of fifty per cent, upon the
price of his license, was made subject to the control and caprice of those, by whom he is looked upon as an innovator, and who cannot be unbiassed [sic] thereby. (16)
6. See note 1 on alternative names for beer shops. 7. To corroborate his opinion that working-class drunkenness is the direct result of poverty and
is thus morally excusable, Engels later cites two physicians, a Dr. Hawkins and a Dr. Kay, who subscribe to the same theory (178,193).
8. However convinced Mayhew may have been that the increased availability of "whole amuse ments" would solve the drink problem, he did acknowledge that on occasion the squalor of
working-class dwellings left them little alternative but to escape through drink. At one point he records a conversation with a tenant of a particularly run-down boarding house who
claims that drinking is the only way to tolerate life amid such conditions. This poor man
insists, '"You must get half-drunk, or your money for your bed is wasted. There's so much
rest owing to you, after a hard day; and bugs and bad air'll prevent its being paid, if you don't
126 VICTORIAN LITERATURE AND CULTURE
lay in some stock of beer, or liquor of some sort, to sleep on'" (115). Such a pragmatic approach to drink echoes the sentiment expressed in
an oft-repeated working-class maxim
of this era: "Drink is the quickest way out of Manchester" (Shiman 3). 9. The penalties for transgressing the
new class-based drinking codes are laid out in several
literary texts of this era. In Thackeray's Vanity Fair, for instance, we learn exactly how taboo
it was for a member of the privileged classes to drink working-class beverages in a
working
class setting. In the process of attempting to secure his place in the will of his wealthy aunt, James Crawley makes the unpardonable blunder of getting drunk at a lowly tavern. As
expected, Miss Crawley subsequently disinherits him, leading the narrator to explain, "Had he drunk a dozen bottles of claret, the old spinster could have pardoned him. Mr. Fox and
Mr. Sheridan drank claret. Gentlemen drank claret. But eighteen glasses of gin consumed
among boxers in an ignoble pothouse ?
it was an odious crime and not to be pardoned
readily" (377). By the time Hardy published Far From the Madding Crowd in 1874, the doctrine of
separate drinks for separate classes had become unmistakably clear. At the harvest home
celebration in Chapter XXXVI of this novel, the reckless Sergeant Troy decides to treat his laborers to brandy instead of their usual beer. Realizing the danger in this, Bathsheba
implores her husband, "No ? don't give it to them
? pray don't Frank. It will only do
them harm" (252, ch.36). Troy persists, however, and, before long, all have passed out from
too much brandy. After describing this scene, the narrator explains, "Having from their
youth up been entirely unaccustomed to any liquor stronger than cider
or mild ale, it was no
wonder that they had succumbed one and all with extraordinary uniformity after the lapse of about an hour" (256, ch. 36).
WORKS CITED
"An Act to amend an Act passed in the First Year of His present Majesty, to permit the general Sale of Beer and Cide by Retail in England." 1834.
Bishop, James. A Defence of the New Beer Trade. London: Dean and Munday, 1838.
Calvert, J. M. The Impropriety of Religious Characters Keeping Beer-Houses. Sheffield: J. Blurton,
1852.
Clark, Peter. The English Alehouse: A Social History 1200-1830. London: Longman, 1983.
Cobbett, William. The Political Register. 27 March 1830.
Davidoff, Leonore and Catherine Hall. Family Fortunes: Men and Women of the English Middle
Class, 1780-1850. Chicago: U of Chicago P, 1987.
Dickens, Charles. "Gin Shops." 1835. Dickens' Journalism: Sketches by Boz and Other Early Papers.
Ed. Michael Slater. Columbus: Ohio State UP, 1994.
Engels, Friedrich. The Condition of the Working Class in England. 1844. London: Penguin, 1987.
Evans, Hilary and Mary Evans. The Man Who Knew the Drunkard's Daughter: The Life and Art of
George Cruikshank 1792-1878. London: Frederick M?ller, 1978. A Few Remarks on the Beer Act. Gloucester: Jew and Wingate, 1833.
French, Richard Valpy. Nineteen Centuries of Drink in England. London: Longmans, Green, 1884.
Gay, John. The Beggar's Opera. 1728. New York: Penguin, 1987.
Gourvish, T. R. and R. G. Wilson. The British Brewing Industry 1830-1980. Cambridge: Cambridge UP, 1994.
Hackwood, Frederick W. Inns, Ales, and Drinking Customs of Old England. London: T. Fisher
Unwin, 1909.
Hamer, John. A Letter to Her Majesty's Ministers on the Operation and Beneficial Results of the New
Beer Act, and the Evils of the Licensing System. Manchester: H. Lowes, 1838.
The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness 127
Hardy, Thomas. Far From the Madding Crowd. 1874. Oxford: Oxford UP, 1993.
Harrison, Brian. Drink and the Victorians: The Temperance Question in England 1815-1872. 2nd edition. Keele: Keele UP, 1994.
-. "Pubs." The Victorian City: Images and Realities. Ed. H. J. Dyos and Michael Wolff. 2 vols.
London: Routledge, 1973.1:161-90.
Holland, Lady. A Memoir of the Reverend Sydney Smith. London: Longmans, Green, 1874.
Hughes, Thomas. Tom Brown's Schooldays. 1857. London: J. M. Dent and Sons, 1949.
Humpherys, Anne. Travels Into the Poor Man's Country: The Work of Henry Mayhew. Athens: U of Georgia P, 1977.
An Inhabitant of Manchester. A Letter to Her Majesty's Ministers on the Operation and Repeal of the New Beer Act. Manchester: Clarke, 1838.
Light, George Evans. "Beer, Cultivated National Identity, and Anglo-Dutch Relations, 1524-1625."
Journal x 2.2 (1998). www.olemiss.edu/depts/english/pubs/jx/2_2/light.html. Mackay, Charles. The Drunkard's Children. London: D. Bogue, 1848.
Mathias, Peter. The Brewing Industry in England 1700-1830. Cambridge: Cambridge UP, 1959.
Mayhew, Henry. London Labour and the London Poor. 1849-52. London: Penguin, 1985.
Mitchell, B. R. British Historical Statistics. Cambridge: Cambridge UP, 1988.
Park, Peter. "Sketches Toward a Political Economy of Drink and Drinking Problems." Journal of Drug Issues 13 (1983): 57-75.
Place, Francis. Improvement of the Working People: Drunkenness ?
Education. London: Charles
Fox, 1834.
Polwhele, Richard. Reminiscences in Prose and Verse. London: J. B. Nichols, 1836.
Pool, Daniel. What Jane Austen Ate and Charles Dickens Knew: From Fox Hunting to Whist ?
the
Facts of Daily Life in Nineteenth-century England. New York: Touchstone, 1993.
Report by the Committee on Intemperance for the Lower House of Convocation of the Province of Canterbury. London: Longman, Green, Reader, and Dyer, 1869.
Report by the Select Committee on the Sale of Beer. In House of Commons Parliamentary Papers 1801-1900. Volume XV (1833). Cambridge: Chadwyck-Healey, 1991.
Report from the Select Committee on Habitual Drunkards. 1872. In British Parliamentary Papers. Social Problems: Drunkenness (Vol. 2). Shannon: Irish UP, 1968.
Report from the Select Committee on Inquiry into Drunkenness. 1834. In British Parliamentary Papers. Social Problems: Drunkenness (Vol. 1). Shannon: Irish UP, 1968.
Reports from the Select Committee of the House of Lords on the Prevalence of Habits of Intemper ance. 3 vols. 1877. In British Parliamentary Papers. Social Problems: Drunkenness (Vol. 3). Shannon: Irish UP, 1968.
Shakespeare, William. The Tragedy of Othello, the Moor of Venice. 1604. The Riverside Shake speare. Boston: Houghton-Mifflin, 1974.
Shiman, Lilian Lewis. Crusade Against Drink in Victorian England. New York: St. Martin's, 1988.
Smith, James. Comic Miscellanies in Prose and Verse. London: Henry Colburn, 1840. Literature
Online, http://lion, chadwyck. com.
Taylor, Anna. Bacchus in Romantic England: Writers and Drink, 1780-1830. New York: St. Mar
tin's, 1999.
Thackeray, William Makepeace. Vanity Fair. 1848. New York: Quality Paperback, 1991. Times. 21 Oct. 1830. 2D.
Webb, Sidney and Beatrice Webb. The History of Liquor Licensing in England Principally from 1700 to 1830. London: Longmans, Green, 1903.
- Article Contents
- p. 109
- p. 110
- p. 111
- p. 112
- p. 113
- p. 114
- p. 115
- p. 116
- p. 117
- p. 118
- p. 119
- p. 120
- p. 121
- p. 122
- p. 123
- p. 124
- p. 125
- p. 126
- p. 127
- Issue Table of Contents
- Victorian Literature and Culture, Vol. 29, No. 1 (2001), pp. i-viii, 1-240
- Front Matter
- Erratum [p. vi-vi]
- Editors' Topic: Constructions of Victorian Classes [Part II]
- The Smell of Class: British Novels of the 1860s [pp. 1-19]
- Down among the Dead: Edwin Chadwick's Burial Reform Discourse in Mid-Nineteenth-Century England [pp. 21-38]
- The Importance of Being an Earnest Improver: Class, Caste, and "Self-Help" in Mid-Victorian England [pp. 39-50]
- Charles Kingsley, the Romantic Legacy, and the Unmaking of the Working-Class Intellectual [pp. 51-65]
- Thomas Carlyle, "Chartism", and the Irish in Early Victorian England [pp. 67-83]
- Instructive Sufficiency: Re-Reading the Governess through "Agnes Grey" [pp. 85-108]
- "The Sovereign People Are in a Beastly State": The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness [pp. 109-127]
- Performing the Voyage out: Victorian Female Emigration and the Class Dynamics of Displacement [pp. 129-146]
- Review Essays
- Review: Victorian Art History: Rap 2 Unwrapped [pp. 149-158]
- Review: Millennial Victoria [pp. 159-170]
- Works in Progress
- Making What Will Suffice: Carlyle's Fetishism [pp. 173-193]
- Wilde's "Salomé" and the Ambiguous Fetish [pp. 195-218]
- Ethnographic Collecting and Travel: Blurring Boundaries, Forming a Discipline [pp. 219-239]
- Back Matter
The Beer Act and Industrialization
Please read: Nicholas Mason, "The Sovereign People Are in a Beastly State: The Beer Act of 1830 and Victorian Discourse on Working-Class Drunkenness,” Victorian Literature and Culture, Vol. 29, No. 1 (2001), pp. 109-127, available in the Readings Folder on JSTOR at: http://www.jstor.org/stable/25058542
A) Discuss how it fits into his history of the Beer Acts. B) State whether this is a primary or secondary source and why.
C) Discuss what this tells us about industrialization.
119
4Propositional Logic
flytosky11/iStock/Thinkstock
Learning Objectives After reading this chapter, you should be able to:
1. Explain key words and concepts from propositional logic.
2. Describe the basic logical operators and how they function in a statement.
3. Symbolize complex statements using logical operators.
4. Generate truth tables to evaluate the validity of truth-functional arguments.
5. Evaluate common logical forms.
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Section 4.1 Basic Concepts in Propositional Logic
Chapter 3 discussed categorical logic and touched on how analyzing an argument’s logical form helps determine its validity. The usefulness of form in determining validity will become even clearer in this chapter’s discussion of what is known as propositional logic, another type of deductive logic. Whereas categorical logic analyzes arguments whose validity is based on quantitative terms like all and some, propositional logic looks at arguments whose validity is based on the way they combine smaller sentences to make larger ones, using connectives like or, and, and not.
In this chapter, we will learn about the symbols and tools that help us analyze arguments and test for validity; we will also examine several common deductive argument forms. Whereas Chapter 3 introduced the idea of form—and thereby, formal logic—this chapter will more thoroughly consider the study of validity based on logical form. We shall see that by adding a couple more symbols to propositional logic, it is also possible to represent the types of state- ments represented in categorical logic, creating the robust and highly applicable discipline known today as predicate logic. (See A Closer Look: Translating Categorical Logic for more on predicate logic.)
4.1 Basic Concepts in Propositional Logic Propositional logic aims to make the concept of validity formal and precise. Remember from Chapter 3 that an argument is valid when the truth of its premises guarantees the truth of its conclusion. Propositional logic demonstrates exactly why certain types of prem- ises guarantee the truth of certain types of conclusions. It does this by breaking down the forms of complex claims into their simple component parts. For example, consider the fol- lowing argument:
Either the maid or the butler did it. The maid did not do it. Therefore, the butler did it.
This argument is valid, but not because of anything about the maid or butler. It is valid because of the way that the sentences combine words like or and not to make a logically valid form. Formal logic is not concerned about the content of arguments but with their form. Recall from Chapter 3, Section 3.2, that an argument’s form is the way it combines its component parts to make an overall pattern of reasoning. In this argument, the component parts are the small sentences “the butler did it” and “the maid did it.” If we give those parts the names P and Q, then our argument has the form:
P or Q. Not P. Therefore, Q.
Note that the expression “not P” means “P is not true.” In this case, since P is “the butler did it,” it follows that “not P” means “the butler did not do it.” An inspection of this form should reveal it is logically valid reasoning.
As the name suggests, propositional logic deals with arguments made up of propositions, just as categorical logic deals with arguments made up of categories (see Chapter 3). In
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Section 4.1 Basic Concepts in Propositional Logic
philosophy, a proposition is the meaning of a claim about the world; it is what that claim asserts. We will refer to the subject of this chapter as “propositional logic” because that is the most common terminology in the field. However, it is sometimes called “sentence logic.” The principles are the same no matter which terminology we use, and in the rest of the chapter we will frequently talk about P and Q as representing sentences (or “statements”) as well.
The Value of Formal Logic This process of making our reasoning more precise by focusing on an argument’s form has proved to be enormously useful. In fact, formal logic provides the theoretical underpinnings for computers. Computers operate on what are called “logic circuits,” and computer programs are based on propositional logic. Computers are able to understand our commands and always do exactly what they are programmed to do because they use formal logic. In A Closer Look: Alan Turing and How Formal Logic Won the War, you will see how the practical applica- tions of logic changed the course of history.
Another value of formal logic is that it adds efficiency, precision, and clarity to our language. Being able to examine the structure of people’s statements allows us to clarify the meanings of complex sentences. In doing so, it creates an exact, structured way to assess reasoning and to discern between formally valid and invalid arguments.
A Closer Look: Alan Turing and How Formal Logic Won the War The idea of a computing machine was conceived over the last few centuries by great thinkers such as Gottfried Leibniz, Blaise Pascal, and Charles Babbage. However, it was not until the first half of the 20th century that philosophers, logicians, mathematicians, and engineers were actually able to create “thinking machines” or “electronic brains” (Davis, 2000).
One pioneer of the computer age was British mathematician, philosopher, and logician Alan Turing. He came up with the concept of a Turing machine, an electronic device that takes input in the form of zeroes and ones, manipulates it according to an algorithm, and creates a new output (BBC News, 1999).
Computers themselves were invented by creating electric cir- cuits that do basic logical operations that you will learn about in this chapter. These electric circuits are called “logic gates” (see Figure 4.2 later in the chapter). By turning logic into cir- cuits, basic “thinking” could be done with a series of fast elec- trical impulses.
Using logical brilliance, Turing was able to design early com- puters for use during World War II. The British used these early computers to crack the Nazis’ very complex Enigma code. The ability to know the German plans in advance gave the Allies a huge advantage. Prime Minister Winston Churchill even said to King George VI, “It was thanks to Ultra [one of the computers used] that we won the war” (as cited in Shaer, 2012).
Science and Society/SuperStock
An Enigma cipher machine, which was widely used by the Nazi Party to encipher and decipher secret military messages during World War II.
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Section 4.1 Basic Concepts in Propositional Logic
Statement Forms As we have discussed, propositional logic clarifies formal reasoning by breaking down the forms of complex claims into the simple parts of which they are composed. It does this by using symbols to represent the smaller parts of complex sentences and showing how the larger sentence results from com- bining those parts in a certain way. By doing so, formal logic clarifies the argu- ment’s form, or the pattern of reason- ing it uses.
Consider what this looks like in mathematics. If you have taken a course in algebra, you will remember statements such as the following:
x + y = y + x
This statement is true no matter what we put for x and for y. That is why we call x and y variables; they do not represent just one number but all numbers. No matter what specific numbers we put in, we will still get a true statement, like the following:
5 + 3 = 3 + 5
7 + 2 = 2 + 7
1,235 + 943 = 943 + 1,235
By replacing the variables in the general equation with these specific values, we get instances (as discussed in Chapter 3) of that general truth. In other words, 5 + 3 = 3 + 5 is an instance of the general statement x + y = y + x. One does not even need to use a calculator to know that the last statement of the three is true, for its truth is not based on the specific numbers used but on the general form of the equation. Formal logic works in the exact same way.
Take the statement “If you have a dog, then you have a dog or you have a cat.” This statement is true, but its truth does not depend on anything about dogs or cats; its truth is based on its logical form—the way the sentence is structured. Here are two other statements with the same logical form: “If you are a miner, then you are a miner or you are a trapper” and “If you are a man, then you are a man or a woman.” These statements are all true not because of their content, but because of their shared logical form.
To help us see exactly what this form is, propositional logic uses variables to represent the different sentences within this form. Just as algebra uses letters like x and y to represent numbers, logicians use letters like P and Q to represent sentences. These letters are therefore called sentence variables.
Bill Long/Cartoonstock
Formal logic uses symbols and statement forms to clarify an argument’s reasoning.
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Sentence: “If you have a dog, then you have a dog AND you have a cat”
Form: If P then P Q and
Section 4.2 Logical Operators
The chief difference between propositional and categorical logic is that, in categorical logic (Chapter 3), variables (like M and S) are used to represent categories of things (like dogs and mammals), whereas variables in propositional logic (like P and Q) represent whole sentences (or propositions).
In our current example, propositional logic enables us to take the statement “If you have a dog, then you have a dog or you have a cat” and replace the simple sentences “You have a dog” and “You have a cat,” with the variables P and Q, respectively (see Figure 4.2). The result, “If P, then P or Q,” is known as the general statement form. Our specific sentence, “If you have a dog, then you have a dog or you have a cat,” is an instance of this general form. Our other example statements—”If you are a miner, then you are a miner or you are a trapper” and “If you are a man, then you are a man or a woman”—are other instances of that same statement form, “If P, then P or Q.” We will talk about more specific forms in the next section.
At first glance, propositional logic can seem intimidating because it can be very mathemati- cal in appearance, and some students have negative associations with math. We encourage you to take each section one step at a time and see the symbols as tools you can use to your advantage. Many students actually find that logic helps them because it presents symbols in a friendlier manner than in math, which can then help them warm up to the use of symbols in general.
4.2 Logical Operators In the prior section, we learned about what constitutes a statement form in propositional logic: a complex sentence structure with propositional variables like P and Q. In addition to the variables, however, there are other words that we used in representing forms, words like and and or. These terms, which connect the variables together, are called logical operators, also known as connectives or logical terms.
Logicians like to express formal precision by replacing English words with symbols that rep- resent them. Therefore, in a statement form, logical operators are represented by symbols. The resulting symbolic statement forms are precise, brief, and clear. Expressing sentences in terms of such forms allows logic students more easily to determine the validity of arguments that include them. This section will analyze some of the most common symbols used for logi- cal operators.
Figure 4.1: Finding the form
In this instance of the statement form, you can see that P and Q relate to the prepositions “you have a dog” and “you have a cat,” respectively.
Sentence: “If you have a dog, then you have a dog AND you have a cat”
Form: If P then P Q and
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Section 4.2 Logical Operators
Conjunction Those of you who have heard the Schoolhouse Rock! song “Conjunction Junction” (what’s your function?)—or recall past English grammar lessons—will recognize that a conjunction is a word used to connect, or conjoin, sentences or concepts. By that definition, it refers to words like and, but, and or. Logic, however, uses the word conjunction to refer only to and sentences. Accordingly, a conjunction is a compound statement in which the smaller component state- ments are joined by and.
For example, the conjunction of “roses are red” and “violets are blue” is the sentence “roses are red and violets are blue.” In logic, the symbol for and is an ampersand (&). Thus, the gen- eral form of a conjunction is P & Q. To get a specific instance of a conjunction, all you have to do is replace the P and the Q with any specific sentences. Here are some examples:
P Q P & Q
Joe is nice. Joe is tall. Joe is nice, and Joe is tall. Mike is sad. Mike is lonely. Mike is sad, and Mike is lonely.
Winston is gone. Winston is not forgotten. Winston is gone and not forgotten.
Notice that the last sentence in the example does not repeat “Winston is” before “forgotten.” That is because people tend to abbreviate things. Thus, if we say “Jim and Mike are on the team,” this is actually an abbreviation for “Jim is on the team, and Mike is on the team.”
The use of the word and has an effect on the truth of the sentence. If we say that P & Q is true, it means that both P and Q are true. For example, suppose we say, “Joe is nice and Joe is tall.” This means that he is both nice and tall. If he is not tall, then the statement is false. If he is not nice, then the statement is false as well. He has to be both for the conjunction to be true. The truth of a complex statement thus depends on the truth of its parts. Whether a proposition is true or false is known as its truth value: The truth value of a true sentence is simply the word true, while the truth value of a false sentence is the word false.
To examine how the truth of a statement’s parts affects the truth of the whole statement, we can use a truth table. In a truth table, each variable (in this case, P and Q) has its own column, in which all possible truth values for those variables are listed. On the right side of the truth table is a column for the complex sentence(s) (in this case the conjunction P & Q) whose truth we want to test. This last column shows the truth value of the statement in question based on the assigned truth values listed for the variables on the left. In other words, each row of the truth table shows that if the letters (like P and Q) on the left have these assigned truth values, then the complex statements on the right will have these resulting truth values (in the complex column).
Here is the truth table for conjunction:
P Q P & Q
T (Joe is nice.) T (Joe is tall.) T (Joe is nice, and Joe is tall.) T (Joe is nice.) F (Joe is not tall.) F (It is not true that Joe is nice and tall.)
F (Joe is not nice.) T (Joe is tall.) F (It is not true that Joe is nice and tall.) F (Joe is not nice.) F (Joe is not tall.) F (It is not true that Joe is nice and tall.)
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Section 4.2 Logical Operators
What the first row means is that if the statements P and Q are both true, then the conjunction P & Q is true as well. The second row means that if P is true and Q is false, then P & Q is false (because P & Q means that both statements are true). The third row means that if P is false and Q is true, then P & Q is false. The final row means that if both statements are false, then P & Q is false as well.
A shorter method for representing this truth table, in which T stands for “true” and F stands for “false,” is as follows:
P Q P & Q
T T T T F F F T F F F F
The P and Q columns represent all of the possible truth combinations, and the P & Q column represents the resulting truth value of the conjunction. Again, within each row, on the left we simply assume a set of truth values (for example, in the second row we assume that P is true and Q is false), then we determine what the truth value of P & Q should be to the right. Therefore, each row is like a formal “if–then” statement: If P is true and Q is false, then P & Q will be false.
Truth tables highlight why propositional logic is also called truth-functional logic. It is truth- functional because, as truth tables demonstrate, the truth of the complex statement (on the right) is a function of the truth values of its component statements (on the left).
Everyday Logic: The Meaning of But
Like the word and, the word but is also a conjunction. If we say, “Mike is rich, but he’s mean,” this seems to mean three things: (1) Mike is rich, (2) Mike is mean, and (3) these things are in contrast with each other. This third part, however, cannot be measured with simple truth values. Therefore, in terms of logic, we simply ignore such conversational elements (like point 3) and focus only on the truth conditions of the sentence. Therefore, strange as it may seem, in proposi- tional logic the word but is taken to be a synonym for and.
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Section 4.2 Logical Operators
Disjunction Disjunction is just like conjunction except that it involves statements connected with an or (see Figure 4.2 for a helpful visualization of the difference). Thus, a statement like “You can either walk or ride the bus” is the disjunction of the statements “You can walk” and “you can ride the bus.” In other words, a disjunction is an or statement: P or Q. In logic the symbol for or is ∨. An or statement, therefore, has the form P ∨ Q.
Here are some examples:
P Q P ∨ Q
Mike is tall. Doug is rich. Mike is tall, or Doug is rich.
You can complain. You can change things. You can complain, or you can change things.
The maid did it. The butler did it. Either the maid or the butler did it.
Notice that, as in the conjunction example, the last example abbreviates one of the clauses (in this case the first clause, “the maid did it”). It is common in natural (nonformal) languages to abbreviate sentences in such ways; the compound sentence actually has two complete com- ponent sentences, even if they are not stated completely. The nonabbreviated version would be “Either the maid did it, or the butler did it.”
The truth table for disjunction is as follows:
P Q P ∨ Q
T T T T F T F T T F F F
Note that or statements are true whenever at least one of the component sentences (the “dis- juncts”) is true. The only time an or statement is false is when P and Q are both false.
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Basic AND Gate
Basic OR Gate
Battery
Battery
P gate: Closed if P is true
P gate: Closed if P is true
Q gate: Closed if Q is true
Q gate: Closed if Q is true
The light goes on only if both P and Q are true.
The light goes if either P or Q is true.
Section 4.2 Logical Operators
Figure 4.2: Simple logic circuits
These diagrams of simple logic circuits (recall the reference to these circuits in A Closer Look: Alan Turing and How Formal Logic Won the War) help us visualize how the rules for conjunctions (AND gate) and disjunctions (OR gate) work. With the AND gate, there is only one path that will turn on the light, but with the OR gate, there are two paths to illumination.
Basic AND Gate
Basic OR Gate
Battery
Battery
P gate: Closed if P is true
P gate: Closed if P is true
Q gate: Closed if Q is true
Q gate: Closed if Q is true
The light goes on only if both P and Q are true.
The light goes if either P or Q is true.
Everyday Logic: Inclusive Versus Exclusive Or
The top line of the truth table for disjunctions may seem strange to some. Some think that the word or is intended to allow only one of the two sentences to be true. They therefore argue for an interpretation of disjunction called exclusive or. An exclusive or is just like the or in the truth table, except that it makes the top row (the one in which P and Q are both true) false.
One example given to justify this view is that of a waiter asking, “Do you want soup or salad?” If you want both, the answer should not be “yes.” Some therefore suggest that the English or should be understood in the exclusive sense.
However, this example can be misleading. The waiter is not asking “Is the statement ‘do you want soup or salad’ true?” The waiter is asking you to choose between the two options. When we ask for the truth value of a sentence of the form P or Q, on the other hand, we are asking whether the sentence is true. Consider it this way: If you wanted both soup and salad, the answer to the waiter’s question would not be “no,” but it would be if you were using an exclusive or.
(continued)
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Section 4.2 Logical Operators
Negation The simplest logical symbol we use on sentences simply negates a claim. Negation is the act of asserting that a claim is false. For every statement P, the negation of P states that P is false. It is symbolized ~P and pronounced “not P.” Here are some examples:
P ~P
Snow is white. Snow is not white.
I am happy. I am not happy.
Either John or Mike got the job. Neither John nor Mike got the job.
Since ~P states that P is not true, its truth value is the opposite of P’s truth value. In other words, if P is true, then ~P is false; if P is false then ~P is true. Here, then, is the truth table:
P ~P
T F F T
Everyday Logic: The Word Not
Sometimes just putting the word not in front of the verb does not quite capture the meaning of negation. Take the statement “Jack and Jill went up the hill.” We could change it to “Jack and Jill did not go up the hill.” This, however, seems to mean that neither Jack nor Jill went up the hill, but the meaning of negation only requires that at least one did not go up the hill. The simplest way to correctly express the negation would be to write “It is not true that Jack and Jill went up the hill” or “It is not the case that Jack and Jill went up the hill.”
Similar problems affect the negation of claims such as “John likes you.” If John does not know you, then this statement is not true. However, if we put the word not in front of the verb, we get “John does not like you.” This seems to imply that John dislikes you, which is not what the negation means (especially if he does not know you). Therefore, logicians will instead write something like, “It is not the case that John likes you.”
When we see the connective or used in English, it is generally being used in the inclusive sense (so called because it includes cases in which both disjuncts are true). Suppose that your tax form states, “If you made more than $20,000, or you are self-employed, then fill out form 201-Z.” Sup- pose that you made more than $20,000, and you are self-employed—would you fill out that form? You should, because the standard or that we use in English and in logic is the inclusive version. Therefore, in logic we understand the word or in its inclusive sense, as seen in the truth table.
Everyday Logic: Inclusive Versus Exclusive Or (continued)
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Section 4.2 Logical Operators
Conditional A conditional is an “if–then” statement. An example is “If it is raining, then the street is wet.” The general form is “If P, then Q,” where P and Q represent any two claims. Within a condi- tional, P—the part that comes between if and then—is called the antecedent; Q—the part after then—is called the consequent. A conditional statement is symbolized P → Q and pro- nounced “if P, then Q.”
Here are some examples:
P Q P → Q
You are rich. You can buy a boat. If you are rich, then you can buy a boat.
You are not satisfied. You can return the product. If you are not satisfied, then you can return the product.
You need bread or milk. You should go to the market. If you need bread or milk, then you should go to the market.
Formulating the truth table for conditional statements is somewhat tricky. What does it take for a conditional statement to be true? This is actually a controversial issue within philosophy. It is actually easier to think of it as: What does it mean for a conditional statement to be false?
Suppose Mike promises, “If you give me $5, then I will wash your car.” What would it take for this statement to be false? Under what conditions, for example, could you accuse Mike of breaking his promise?
It seems that the only way for Mike to break his promise is if you give him the $5, but he does not wash the car. If you give him the money and he washes the car, then he kept his word. If
Everyday Logic: Other Instances of Conditionals
Sometimes conditionals are expressed in other ways. For example, sometimes people leave out the then. They say things like, “If you are hungry, you should eat.” In many of these cases, we have to be clever in determining what P and Q are.
Sometimes people even put the consequent first: for example, “You should eat if you are hungry.” This state- ment means the same thing as “If you are hungry, then you should eat”; it is just ordered differently. In both cases the antecedent is what comes after the if in the English sentence (and prior to the → in the logical form). Thus, “If P then Q” is translated “P → Q,” and “P if Q” is translated “Q → P.”
Monkey Business/Thinkstock
People use conditionals frequently in real life. Think of all the times someone has said, “Get some rest if you are tired” or “You don’t have to do something if you don’t want to.”
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Section 4.2 Logical Operators
you did not give him the money, then his word was simply not tested (with no payment on your part, he is under no obligation). If you do not pay him, he may choose to wash the car anyway (as a gift), or he may not; neither would make him a liar. His promise is only broken in the case in which you give him the money but he does not wash it. Therefore, in general, we call conditional statements false only in the case in which the antecedent is true and the consequent is false (in this case, if you give him the money, but he still does not wash the car). This results in the following truth table:
P Q P → Q
T T T T F F F T T F F T
Some people question the bottom two lines. Some feel that the truth value of those rows should depend on whether he would have washed the car if you had paid him. However, this sophisticated hypothetical is beyond the power of truth-functional logic. The truth table is as close as we can get to the meaning of “if . . . then . . .” with a simple truth table; in other words, it is best we can do with the tool at hand.
Finally, some feel that the third row should be false. That, however, would mean that Mike choosing to wash the car of a person who had no money to give him would mean that he broke his promise. That does not appear, however, to be a broken promise, only an act of generosity on his part. It therefore does not appear that his initial statement “If you give me $5, then I will wash your car” commits to washing the car only if you give him $5. This is instead a variation on the conditional theme known as “only if.”
Only If So what does it mean to say “P only if Q”? Let us take a look at another example: “You can get into Harvard only if you have a high GPA.” This means that a high GPA is a requirement for getting in. Note, however, that that is not the same as saying, “You can get into Harvard if you have a high GPA,” for there might be other requirements as well, like having high test scores, good letters of recommendation, and a good essay.
Thus, the statement “You can get into Harvard only if you have a high GPA” means:
You can get into Harvard → You have a high GPA
However, this does not mean the same thing as “You have a high GPA → You can get into Harvard.”
In general, “P only if Q” is translated P → Q. Notice that this is the same as the translation of “If P, then Q.” However, it is not the same as “P if Q,” which is translated Q → P. Here is a summary of the rules for these translations:
P only if Q is translated: P → Q
P if Q is translated: Q → P
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Section 4.2 Logical Operators
Thus, “P if Q” and “P only if Q” are the converse of each other. Recall the discussion of conver- sion in Chapter 3; the converse is what you get when you switch the order of the elements within a conditional or categorical statement.
To say that P → Q is true is to assert that the truth of Q is necessary for the truth of P. In other words, Q must be true for P to be true. To say that P → Q is true is also to say that the truth of P is sufficient for the truth of Q. In other words, knowing that P is true is enough information to conclude that Q is also true.
In our earlier example, we saw that having a high GPA is necessary but not sufficient for get- ting into Harvard, because one must also have high test scores and good letters of recommen- dation. Further discussion of the concepts of necessary and sufficient conditions will occur in Chapter 5.
In some cases P is both a necessary and a sufficient condition for Q. This is called a biconditional.
Biconditional A biconditional asserts an “if and only if ” statement. It states that if P is true, then Q is true, and if Q is true, then P is true. For example, if I say, “I will go to the party if you will,” this means that if you go, then I will too (P → Q), but it does not rule out the possibility that I will go without you. To rule out that possibility, I could state “I will go to the party only if you will” (Q → P). If we want to assert both conditionals, I could say, “I will go to the party if and only if you will.” This is a biconditional.
The statement “P if and only if Q” literally means “P if Q and P only if Q.” Using the translation methods for if and only if, this is translated “(Q → P) & (P → Q).” Because the biconditional makes the arrow between P and Q go both ways, it is symbolized: P ↔ Q.
Here are some examples:
P Q P ↔ Q
You can go to the party. You are invited. You can go to the party if and only if you are invited.
You will get an A. You get above a 92%. You will get an A if and only if you get above a 92%.
You should propose. You are ready to marry her. You should propose if and only if you are ready to marry her.
There are other phrases that people sometimes use instead of “if and only if.” Some people say “just in case” or something else like it. Mathematicians and philosophers even use the abbreviation iff to stand for “if and only if.” Sometimes people even simply say “if ” when they really mean “if and only if.” One must be clever to understand what people really mean when they speak in sloppy, everyday language. When it comes to precision, logic is perfect; English is fuzzy!
Here is how we do the truth table: For the biconditional P ↔ Q to be true, it must be the case that if P is true then Q is true and vice versa. Therefore, one cannot be true when the other one is false. In other words, they must both have the same truth value. That means the truth table looks as follows:
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Section 4.2 Logical Operators
P Q P ↔ Q
T T T T F F F T F F F T
The biconditional is true in exactly those cases in which P and Q have the same truth value.
Practice Problems 4.1
Complete the following identifications.
1. “I am tired and hungry.” This statement is a __________. a. conjunction b. disjunction c. conditional d. biconditional
2. “If we learn logic, then we will be able to evaluate arguments.” This statement is a __________. a. conjunction b. disjunction c. conditional d. biconditional
3. “We can learn logic if and only if we commit ourselves to intense study.” This state- ment is a __________. a. conjunction b. disjunction c. conditional d. biconditional
4. “We either attack now, or we will lose the war.” This statement is a __________. a. conjunction b. disjunction c. conditional d. biconditional
5. “The tide will rise only if the moon’s gravitational pull acts on the ocean.” This state- ment is a __________. a. conjunction b. disjunction c. conditional d. biconditional
6. “If I am sick or tired, then I will not go to the interpretive dance competition.” This statement is a __________. a. conjunction b. disjunction c. conditional d. biconditional
(continued)
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Section 4.3 Symbolizing Complex Statements
4.3 Symbolizing Complex Statements We have learned the basic logical operators and their corresponding symbols and truth tables. However, these basic symbols also allow us analyze much more complicated state- ments. Within the statement form P → Q, what if either P or Q itself is a complex statement? For example:
P Q P → Q
You are hungry or thirsty. We should go to the diner. If you are hungry or thirsty, then we should go to the diner.
In this example, the antecedent, P, states, “You are hungry or thirsty,” which can be symbolized H ∨ T, using the letter H for “You are hungry” and T for “You are thirsty.” If we use the letter D for “We should go to the diner,” then the whole statement can be symbolized (H ∨ T) → D.
Notice the use of parentheses. Parentheses help specify the order of operations, just like in arithmetic. For example, how would you evaluate the quantity 3 + (2 × 5)? You would execute the mathematical operation within the parentheses first. In this case you would first multiply 2 and 5 and then add 3, getting 13. You would not add the 3 and the 2 first and then multiply by 5 to get 25. This is because you know to evaluate what is within the parentheses first.
7. “One can surf monster waves if and only if one has experience surfing smaller waves.” This statement is a __________. a. conjunction b. disjunction c. conditional d. biconditional
8. “The economy is recovering, and people are starting to make more money.” This statement is a __________. a. conjunction b. disjunction c. conditional d. biconditional
9. “If my computer crashes again, then I am going to buy a new one.” This statement is a __________. a. conjunction b. disjunction c. conditional d. biconditional
10. “You can post responses on 2 days or choose to write a two-page paper.” This state- ment is a __________. a. conjunction b. disjunction c. conditional d. biconditional
Practice Problems 4.1 (continued)
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Section 4.3 Symbolizing Complex Statements
It is the exact same way with logic. In the statement (H ∨ T) → D, because of the parentheses, we know that this statement is a conditional (not a disjunction). It is of the form P → Q, where P is replaced by H ∨ T and Q is replaced by D.
Here is another example:
N & S G (N & S) → G
He is nice and smart. You should get to know him. If he is nice and smart, then you should get to know him.
This example shows a complex way to make a sentence out of three component sentences. N is “He is nice,” S is “he is smart,” and G is “you should get to know him.” Here is another:
R (S & C) R → (S & C)
You want to be rich. You should study hard and go to college.
If you want to be rich, then you should study hard and go to
college.
If R is “You want to be rich,” S is “You should study hard,” and C is “You should go to college,” then the whole statement in this final example, symbolized R → (S & C), means “If you want to be rich, then you should study hard and go to college.”
Complex statements can be created in this manner for every form. Take the statement (~A & B) ∨ (C → ~D). This statement has the general form of a disjunction. It has the form P ∨ Q, where P is replaced with ~A & B, and Q is replaced with C → ~D.
Everyday Logic: Complex Statements in Ordinary Language
It is not always easy to determine how to translate complex, ordinary language statements into logic; one sometimes has to pick up on clues within the statement.
For instance, notice in general that neither P nor Q is translated ~(P ∨ Q). This is because P ∨ Q means that either one is true, so ~(P ∨ Q) means that neither one is true. It happens to be equiva- lent to saying ~P & ~Q (we will talk about logical equivalence later in this chapter).
Here are some more complex examples:
Statement Translation
If you don’t eat spinach, then you will neither be big nor strong.
~S → ~(B ∨ S)
Either he is strong and brave, or he is both reckless and foolish.
(S & B) ∨ (R & F)
Come late and wear wrinkled clothes only if you don’t want the job.
(L & W) → ~J
He is strong and brave, and if he doesn’t like you, he will let you know.
(S & B) & (~L → K)
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Section 4.3 Symbolizing Complex Statements
Truth Tables With Complex Statements We have managed to symbolize complex statements by seeing how they are systematically constructed out of their parts. Here we use the same principle to create truth tables that allow us to find the truth values of complex statements based on the truth values of their parts. It will be helpful to start with a summary of the truth values of sentences constructed with the basic truth-functional operators:
P Q ~P P & Q P ∨ Q P → Q P ↔ Q T T F T T T T T F F F T F F F T T F T T F F F T F F T T
The truth values of more complex statements can be discovered by applying these basic for- mulas one at a time. Take a complex statement like (A ∨ B) → (A & B). Do not be intimidated by its seemingly complex form; simply take it one operator at a time. First, notice the main form of the statement: It is a conditional (we know this because the other operators are within parentheses). It therefore has the form P → Q, where P is “A ∨ B” and Q is “A & B.”
The antecedent of the conditional is A ∨ B; the consequent is A & B. The way to find the truth values of such statements is to start inside the parentheses and find those truth values first, and then work our way out to the main operator—in this case →.
Here is the truth table for these components:
A B A ∨ B A & B
T T T T T F T F F T T F F F F F
Now we take the truth tables for these components to create the truth table for the overall conditional:
A B A ∨ B A & B (A ∨ B) → (A & B) T T T T T T F T F F F T T F F F F F F T
In this way the truth values of very complex statements can be determined from the values of their parts. We may refer to these columns (in this case A ∨ B and A & B) as helper columns, because they are there just to assist us in determining the truth values for the more complex statement of which they are a part.
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Section 4.3 Symbolizing Complex Statements
Here is another one: (A & ~B) → ~(A ∨ B). This one is also a conditional, where the anteced- ent is A & ~B and the consequent is ~(A ∨ B). We do these components first because they are inside parentheses. However, to find the truth table for A & ~B, we will have to fill out the truth table for ~B first (as a helper column).
A B ~B A & ~B
T T F F T F T T F T F F F F T F
We found ~B by simply negating B. We then found A & ~B by applying the truth table for con- junctions to the column for A and the column for ~B.
Now we can fill out the truth table for A ∨ B and then use that to find the values of ~(A ∨ B):
A B A ∨ B ~(A ∨ B)
T T T F T F T F F T T F F F F T
Finally, we can now put A & ~B and ~(A ∨ B) together with the conditional to get our truth table:
A B A & ~B ~(A ∨ B) (A & ~B) → ~(A ∨ B) T T F F T T F T F F F T F F T F F F T T
Although complicated, it is not hard when one realizes that one has to apply only a series of simple steps in order to get the end result.
Here is another one: (A → ~B) ∨ ~(A & B). First we will do the truth table for the left part of the disjunction (called the left disjunct), A → ~B:
A B ~B A → ~B T T F F T F T T F T F T F F T T
Of course, the last column is based on combining the first column, A, with the third column, ~B, using the conditional. Now we can work on the right disjunct, ~(A & B):
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Section 4.3 Symbolizing Complex Statements
A B A & B ~(A & B)
T T T F T F F T F T F T F F F T
The final truth table, then, is:
A B A→~B ~(A & B) (A→~B) ∨ ~(A & B) T T F F F T F T T T F T T T T F F T T T
You may have noticed that three formulas in the truth table have the exact same values on every row. That means that the formulas are logically equivalent. In propositional logic, two formulas are logically equivalent if they have the same truth values on every row of the truth table. Logically equivalent formulas are therefore true in the exact same circumstances. Logicians consider this important because two formulas that are logically equivalent, in the logical sense, mean the same thing, even though they may look quite different. The conditions for their truth and falsity are identical.
The fact that the truth value of a complex statement follows from the truth values of its compo- nent parts is why these operators are called truth-functional. The operators, &, ∨, ~, →, and ↔, are truth-functions, meaning that the truth of the whole sentence is a function of the truth of the parts.
Because the validity of argument forms within propositional logic is based on the behavior of the truth-functional operators, another name for propositional logic is truth-functional logic.
Truth Tables With Three Letters In each of the prior complex statement examples, there were only two letters (variables like P and Q or constants like A and B) in the top left of the truth table. Each truth table had only four rows because there are only four possible combinations of truth values for two variables (both are true, only the first is true, only the second is true, and both are false).
It is also possible to do a truth table for sentences that contain three or more variables (or constants). Recall one of the earlier examples: “Come late and wear wrinkled clothes only if you don’t want the job,” which we represented as (L & W) → ~J. Now that there are three let- ters, how many possible combinations of truth values are there for these letters?
The answer is that a truth table with three variables (or constants) will have eight lines. The general rule is that whenever you add another letter to a truth table, you double the number of possible combinations of truth values. For each earlier combination, there are now two: one in which the new letter is true and one in which it is false. Therefore, to make a truth table
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P
T
T
F
F
T
F
T F
T F
T F
T F
T
F
Q R
Section 4.3 Symbolizing Complex Statements
with three letters, imagine the truth table for two letters and imagine each row splitting in two, as follows:
The resulting truth table rows would look like this:
P Q R
T T T T T F T F T T F F F T T F T F F F T F F F
The goal is to have a row for every possible truth value combination. Generally, to fill in the rows of any truth table, start with the last letter and simply alternate T, F, T, F, and so on, as in the R column. Then move one letter to the left and do twice as many Ts followed by twice as many Fs (two of each): T, T, F, F, and so on, as in the Q column. Then move another letter to the left and do twice as many of each again (four each), in this case T, T, T, T, F, F, F, F, as in the P column. If there are more letters, then we would repeat the process, adding twice as many Ts for each added letter to the left.
With three letters, there are eight rows; with four letters, there are sixteen rows, and so on. This chapter does not address statements with more than three letters, so another way to ensure you have enough rows is to memorize this pattern.
P
T
T
F
F
T
F
T F
T F
T F
T F
T
F
Q R
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Section 4.3 Symbolizing Complex Statements
The column with the forms is filled out the same way as when there were two letters. The fact that they now have three letters makes little difference, because we work on only one operator, and therefore at most two columns of letters, at a time. Let us start with the example of P → (Q & R). We begin by solving inside the parentheses by determining the truth values for Q & R, then we create the conditional between P and that result. The table looks like this:
P Q R Q & R P → (Q & R) T T T T T T T F F F T F T F F T F F F F F T T T T F T F F T F F T F T F F F F T
The rules for determining the truth values of Q & R and then of P → (Q & R) are exactly the same as the rules for & and → that we used in the two-letter truth tables earlier; now we just use them for more rows. It is a formal process that generates truth values by the same strict algorithms as in the two-letter tables.
Practice Problems 4.2
Symbolize the following complex statements using the symbols that you have learned in this chapter.
1. One should be neither a borrower nor a lender.
2. Atomic bombs are dangerous and destructive.
3. If we go to the store, then I need to buy apples and lettuce.
4. Either Microsoft enhances its product and Dell’s sales decrease, or Gateway will start making computers again.
5. If Hondas have better gas mileage than Range Rovers and you are looking for some- thing that is easy to park, I recommend that you buy the Honda.
6. Global warming will decrease if and only if emissions decrease in China and other major polluters around the world.
7. One cannot be both happy and successful in our society, but one can be happy or successful.
8. I will pass this course if and only if I study hard and practice regularly, if I have the time and energy to do so.
9. God can only exist if evil does not exist, if it is true that God is both all-powerful and all-good.
10. The conflict in Israel will end only if the Palestinians feel that they can live outside the supervision of the Israelis and the two sides stop attacking one another.
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Section 4.4 Using Truth Tables to Test for Validity
4.4 Using Truth Tables to Test for Validity Truth tables serve many valuable purposes. One is to help us better understand how the logi- cal operators work. Another is to help us understand how truth is determined within formally structured sentences. One of the most valuable things truth tables offer is the ability to test argument forms for validity. As mentioned at the beginning of this chapter, one of the main pur- poses of formal logic is to make the concept of validity precise. Truth tables help us do just that.
As mentioned in previous chapters, an argument is valid if and only if the truth of its premises guarantees the truth of its conclusion. This is equivalent to saying that there is no way that the premises can be true and the conclusion false.
Truth tables enable us to determine precisely if there is any way for all of the premises to be true and the conclusion false (and therefore whether the argument is valid): We simply create a truth table for the premises and conclusion and see if there is any row on which all of the premises are true and the conclusion is false. If there is, then the argument is invalid, because that row shows that it is possible for the premises to be true and the conclusion false. If there is no such line, then the argument is valid:
Since the rows of a truth table cover all possibilities, if there is no row on which all of the premises are true and the conclusion is false, then it is impossible, so the argument is valid.
Let us start with a simple example—note that the ∴ symbol means “therefore”:
P ∨ Q ~Q ∴ P
This argument form is valid; if there are only two options, P and Q, and one of them is false, then it follows that the other one must be true. However, how can we formally demonstrate its validity? One way is to create a truth table to find out if there is any possible way to make all of the premises true and the conclusion false.
Here is how to set up the truth table, with a column for each premise (P1 and P2) and the conclusion (C):
P1 P2 C
P Q P ∨ Q ~Q P
T T T F F T F F
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Section 4.4 Using Truth Tables to Test for Validity
We then fill in the columns, with the correct truth values:
P1 P2 C
P Q P ∨ Q ~Q P
T T T F T T F T T T F T T F F F F F T F
We then check if there are any rows in which all of the premises are true and the conclusion is false. A brief scan shows that there are no such lines. The first two rows have true conclusions, and the remaining two rows each have at least one false premise. Since the rows of a truth table represent all possible combinations of truth values, this truth table therefore demon- strates that there is no possible way to make all of the premises true and the conclusion false. It follows, therefore, that the argument is logically valid.
To summarize, the steps for using the truth table method to determine an argument’s validity are as follows:
1. Set up the truth table by creating rows for each possible combination of truth values for the basic letters and a column for each premise and the conclusion.
2. Fill out the truth table by filling out the truth values in each column according to the rules for the relevant operator (~, &, ∨, →, ↔).
3. Use the table to evaluate the argument’s validity. If there is even one row on which all of the premises are true and the conclusion is false, then the argument is invalid; if there is no such row, then the argument is valid.
This truth table method works for all arguments in propositional logic: Any valid proposi- tional logic argument will have a truth table that shows it is valid, and every invalid proposi- tional logic argument will have a truth table that shows it is invalid. Therefore, this is a perfect test for validity: It works every time (as long as we use it accurately).
Examples With Arguments With Two Letters Let us do another example with only two letters. This argument will be slightly more complex but will still involve only two letters, A and B.
Example 1 A → B ~(A & B) ∴ ~(B ∨ A)
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Section 4.4 Using Truth Tables to Test for Validity
To test this symbolized argument for validity, we first set up the truth table by creating rows with all of the possible truth values for the basic letters on the left and then create a column for each premise (P1 and P2) and conclusion (C), as follows:
P1 P2 C
A B A → B ~(A & B) ~(B ∨ A)
T T T F F T F F
Second, we fill out the truth table using the rules created by the basic truth tables for each operator. Remember to use helper columns where necessary as steps toward filling in the columns of complex formulas. Here is the truth table with only the helper columns filled in:
P1 P2 C
A B A → B A & B ~(A & B) B ∨ A ~(B ∨ A)
T T T T T F F T F T F T F F F F
Here is the truth table with the rest of the columns filled in:
P1 P2 C
A B A → B A & B ~(A & B) B ∨ A ~(B ∨ A)
T T T T F T F T F F F T T F F T T F T T F F F T F T F T
Finally, to evaluate the argument’s validity, all we have to do is check to see if there are any lines in which all of the premises are true and the conclusion is false. Again, if there is such a line, since we know it is possible for all of the premises to be true and the conclusion false, the argument is invalid. If there is no such line, then the argument is valid.
It does not matter what other rows may exist in the table. There may be rows in which all of the premises are true and the conclusion is also true; there also may be rows with one or more false premises. Neither of those types of rows determine the argument’s validity; our only concern is whether there is any possible row on which all of the premises are true and the conclusion false. Is there such a line in our truth table? (Remember: Ignore the helper columns and just focus on the premises and conclusion.)
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Section 4.4 Using Truth Tables to Test for Validity
The answer is yes, all of the premises are true and the conclusion is false in the third row. This row supplies a proof that this argument’s form is invalid. Here is the line:
P1 P2 C
A B A → B ~(A & B) ~(B ∨ A)
F T T T F
Again, it does not matter what is on the other row. As long as there is (at least) one row in which all of the premises are true and the conclusion false, the argument is invalid.
Example 2 A → (B & ~A) A ∨ ~B ∴ ~(A ∨ B)
First we set up the truth table:
P1 P2 C
A B ~A B & ~A A → (B & ~A) ~B A ∨ ~B A ∨ B ~(A ∨ B)
T T T F F T F F
Next we fill in the values, filling in the helper columns first:
P1 P2 C
A B ~A B & ~A A → (B & ~A) ~B A ∨ ~B A ∨ B ~(A ∨ B)
T T F F F T T F F F T T F T T T F T F F T F T F
Now that the helper columns are done, we can fill in the rest of the table’s values:
P1 P2 C
A B ~A B & ~A A → (B & ~A) ~B A ∨ ~B A ∨ B ~(A ∨ B)
T T F F F F T T F T F F F F T T T F F T T T T F F T F F F T F T T T F T
Finally, we evaluate the table for validity. Here we see that there are no lines in which all of the premises are true and the conclusion is false. Therefore, there is no possible way to make all of the premises true and the conclusion false, so the argument is valid.
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Section 4.4 Using Truth Tables to Test for Validity
The earlier examples each had two premises. The following example has three premises. The steps of the truth table test are identical.
Example 3 ~(M ∨ B) M → ~B B ∨ ~M ∴ ~M & B)
First we set up the truth table. This table already has the helper columns filled in.
P1 P2 P3 C
M B M ∨ B ~(M ∨ B) ~B M → ~B ~M B ∨ ~M ~M & B
T T T F F T F T T F F T T F T F F F T T
Now we fill in the rest of the columns, using the helper columns to determine the truth values of our premises and conclusion on each row:
P1 P2 P3 C
M B M ∨ B ~(M ∨ B) ~B M → ~B ~M B ∨ ~M ~M & B
T T T F F F F T F T F T F T T F F F F T T F F T T T T F F F T T T T T F
Now we look for a line in which all of the premises are true and the conclusion false. The final row is just such a line. This demonstrates conclusively that the argument is invalid.
Examples With Arguments With Three Letters The last example had three premises, but only two letters. These next examples will have three letters. As explained earlier in the chapter, the presence of the extra letter doubles the number of rows in the truth table.
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Section 4.4 Using Truth Tables to Test for Validity
Example 1 A → (B ∨ C) ~(C & B) ∴ ~(A & B)
First we set up the truth table. Note, as mentioned earlier, now there are eight possible com- binations on the left.
P1 P2 C
A B C B ∨ C A → (B ∨ C) C & B ~(C & B) A & B ~(A & B)
T T T T T F T F T T F F F T T F T F F F T F F F
Then we fill the table out. Here it is with just the helper columns:
P1 P2 C
A B C B ∨ C A → (B ∨ C) C & B ~(C & B) A & B ~(A & B)
T T T T T T T T F T F T T F T T F F T F F F F F F T T T T F F T F T F F F F T T F F F F F F F F
Here is the full truth table:
P1 P2 C
A B C B ∨ C A → (B ∨ C) C & B ~ (C & B) A & B ~(A & B)
T T T T T T F T F T T F T T F T T F T F T T T F T F T T F F F F F T F T F T T T T T F F T F T F T T F T F T F F T T T F T F T F F F F T F T F T
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Section 4.4 Using Truth Tables to Test for Validity
Finally, we evaluate; that is, we look for a line in which all of the premises are true and the conclusion false. This is the case with the second line. Once you find such a line, you do not need to look any further. The existence of even one line in which all of the premises are true and the conclusion is false is enough to declare the argument invalid.
Let us do another one with three letters:
Example 2 A → ~B B ∨ C ∴ A → C
We begin by setting up the table:
P1 P2 C
A B C ~B A → ~B B ∨ C A → C
T T T T T F T F T T F F F T T F T F F F T F F F
Now we can fill in the rows, beginning with the helper columns:
P1 P2 C
A B C ~B A → ~B B ∨ C A → C
T T T F F T T T T F F F T F T F T T T T T T F F T T F F F T T F T T T F T F F T T T F F T T T T T F F F T T F T
Here, when we look for a line in which all of the premises are true and the conclusion false, we do not find one. There is no such line; therefore the argument is valid.
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Section 4.4 Using Truth Tables to Test for Validity
Practice Problems 4.3
Answer these questions about truth tables.
1. A truth table with two variables has how many lines? a. 1 b. 2 c. 4 d. 8
2. A truth table with three variables has how many lines? a. 1 b. 2 c. 4 d. 8
3. In order to prove that an argument is invalid using a truth table, one must __________. a. find a line in which all premises and the conclusion are false b. find a line in which the premises are true and the conclusion is false c. find a line in which the premises are false and the conclusion is true d. find a line in which the premises and the conclusion are true
4. This is how one can tell if an argument is valid using a truth table: a. There is a line in which the premises and the conclusion are true. b. There is no line in which the premises are false. c. There is no line in which the premises are true and the conclusion is false. d. All of the above e. None of the above
5. When two statements have the same truth values in all circumstances, they are said to be __________. a. logically contradictory b. logically equivalent c. logically cogent d. logically valid
6. An if–then statement is called a __________. a. conjunction b. disjunction c. conditional d. biconditional
7. An if and only if statement is called a __________. a. conjunction b. disjunction c. conditional d. biconditional
8. An and statement is called a __________. a. conjunction b. disjunction c. conditional d. biconditional
(continued)
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Section 4.4 Using Truth Tables to Test for Validity
Utilize truth tables to determine the validity of the following arguments.
9. J → K J ∴ K
10. H → G G ∴ H
11. K→ K ∴ K
12. ~(H & Y) Y ∨~H ∴ ~H
13. W → Q ~W ∴ ~Q
14. A → B B → C ∴ A → C
15. ~(P ↔ U) ∴ ~(P → U)
16. ~S ∨ H ~S ∴ ~H
17. ~K → ~L J → ~K ∴ J → ~L
18. Y & P P ∴ ~Y
19. A → ~G V → ~G ∴ A → V
20. B & K & I ∴ K
Practice Problems 4.3 (continued)
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Section 4.5 Some Famous Propositional Argument Forms
4.5 Some Famous Propositional Argument Forms Using the truth table test for validity, we have seen that we can determine the validity or inva- lidity of all propositional argument forms. However, there are some basic argument forms that are so common that it is worthwhile simply to memorize them and whether or not they are valid. We will begin with five very famous valid argument forms and then cover two of the most famous invalid argument forms.
Common Valid Forms It is helpful to know some of the most commonly used valid argument forms. Those presented in this section are used so regularly that, once you learn them, you may notice people using them all the time. They are also used in what are known as deductive proofs (see A Closer Look: Deductive Proofs).
A Closer Look: Deductive Proofs A big part of formal logic is constructing proofs. Proofs in logic are a lot like proofs in mathematics. We start with certain premises and then use certain rules—called rules of inference—in a step-by-step way to arrive at the con- clusion. By using only valid rules of inference and apply- ing them carefully, we make certain that every step of the proof is valid. Therefore, if there is a logical proof of the conclusion from the premises, then we can be certain that the argument itself is valid.
The rules of inference used in deductive proofs are actually just simple valid argument forms. In fact, the valid argument forms covered here—including modus ponens, hypothetical syllogisms, and disjunctive syllogisms—are examples of argument forms that are used as inference rules in logical proofs. Using these and other formal rules, it is possible to give a logical proof for every valid argument in propositional logic (Kennedy, 2012).
Logicians, mathematicians, philosophers, and computer scientists use logical proofs to show that the validity of certain inferences is absolutely certain and founded on
the most basic principles. Many of the inferences we make in daily life are of limited certainty; however, the validity of inferences that have been logically proved is considered to be the most certain and uncontroversial of all knowledge because it is derivable from pure logic.
Covering how to do deductive proofs is beyond the scope of this book, but readers are invited to peruse a book or take a course on formal logic to learn more about how deductive proofs work.
Mark Wragg/iStock/Thinkstock
Rather than base decisions on chance, people use the information around them to make deductive and inductive inferences with varying degrees of strength and validity. Logicians use proofs to show the validity of inferences.
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Section 4.5 Some Famous Propositional Argument Forms
Modus Ponens Perhaps the most famous propositional argument form of all is known as modus ponens— Latin for “the way of putting.” (You may recognize this form from the earlier section on the truth table method.) Modus ponens has the following form:
P → Q P ∴ Q
You can see that the argument is valid just from the meaning of the conditional. The first premise states, “If P is true, then Q is true.” It would logically follow that if P is true, as the second premise states, then Q must be true. Here are some examples:
If you want to get an A, you have to study. You want to get an A. Therefore, you have to study.
If it is raining, then the street is wet. It is raining. Therefore, the street is wet.
If it is wrong, then you shouldn’t do it. It is wrong. Therefore, you shouldn’t do it.
A truth table will verify its validity.
P1 P2 C
P Q P → Q P Q
T T T T T T F F T F F T T F T F F T F F
There is no line in which all of the premises are true and the conclusion false, verifying the validity of this important logical form.
Modus Tollens A closely related form has a closely related name. Modus tollens—Latin for “the way of tak- ing”—has the following form:
P → Q ~Q ∴ ~P
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Section 4.5 Some Famous Propositional Argument Forms
A truth table can be used to verify the validity of this form as well. However, we can also see its validity by simply thinking it through. Suppose it is true that “If P, then Q.” Then, if P were true, it would follow that Q would be true as well. But, according to the second premise, Q is not true. It follows, therefore, that P must not be true; otherwise, Q would have been true. Here are some examples of arguments that fit this logical form:
In order to get an A, I must study. I will not study. Therefore, I will not get an A.
If it rained, then the street would be wet. The street is not wet. Therefore, it must not have rained.
If the ball hit the window, then I would hear glass shattering. I did not hear glass shattering. Therefore, the ball must not have hit the window.
For practice, construct a truth table to demonstrate the validity of this form.
Disjunctive Syllogism A disjunctive syllogism is a valid argument form in which one premise states that you have two options, and another premise allows you to rule one of them out. From such premises, it follows that the other option must be true. Here are two versions of it formally (both are valid):
P ∨ Q ~P ∴ Q
P ∨ Q ~Q ∴ P
In other words, if you have “P or Q” and not Q, then you may infer P. Here is another example: “Either the butler or the maid did it. It could not have been the butler. Therefore, it must have been the maid.” This argument form is quite handy in real life. It is frequently useful to con- sider alternatives and to rule one out so that the options are narrowed down to one.
Ruth Black/iStock/Thinkstock
Evaluate this argument form for validity: If the cake is made with sugar, then the cake is sweet. The cake is not sweet. Therefore, the cake is not made with sugar.
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Section 4.5 Some Famous Propositional Argument Forms
Hypothetical Syllogism One of the goals of a logically valid argument is for the premises to link together so that the conclusion follows smoothly, with each premise providing a link in the chain. Hypothetical syllogism provides a nice demonstration of just such premise linking. Hypothetical syllogism takes the following form:
P → Q Q → R ∴ P → R
For example, “If you lose your job, then you will have no income. If you have no income, then you will starve. Therefore, if you lose your job, then you will starve!”
Double Negation Negating a sentence (putting a ~ in front of it) makes it say the opposite of what it originally said. However, if we negate it again, we end up with a sentence that means the same thing as our original sentence; this is called double negation.
Imagine that our friend Johnny was in a race, and you ask me, “Did he win?” and I respond, “He did not fail to win.” Did he win? It would appear so. Though some languages allow double negations to count as negative statements, in logic a double negation is logically equivalent to the original statement. Both of these forms, therefore, are valid:
P ∴ ~~P
~~P ∴ P
A truth table will verify that each of these forms is valid; both P and ~~P have the same truth values on every row of the truth table.
Common Invalid Forms Both modus ponens and modus tollens are logically valid forms, but not all famous logical forms are valid. The last two forms we will discuss—denying the antecedent and affirming the consequent—are famous invalid forms that are the evil twins of the previous two.
Denying the Antecedent Take a look at the following argument:
If you give lots of money to charity, then you are nice. You do not give lots of money to charity. Therefore, you must not be nice.
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Section 4.5 Some Famous Propositional Argument Forms
This might initially seem like a valid argument. However, it is actually invalid in its form. To see that this argument is logically invalid, take a look at the following argument with the same form:
If my cat is a dog, then it is a mammal. My cat is not a dog. Therefore, my cat is not a mammal.
This second example is clearly invalid since the premises are true and the conclusion is false. Therefore, there must be something wrong with the form. Here is the form of the argument:
P → Q ~P ∴ ~Q
Because this argument form’s second premise rejects the antecedent, P, of the conditional in the first premise, this argument form is referred to as denying the antecedent. We can con- clusively demonstrate that the form is invalid using the truth table method.
Here is the truth table:
P1 P2 C
P Q P → Q ~P ~Q
T T T F F T F F F T F T T T F F F T T T
We see on the third line that it is possible to make both premises true and the conclusion false, so this argument form is definitely invalid. Despite its invalidity, we see this form all the time in real life. Here some examples:
If you are religious, then you believe in living morally. Jim is not religious, so he must not believe in living morally.
Plenty of people who are not religious still believe in living morally. Here is another one:
If you are training to be an athlete, then you should stay in shape. You are not training to be an athlete. Thus, you should not stay in shape.
There are plenty of other good reasons to stay in shape.
If you are Republican, then you support small government. Jack is not Republican, so he must not support small government.
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Section 4.5 Some Famous Propositional Argument Forms
Libertarians, for example, are not Republicans, yet they support small government. These examples abound; we can generate them on any topic.
Because this argument form is so common and yet so clearly invalid, denying the antecedent is a famous fallacy of formal logic.
Affirming the Consequent Another famous formal logical fallacy also begins with a conditional. However, the other two lines are slightly different. Here is the form:
P → Q Q ∴ P
Because the second premise states the consequent of the conditional, this form is called affirming the consequent. Here is an example:
If you get mono, you will be very tired. You are very tired. Therefore, you have mono.
The invalidity of this argument can be seen in the following argument of the same form:
If my cat is a dog, then it is a mammal. My cat is a mammal. Therefore, my cat is a dog.
Clearly, this argument is invalid because it has true premises and a false conclusion. There- fore, this must be an invalid form. A truth table will further demonstrate this fact:
P1 P2 C
P Q P → Q Q P
T T T T T T F F F T F T T T F F F T F F
The third row again demonstrates the possibility of true premises and a false conclusion, so the argument form is invalid. Here are some examples of how this argument form shows up in real life:
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Section 4.5 Some Famous Propositional Argument Forms
In order to get an A, I have to study. I am going to study. Therefore, I will get an A.
There might be other requirements to get an A, like showing up for the test.
If it rained, then the street would be wet. The street is wet. Therefore, it must have rained.
Sprinklers may have done the job instead.
If he committed the murder, then he would have had to have motive and opportunity. He had motive and opportunity. Therefore, he committed the murder.
This argument gives some evidence for the conclusion, but it does not give proof. It is possible that someone else also had motive and opportunity.
The reader may have noticed that in some instances of affirming the consequent, the prem- ises do give us some reason to accept the conclusion. This is because of the similarity of this form to the inductive form known as inference to the best explanation, which is covered in more detail in Chapter 6. In such inferences we create an “if–then” statement that expresses something that would be the case if a certain assumption were true. These things then act as symptoms of the truth of the assumption. When those symptoms are observed, we have some evidence that the assumption is true. Here are some examples:
If you have measles, then you would present the following symptoms. . . . You have all of those symptoms. Therefore, it looks like you have measles.
If he is a faithful Catholic, then he would go to Mass. I saw him at Mass last Sunday. Therefore, he is probably a faithful Catholic.
All of these seem to supply decent evidence for the conclusion; however, the argument form is not logically valid. It is logically possible that another medical condition could have the same symptoms or that a person could go to Mass out of curiosity. To determine the (inductive) inferential strength of an argument of that form, we need to think about how likely Q is under different assumptions.
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Section 4.5 Some Famous Propositional Argument Forms
A Closer Look: Translating Categorical Logic The chapter about categorical logic seems to cover a completely different type of reasoning than this chapter on propositional logic. However, logical advancements made just over a century ago by a man named Gottlob Frege showed that the two types of logic can be com- bined in what has come to be known as quantificational logic (also known as predicate logic) (Frege, 1879).
In addition to truth-functional logic, quantificational logic allows us to talk about quantities by including logical terms for all and some. The addition of these terms dramatically increases the power of our logical language and allows us to represent all of categorical logic and much more. Here is a brief overview of how the basic sentences of categorical logic can be repre- sented within quantificational logic.
The statement “All dogs are mammals” can be understood to mean “If you are a dog, then you are a mammal.” The word you in this sentence applies to any individual. In other words, the sentence states, “For all individuals, if that individual is a dog, then it is a mammal.” In general, statements of the form “All S is M” can be represented as “For all things, if that thing is S, then it is M.”
The statement “Some dogs are brown” means that there exist dogs that are brown. In other words, there exist things that are both dogs and brown. Therefore, statements of the form “Some S is M” can be represented as “There exists a thing that is both S and M” (propositions of the form “Some S are not M” can be represented by simply adding a negation in front of the M).
Statements like “No dogs are reptiles” can be understood to mean that all dogs are not reptiles. In general, statements of the form “No S are M” can be represented as “For all things, if that thing is an S, then it is not M.”
Quantificational logic allows us to additionally represent the meanings of statements that go well beyond the AEIO propositions of categorical logic. For example, complex statements like “All dogs that are not brown are taller than some cats” can also be represented with the power of quantificational logic though they are well beyond the capacity of categorical logic. The additional power of quantificational logic enables us to represent the meaning of vast stretches of the English language as well as statements used in formal disciplines like math- ematics. More instruction in this interesting area can be found in a course on formal logic.
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Section 4.5 Some Famous Propositional Argument Forms
Practice Problems 4.4
Each of the following arguments is a deductive form. Identify the valid form under which the example falls. If the example is not a valid form, select “not a valid form.”
1. If we do not decrease poverty in society, then our society will not be an equal one. We are not going to decrease poverty in society. Therefore, our society will not be an equal one. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
2. If we do not decrease poverty in society, then our society will not be an equal one. Our society will be an equal one. Therefore, we will decrease poverty in society. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
3. If the moon is full, then it is a good time for night fishing. If it’s a good time for night fishing, then we should go out tonight. Therefore, if the moon is full, then we should go out tonight. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
4. Either the Bulls or the Knicks will lose tonight. The Bulls are not going to lose. Therefore, the Knicks will lose. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
5. If the battery is dead, then the car won’t start. The car won’t start. Therefore, the bat- tery is dead. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
6. If I take this new job, then we will have to move to Alaska. I am not going to take the new job. Therefore, we will not have to move to Alaska. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
(continued)
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Summary and Resources
7. If human perception conditions reality, then humans cannot know things in them- selves. If humans cannot know things in themselves, then they cannot know the truth. Therefore, if human perceptions conditions reality, then humans cannot know the truth. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
8. We either adopt the plan or we will be in danger of losing our jobs. We are not going to adopt the plan. Therefore, we will be in danger of losing our jobs. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
9. If media outlets are owned by corporations with advertising interests, then it will be difficult for them to be objective. Media outlets are owned by corporations with advertising interests. Therefore, it will be difficult for them to be objective. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
10. If you eat too much aspartame, you will get a headache. You do not have a headache. Therefore, you did not eat too much aspartame. a. modus ponens b. modus tollens c. disjunctive syllogism d. hypothetical syllogism e. not a valid form
Practice Problems 4.4 (continued)
Summary and Resources
Chapter Summary Propositional logic shows how the truth values of complex statements can be systematically derived from the truth values of their parts. Words like and, or, not, and if . . . then . . . each have truth tables that demonstrate the algorithms for determining these truth values. Once we have found the logical form of an argument, we can determine whether it is logically valid by using the truth table method. This method involves creating a truth table that represents all possible truth values of the component parts and the resulting values for the premises and conclusion
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Summary and Resources
of the argument. If there is even one row of the truth table in which all of the premises are true and the conclusion is false, then the argument is invalid; if there is no such row, then it is valid.
Knowledge of propositional logic has proved very valuable to humankind: It allows us to formally demonstrate the validity of different types of reasoning; it helps us precisely under- stand the meaning of certain types of terms in our language; it enables us to determine the truth conditions of formally complex statements; and it forms the basis for computing.
Critical Thinking Questions
1. Symbolizing arguments makes them easier to visualize and examine in the realm of propositional logic. Do you find that the symbols make things easier to visualize or more confusing? If logicians use these methods to make things easier, then what does that mean if you think that using these symbols is confusing?
2. In your own words, what is the difference between categorical logic and proposi- tional logic? How do they relate to one another? How do they differ?
3. How does understanding how to symbolize statements and complete truth tables relate to your everyday life? What is the practical importance of understanding how to use these methods to determine validity?
4. If you were at work or with your friends and someone presented an argument, do you think you could evaluate it using the methods you have learned thus far in this book? Is it important to evaluate arguments, or is this just something academics do in their spare time? Why do you believe this is (or is not) the case?
5. How would you now explain the concept of validity to someone with whom you interact on a daily basis who might not have an understanding of logic? How would you explain how validity differs from truth?
Web Resources http://www.manyworldsof logic.com/exercises/quizTruthFunctional.html Test your understanding of propositional, or truth-functional, logic by taking the quizzes available at philosophy professor Paul Herrick’s Many Worlds of Logic website.
https://www.youtube.com/watch?v=moHkk_89UZE Watch a video that walks you through how to construct a truth table.
https://www.youtube.com/watch?feature=player_embedded&v=83xPkTqoulE Watch Ashford University professor Justin Harrison explain how to construct a conjunction truth table.
Key Terms
affirming the consequent An argument with two premises, one of which is a condi- tional and the other of which is the conse- quent of that conditional. It has the form P → Q, Q, therefore P. It is invalid.
antecedent The part of a conditional state- ment that occurs after the if; it is the P in P → Q.
biconditional A statement of the form P ↔ Q (P if and only if Q).
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Summary and Resources
conditional An “if–then” statement. It is symbolized P → Q.
conjunction A statement in which two sen- tences are joined with an and. It is symbol- ized P & Q. Also, an inference rule that allows us to infer P & Q from premises P and Q.
connectives See operators.
consequent The part of a conditional state- ment that occurs after the then; it is the Q in P → Q.
converse The result of switching the order of the terms within a conditional or cat- egorical statement. The converse of P → Q is Q → P. The converse of “All S are M” is “All M are S.”
denying the antecedent An argument with two premises, one of which is a conditional and the other of which is the negation of the antecedent of that conditional. It has the form P → Q, ~P, therefore ~Q. It is invalid.
disjunction A sentence in which two smaller sentences are joined with an or. It is symbolized P ∨ Q.
disjunctive syllogism An inference rule that allows us to infer one disjunct from the nega- tion of the other disjunct. If you have “P or Q” and you have not P, then you may infer Q. If you have “P or Q” and not Q, then you may infer P.
double negation The result of negating a sentence that has already been negated (one that already has a ~ in front of it). The result- ing sentence means the same thing as the original, non-negated sentence.
hypothetical syllogism An inference rule that allows us to infer P → R from P → Q and Q → R.
logically equivalent Two statements are logically equivalent if they have the same values on every row of a truth table. That means they are true in the exact same circumstances.
modus ponens An argument that affirms the antecedent of its conditional premise. It has the form P → Q, P, therefore Q.
modus tollens An argument that denies the consequent of its conditional premise. It has the form P → Q, ~Q, therefore ~P.
negation A statement that asserts that another statement, P, is false. It is symbol- ized ~P and pronounced “not P.”
operators Words (like and, or, not, and if . . . then . . . ) used to make complex state- ments whose truth values are functions of the truth values of their parts. Also known as connectives when they are used to link two sentences.
proposition The meaning expressed by a claim that asserts something is true or false.
propositional logic A way of clarifying reasoning by breaking down the forms of complex claims into the simple propositions of which they are composed, connected with truth-functional operators. Also known as sentence logic, sentential logic, statement logic, and truth-functional logic.
sentence variables Letters like P and Q that are used in forms to represent any sentence at all, just as a variable in algebra represents any number.
statement form The result of replacing the component statements in a sentence with statement variables (like P and Q), con- nected with logical operators.
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Summary and Resources
truth table A table in which columns to the right show the truth values of complex sentences based on each combination of truth values of their component sentences on the left.
truth value An indicator of whether a state- ment is true on a given row of a truth table. A statement’s truth value is true (abbrevi- ated T) if the statement is true; it is false (abbreviated F) if the statement is false.
Answers to Practice Problems Practice Problems 4.1
1. a 2. b 3. d 4. b 5. c
6. c 7. d 8. a 9. c
10. b
Practice Problems 4.2
1. ~(B ∨ L) 2. D & S 3. G → (A & L) 4. (M & D) ∨ G 5. (H & L) → R
6. G ↔ (C & M) 7. ~(H & S) & (H ∨ S) 8. (T & E) → [P ↔ (H & P)] 9. (P & G) → (X → ~E)
10. C→ (P & T)
Practice Problems 4.3
1. c 2. d 3. b 4. c
5. b 6. c 7. d 8. a
9. valid J K J → K J K
T T T T T T F F T F F T T F T F F T F F
10. invalid H G H → G G H
T T T T T T F F F T F T T T F F F T F F
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Summary and Resources
11. valid K K → K K
T T T F T F
12. valid H Y H & Y ~(H & Y) Y ∨ ~H ~H
T T T F T F T F F T T F F T F T T T F F F T T T
13. invalid W Q W → Q ~W ~Q
T T T F F T F F F T F T T T F F F T T T
14. valid A B C A → B B → C A → C
T T T T T T T T F T F T T F T F T T T F F F T F F T T T T T F T F T F T F F T T T T F F F T T T
15. invalid P U P ↔ U ~(P ↔ U) P → U ~(P → U)
T T T F T F T F F T F T F T F T T F F F T F T F
16. invalid S H ~S ∨ H ~S H
T T T F T T F F F F F T T T T F F T T F
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Summary and Resources
17. valid J K L ~K → ~L J → ~K P → ~L
T T T T F F T T F T F T T F T F T F T F F T T T F T T T T T F T F T T T F F T F T T F F F T T T
18. invalid Y P Y & P P ~Y
T T T T F T F F F F F T F T T F F F F T
19. invalid A G V A → ~G V → ~G A → V
T T T F F T T T F F T F T F T T T T T F F T T T F T T T F T F T F T T T F F T T T T F F F T T T
20. valid B I K K & I B & (K & I) K
T T T T T T T T F F F F T F T F F T T F F F F F F T T T F T F T F F F F F F T F F T F F F F F F
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Summary and Resources
Practice Problems 4.4
1. a 2. b 3. d 4. d 5. e
6. e 7. d 8. c 9. a
10. b
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59
3Deductive Reasoning
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Learning Objectives After reading this chapter, you should be able to:
1. Define basic key terms and concepts within deductive reasoning.
2. Use variables to represent an argument’s logical form.
3. Use the counterexample method to evaluate an argument’s validity.
4. Categorize different types of deductive arguments.
5. Analyze the various statements—and the relationships between them—in categorical arguments.
6. Evaluate categorical syllogisms using the rules of the syllogism and Venn diagrams.
7. Differentiate between sorites and enthymemes.
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Section 3.1 Basic Concepts in Deductive Reasoning
By now you should be familiar with how the field of logic views arguments: An argument is just a collection of sentences, one of which is the conclusion and the rest of which, the prem- ises, provide support for the conclusion. You have also learned that not every collection of sentences is an argument. Stories, explanations, questions, and debates are not arguments, for example. The essential feature of an argument is that the premises support, prove, or give evidence for the conclusion. This relationship of support is what makes a collection of sen- tences an argument and is the special concern of logic. For the next four chapters, we will be taking a closer look at the ways in which premises might support a conclusion. This chapter discusses deductive reasoning, with a specific focus on categorical logic.
3.1 Basic Concepts in Deductive Reasoning As noted in Chapter 2, at the broadest level there are two types of arguments: deductive and inductive. The difference between these types is largely a matter of the strength of the con- nection between premises and conclusion. Inductive arguments are defined and discussed in Chapter 5; this chapter focuses on deductive arguments. In this section we will learn about three central concepts: validity, soundness, and deduction.
Validity Deductive arguments aim to achieve validity, which is an extremely strong connection between the premises and the conclusion. In logic, the word valid is only applied to argu- ments; therefore, when the concept of validity is discussed in this text, it is solely in reference to arguments, and not to claims, points, or positions. Those expressions may have other uses in other fields, but in logic, validity is a strict notion that has to do with the strength of the connection between an argument’s premises and conclusion.
To reiterate, an argument is a collection of sentences, one of which (the conclusion) is sup- posed to follow from the others (the premises). A valid argument is one in which the truth of the premises absolutely guarantees the truth of the conclusion; in other words, it is an argu- ment in which it is impossible for the premises to be true while the conclusion is false. Notice that the definition of valid does not say anything about whether the premises are actually true, just whether the conclusion could be false if the premises were true. As an example, here is a silly but valid argument:
Everything made of cheese is tasty. The moon is made of cheese. Therefore, the moon is tasty.
No one, we hope, actually thinks that the moon is made of cheese. You may or may not agree that everything made of cheese is tasty. But you can see that if everything made of cheese were tasty, and if the moon were made of cheese, then the moon would have to be tasty. The truth of that conclusion simply logically follows from the truth of the premises.
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Section 3.1 Basic Concepts in Deductive Reasoning
Here is another way to better understand the strictness of the concept of validity: You have probably seen some far-fetched movies or read some bizarre books at some point. Books and movies have magic, weird science fiction, hallucinations, and dream sequences—almost any- thing can happen. Imagine that you were writing a weird, bizarre novel, a novel as far removed from reality as possible. You certainly could write a novel in which the moon was made of cheese. You could write a novel in which everything made of cheese was tasty. But you could not write a novel in which both of these premises were true, but in which the moon turned out not to be tasty. If the moon were made of cheese but was not tasty, then there would be at least one thing that was made of cheese and was not tasty, making the first premise false.
Therefore, if we assume, even hypothetically, that the premises are true (even in strange hypothetical scenarios), it logically follows that the conclusion must be as well. Therefore, the argument is valid. So when thinking about whether an argument is valid, think about whether it would be possible to have a movie in which all the premises were true but the conclusion was false. If it is not possible, then the argument is valid.
Here is another, more realistic, example:
All whales are mammals. All mammals breathe air. Therefore, all whales breathe air.
Is it possible for the premises to be true and the conclusion false? Well, imagine that the conclu- sion is false. In that case there must be at least one whale that does not breathe air. Let us call that whale Fred. Is Fred a mammal? If he is, then there is at least one mammal that does not breathe air, so the second premise would be false. If he isn’t, then there is at least one whale that is not a mammal, so the first premise would be false. Again, we see that it is impossible for the conclusion to be false and still have all the premises be true. Therefore, the argument is valid.
Here is an example of an invalid argument:
All whales are mammals. No whales live on land. Therefore, no mammals live on land.
In this case we can tell that the truth of the conclusion is not guaranteed by the premises because the premises are actually true and the conclusion is actually false. Because a valid argument means that it is impossible for the premises to be true and the conclusion false, we can be sure that an argument in which the premises are actually true and the conclusion is actually false must be invalid. Here is a trickier example of the same principle:
All whales are mammals. Some mammals live in the water. Therefore, some whales live in the water.
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Section 3.1 Basic Concepts in Deductive Reasoning
This one is trickier because both prem- ises are true, and the conclusion is true as well, so many people may be tempted to call it valid. However, what is important is not whether the prem- ises and conclusion are actually true but whether the premises guarantee that the conclusion is true. Think about making a movie: Could you make a movie that made this argument’s prem- ises true and the conclusion false?
Suppose you make a movie that is set in a future in which whales move back onto land. It would be weird, but not any weirder than other ideas movies have presented. If seals still lived in the water in this movie, then both prem- ises would be true, but the conclusion would be false, because all the whales would live on land.
Because we can create a scenario in which the premises are true and the conclusion is false, it follows that the argument is invalid. So even though the conclusion isn’t actually false, it’s enough that it is possible for it to be false in some situation that would make the premises true. This mere possibility means the argument is invalid.
Soundness Once you understand what valid means in logic, it is very easy to understand the concept of soundness. A sound argument is just a valid argument in which all the premises are true. In defining validity, we saw two examples of valid arguments; one of them was sound and the other was not. Since both examples were valid, the one with true premises was the one that was sound.
We also saw two examples of invalid arguments. Both of those are unsound simply because they are invalid. Sound arguments have to be valid and have all true premises. Notice that since only arguments can be valid, only arguments can be sound. In logic, the concept of soundness is not applied to principles, observations, or anything else. The word sound in logic is only applied to arguments.
Here is an example of a sound argument, similar to one you may recall seeing in Chapter 2:
All men are mortal. Bill Gates is a man. Therefore, Bill Gates is mortal.
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Consider the following argument: “If it is raining, then the streets are wet. The streets are wet. Therefore, it is raining.” Is this a valid argument? Could there be another reason why the road is wet?
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Section 3.1 Basic Concepts in Deductive Reasoning
There is no question about the argument’s validity. Therefore, as long as these premises are true, it follows that the conclusion must be true as well. Since the premises are, in fact, true, we can reason the conclusion is too.
It is important to note that having a true conclusion is not part of the definition of soundness. If we were required to know that the conclusion was true before deciding whether the argu- ment is sound, then we could never use a sound argument to discover the truth of the conclu- sion; we would already have to know that the conclusion was true before we could judge it to be sound. The magic of how deductive reasoning works is that we can judge whether the reasoning is valid independent of whether we know that the premises or conclusion are actu- ally true. If we also notice that the premises are all true, then we may infer, by the power of pure reasoning, the truth of the conclusion.
Therefore, knowledge of the truth of the premises and the ability to reason validly enable us to arrive at some new information: that the conclusion is true as well. This is the main way that logic can add to our bank of knowledge.
Although soundness is central in considering whether to accept an argument’s conclusion, we will not spend much time worrying about it in this book. This is because logic really deals with the connections between sentences rather than the truth of the sentences themselves. If some- one presents you with an argument about biology, a logician can help you see whether the argu- ment is valid—but you will need a biologist to tell you whether the premises are true. The truth of the premises themselves, therefore, is not usually a matter of logic. Because the premises can come from any field, there would be no way for logic alone to determine whether such premises are true or false. The role of logic—specifically, deductive reasoning—is to determine whether the reasoning used is valid.
Deduction You have likely heard the term deduction used in other contexts: As Chapter 2 noted, the detective Sherlock Holmes (and others) uses deduction to refer to any process by which we infer a conclusion from pieces of evidence. In rhetoric classes and other places, you may hear deduction used to refer to the process of reasoning from general principles to a specific conclusion. These are all acceptable uses of the term in their respective contexts, but they do not reflect how the concept is defined in logic.
In logic, deduction is a technical term. Whatever other meanings the word may have in other contexts, in logic, it has only one meaning: A deductive argument is one that is presented as being valid. In other words, a deductive argument is one that is trying to be valid. If an argu- ment is presented as though it is supposed to be valid, then we may infer it is deductive. If an argument is deductive, then the argument can be evaluated in part on whether it is, in fact, valid. A deductive argument that is not found to be valid has failed in its purpose of demon- strating its conclusion to be true.
In Chapters 5 and 6, we will look at arguments that are not trying to be valid. Those are induc- tive arguments. As noted in Chapter 2, inductive arguments simply attempt to establish their conclusion as probable—not as absolutely guaranteed. Thus, it is not important to assess whether inductive arguments are valid, since validity is not the goal. However, if a deductive
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Consider the following argument: “If it is raining, then the streets are wet. The streets are wet. Therefore, it is raining.” Is this a valid argument? Could there be another reason why the road is wet?
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Section 3.1 Basic Concepts in Deductive Reasoning
argument is not valid, then it has failed in its goal; therefore, for deductive reasoning, validity is a primary concern.
Consider someone arguing as follows:
All donuts have added sugar. All donuts are bad for you. Therefore, everything with added sugar is bad for you.
Even though the argument is invalid— exactly why this is so will be clearer in the next section—it seems clear that the person thinks it is valid. She is not merely suggesting that maybe things with added sugar might be bad for you. Rather, she is presenting the reasoning as though the premises guarantee the truth of the conclusion. Therefore, it appears to be an attempt at deductive reasoning, even though this one happens to be invalid.
Because our definition of validity depends on understanding the author’s intention, this means that deciding whether something is a deductive argu- ment requires a bit of interpretation— we have to figure out what the person giving the argument is trying to do. As
noted briefly in Chapter 2, we ought to seek to provide the most favorable possible interpreta- tion of the author’s intended reasoning. Once we know that an argument is deductive, the next question in evaluating it is whether it is valid. If it is deductive but not valid, we really do not need to consider anything further; the argument fails to demonstrate the truth of its conclusion in the intended sense.
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Interpreting the intention of the person making an argument is a key step in determining whether the argument is deductive.
Practice Problems 3.1
Examine the following arguments. Then determine whether they are deductive argu- ments or not.
1. Charles is hard to work with, since he always interrupts others. Therefore, I do not want to work with Charles in the development committee.
2. No physical object can travel faster than light. An electron is a physical object. So an electron cannot travel faster than light.
3. The study of philosophy makes your soul more slender, healthy, and beautiful. You should study philosophy.
(continued)
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Section 3.1 Basic Concepts in Deductive Reasoning
4. We should go to the beach today. It’s sunny. The dolphins are out, and I have a bottle of fine wine.
5. Triangle A is congruent to triangle B. Triangle A is an equilateral triangle. Therefore, triangle B is an equilateral triangle.
6. The farmers in Poland have produced more than 500 bushels of wheat a year on average for the past 10 years. This year they will produce more than 500 bushels of wheat.
7. No dogs are fish. Some guppies are fish. Therefore, some guppies are not dogs.
8. Paying people to mow your lawn is not a good policy. When people mow their own lawns, they create self-discipline. In addition, they are able to save a lot of money over time.
9. If Mount Roosevelt was completed in 1940, then it’s only 73 years old. Mount Roos- evelt is not 73 years old. Therefore, Mount Roosevelt was not completed in 1940.
10. You’re either with me, or you’re against me. You’re not with me. Therefore, you’re against me.
11. The worldwide use of oil is projected to increase by 33% over the next 5 years. How- ever, reserves of oil are dwindling at a rapid rate. That means that the price of oil will drastically increase over the next 5 years.
12. A nation is only as great as its people. The people are reliant on their leaders. Leaders create the laws in which all people can flourish. If those laws are not created well, the people will suffer. This is why the people of the United States are currently suffering.
13. If we save up money for a house, then we will have a place to stay with our children. However, we haven’t saved up any money for a house. Therefore, we won’t have a place to stay with our children.
14. We have to focus all of our efforts on marketing because right now; we don’t have any idea of who our customers are.
15. Walking is great exercise. When people exercise they are happier and they feel better about themselves. I’m going to start walking 4 miles every day.
16. Because all libertarians believe in more individual freedom, all people who believe in individual freedom are libertarians.
17. Our dogs are extremely sick. I have to work every day this week, and our house is a mess. There’s no way I’m having my family over for Festivus.
18. Pigs are smarter than dogs. Animals that are easier to train are smarter than other animals. Pigs are easier to train than dogs.
19. Seventy percent of the students at this university come from upper class families. The school budget has taken a hit since the economic downturn. We need funding for the three new buildings on campus. I think it’s time for us to start a phone cam- paign to raise funds so that we don’t plunge into bankruptcy.
20. If she wanted me to buy her a drink, she would’ve looked over at me. But she never looked over at me. So that means that she doesn’t want me to buy her a drink.
Practice Problems 3.1 (continued)
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Section 3.2 Evaluating Deductive Arguments
3.2 Evaluating Deductive Arguments If validity is so critical in evaluating deductive argu- ments, how do we go about determining whether an argument is valid or invalid? In deductive reason- ing, the key is to look at the pattern of an argument , which is called its logical form. As an example, see if you can tell whether the following argument is valid:
All quidnuncs are shunpikers. All shunpikers are flibbertigibbets. Therefore, all quidnuncs are flibbertigibbets.
You could likely tell that the argument is valid even though you do not know the meanings of the words. This is an important point. We can often tell whether an argument is valid even if we are not in a posi- tion to know whether any of its propositions are true or false. This is because deductive validity typi- cally depends on certain patterns of argument. In fact, even nonsense arguments can be valid. Lewis Carroll (a pen name for C. L. Dodgson) was not only the author of Alice’s Adventures in Wonderland, but also a clever logician famous for both his use of non- sense words and his tricky logic puzzles.
We will look at some of Carroll’s puzzles in this chapter’s sections on categorical logic, but for now, let us look at an argument using nonsense words from his poem “Jabberwocky.” See if you can tell whether the following argument is valid:
All bandersnatches are slithy toves. All slithy toves are uffish. Therefore, all bandersnatches are uffish.
If you could tell the argument about quidnuncs was valid, you were probably able to tell that this argument is valid as well. Both arguments have the same pattern, or logical form.
Representing Logical Form Logical form is generally represented by using variables or other symbols to highlight the pat- tern. In this case the logical form can be represented by substituting capital letters for certain parts of the propositions. Our argument then has the form:
All S are M. All M are P. Therefore, all S are P.
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In addition to his well-known literary works, Lewis Carroll wrote several mathematical works, including three books on logic: Symbolic Logic Parts 1 and 2, and The Game of Logic, which was intended to introduce logic to children.
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Section 3.2 Evaluating Deductive Arguments
Any argument that follows this pattern, or form, is valid. Try it for yourself. Think of any three plural nouns; they do not have to be related to each other. For example, you could use sub- marines, candy bars, and mountains. When you have thought of three, substitute them for the letters in the pattern given. You can put them in any order you like, but the same word has to replace the same letter. So you will put one noun in for S in the first and third lines, one noun for both instances of M, and your last noun for both cases of P. If we use the suggested nouns, we would get:
All submarines are candy bars. All candy bars are mountains. Therefore, all submarines are mountains.
This argument may be close to nonsense, but it is logically valid. It would not be possible to make up a story in which the premises were true but the conclusion was false. For example, if one wizard turns all submarines into candy bars, and then a second wizard turns all candy bars into mountains, the story would not make any sense (nor would it be logical) if, in the end, all submarines were not mountains. Any story that makes the premises true would have to also make the conclusion true, so that the argument is valid.
As mentioned, the form of an argument is what you get when you remove the specific mean- ing of each of the nonlogical words in the argument and talk about them in terms of variables. Sometimes, however, one has to change the wording of a claim to make it fit the required form. For example, consider the premise “All men like dogs.” In this case the first category would be “men,” but the second category is not represented by a plural noun but by a predi- cate phrase, “like dogs.” In such cases we turn the expression “like dogs” into the noun phrase “people who like dogs.” In that case the form of the sentence is still “All A are B,” in which B is “people who like dogs.” As another example, the argument:
All whales are mammals. Some mammals live in the water. Therefore, at least some whales live in the water.
can be rewritten with plural nouns as:
All whales are mammals. Some mammals are things that live in the water. Therefore, at least some whales are things that live in the water.
and has the form:
All A are B. Some B are C. Therefore, at least some A are C.
The variables can represent anything (anything that fits grammatically, that is). When we substitute specific expressions (of the appropriate grammatical category) for each of the vari- ables, we get an instance of that form. So another instance of this form could be made by replacing A with Apples, B with Bananas, and C with Cantaloupes. This would give us
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Section 3.2 Evaluating Deductive Arguments
All Apples are Bananas. Some Bananas are Cantaloupes. Therefore, at least some Apples are Cantaloupes.
It does not matter at this stage whether the sentences are true or false or whether the reason- ing is valid or invalid. All we are concerned with is the form or pattern of the argument.
We will see many different patterns as we study deductive logic. Different kinds of deductive arguments require different kinds of forms. The form we just used is based on categories; the letters represented groups of things, like dogs, whales, mammals, submarines, or candy bars. That is why in these cases we use plural nouns. Other patterns will require substituting entire sentences for letters. We will study forms of this type in Chapter 4. The patterns you need to know will be introduced as we study each kind of argument, so keep your eyes open for them.
Using the Counterexample Method By definition, an argument form is valid if and only if all of its instances are valid. Therefore, if we can show that a logical form has even one invalid instance, then we may infer that the argument form is invalid. Such an instance is called a counterexample to the argument form’s validity; thus, the counterexample method for showing that an argument form is invalid involves creating an argument with the exact same form but in which the premises are true and the conclusion is false. (We will examine other methods in this chapter and in later chapters.) In other words, finding a counterexample demonstrates the invalidity of the argument’s form.
Consider the invalid argument example from the prior section:
All donuts have added sugar. All donuts are bad for you. Therefore, everything with added sugar is bad for you.
By replacing predicate phrases with noun phrases, this argument has the form:
All A are B. All A are C. Therefore, all B are C.
This is the same form as that of the following, clearly invalid argument:
All birds are animals. All birds have feathers. Therefore, all animals have feathers.
Because we can see that the premises of this argument are true and the conclusion is false, we know that the argument is invalid. Since we have identified an invalid instance of the form, we know that the form is invalid. The invalid instance is a counterexample to the form. Because we have a counterexample, we have good reason to think that the argument about donuts is not valid.
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Section 3.2 Evaluating Deductive Arguments
One of our recent examples has the form:
All A are B. Some B are C. Therefore, at least some A are C.
Here is a counterexample that challenges this argu- ment form’s validity:
All dogs are mammals. Some mammals are cats. Therefore, at least some dogs are cats.
By substituting dogs for A, mammals for B, and cats for C, we have found an example of the argument’s form that is clearly invalid because it moves from true premises to a false conclusion. Therefore, the argument form is invalid.
Here is another example of an argument:
All monkeys are primates. No monkeys are reptiles. Therefore, no primates are reptiles.
The conclusion is true in this example, so many may mistakenly think that the reasoning is valid. However, to better investigate the validity of the reasoning, it is best to focus on its form. The form of this argument is:
All A are B. No A are C. Therefore, no B are C.
To demonstrate that this form is invalid, it will suffice to demonstrate that there is an argu- ment of this exact form that has all true premises and a false conclusion. Here is such a counterexample:
All men are human. No men are women. Therefore, no humans are women.
Clearly, there is something wrong with this argument. Though this is a different argument, the fact that it is clearly invalid, even though it has the exact same form as our original argument, means that the original argument’s form is also invalid.
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Can you think of a counterexample that can prove this dog’s argument is invalid?
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Section 3.3 Types of Deductive Arguments
3.3 Types of Deductive Arguments Once you learn to look for arguments, you will see them everywhere. Deductive arguments play very important roles in daily reasoning. This section will discuss some of the most impor- tant types of deductive arguments.
Mathematical Arguments Arguments about or involving mathematics generally use deductive reasoning. In fact, one way to think about deductive reasoning is that it is reasoning that tries to establish its conclu- sion with mathematical certainty. Let us consider some examples.
Suppose you are splitting the check for lunch with a friend. In calculating your portion, you reason as follows:
I had the chicken sandwich plate for $8.49. I had a root beer for $1.29. I had nothing else. $8.49 + $1.29 = $9.78. Therefore, my portion of the bill, excluding tip and tax, is $9.78.
Notice that if the premises are all true, then the conclusion must be true also. Of course, you might be mistaken about the prices, or you might have forgotten that you had a piece of pie for dessert. You might even have made a mistake in how you added up the prices. But these are all premises. So long as your premises are correct and the argument is valid, then the conclusion is certain to be true.
But wait, you might say—aren’t we often mistaken about things like this? After all, it is common for people to make mistakes when figuring out a bill. Your friend might even disagree with one of your premises: For example, he might think the chicken sandwich plate was really $8.99. How can we say that the conclusion is established with mathematical certainty if we are willing to admit that we might be mistaken?
These are excellent questions, but they pertain to our certainty of the truth of the premises. The important feature of valid arguments is that the reasoning is so strong that the conclu- sion is just as certain to be true as the premises. It would be a very strange friend indeed who
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A mathematical proof is a valid deductive argument that attempts to prove the conclusion. Because mathematical proofs are deductively valid, mathematicians establish mathematical truth with complete certainty (as long as they agree on the premises).
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Section 3.3 Types of Deductive Arguments
agreed with all of your premises and yet insisted that your portion of the bill was something other than $9.78. Still, no matter how good our reasoning, there is almost always some pos- sibility that we are mistaken about our premises.
Arguments From Definitions Another common type of deductive argument is argument from definition. This type of argument typically has two premises. One premise gives the definition of a word; the second premise says that something meets the definition. Here is an example:
Bachelor means “unmarried male.” John is an unmarried male. Therefore, John is a bachelor.
Notice that as with arguments involving math, we may disagree with the premises, but it is very hard to agree with the premises and disagree with the conclusion. When the argument is set out in standard form, it is typically relatively easy to see that the argument is valid.
On the other hand, it can be a little tricky to tell whether the argument is sound. Have we really gotten the definition right? We have to be very careful, as definitions often sound right even though they are a little bit off. For example, the stated definition of bachelor is not quite right. At the very least, the definition should apply only to human males, and probably only adult ones. We do not normally call children or animals “bachelors.”
Chris Madden/Cartoonstock
When crafting or evaluating a deductive argument via definition, special attention should be paid to the clarity of the definition.
An interesting feature of definitions is that they can be understood as going both ways. In other words, if bachelor means “unmarried male,” then we can reason either from the man being an unmarried male to his being a bach- elor, as in the previous example, or from the man being a bachelor to his being an unmar- ried male, as in the following example.
Bachelor means “unmarried male.” John is a bachelor. Therefore, John is an unmar- ried male.
Arguments from definition can be very power- ful, but they can also be misused. This typically happens when a word has two meanings or when the definition is not fully accurate. We will learn more about this when we study fallacies in Chapter 7, but here is an example to consider:
Murder is the taking of an inno- cent life. Abortion takes an innocent life. Therefore, abortion is murder.
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Section 3.3 Types of Deductive Arguments
This is an argument from definition, and it is valid—the premises guarantee the truth of the conclusion. However, are the premises true? Both premises could be disputed, but the first premise is probably not right as a definition. If the word murder really just meant “taking an innocent life,” then it would be impossible to commit murder by killing someone who was not innocent. Furthermore, there is nothing in this definition about the victim being a human or the act being intentional. It is very tricky to get definitions right, and we should be very care- ful about reaching conclusions based on oversimplified definitions. We will come back to this example from a different angle in the next section when we study syllogisms.
Categorical Arguments Historically, some of the first arguments to receive a detailed treatment were categorical arguments, having been thoroughly explained by Aristotle himself (Smith, 2014). Categorical arguments are arguments whose premises and conclusions are statements about categories of things. Let us revisit an example from earlier in this chapter:
All whales are mammals. All mammals breathe air. Therefore, all whales breathe air.
In each of the statements of this argument, the membership of two categories is compared. The categories here are whales, mammals, and air breathers. As discussed in the previous section on evaluating deductive arguments, the validity of these arguments depends on the repetition of the category terms in certain patterns; it has nothing to do with the specific cat- egories being compared. You can test this by changing the category terms whales, mammals, and air breathers with any other category terms you like. Because this argument’s form is valid, any other argument with the same form will be valid. The branch of deductive reason- ing that deals with categorical arguments is known as categorical logic. We will discuss it in the next two sections.
Propositional Arguments Propositional arguments are a type of reasoning that relates sentences to each other rather than relating categories to each other. Consider this example:
Either Jill is in her room, or she’s gone out to eat. Jill is not in her room. Therefore, she’s gone out to eat.
Notice that in this example the pattern is made by the sentences “Jill is in her room” and “she’s gone out to eat.” As with categorical arguments, the validity of propositional arguments can be determined by examining the form, independent of the specific sentences used. The branch of deductive reasoning that deals with propositional arguments is known as proposi- tional logic, which we will discuss in Chapter 4.
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Section 3.4 Categorical Logic: Introducing Categorical Statements
3.4 Categorical Logic: Introducing Categorical Statements The field of deductive logic is a rich and productive one; one could spend an entire lifetime studying it. (See A Closer Look: More Complicated Types of Deductive Reasoning.) Because the focus of this book is critical thinking and informal logic (rather than formal logic), we will only look closely at categorical and propositional logic, which focus on the basics of argument. If you enjoy this introductory exposure, you might consider looking for more books and courses in logic.
Categorical arguments have been studied extensively for more than 2,000 years, going back to Aristotle. Categorical logic is the logic of argument made up of categorical statements. It is a logic that is concerned with reasoning about certain relationships between categories of things. To learn more about how categorical logic works, it will be useful to begin by analyz- ing the nature of categorical statements, which make up the premises and conclusions of categorical arguments. A categorical statement talks about two categories or groups. Just to keep things simple, let us start by talking about dogs, cats, and animals.
A Closer Look: More Complicated Types of Deductive Reasoning As noted, deductive logic deals with a precise kind of reasoning in which logical validity is based on logical form. Within logical forms, we can use letters as variables to replace English words. Logicians also frequently replace other words that occur within arguments—such as all, some, or, and not—to create a kind of symbolic language. Formal logic represented in this type of symbolic language is called symbolic logic.
Because of this use of symbols, courses in symbolic logic end up looking like math classes. An introductory course in symbolic logic will typically begin with propositional logic and then move to something called predicate logic. Predicate logic combines everything from categori- cal and propositional logic but allows much more flexibility in the use of some and all. This flexibility allows it to represent much more complex and powerful statements.
Predicate logic forms the basis for even more advanced types of logic. Modal logic, for example, can be used to represent many deductive arguments about possibility and necessity that can- not be symbolized using predicate logic alone. Predicate logic can even help provide a foun- dation for mathematics. In particular, when predicate logic is combined with a mathematical field called set theory, it is possible to prove the fundamental truths of arithmetic. From there it is possible to demonstrate truths from many important fields of mathematics, including cal- culus, without which we could not do physics, engineering, or many other fascinating and use- ful fields. Even the computers that now form such an essential part of our lives are founded, ultimately, on deductive logic.
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Section 3.4 Categorical Logic: Introducing Categorical Statements
One thing we can say about these groups is that all dogs are animals. Of course, all cats are animals, too. So we have the following true categorical statements:
All dogs are animals.
All cats are animals.
In categorical statements, the first group name is called the subject term; it is what the sentence is about. The second group name is called the predicate term. In the categorical sentences just mentioned, dogs and cats are both in the subject position, and animals is in the predicate position. Group terms can go in either position, but of course, the sentence might be false. For example, in the sentence “All animals are dogs” the term dogs is in the predicate position.
You may recall that we can represent the logical form of these types of sentences by replacing the category terms with single letters. Using this method, we can represent the form of these categorical statements in the following way:
All D are A.
All C are A.
Another true statement we can make about these groups is “No dogs are cats.” Which term is in subject position, and which is in predicate position? If you said that dogs is the subject and cats is the predicate, you’re right! The logical form of “No dogs are cats” can be given as “No D are C.”
We now have two sentences in which the category dogs is the subject: “All dogs are animals” and “No dogs are cats.” Both of these statements tell us something about every dog. The first, which starts with all, tells us that each dog is an animal. The second, which begins with no, tells us that each dog is not a cat. We say that both of these types of sentences are universal because they tell us something about every member of the subject class.
Not all categorical statements are universal. Here are two statements about dogs that are not universal:
Some dogs are brown.
Some dogs are not tall.
Statements that talk about some of the things in a category are called particular statements. The distinction between a statement being universal or particular is a distinction of quantity.
Another distinction is that we can say that the things mentioned are in or not in the predi- cate category. If we say the things are in that category, our statement is affirmative. If we say the things are not in that category, our statement is negative. The distinction between
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Section 3.4 Categorical Logic: Introducing Categorical Statements
a statement being affirmative or negative is a distinction of quality. For example, when we say “Some dogs are brown,” the thing mentioned (dogs) is in the predicate category (brown things), making this an affirmative statement. When we say “Some dogs are not tall,” the thing mentioned (dogs) is not in the predicate category (tall things), and so this is a nega- tive statement.
Taking both of these distinctions into account, there are four types of categorical statements: universal affirmative, universal negative, particular affirmative, and particular negative. Table 3.1 shows the form of each statement along with its quantity and quality.
Table 3.1: Types of categorical statements
Quantity Quality
All S is P Universal Affirmative
No S is P Universal Negative
Some S is P Particular Affirmative
Some S is not P Particular Negative
To abbreviate these categories of statement even further, logicians over the millennia have used letters to represent each type of statement. The abbreviations are as follows:
A: Universal affirmative (All S is P)
E: Universal negative (No S is P)
I: Particular positive (Some S is P)
O: Particular negative (Some S is not P)
Accordingly, the statements are known as A propositions, E propositions, I propositions, and O propositions. Remember that the single capital letters in the statements themselves are just placeholders for category terms; we can fill them in with any category terms we like. Figure 3.1 shows a traditional way to arrange the four types of statements by quantity and quality.
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A All S is P No S is P
E
I Some S is P
Co nt
ra di ct or
ie s Contradictories
Some S is not P O
Section 3.4 Categorical Logic: Introducing Categorical Statements
Now we need to get just a bit clearer on what the four statements mean. Granted, the meaning of categorical statements seems clear: To say, for example, that “no dogs are reptiles” simply means that there are no things that are both dogs and reptiles. However, there are certain cases in which the way that logicians understand categorical statements may differ some- what from how they are commonly understood in everyday language. In particular, there are two specific issues that can cause confusion.
Clarifying Particular Statements The first issue is with particular statements (I and O propositions). When we use the word some in everyday life, we typically mean more than one. For example, if someone says that she has some apples, we generally think that this means that she has more than one. How- ever, in logic, we take the word some simply to mean at least one. Therefore, when we say that some S is P, we mean only that at least one S is P. For example, we can say “Some dogs live in the White House” even if only one does.
Clarifying Universal Statements The second issue involves universal statements (A and E propositions). It is often called the “issue of existential presupposition”—the issue concerns whether a universal statement
Figure 3.1: The square of opposition
The square of opposition serves as a quick reference point when evaluating categorical statements. Note that A statements and O statements always contradict one another; when one is true, the other is false. The same is true of E statements and I statements.
A All S is P No S is P
E
I Some S is P
Co nt
ra di ct or
ie s Contradictories
Some S is not P O
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Section 3.4 Categorical Logic: Introducing Categorical Statements
implies a particular statement. For example, does the fact that all dogs are animals imply that some dogs are animals? The question really becomes an issue only when we talk about things that do not really exist. For example, consider the claim that all the survivors of the Civil War live in New York. Given that there are no survivors of the Civil War anymore, is the statement true or not?
The Greek philosopher Aristotle, the inventor of categorical logic, would have said the state- ment is false. He thought that “All S is P” could only be true if there was at least one S (Parsons, 2014). Modern logicians, however, hold that that “All S is P” is true even when no S exists. The reasons for the modern view are somewhat beyond the scope of this text—see A Closer Look: Existential Import for a bit more of an explanation—but an example will help support the claim that universal statements are true when no member of the subject class exists.
Suppose we are driving somewhere and stop for snacks. We decide to split a bag of M&M’s. For some reason, one person in our group really wants the brown M&M’s, so you promise that he can have all of them. However, when we open the bag, it turns out that there are no brown candies in it. Since this friend did not get any brown M&M’s, did you break your promise? It seems clear that you did not. He did get all of the brown M&M’s that were in the bag; there just weren’t any. In order for you to have broken your promise, there would have to be a brown M&M that you did not let your friend have. Therefore, it is true that your friend got all the brown M&M’s, even though he did not get any.
This is the way that modern logicians think about universal propositions when there are no members of the subject class. Any universal statement with an empty subject class is true, regardless of whether the statement is positive or negative. It is true that all the brown M&M’s were given to your friend and also true that no brown M&M’s were given to your friend.
A Closer Look: Existential Import It is important to remember that particular statements in logic (I and O propositions) refer to things that actually exist. The statement “Some dogs are mammals” is essentially saying, “There is at least one dog that exists in the universe, and that dog is a mammal.” The way that logicians refer to this attribute of I and O statements is that they have “existential import.” This means that for them to be true, there must be something that actually exists that has the property mentioned in the statement.
The 19th-century mathematician George Boole, however, presented a problem. Boole agreed with Aristotle that the existential statements I and O had to refer to existing things to be true. Also, for Aristotle, all A statements that are true necessarily imply the truth of their corre- sponding I statements. The same goes with E and O statements.
Boole pointed out that some true A and E statements refer to things that do not actually exist. Consider the statement “All vampires are creatures that drink blood.” This is a true statement. That means that the corresponding I statement, “Some vampires are creatures that drink blood,” would also be true, according to Aristotle. However, Boole noted that there are no exist- ing things that are vampires. If vampires do not exist, then the I statement, “Some vampires are creatures that drink blood,” is not true: The truth of this statement rests on the idea that there is an actually existing thing called a vampire, which, at this point, there is no evidence of.
(continued)
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Section 3.4 Categorical Logic: Introducing Categorical Statements
Boole reasoned that Aristotle’s ideas did not work in cases where A and E statements refer to nonexisting classes of objects. For example, the E statement “No vampires are time machines” is a true statement. However, both classes in this statement refer to things that do not actually exist. Therefore, the statement “Some vampires are not time machines” is not true, because this statement could only be true if vampires and time machines actually existed.
Boole reasoned that Aristotle’s claim that true A and E state- ments led necessarily to true I and O statements was not uni- versally true. Hence, Boole claimed that there needed to be a revision of the forms of categorical syllogisms that are consid- ered valid. Because one cannot generally claim that an exis- tential statement (I or O) is true based on the truth of the cor- responding universal (A or E), there were some valid forms of syllogisms that had to be excluded under the Boolean (mod- ern) perspective. These syllogisms were precisely those that reasoned from universal premises to a particular conclusion.
Of course, we all recognize that in everyday life we can logi- cally infer that if all dogs are mammals, then it must be true that some dogs are mammals. That is, we know that there is at least one existing dog that is a mammal. However, because our logical rules of evaluation need to apply to all instances of syllogisms, and because there are other instances where universals do not lead of necessity to the truth of particulars, the rules of evaluation had to be reformed after Boole presented his analysis. It is important to avoid committing the exis- tential fallacy, or assuming that a class has members and then drawing an inference about an actually existing member of the class.
Science and Society/SuperStock
George Boole, for whom Boolean logic is named, challenged Aristotle’s assertion that the truth of A statements implies the truth of corresponding I statements. Boole suggested that some valid forms of syllogisms had to be excluded.
A Closer Look: Existential Import (continued)
Accounting for Conversational Implication These technical issues likely sound odd: We usually assume that some implies that there is more than one and that all implies that something exists. This is known as conversational implication (as opposed to logical implication). It is quite common in everyday life to make a conversational implication and take a statement to suggest that another statement is true as well, even though it does not logically imply that the other must be true. In logic, we focus on the literal meaning.
One of the common reasons that a statement is taken to conversationally imply another is that we are generally expected to make the most fully informative statement that we can in response to a question. For example, if someone asks what time it is and you say, “Sometime after 3,” your statement seems to imply that you do not know the exact time. If you knew it was 3:15 exactly, then you probably should have given this more specific information in response to the question.
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Section 3.4 Categorical Logic: Introducing Categorical Statements
For example, we all know that all dogs are animals. Suppose, however, someone says, “Some dogs are animals.” That is an odd thing to say: We generally would not say that some dogs are animals unless we thought that some of them are not animals. However, that would be mak- ing a conversational implication, and we want to make logical implications. For the purposes of logic, we want to know whether the statement “some dogs are animals” is true or false. If we say it is false, then we seem to have stated it is not true that some dogs are animals; this, however, would seem to mean that there are no dogs that are animals. That cannot be right. Therefore, logicians take the statement “Some dogs are animals” simply to mean that there is at least one dog that is an animal, which is true. The statement “Some dogs are not animals” is not part of the meaning of the statement “Some dogs are animals.” In the language of logic, the statement that some S are not P is not part of the meaning of the statement that some S are P.
Of course, it would be odd to make the less informative statement that some dogs are animals, since we know that all dogs are animals. Because we tend to assume someone is making the most informative statement possible, the statement “Some dogs are animals” may conversa- tionally imply that they are not all animals, even though that is not part of the literal meaning of the statement.
In short, a particular statement is true when there is at least one thing that makes it true, even if the universal statement would also be true. In fact, sometimes we emphasize that we are not talking about the whole category by using the words at least, as in, “At least some planets orbit stars.” Therefore, it appears to be nothing more than conversational implication, not lit- eral meaning, that leads our statement “Some dogs are animals” to suggest that some also are not. When looking at categorical statements, be sure that you are thinking about the actual meaning of the sentence rather than what might be conversationally implied.
Practice Problems 3.2
Complete the following problems.
1. “All dinosaurs are things that are extinct.” Which of the following is the subject term in this statement? a. dinosaurs b. things that are extinct
2. “No Honda Civics are Lamborghinis.” Which of the following is the predicate term in this statement? a. Lamborghinis b. Honda Civics
3. “Some authors are people who write horror.” Which of the following is the predicate term in this statement? a. authors b. people who write horror
4. “Some politicians are not people who can be trusted.” Which of the following is the subject term in this statement? a. politicians b. people who can be trusted
(continued)
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Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
5. “All mammals are pieces of cheese.” Which of the following is the predicate term in this statement? a. pieces of cheese b. mammals
6. What is the quantity of the following statement? “All dinosaurs are things that are extinct.” a. universal b. particular c. affirmative d. negative
7. What is the quality of the following statement? “No Honda Civics are Lamborghinis.” a. universal b. particular c. affirmative d. negative
8. What is the quality of the following statement? “Some authors are people who write horror.” a. universal b. particular c. affirmative d. negative
9. What is the quantity of the following statement? “Some politicians are not people who can be trusted.” a. universal b. particular c. affirmative d. negative
10. What is the quality of the following statement? “All mammals are pieces of cheese.” a. universal b. particular c. affirmative d. negative
Practice Problems 3.2 (continued)
3.5 Categorical Logic: Venn Diagrams as Pictures of Meaning
Given that it is sometimes tricky to parse out the meaning and implications of categorical statements, a logician named John Venn devised a method that uses diagrams to clarify the literal meanings and logical implications of categorical claims. These diagrams are appro- priately called Venn diagrams (Stapel, n.d.). Venn diagrams not only give a visual picture of the meanings of categorical statements, they also provide a method by which we can test the validity of many categorical arguments.
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E
H
E
Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
Drawing Venn Diagrams Here is how the diagramming works: Imagine we get a bunch of people together and all go to a big field. We mark out a big circle with rope on the field and ask everyone with brown eyes to stand in the circle. Would you stand inside the circle or outside it? Where would you stand if we made another circle and asked everyone with brown hair to stand inside? If your eyes or hair are sort of brownish, just pick whether you think you should be inside or outside the circles. No standing on the rope allowed! Remember your answers to those two questions.
Here is an image of the brown-eye circle, labeled “E” for “eyes”; touch inside or outside the circle indicating where you would stand.
Here is a picture of the brown-hair circle, labeled “H” for “hair”; touch inside or outside the circle indicating where you would stand.
Notice that each circle divides the people into two groups: Those inside the circle have the feature we are interested in, and those outside the circle do not.
Where would you stand if we put both circles on the ground at the same time?
E
H
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E H
E H
Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
As long as you do not have both brown eyes and brown hair, you should be able to figure out where to stand. But where would you stand if you have brown eyes and brown hair? There is not any spot that is in both circles, so you would have to choose. In order to give brown-eyed, brown-haired people a place to stand, we have to overlap the circles.
Now there is a spot where people who have both brown hair and brown eyes can stand: where the two circles overlap. We noted earlier that each circle divides our bunch of people into two groups, those inside and those outside. With two circles, we now have four groups. Figure 3.2 shows what each of those groups are and where people from each group would stand.
E H
E H
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All S is P
S P
No S is P
S P
Some S is P
S P
Some S is not P
S P
Neither brown eyes nor brown hair
Brown eyes, not
brown hair
Brown hair, not
brown eyes
Brown eyes and
brown hair
Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
With this background, we can now draw a picture for each categorical statement. When we know a region is empty, we will darken it to show there is nobody there. If we know for sure that someone is in a region, we will put an x in it to represent a person standing there. Figure 3.3 shows the pictures for each of the four kinds of statements.
Figure 3.3: Venn diagrams of categorical statements
Each of the four categorical statements can be represented visually with a Venn diagram.
All S is P
S P
No S is P
S P
Some S is P
S P
Some S is not P
S P
Figure 3.2: Sample Venn diagram
Neither brown eyes nor brown hair
Brown eyes, not
brown hair
Brown hair, not
brown eyes
Brown eyes and
brown hair
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Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
In drawing these pictures, we adopt the convention that the subject term is on the left and the predicate term is on the right. There is nothing special about this way of doing it, but diagrams are easier to understand if we draw them the same way as much as possible. The important thing to remember is that a Venn diagram is just a picture of the meaning of a state- ment. We will use this fact in our discussion of inferences and arguments.
Drawing Immediate Inferences As mentioned, Venn diagrams help us determine what inferences are valid. The most basic of such inferences, and a good place to begin, is something called immediate inference. Immedi- ate inferences are arguments from one categorical statement as premise to another as con- clusion. In other words, we immediately infer one statement from another. Despite the fact that these inferences have only one premise, many of them are logically valid. This section will use Venn diagrams to help discern which immediate inferences are valid.
The basic method is to draw a diagram of the premises of the argument and determine if the diagram thereby shows the conclusion is true. If it does, then the argument is valid. In other words, if drawing a diagram of just the premises automatically creates a diagram of the con- clusion, then the argument is valid. The diagram shows that any way of making the premises true would also make the conclusion true; it is impossible for the conclusion to be false when the premises are true. We will see how to use this method with each of the immediate infer- ences and later extend the method to more complicated arguments.
Conversion Conversion is just a matter of switching the positions of the subject and predicate terms. The resulting statement is called the converse of the original statement. Table 3.2 shows the converse of each type of statement.
Table 3.2: Conversion
Statement Converse
All S is P. All P is S.
No S is P. No P is S.
Some S is P. Some P is S.
Some S is not P. Some P is not S.
Forming the converse of a statement is easy; just switch the subject and predicate terms with each other. The question now is whether the immediate inference from a categorical state- ment to its converse is valid or not. It turns out that the argument from a statement to its converse is valid for some statement types, but not for others. In order to see which, we have to check that the converse is true whenever the original statement is true.
An easy way to do this is to draw a picture of the two statements and compare them. Let us start by looking at the universal negative statement, or E proposition, and its converse. If we form an argument from this statement to its converse, we get the following:
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No S is P
S P
No P is S
S P
Some S is P
S P
Some P is S
S P
Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
No S is P. Therefore, no P is S.
Figure 3.4 shows the Venn diagrams for these statements.
As you can see, the same region is shaded in both pictures—the region that is inside both circles. It does not matter which order the circles are in, the picture is the same. This means that the two statements have the same meaning; we call such statements equivalent.
The Venn diagrams for these statements demonstrate that all of the information in the con- clusion is present in the premise. We can therefore infer that the inference is valid. A shorter way to say it is that conversion is valid for universal negatives.
We see the same thing when we look at the particular affirmative statement, or I proposition.
In the case of particular affirmatives as well, we can see that all of the information in the conclusion is contained within the premises. Therefore, the immediate inference is valid. In fact, because the diagram for “Some S is P” is the same as the diagram for its converse, “Some P is S” (see Figure 3.5), it follows that these two statements are equivalent as well.
Figure 3.4: Universal negative statement and its converse
In this representation of “No S is P. Therefore, no P is S,” the areas shaded are the same, meaning the statements are equivalent.
No S is P
S P
No P is S
S P
Figure 3.5: Particular affirmative statement and its converse
As with the E proposition, all of the information contained in the conclusion of the I proposition is also contained within the premises, making the inference valid.
Some S is P
S P
Some P is S
S P
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Some S is P
S P
Some P is S
S P
Some S is not P
S P
Some P is not S
S P
Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
However, there will be a big difference when we draw pictures of the universal affirmative (A proposition), the particular negative (O proposition), and their converses (see Figure 3.6 and Figure 3.7).
In these two cases we get different pictures, so the statements do not mean the same thing. In the original statements, the marked region is inside the S circle but not in the P circle. In the converse statements, the marked region is inside the P circle but not in the S circle. Because there is information in the conclusions of these arguments that is not present in the premises, we may infer that conversion is invalid in these two cases.
Figure 3.6: Universal affirmative statement and its converse
Unlike Figures 3.4 and 3.5 where the diagrams were identical, we get two different diagrams for A propositions. This tells us that there is information contained in the conclusion that was not included in the premises, making the inference invalid.
Some S is P
S P
Some P is S
S P
Figure 3.7: Particular negative statement and its converse
As with A propositions, O propositions present information in the conclusion that was not present in the premises, rendering the inference invalid.
Some S is not P
S P
Some P is not S
S P
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Non-brown eyes
Non-brown hair
Non-brown eyes and
non-brown hair
Brown eyes and brown hair
Non-brown eyes
Non-brown hair
Non-brown eyes and
non-brown hair
Brown eyes and brown hair
Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
Let us consider another type of immediate inference.
Contraposition Before we can address contraposition, it is necessary to introduce the idea of a complement class. Remember that for any category, we can divide things into those that are in the category and those that are out of the category. When we imagined rope circles on a field, we asked all the brown-haired people to step inside one of the circles. That gave us two groups: the brown- haired people inside the circle, and the non-brown-haired people outside the circle. These two groups are complements of each other. The complement of a group is everything that is not in the group. When we have a term that gives us a category, we can just add non- before the term to get a term for the complement group. The complement of term S is non-S, the complement of term animal is nonanimal, and so on. Let us see what complementing a term does to our Venn diagrams.
Recall the diagram for brown-eyed people. You were inside the circle if you have brown eyes, and outside the circle if you do not. (Remember, we did not let people stand on the rope; you had to be either in or out.) So now consider the diagram for non-brown-eyed people.
If you were inside the brown-eyed circle, you would be outside the non-brown-eyed circle. Similarly, if you were outside the brown-eyed circle, you would be inside the non-brown-eyed circle. The same would be true for complementing the brown-haired circle. Complementing just switches the inside and outside of the circle.
Do you remember the four regions from Figure 3.2? See if you can find the regions that would have the same people in the complemented picture. Where would someone with blue eyes and brown hair stand in each picture? Where would someone stand if he had red hair and green eyes? How about someone with brown hair and brown eyes?
In Figure 3.8, the regions are colored to indicate which ones would have the same people in them. Use the diagram to help check your answers from the previous paragraph. Notice that the regions in both circles and outside both circles trade places and that the region in the left circle only trades places with the region in the right circle.
Non-brown eyes
Non-brown hair
Non-brown eyes and
non-brown hair
Brown eyes and brown hair
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S P Non-S Non-P
Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
Now that we know what a complement is, we are ready to look at the immediate infer- ence of contraposition. Contraposition combines conversion and complementing; to get the contrapositive of a statement, we first get the converse and then find the complement of both terms.
Let us start by considering the universal affirmative statement, “All S is P.” First we form its converse, “All P is S,” and then we complement both class terms to get the contrapositive, “All non-P is non-S.” That may sound like a mouthful, but you should see that there is a simple, straightforward process for getting the contrapositive of any statement. Table 3.3 shows the process for each of the four types of categorical statements.
Table 3.3: Contraposition
Original Converse Contrapositive
All S is P. All P is S. All non-P is non-S.
No S is P. No P is S. No non-P is non-S.
Some S is P. Some P is S. Some non-P is non-S.
Some S is not P. Some P is not S. Some non-P is not non-S.
Figure 3.9 shows the diagrams for the four statement types and their contrapositives, colored so that you can see which regions represent the same groups.
Figure 3.8: Complement class
S P Non-S Non-P
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All S is P
S P
All non-P is non-S
S P
No S is P
S P
No non-P is non-S
S P
Some S is P
S P
Some non-P is non-S
S P
Some S is not P
S P
Some non-P is not non-S
S P
Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
Figure 3.9: Contrapositive Venn diagrams
Using the converse and contrapositive diagrams, you can infer the original statement.
All S is P
S P
All non-P is non-S
S P
No S is P
S P
No non-P is non-S
S P
Some S is P
S P
Some non-P is non-S
S P
Some S is not P
S P
Some non-P is not non-S
S P
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Section 3.5Categorical Logic: Venn Diagrams as Pictures of Meaning
As you can see, contraposition preserves meaning in universal affirmative and particular nega- tive statements. So from either of these types of statements, we can immediately infer their contrapositive, and from the contrapositive, we can infer the original statement. In other words, these statements are equivalent; therefore, in those two cases, the contrapositive is valid.
In the other cases, particular affirmative and universal negative, we can see that there is infor- mation in the conclusion that is not present in diagram of the premise; these immediate infer- ences are invalid.
There are more immediate inferences that can be made, but our main focus in this chapter is on arguments with multiple premises, which tend to be more interesting, so we are going to move on to syllogisms.
Practice Problems 3.3
Answer the following questions about conversion and contraposition.
1. What is the converse of the statement “No humperdinks are picklebacks”? a. No humperdinks are picklebacks. b. All picklebacks are humperdinks. c. Some humperdinks are picklebacks. d. No picklebacks are humperdinks.
2. What is the converse of the statement “Some mammals are not dolphins”? a. Some dolphins are mammals. b. Some dolphins are not mammals. c. All dolphins are mammals. d. No dolphins are mammals.
3. What is the contrapositive of the statement “All couches are pieces of furniture”? a. All non-couches are non-pieces of furniture. b. All pieces of furniture are non-couches. c. All non-pieces of furniture are couches. d. All non-pieces of furniture are non-couches.
4. What is the contrapositive of the statement “Some apples are not vegetables”? a. Some non-apples are not non-vegetables. b. Some non-vegetables are not non-apples. c. Some non-vegetables are non-apples. d. Some non-vegetables are apples.
5. What is the converse of the statement “Some men are bachelors”? a. Some bachelors are men. b. Some bachelors are non-men. c. All bachelors are men. d. No women are bachelors.
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Section 3.6 Categorical Logic: Categorical Syllogisms
3.6 Categorical Logic: Categorical Syllogisms Whereas contraposition and conversion can be seen as arguments with only one premise, a syl- logism is a deductive argument with two premises. The categorical syllogism, in which a conclu- sion is derived from two categorical premises, is perhaps the most famous—and certainly one of the oldest—forms of deductive argument. The categorical syllogism—which we will refer to here as just “syllogism”—presented by Aristotle in his Prior Analytics (350 BCE/1994), is a very spe- cific kind of deductive argument and was subsequently studied and developed extensively by logicians, mathematicians, and philosophers.
Ron Morgan/Cartoonstock
Aristotle’s categorical syllogism uses two categorical premises to form a deductive argument.
Terms We will first discuss the syllogism’s basic outline, following Aristotle’s insistence that syllogisms are arguments that have two premises and a conclu- sion. Let us look again at our standard example:
All S are M. All M are P. Therefore, all S are P.
There are three total terms here: S, M, and P. The term that occurs in the predicate position in the conclusion (in this case, P) is the major term. The term that occurs in the subject position in the con- clusion (in this case, S) is the minor term. The other term, the one that occurs in both premises but not the conclusion, is the middle term (in this case, M).
The premise that includes the major term is called the major premise. In this case it is the first premise. The premise that includes the minor term, the second one here, is called the minor premise. The conclusion will present the relationship between the predicate term of the major premise (P) and the subject term of the minor premise (S) (Smith, 2014).
There are 256 possible different forms of syllogisms, but only a small fraction of those are valid, which can be shown by testing syllogisms through the traditional rules of the syllogism or by using Venn diagrams, both of which we will look at later in this section.
Distribution As Aristotle understood logical propositions, they referred to classes, or groups: sets of things. So a universal affirmative (type A) proposition that states “All Clydesdales are horses” refers to the group of Clydesdales and says something about the relationship between all of the members of that group and the members of the group “horses.” However, nothing at all is said
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Section 3.6 Categorical Logic: Categorical Syllogisms
about those horses that might not be Clydesdales, so not all members of the group of horses are referred to. The idea of referring to members of such groups is the basic idea behind dis- tribution: If all of the members of a group are referred to, the term that refers to that group is said to be distributed.
Using our example, then, we can see that the proposition “All Clydesdales are horses” refers to all the members of that group, so the term Clydesdales is said to be distributed. Universal affirmatives like this one distribute the term that is in the first, or subject, position.
However, what if the proposition were a universal negative (type E) proposition, such as “No koala bears are carnivores”? Here all the members of the group “koala bears” (the subject term) are referred to, but all the members of the group “carnivores” (the predicate term) are also referred to. When we say that no koala bears are carnivores, we have said something about all koala bears (that they are not carnivores) and also something about all carnivores (that they are not koala bears). So in this universal negative proposition, both of its terms are distributed.
To sum up distribution for the universal propositions, then: Universal affirmative (A) proposi- tions distribute only the first (subject) term, and universal negative (E) propositions distrib- ute both the first (subject) term and the second (predicate) term.
The distribution pattern follows the same basic idea for particular propositions. A particular affirmative (type I) proposition, such as “Some students are football players,” refers only to at least one member of the subject class (“students”) and only to at least one member of the predicate class (“football players”). Thus, remembering that some is interpreted as meaning “at least one,” the particular affirmative proposition distributes neither term, for this proposi- tion does not refer to all the members of either group.
Finally, a particular negative (type O) proposition, such as “Some Floridians are not surfers,” only refers to at least one Floridian—but says that at least one Floridian does not belong to the entire class of surfers or is excluded from the entire class of surfers. In this way, the particular negative proposition distributes only the term that refers to surfers, or the predicate term.
To sum up distribution for the particular propositions, then: particular affirmative (I) propo- sitions distribute neither the first (subject) nor the second (predicate) term, and particular negative (O) propositions distribute only the second (predicate) term. This is a lot of detail, to be sure, but it is summarized in Table 3.4.
Table 3.4: Distribution
Proposition Subject Predicate
A Distributed Not
E Distributed Distributed
I Not Not
O Not Distributed
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Section 3.6 Categorical Logic: Categorical Syllogisms
Once you understand how distribution works, the rules for determining the validity of syl- logisms are fairly straightforward. You just need to see that in any given syllogism, there are three terms: a subject term, a predicate term, and a middle term. But there are only two posi- tions, or “slots,” a term can appear in, and distribution relates to those positions.
Rules for Validity Once we know how to determine whether a term is distributed, it is relatively easy to learn the rules for determining whether a categorical syllogism is valid. The traditional rules of the syllogism are given in various ways, but here is one standard way:
Rule 1: The middle term must be distributed at least once.
Rule 2: Any term distributed in the conclusion must be distributed in its corresponding premise.
Rule 3: If the syllogism has a negative premise, it must have a negative conclusion, and if the syllogism has a negative conclusion, it must have a negative premise.
Rule 4: The syllogism cannot have two negative premises.
Rule 5: If the syllogism has a particular premise, it must have a particular conclusion, and if the syllogism has a particular conclusion, it must have a particular premise.
A syllogism that satisfies all five of these rules will be valid; a syllogism that does not will be invalid. Perhaps the easiest way of seeing how the rules work is to go through a few examples. We can start with our standard syllogism with all universal affirmatives:
All M are P. All S are M. Therefore, all S are P.
Rule 1 is satisfied: The middle term is distributed by the first premise; a universal affirmative (A) proposition distributes the term in the first (subject) position, which here is M. Rule 2 is satisfied because the subject term that is distributed by the conclusion is also distributed by the second premise. In both the conclusion and the second premise, the universal affirmative proposition distributes the term in the first position. Rule 3 is also satisfied because there is not a negative premise without a negative conclusion, or a negative conclusion without a neg- ative premise (all the propositions in this syllogism are affirmative). Rule 4 is passed because both premises are affirmative. Finally, Rule 5 is passed as well because there is a universal conclusion. Since this syllogism passes all five rules, it is valid.
These get easier with practice, so we can try another example:
Some M are not P. All M are S. Therefore, some S are not P.
Rule 1 is passed because the second premise distributes the middle term, M, since it is the sub- ject in the universal affirmative (A) proposition. Rule 2 is passed because the major term, P, that
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Section 3.6 Categorical Logic: Categorical Syllogisms
is distributed in the O conclusion is also distributed in the corresponding O premise (the first premise) that includes that term. Rule 3 is passed because there is a negative conclusion to go with the negative premise. Rule 4 is passed because there is only one negative premise. Rule 5 is passed because the first premise is a particular premise (O). Since this syllogism passes all five rules, it is valid; there is no way that all of its premises could be true and its conclusion false.
Both of these have been valid; however, out of the 256 possible syllogisms, most are invalid. Let us take a look at one that violates one or more of the rules:
No P are M. Some S are not M. Therefore, all S are P.
Rule 1 is passed. The middle term is distributed in the first (major) premise. However, Rule 2 is violated. The subject term is distributed in the conclusion, but not in the corresponding second (minor) premise. It is not necessary to check the other rules; once we know that one of the rules is violated, we know that the argument is invalid. (However, for the curious, Rule 3 is violated as well, but Rules 4 and 5 are passed).
Venn Diagram Tests for Validity Another value of Venn diagrams is that they provide a nice method for evaluating the validity of a syllogism. Because every valid syllogism has three categorical terms, the diagrams we use must have three circles:
The idea in diagramming a syllogism is that we diagram each premise and then check to see if the conclusion has been automatically diagrammed. In other words, we determine whether the conclusion must be true, according to the diagram of the premises.
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
It is important to remember that we never draw a diagram of the conclusion. If the argu- ment is valid, diagramming the premises will automatically provide a diagram of the conclu- sion. If the argument is invalid, diagramming the premises will not provide a diagram of the conclusions.
Diagramming Syllogisms With Universal Statements Particular statements are slightly more difficult in these diagrams, so we will start by looking at a syllogism with only universal statements. Consider the following syllogism:
All S is M. No M is P. Therefore, no S is P.
Remember, we are only going to diagram the two premises; we will not diagram the conclusion. The easiest way to diagram each premise is to temporarily ignore the circle that is not relevant to the premise. Looking just at the S and M circles, we diagram the first premise like this:
M
S P
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M
M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Here is what the diagram for the second premise looks like:
Now we can take those two diagrams and superimpose them, so that we have one diagram of both premises:
M
M
S P
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Now we can check whether the argument is valid. To do this, we see if the conclusion is true according to our diagram. In this case our conclusion states that no S is P; is this statement true, according to our diagram? Look at just the S and P circles; you can see that the area between the S and P circles (outlined) is fully shaded. So we have a diagram of the conclu- sion. It does not matter if the S and P circles have some extra shading in them, so long as the diagram has all the shading needed for the truth of the conclusion.
Let us look at an invalid argument next.
All S is M. All P is M. Therefore, all S is P.
Again, we diagram each premise and look to see if we have a diagram of the conclusion. Here is what the diagram of the premises looks like:
M
S P
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Now we check to see whether the conclusion must be true, according to the diagram. Our conclusion states that all S is P, meaning that no unshaded part of the S circle can be outside of the P circle. In this case you can see that we do not have a diagram of the conclusion. Since we have an unshaded part of S outside of P (outlined), the argument is invalid.
Let us do one more example with all universals.
All M are P. No M is S. Therefore, no S is P.
Here is how to diagram the premises:
M
S P
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Is the conclusion true in this diagram? In order to know that the conclusion is true, we would need to know that there are no S that are P. However, we see in this diagram that there is room for some S to be P. Therefore, these premises do not guarantee the truth of this conclusion, so the argument is invalid.
Diagramming Syllogisms With Particular Statements Particular statements (I and O) are a bit trickier, but only a bit. The problem is that when you diagram a particular statement, you put an x in a region. If that region is further divided by a third circle, then the single x will end up in one of those subregions even though we do not know which one it should go in. As a result, we have to adopt a convention to indicate that the x may be in either of them. To do this, we will draw an x in each subregion and connect them with a line to show that we mean the individual might be in either subregion. To see how this works, let us consider the following syllogism.
M
S P
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Some S is not M. All P are M. Therefore, some S is not P.
We start by diagramming the first premise:
Then we add the diagram for the second premise:
M
S P
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Notice that in diagramming the second premise, we shaded over one of the linked x’s. This leaves us with just one x. When we look at just the S and P circles, we can see that the remain- ing is inside the S circle but outside the P circle.
To see if the argument is valid, we have to determine whether the conclusion must be true according to this diagram. The truth of our conclusion depends on there being at least one S that is not P. Here we have just such an entity: The remaining x is in the S circle but not in the P circle, so the conclusion must be true. This shows that the conclusion validly follows from the premises.
Here is an example of an invalid syllogism.
Some S is M. Some M is P. Therefore, some S is P.
M
S P
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Here is the diagram with both premises represented:
Now it seems we have x’s all over the place. Remember, our job now is just to see if the conclu- sion is already diagrammed when we diagram the premises. The diagram of the conclusion would have to have an x that was in the region between where the S and P circles overlap. We can see that there are two in that region, each linked to an x outside the region. The fact that they are linked to other x’s means that neither x has to be in the middle region; they might both be at the other end of the link. We can show this by carefully erasing one of each pair of linked x’s. In fact, we will erase one x from each linked pair, trying to do so in a way that makes the conclusion false. First we erase the right-hand x from the pair in the S circle. Here is what the diagram looks like now:
M
S P
M
S P
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Here is the diagram with both premises represented:
Now it seems we have x’s all over the place. Remember, our job now is just to see if the conclu- sion is already diagrammed when we diagram the premises. The diagram of the conclusion would have to have an x that was in the region between where the S and P circles overlap. We can see that there are two in that region, each linked to an x outside the region. The fact that they are linked to other x’s means that neither x has to be in the middle region; they might both be at the other end of the link. We can show this by carefully erasing one of each pair of linked x’s. In fact, we will erase one x from each linked pair, trying to do so in a way that makes the conclusion false. First we erase the right-hand x from the pair in the S circle. Here is what the diagram looks like now:
M
S P
M
S P
Now we erase the left-hand x from the remaining pair. Here is the final diagram:
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Notice that there are no x’s remaining in the overlapped region of S and P. This modification of the diagram still makes both premises true, but it also makes the conclusion false. Because this combination is possible, that means that the argument must be invalid.
Here is a more common example of an invalid categorical syllogism:
All S are M. Some M are P. Therefore, some S are P.
This argument form looks valid, but it is not. One way to see that is to notice that Rule 1 is vio- lated: The middle term does not distribute in either premise. That is why this argument form represents an example of the common deductive error in reasoning known as the “undistrib- uted middle.”
M
S P
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M
S P
Section 3.6 Categorical Logic: Categorical Syllogisms
A perhaps more intuitive way to see why it is invalid is to look at its Venn diagram. Here is how we diagram the premises:
The two x’s represent the fact that our particular premise states that some M are P and does not state whether or not they are in the S circle, so we represent both possibilities here. Now we simply need to check if the conclusion is necessarily true.
We can see that it is not, because although one x is in the right place, it is linked with another x in the wrong place. In other words, we do not know whether the x in “some M are P” is inside or outside the S boundary. Our conclusion requires that the x be inside the S boundary, but we do not know that for certain whether it is. Therefore, the argument is invalid. We could, for example, erase the linked x that is inside of the S circle, and we would have a diagram that makes both premises be true and the conclusion false.
M
S P
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S P
Section 3.6 Categorical Logic: Categorical Syllogisms
Because this diagram shows that it is possible to make the premises true and the conclusion false, it follows that the argument is invalid.
A final way to understand why this form is invalid is to use the counterexample method and consider that it has the same form as the following argument:
All dogs are mammals. Some mammals are cats. Therefore, some dogs are cats.
This argument has the same form and has all true premises and a false conclusion. This coun- terexample just verifies that our Venn diagram test got the right answer. If applied correctly, the Venn diagram test works every time. With this example, all three methods agree that our argument is invalid.
Moral of the Story: The Venn Diagram Test for Validity Here, in summary, are the steps for doing the Venn diagram test for validity:
1. Draw the three circles, all overlapping. 2. Diagram the premises.
a. Shade in areas where nothing exists. b. Put an x for areas where something exists. c. If you are not sure what side of a line the x should be in, then put two linked x’s,
one on each side.
3. Check to see if the conclusion is (must be) true in this diagram. a. If there are two linked x’s, and one of them makes the conclusion true and the
other does not, then the argument is invalid because the premises do not guar- antee the truth of the conclusion.
b. If the conclusion must be true in the diagram, then the argument is valid; other- wise it is not.
M
S P
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Section 3.6 Categorical Logic: Categorical Syllogisms
Because this diagram shows that it is possible to make the premises true and the conclusion false, it follows that the argument is invalid.
A final way to understand why this form is invalid is to use the counterexample method and consider that it has the same form as the following argument:
All dogs are mammals. Some mammals are cats. Therefore, some dogs are cats.
This argument has the same form and has all true premises and a false conclusion. This coun- terexample just verifies that our Venn diagram test got the right answer. If applied correctly, the Venn diagram test works every time. With this example, all three methods agree that our argument is invalid.
Moral of the Story: The Venn Diagram Test for Validity Here, in summary, are the steps for doing the Venn diagram test for validity:
1. Draw the three circles, all overlapping. 2. Diagram the premises.
a. Shade in areas where nothing exists. b. Put an x for areas where something exists. c. If you are not sure what side of a line the x should be in, then put two linked x’s,
one on each side.
3. Check to see if the conclusion is (must be) true in this diagram. a. If there are two linked x’s, and one of them makes the conclusion true and the
other does not, then the argument is invalid because the premises do not guar- antee the truth of the conclusion.
b. If the conclusion must be true in the diagram, then the argument is valid; other- wise it is not.
Practice Problems 3.4
Answer the following questions. Note that some questions may have more than one answer.
1. Which rules does the following syllogism pass?
All M are P. Some M are S. Therefore, some S are P.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclusion,
and if the syllogism has a negative conclusion, it must have a negative premise. d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. All the rules
2. Which rules does the following syllogism fail?
No P are M. All S are M. Therefore, all S are P.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclusion,
and if the syllogism has a negative conclusion, it must have a negative premise. (continued)
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Section 3.6 Categorical Logic: Categorical Syllogisms
d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. All the rules
3. Which rules does the following syllogism fail?
Some M are P. Some S are not M. Therefore, some S are not P.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclu-
sion, and if the syllogism has a negative conclusion, it must have a negative premise.
d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. All the rules
4. Which rules does the following syllogism fail?
No P are M. No M are S. Therefore, no S are P.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclu-
sion, and if the syllogism has a negative conclusion, it must have a negative premise.
d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. All the rules
5. Which rules does the following syllogism fail?
All M are P. Some M are not S. Therefore, no S are P.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclu-
sion, and if the syllogism has a negative conclusion, it must have a negative premise.
Practice Problems 3.4 (continued)
(continued)
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Section 3.6 Categorical Logic: Categorical Syllogisms
d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. All the rules
6. Which rules does the following syllogism fail?
All humans are dogs. Some dogs are mammals. Therefore, no humans are mammals.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclu-
sion, and if the syllogism has a negative conclusion, it must have a negative premise.
d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. None of the rules
7. Which rules does the following syllogism fail?
Some books are hardbacks. All hardbacks are published materials. Therefore, some books are published materials.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclu-
sion, and if the syllogism has a negative conclusion, it must have a negative premise.
d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. None of the rules
8. Which rules does the following syllogism fail?
No politicians are liars. Some politicians are men. Therefore, some men are not liars.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclusion,
and if the syllogism has a negative conclusion, it must have a negative premise.
Practice Problems 3.4 (continued)
(continued)
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Section 3.6 Categorical Logic: Categorical Syllogisms
d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. None of the rules
9. Which rules does the following syllogism fail?
Some Macs are computers. No PCs are Macs. Therefore, all PCs are computers.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclu-
sion, and if the syllogism has a negative conclusion, it must have a negative premise.
d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. None of the rules
10. Which rules does the following syllogism fail?
All media personalities are people who manipulate the masses. No professors are media personalities. Therefore, no professors are people who manipulate the masses.
a. Rule 1: The middle term must be distributed at least once. b. Rule 2: Any term distributed in the conclusion must be distributed in its corre-
sponding premise. c. Rule 3: If the syllogism has a negative premise, it must have a negative conclu-
sion, and if the syllogism has a negative conclusion, it must have a negative premise.
d. Rule 4: The syllogism cannot have two negative premises. e. Rule 5: If the syllogism has a particular premise, it must have a particular con-
clusion, and if the syllogism has a particular conclusion, it must have a particular premise.
f. None of the rules
11. Examine the following syllogisms. In the first pair, the terms that are distributed are marked in bold. Can you explain why? The second pair is left for you to determine which terms, if any, are distributed. Some P are M. Some M are not S. Therefore, some S are not P.
No P are M. All M are S. Therefore, no S are P.
All M are P. All M are S. Therefore, all S are P.
Some P are not M. No S are M. Therefore, no S are P.
Practice Problems 3.4 (continued)
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Section 3.7 Categorical Logic: Types of Categorical Arguments
3.7 Categorical Logic: Types of Categorical Arguments Many examples of deductively valid arguments that we have considered can seem quite sim- ple, even if the theory and rules behind them can be a bit daunting. You might even wonder how important it is to study deduction if even silly arguments about the moon being tasty are considered valid. Remember that this is just a brief introduction to deductive logic. Deductive arguments can get quite complex and difficult, even though they are built from smaller pieces such as those we have covered in this chapter. In the same way, a brick is a very simple thing, interesting in its form, but not much use all by itself. Yet someone who knows how to work with bricks can make a very complex and sturdy building from them.
Thus, it will be valuable to consider some of the more complex types of categorical argu- ments, sorites and enthymemes. Both of these types of arguments are often encountered in everyday life.
Sorites A sorites is a specific kind of argument that strings together several subarguments. The word sorites comes from the Greek word meaning a “pile” or a “heap”; thus, a sorites-style argu- ment is a collection of arguments piled together. More specifically, a sorites is any categorical argument with more than two premises; the argument can then be turned into a string of categorical syllogisms. Here is one example, taken from Lewis Carroll’s book Symbolic Logic (1897/2009):
The only animals in this house are cats; Every animal is suitable for a pet, that loves to gaze at the moon; When I detest an animal, I avoid it; No animals are carnivorous, unless they prowl at night; No cat fails to kill mice; No animals ever take to me, except what are in this house; Kangaroos are not suitable for pets; None but carnivora kill mice; I detest animals that do not take to me; Animals, that prowl at night, always love to gaze at the moon. Therefore, I always avoid kangaroos. (p. 124)
Figuring out the logic in such complex sorites can be challenging and fun. However, it is easy to get lost in sorites arguments. It can be difficult to keep all the premises straight and to make sure the appropriate relationships are established between each premise in such a way that, ultimately, the conclusion follows.
Carroll’s sorites sounds ridiculous, but as discussed earlier in the chapter, many of us develop complex arguments in daily life that use the conclusion of an earlier argument as the premise of the next argument. Here is an example of a relatively short one:
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Section 3.7 Categorical Logic: Types of Categorical Arguments
All of my friends are going to the party. No one who goes to the party is boring. People that are not boring interest me. Therefore, all of my friends interest me.
Here is another example that we might reason through when thinking about biology:
All lizards are reptiles. No reptiles are mammals. Only mammals nurse their young. Therefore, no lizards nurse their young.
There are many examples like these. It is possible to break them into smaller syllogistic sub- arguments as follows:
All lizards are reptiles. No reptiles are mammals. Therefore, no lizards are mammals. No lizards are mammals. Only mammals nurse their young. Therefore, no lizards nurse their young.
Breaking arguments into components like this can help improve the clarity of the overall reasoning. If a sorites gets too long, we tend to lose track of what is going on. This is part of what can make some arguments hard to understand. When constructing your own argu- ments, therefore, you should beware of bunching premises together unnecessarily. Try to break a long argument into a series of smaller arguments instead, including subarguments, to improve clarity.
Enthymemes While sorites are sets of arguments strung together into one larger argument, a related argu- ment form is known as an enthymeme, a syllogistic argument that omits either a premise or a conclusion. There are also many nonsyllogistic arguments that leave out premises or con- clusions; these are sometimes also called enthymemes as well, but here we will only consider enthymemes based on syllogisms.
A good question is why the arguments are missing premises. One reason that people may leave a premise out is that it is considered to be too obvious to mention. Here is an example:
All dolphins are mammals. Therefore, all dolphins are animals.
Here the suppressed premise is “All mammals are animals.” Such a statement probably does not need to be stated because it is common knowledge, and the reader knows how to fill it in to get to the conclusion. Technically speaking, we are said to “suppress” the premise that does not need to be stated.
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Section 3.7 Categorical Logic: Types of Categorical Arguments
Sometimes people even leave out conclusions if they think that the inference involved is so clear that no one needs the conclusion stated explicitly. Arguments with unstated conclusions are considered enthymematic as well. Let us suppose a baseball fan complains, “You have to be rich to get tickets to game 7, and none of my friends is rich.” What is the implied conclu- sion? Here is the argument in standard form:
Everyone who can get tickets to game 7 is rich. None of my friends is rich. Therefore, ???
In this case we may validly infer that none of the fan’s friends can get tickets to game 7.
To be sure, you cannot always assume your audience has the required background knowl- edge, and you must attempt to evaluate whether a premise or conclusion does need to be stated explicitly. Thus, if you are talking about math to professional physicists, you do not need to spell out precisely what the hypotenuse of an angle is. However, if you are talking to third graders, that is certainly not a safe assumption. Determining the background knowledge of those with whom one is talking—and arguing—is more of an art than a science.
Validity in Complex Arguments Recall that a valid argument is one whose premises guarantee the truth of the conclusion. Sorites are illustrations of how we can “stack” smaller valid arguments together to make larger valid arguments. Doing so can be as complicated as building a cathedral from bricks, but so long as each piece is valid, the structure as a whole will be valid.
How do we begin to examine a complex argument’s validity? Let us start by looking at another example of sorites from Lewis Carroll’s book Symbolic Logic (1897/2009):
Babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons are despised. Therefore, no babies can manage a crocodile. (p. 112)
Is this argument valid? We can see that it is by breaking it into a pair of syllogisms. Start by considering the first and third premises. We will rewrite them slightly to show the All that Carroll has assumed. With those two premises, we can build the following valid syllogism:
All babies are illogical. All illogical persons are despised. Therefore, all babies are despised.
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Section 3.7 Categorical Logic: Types of Categorical Arguments
Using the tools from this chapter (the rules, Venn diagrams, or just by thinking it through carefully), we can check that the syllogism is valid. Now we can use the conclusion of our syllogism along with the remaining premise and conclusion from the original argument to construct another syllogism.
All babies are despised. No despised persons can manage a crocodile. Therefore, no babies can manage a crocodile.
Again, we can check that this syllogism is valid using the tools from this chapter. Since both of these arguments are valid, the string that combines them is valid as well. Therefore, the original argument (the one with three premises) is valid.
This process is somewhat like how we might approach adding a very long list of numbers. If you need to add a list of 100 numbers (suppose you are checking a grocery bill), you can do it by adding them together in groups of 10, and then adding the subtotals together. As long as you have done the addition correctly at each stage, your final answer will be the correct total. This is one reason validity is important. It allows us to have confidence in complex arguments by examining the smaller arguments from which they are, or can be, built. If one of the smaller arguments was not valid, then we could not have complete confidence in the larger argument.
But what about soundness? What use is the argument about babies managing crocodiles when we know that babies are not generally despised? Again, let us make a comparison to adding up your grocery bill. Arithmetic can tell you if your bill is added correctly, but it can- not tell you if the prices are correct or if the groceries are really worth the advertised price. Similarly, logic can tell you whether a conclusion validly follows from a set of premises, but it cannot generally tell you whether the premises are true, false, or even interesting. By them- selves, random deductive arguments are as useful as sums of random numbers. They may be good practice for learning a skill, but they do not tell us much about the world unless we can somehow verify that their premises are, in fact, true. To learn about the world, we need to apply our reasoning skills to accurate facts (usually outside of arithmetic and logic) known to be true about the world.
This is why logicians are not as concerned with soundness as they are with validity, and why a mathematician is only concerned with whether you added correctly, and not with whether the prices were correctly recorded. Logic and mathematics give us skills to apply valid reason- ing to the information around us. It is up to us, and to other fields, to make sure the informa- tion that we use in the premises is correct.
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Summary and Resources
Practice Problems 3.5
Answer the following questions.
1. This is the name that is given to an argument that has two premises and one conclusion. a. syllogism b. creative syllogism c. enthymeme d. sorites e. none of the above
2. The discovery of categorical logic is often attributed to this philosopher. a. Plato b. Boole c. Aristotle d. Kant e. Hume
3. Which of the following is a type of deductive argument? a. generalization b. categorical syllogism c. argument by analogy d. modus spartans e. none of the above
4. All categorical statements have which of the following? a. mood and placement b. figure and form c. number and validity d. quantity and quality e. all of the above
5. The premise that contains the predicate term of the conclusion in a categorical syl- logism is __________. a. the minor premise b. the major premise c. the necessary premise d. the conclusion e. none of the above
Summary and Resources
Chapter Summary Validity is the central concept of deductive reasoning. An argument is valid when the truth of the premises absolutely guarantees the truth of the conclusion. For valid arguments, if the premises are true, then the conclusion must be true also. Valid arguments need not have true premises, but if they do, then they are sound arguments. Because they use valid reason- ing and have true premises, sound arguments are guaranteed to have true conclusions.
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Summary and Resources
Deductive arguments can include mathematical arguments, arguments from definitions, cat- egorical arguments, and propositional arguments. Categorical arguments allow us to reason about things based on their properties. Categorical arguments with two premises are called syllogisms. The validity of syllogisms can be evaluated either with a system of rules or by using Venn diagrams.
Syllogisms often leave one premise or the conclusion unstated. These are called enthymemes. Sometimes strings of syllogisms are combined into a larger argument called a sorites. If we have a string of valid arguments that are combined to make a larger argument, then we may infer that the long argument composed of these parts is valid as well.
The process of using subarguments to create longer ones allows us to make rather complex valid arguments out of simple parts. This is an important motivation for studying deductive logic. As with arithmetic, computer programming, and structural engineering, combining smaller steps in a careful way allows us to create complex structures that are fully reliable because they are built out of reliable parts.
Critical Thinking Questions
1. How does the logical definition of validity differ from the way that the term valid is used in everyday speech? How do you plan on differentiating the two as you con- tinue studying logic?
2. In the chapter, you read a section about the importance of having evidence that sup- ports your arguments. Is it important to claim to believe things only when one has evidence, or are there some things that people can justifiably believe without evi- dence? Why?
3. How would you describe what a deductive argument is to someone who does not know the technical terms that apply to arguments? What examples would you use to demonstrate deduction?
4. What is the point of being able to understand if a deductive argument is valid or sound? Why is it important to be able to determine these things? If you do not think it is important, how would you justify your claims that it is not important to be able to determine validity?
5. Has there ever been a time that you presented an argument in which you had little or no evidence to support your claims? What types of claims did you use in the place of premises? What types of techniques did you use to try to present an argument with no information to back up your conclusion(s)? What is a better method to use in the future?
Web Resources http://www.philosophyexperiments.com/validorinvalid/Default.aspx This game at the Philosophy Experiments website tests your ability to determine whether an argument is valid.
http://www.thefirstscience.org/syllogistic-machine This professor’s blog includes an online syllogism solver that allows you to explore fallacies, figures, terms, and modes of syllogisms. Click on “Notes on Syllogistic Logic” for more cover- age of topics discussed in this chapter.
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Summary and Resources
Key Terms
argument from definition An argument in which one premise is a definition.
categorical argument An argument entirely composed of categorical statements.
categorical logic The branch of deduc- tive logic that is concerned with categorical arguments.
categorical statement A statement that relates one category or class to another. Spe- cifically, if S and P are categories, the cate- gorical statements relating them are: All S is P, No S is P, Some S is P, and Some S is not P.
complement class For a given class, the complement class consists of all things that are not in the given class. For example, if S is a class, its complement class is non-S.
contraposition The immediate inference obtained by switching the subject and predi- cate terms with each other and complement- ing them both.
conversion The immediate inference obtained by switching the subject and predi- cate terms with each other.
counterexample method The method of proving an argument form to be not valid by constructing an instance of it with true premises and a false conclusion.
deductive argument An argument that is presented as being valid—if the primary evaluative question about the argument is whether it is valid.
distribution Referring to members of groups. If all the members of a group are referred to, the term that refers to that group is said to be distributed.
enthymeme An argument in which one or more claims are left unstated.
immediate inferences Arguments from one categorical statement as premise to another as conclusion. In other words, we immedi- ately infer one statement from another.
instance A term in logic that describes the sentence that results from replacing each variable within the form with specific sentences.
logical form The pattern of an argument or claim.
predicate term The second term in a cat- egorical proposition.
quality In logic, the distinction between a statement being affirmative or negative.
quantity In logic, the distinction between a statement being universal or particular.
sorites A categorical argument with more than two premises.
sound Describes an argument that is valid and in which all of the premises are true.
subject term The first term in a categorical proposition.
syllogism A deductive argument with exactly two premises.
valid An argument in which the premises absolutely guarantee the conclusion, such that is impossible for the premises to be true while the conclusion is false.
Venn diagram A diagram constructed of overlapping circles, with shaded areas or x’s, which shows the relationships between the represented groups.
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Summary and Resources
Answers to Practice Problems Practice Problems 3.1
1. not deductive 2. deductive 3. not deductive 4. not deductive 5. deductive 6. not deductive 7. deductive 8. not deductive 9. deductive
10. deductive 11. not deductive
12. not deductive 13. deductive 14. not deductive 15. not deductive 16. deductive 17. not deductive 18. deductive 19. not deductive 20. deductive
Practice Problems 3.2
1. a 2. a 3. b 4. a 5. a
6. a 7. d 8. c 9. b
10. c
Practice Problems 3.3
1. d 2. b 3. d
4. b 5. a
Practice Problems 3.4
1. e 2. c 3. b 4. d 5. b 6. a, b, and c 7. e 8. e
9. c 10. b 11. All M are P.
All M are S. Therefore, all S are P.
Some P are not M. No S are M. Therefore, no S are P.
Practice Problems 3.5
1. a 2. c 3. b
4. d 5. a
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25
2The Argument
Rolphot/iStock/Thinkstock
Learning Objectives After reading this chapter, you should be able to:
1. Articulate a clear definition of logical argument.
2. Name premise and conclusion indicators.
3. Extract an argument in the standard form from a speech or essay with the aid of paraphrasing.
4. Diagram an argument.
5. Identify two kinds of arguments—deductive and inductive.
6. Distinguish an argument from an explanation.
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Section 2.1 Arguments in Logic
Chapter 1 defined logic as the study of arguments that provides us with the tools for arriving at warranted judgments. The concept of argument is indeed central to this definition. In this chapter, then, our focus shall be entirely on defining arguments—what they are, how their component parts function, and how learning about arguments helps us lead better lives. Most especially, in this chapter we will introduce the standard argument form, which is the struc- ture that helps us identify arguments and distinguish good ones from bad ones.
2.1 Arguments in Logic Chapter 1 provisionally defined argument as a methodical defense of a position. We referred to this as the commonsense understanding of the way the word argument is employed in logic. The commonsense definition is very useful in helping us recognize a unique form of expression in ordinary human communication. It is part of the human condition to differ in opinion with another person and, in response, to attempt to change that person’s opinion. We may attempt, for example, to provide good reasons for seeing a particular movie or to show that our preferred kind of music is the best. Or we may try to show others that smoking or heavy drinking is harmful. As you will see, these are all arguments in the commonsense understanding of the term.
In Chapter 1 we also distinguished the commonsense understanding of argument from the meaning of argument in ordinary use. Arguments in ordinary use require an exchange between at least two people. As clarified in Chapter 1, commonsense arguments do not neces- sarily involve a dialogue and therefore do not involve an exchange. In fact, one could develop a methodical defense of a position—that is, a commonsense argument—in solitude, simply to examine what it would require to advocate for a particular position. In contrast, arguments, as understood in ordinary use, are characterized by verbal disputes between two or more people and often contain emotional outbursts. Commonsense arguments are not character- ized by emotional outbursts, since unbridled emotions present an enormous handicap for the development of a methodical defense of a position.
In logic an argument is a set of claims in which some, called the premises, serve as support for another claim, called the conclusion. The conclusion is the argument’s main claim. For the most part, this technical definition of argument is what we shall employ in the remainder of this book, though we may use the commonsense definition when talking about less technical examples. Table 2.1 should help clarify which meanings are acceptable within logic. Take a moment to review the table and fix these definitions in your mind.
Table 2.1: Comparing meanings for the term argument
Meaning in ordinary use Commonsense meaning Technical meaning in logic
A verbal quarrel or disagree- ment, often characterized by raised voices and flaring emotions.
The methodical and well- researched defense of a position or point of view advanced in relation to a disputed issue.
A set of claims in which some, called premises, serve as support for another claim, called the conclusion.
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Section 2.1 Arguments in Logic
Arguments in the technical sense are a primary way in which we can defend a position. Accordingly, we can find the structure of logical arguments in commonsense arguments all around us: in letters to the editor, social media, speeches, advertisements, sales pitches, pro- posals submitted for grant funds or bank loans, job applications, requests for a raise, commu- nications of values to children, marriage proposals, and so on. Arguments often provide the basis on which most of our decisions are made. We read or hear an argument, and if we are convinced by it, then we accept its conclusion. For example, consider the following argument:
“I’m just not a math person.” We hear this all the time from anyone who found high school math challenging. . . . In high school math at least, inborn talent is less important than hard work, preparation, and self-confidence. This is what high school math teachers, college professors, and private tutors have observed as the pattern of those who become good in high school math. They point out that in any given class, students fall in a wide range of levels of math preparation. This is not due to genetic predisposition. What is rarely observed is that some children come from households in which parents introduce them to math early on and encourage them to practice it. These students will imme- diately obtain perfect scores while the rest do not. As a result, the students without previous preparation in math immediately assume that those with perfect scores have a natural math talent, without knowing about the prepa- ration that these students had in their homes. In turn, the students who obtain perfect scores assume that they have a natural math talent given their scores relative to the rest of the class, so they are motivated to continue honing their math skills and, by doing this, they cement their top of the class standing. Thus, the belief that math ability cannot change becomes a self-fulfilling prophecy. (Kimball & Smith, 2013)
In this argument, the position defended by the authors is that the belief that math ability can- not change becomes a self-fulfilling prophecy. The authors support this claim with reasons that show good performance in math is not typically the result of a natural ability but of hav- ing a family support system for learning, a prior preparation in math from home, and continu- ous practice. It makes the case that it is hard work and preparation that lead to a person’s proficiency in math and other subjects, not genetic predisposition. This argument helps us recognize that we frequently accept oft-repeated information as fact without even question- ing the basis. As you can see, an argument such as this can provide a solid basis for our every- day decisions, such as encouraging our children to work hard and practice in the subjects they find most difficult or deciding to obtain a university degree with confidence later in life.
To understand the more technical definition of an argument as a set of premises that support a conclusion, consider the following presentation of the reasoning from the commonsense argument we have just examined.
Good performance in math is not due to genetics.
Good performance in math only requires preparation and continuous practice.
Students who do well initially assume they have natural talent and practice more.
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Section 2.1 Arguments in Logic
Students who do less well initially assume they do not have natural talent and practice less.
Therefore, believing that one’s math ability cannot change becomes a self- fulfilling prophecy.
Presenting the reasoning this way can do a great deal to clarify the argument and allow us to examine its central claims and reasoning. This is why the field of logic adopts the more techni- cal definition of argument for much of its work.
Regardless of what we think about math, an important contribution of this argument is that it makes the case that it is hard work and preparation that lead to our proficiency in math, and not the factor of genetic predisposition. Logic is much the same way. If you find some concepts difficult, don’t assume that you just lack talent and that you aren’t a “logic person.” With prac- tice and persistence, anyone can be a logic person.
On your way to becoming a logic person, it is important to remember that not everything that presents a point of view is an argument (see Table 2.2 for examples of arguments and nonar- guments). Consider that when one expresses a complaint, command, or explanation, one is indeed expressing a point of view. However, none of these amount to an argument.
Table 2.2: Is it an argument?
Argument Not an argument
Reprinted with permission from The Hill Times.
Why? This presents a defense of a position. But not all letters to the editor contain arguments.
©Bettmann/Corbis
Why not? This only reports a news story. It informs us of the role of the university but does not offer a defense of a position.
(continued)
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Section 2.1 Arguments in Logic
Argument Not an argument
Greg Gibson/Associated Press
Why? This is a photo of former president Bill Clin- ton making a speech, in which he defends his posi- tion that the facts are different than those reported by the media. Not all speeches contain arguments, only those that defend a position.
©MIKE SEGAR/Reuters/Corbis
Why not? This is a debate between two presidential candidates. Although each candidate may present various arguments, the debate as a whole is not an argument. It is not a defense of a position; it is an exchange between two people on various subjects.
Emmanuel Dunand/AFP/Getty Images
Why? This ad makes a claim and offers a reason for why viewers should take notice.
©James Lawrence/Transtock/Corbis
Why not? This ad has no words, so it makes no specific claim. Even if we try to interpret it to make a claim, no defense is offered.
To help us properly identify logical arguments, we need clear criteria for what a logical argu- ment is. Let us start unpacking what is involved in arguments by addressing their smallest element: the claim.
Claims A claim is an assertion that something is or is not the case. Claims take the form of declara- tive sentences. It is important to note that each premise or conclusion consists of one single claim. In other words, each premise or conclusion consists of one single declarative sentence.
Claims can be either true or false. This means that if what is asserted is actually the case, then the claim is true. If the claim does not correspond to what is actually the case, then the claim is false. For example, the claim “milk is in the refrigerator” predicates that the subject of the
Table 2.2: Is it an argument? (continued)
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Section 2.1 Arguments in Logic
claim, milk, is in the refrigerator. If this claim corresponds to the facts (if the refrigerator con- tains milk), then this claim is true. If it does not correspond to the facts (if the refrigerator does not contain milk), then the claim is false.
Not all claims, however, can be easily checked for truth or falsity. For exam- ple, the truth of the claim “Jacob has the best wife in the world” cannot be settled easily, even if Jacob is the one asserting this claim (“I have the best wife in the world”). In order to under- stand what he could possibly mean by “best wife in the world,” we would have to propose the criteria for what makes a good wife in the first place, and as if this were not challenging enough, we would then have to establish a method or procedure to make comparisons among good wives. Of course, Jacob could merely mean “I like being mar- ried to my wife,” in which case he is not stating a claim about his wife being the
best in the world but merely stating a feeling. It is not uncommon to hear people state things that sound like claims but are actually just expressions of preference or affection, and distin- guishing between these is often challenging because we are not always clear in the way we employ language. Nonetheless, it is important to note that we often make claims from a par- ticular point of view, and these claims are different from factual claims. Claims that advance a point of view, such as the example of Jacob’s wife—and especially claims about morality and ethicality—are indeed more challenging to settle as true or false than factual claims, such as “The speed limit here is 55.”
The important point is that both kinds of claims—the factual claim and the point-of-view claim—assert that something is or is not the case, affirm or deny a particular predicate of a subject, and can be either true or false. The following sentences are examples of claims that meet these criteria.
• There is a full moon tonight. • Pecans are better than peanuts. • All flights to Paris are full. • BMWs are expensive to maintain. • Lola is my sister.
The following are not claims:
• Is it raining? Why? Because questions are not, and cannot be, assertions that some- thing is the case.
• Oh, to be in Paris in the springtime! Why? Because this expresses a sentiment but does not state that anything might be true or false.
• Buy a BMW! Why? Because a command is not an assertion that something is the case.
Image Source Pink/Image Source/Thinkstock
What factual claims can you make about this image?
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Section 2.1 Arguments in Logic
We often intend to advance claims in ways that do not present our claims clearly and properly— for example, by means of rhetorical questions, vague expressions of affection, and commands or metaphors that demand interpretation. But it is important to recognize that intention is not sufficient when communicating with others. In order for our intended claims to be identified as claims, they should meet the three criteria previously mentioned.
Claims are sometimes called propositions. We will use the terms claims and propositions inter- changeably in this book. In this chapter we will stick to the word claim, but in subsequent chapters, we will move to the more formal terminology of propositions.
The Standard Argument Form In informal logic the main method for identifying, constructing, or examining arguments is to extract what we hear or read as arguments and put this in what is known as the standard argument form. It consists of claims, some of which are called premises and one of which is called the conclusion. In the standard argument form, premises are listed first, each on a separate line, with the conclusion on the line after the last premise. There are various meth- ods for displaying standard form. Some methods number the premises; others separate the conclusion with a line. We will generally use the following method, prefacing the conclusion with the word therefore:
Premise Premise Therefore, Conclusion
The number of premises can be as few as one and as many as needed. We must approach either extreme with caution given that, on the one hand, a single premise can offer only very limited support for the conclusion, and on the other hand, many premises risk error or confu- sion. However, there are certain kinds of arguments that, because of their formal structure, may contain only a limited number of premises.
In the standard argument form, each premise or conclusion should be only one sentence long, and premises and conclusions should be stated as clearly and briefly as possible. Accordingly, we must avoid premises or conclusions that have multiple sentences or single sentences with multiple claims. The following example shows what not to do:
I live in Boston, and I like clam chowder. My family also lives in Boston. They also like clam chowder. My friends live in Boston. They all like clam chowder, too. Therefore, everyone I know in Boston likes clam chowder.
If you want to make more than one claim about the same subject, then you can break your declarative sentences into several sentences that each contain only one claim. The clam chow- der argument can then be rewritten as follows:
I live in Boston. I like clam chowder. My family lives in Boston. My family likes clam chowder.
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Section 2.1 Arguments in Logic
My friends live in Boston. My friends like clam chowder. Therefore, everyone I know in Boston likes clam chowder.
The relationship between premises and the conclusion is that of inference—the process of drawing a claim (the conclusion) from the reasons offered in the premises. The act of reason- ing from the premises serves as the glue connecting the premises with the conclusion.
Practice Problems 2.1
Determine whether the following sentences are claims (propositions) or nonclaims (nonpropositions).
1. Moby Dick is a great novel.
2. Computers have made our lives easier.
3. If we go to the movies, we will need to drive the minivan.
4. Do you want to drive the minivan to the movies?
5. Drive the minivan.
6. Either I am a human or I am a dog.
7. Michael Jordan was a great football player.
8. Was it time for you to leave?
9. Private property is a right of every American.
10. Universalized health care is communism.
11. Don’t you dare vote for universalized health care.
12. Nietzsche collapsed in a square upon seeing a man beat a horse.
13. Hooray!
14. Those who reject equality seek tyranny.
15. How many feet are in a mile?
16. If you cannot understand the truth value of a claim, then it is not a claim.
17. Something is a claim if and only if it has a truth value.
18. Treat your boss with respect.
19. Men are much less likely to have osteoporosis than women are.
20. Why are women less likely to have heart attacks?
21. Do as we say.
22. I believe that you should do as your parents say.
23. Socrates is mortal. (continued)
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Section 2.2 Putting Arguments in the Standard Form
24. Why did Freud hold such strange beliefs about parent–child relationships?
25. A democracy exists if and only if its citizens participate in autonomous elections.
26. Do your best.
27. The unexamined life is not worth living.
28. Ayn Rand believed that selfishness was a virtue.
29. Is selfishness a virtue?
30. What people love is not the object of desire, but desire itself.
31. Hey!
32. Those who cannot support themselves should not be supported by taxpayer dollars.
33. Particle and wave behavior are properties of light.
34. Why do we feed so many pounds of plants to animals each year?
35. Go and give your brother a kiss.
36. Because the mind conditions reality, it is impossible to know the thing as such.
37. The library at the local university has more than 300,000 books.
38. Does the nature of reality consist of an ultimately creative impulse?
39. You are taking a quiz.
40. Are you taking a quiz?
Practice Problems 2.1 (continued)
2.2 Putting Arguments in the Standard Form Presenting arguments in the standard argument form is crucial because it provides us with a dispassionate method that will allow us to find out whether the argument is good, regardless of how we feel about the subject matter. The first step is to identify the fundamental argument being presented.
At first it might seem a bit daunting to identify an argument, because arguments typically do not come neatly presented in the standard argument form. Instead, they may come in confus- ing and unclear language, much like this statement by Special Prosecutor Francis Schmitz of Wisconsin regarding Governor Scott Walker:
Governor Walker was not a target of the investigation. At no time has he been served with a subpoena. . . . While these documents outlined the prosecutor’s legal theory, they did not establish the existence of a crime; rather, they were arguments in support of further investigation to determine if criminal charges against any person or entity are warranted. (Crocker, 2014, para. 7 & 10)
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Section 2.2 Putting Arguments in the Standard Form
This was a position presented in regard to the investigation of an alleged illegal campaign finance coordination during the 2011–2012 recall elections (Stein, 2014). Does it claim a vin- dication of Walker? Or does it suggest that there may be sufficient evidence to make Walker a central figure in the investigation? How would you even begin to make heads or tails of such a confusing argument? Do not despair. The remainder of this section will show you exactly what to look for in order to make sense of the most complicated argument. With a little prac- tice, you will be able to do this without much effort.
Find the Conclusion First Although the conclusion is last in the standard form, the conclusion is the first thing to find because the conclu- sion is the main claim in an argument. The other claims—the premises—are present for the sole purpose of support- ing the conclusion. Accordingly, if you are able to find the conclusion, then you should be able to find the premises.
The good news is that language is not only a means for expressing ideas; it also offers a road map for the ideas presented. Chapter 1 underscored the fundamental importance of clear, pre- cise, and correct language in logical reasoning. When used properly, lan- guage also offers structures and direc- tions for communicating meaning, thus facilitating our understanding of what others are saying. One punctua- tion mark—the question mark—tells us that we are confronting a question. A different punctuation mark—the parentheses—tells us that we are being given relevant information but only as an aside or afterthought to the main point; if removed, the parenthetical information would not alter the main point. In the case of arguments, some words serve as signposts identifying conclusions. Take the following example of an argument in the standard argument form:
All men are mortal. Socrates is a man. Therefore, Socrates is mortal.
The word therefore indicates that the sentence is a conclusion. In fact, the word therefore is the standard conclusion indicator we will use when constructing arguments in the stan- dard argument form. However, there are other conclusion indicators that are used in ordinary arguments, including:
• Consequently . . . • So . . . • Hence . . . • Thus . . .
Xtock Images/iStock/Thinkstock
Punctuation, parentheses, and conclusion indicators all serve as signposts to assist us when deconstructing an argument. They provide important clues about where to find the conclusion as well as supporting claims.
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Section 2.2 Putting Arguments in the Standard Form
• Wherefore . . . • As a result . . . • It follows that . . . • For these reasons . . . • We may conclude that . . .
When a conclusion indicator is present, it can help identify the conclusion in an argument. Unfortunately, many arguments do not come with conclusion indicators. In such cases start by trying to identify the main point. If you can clearly identify a single main point, then that is likely to be the conclusion. But sometimes you will have to look at a passage closely to find the conclusion. Suppose you encounter the following argument:
Don’t you know that driving without a seat belt is dangerous? Statistics show that you are 10 times more likely to be injured in an accident if you are not wearing one. Besides, in our state you can get fined $100 if you are caught not wearing one. You ought to wear one even if you are driving a short distance.
Arguments are often longer and more complicated than this one, but let us work with this simple case before trying more complicated examples. You know that the first thing you need to do is to look for the conclusion. The problem is that the author of the argument does not use a conclusion indicator. Now what? Nothing to worry about. Just remember that the con- clusion is the main claim, so the thing to look for is what the author may be trying to defend. Although the first sentence is stated as a question—remember, punctuation marks give us important clues—the author seems to intend to assert that driving without a seat belt is dan- gerous. In fact, the second sentence offers evidence in support of this claim. On the other hand, the third sentence seems to be important, yet it does not speak to driving without a seat belt being dangerous, only expensive. In the final sentence, we find a claim that is supported by all the others. Because of this, the final sentence presents the conclusion.
Now, it so happens that in this case, the conclusion is at the end of this short argument, but keep in mind that conclusions can be found in various places in essays, such as the beginning or sometimes in the middle. Now that you have identified your first piece of the puzzle, we have this:
Premise 1: ? Premise 2: ? Premise 3: ? Therefore, you ought to wear a seat belt whenever you drive.
You might have noticed that the conclusion does not appear as it did in the essay. The origi- nal sentence is “You ought to wear one even if you are driving a short distance.” Why did we modify it? Once again, clarity is of the essence in logical reasoning. Conclusions should make the subject clear, so the pronoun one was replaced with the actual subject to which the author is referring: seat belt. In addition, the predicate “even if you are driving a short distance” was rewritten to reflect the more inclusive point that the author seems to be making: that you should wear a seat belt whenever you drive.
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Section 2.2 Putting Arguments in the Standard Form
This modification of language, known as paraphrasing, is part of the construction of argu- ments in the standard argument form. The act of extracting an argument from a longer piece to its fundamental claims in the standard argument form necessarily involves paraphrasing the original language to the clearest and most precise form possible. This concept will be addressed in greater detail later in this section.
Find the Premises Next After identifying the conclusion, the next thing to do is look for the reasons the author offers in defense of his or her position. These are the premises. As with conclusions, there are prem- ise indicators that serve as signposts that reasons are being offered for the main claim or conclusion. Some examples of premise indicators are:
• Since . . . • For . . . • Given that . . . • Because . . . • As . . . • Owing to . . . • Seeing that . . . • May be inferred from . . .
To practice identifying premises, let us return to our seat belt example:
Don’t you know that driving without a seat belt is dangerous? Statistics show that you are 10 times more likely to be injured in an accident if you are not wearing one. Besides, in our state you can get fined $100 if you are caught not wearing one. You ought to wear one even if you are driving a short distance.
Notice again that this argument starts with a question: “Don’t you know that driving without a seat belt is danger- ous?” The author is not really asking whether you know that driving with- out a seat belt is dangerous. Rather, the author seems to be asking a rhe- torical question—a question that does not actually demand an answer—to assert that driving without a seat belt is dangerous. You should avoid asking rhetorical questions in the essays that you write, because the outcome can be highly uncertain. The success of a rhe- torical question depends on the reader or listener first understanding the hid- den meaning behind the rhetorical question and then correctly articulat- ing the answer you have in mind. This does not always work.
Hkeita/iStock/Thinkstock
Much like a map will get you from point A to point B, putting an argument into the standard argument form will help you navigate from the conclusion to the premises and vice versa.
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Section 2.2 Putting Arguments in the Standard Form
For the sake of this example, however, let us do our best to try to get at the author’s inten- tion. We could paraphrase the first premise to the following claim: Driving without a seat belt is dangerous. Does this paraphrased claim serve as a premise in support of the conclu- sion? In order to answer this, we need to put the conclusion in the form of a question. Again, premises are reasons offered in support of the conclusion, so if we have a well-constructed argument, then the premises should answer why the conclusion is the case. This is what we would have:
Question: Why must you wear a seat belt whenever you drive?
Answer: Because driving without a seat belt is dangerous.
This works, so the paraphrased claim that we drew from the author’s rhetorical question is indeed a reason in defense of the conclusion. So now we have one more piece of the puzzle:
Premise 1: Driving without a seat belt is dangerous. Premise 2: ? Premise 3: ? Therefore, you ought to wear a seat belt whenever you drive.
Let us now move to the next sentence: “Statistics show that you are 10 times more likely to be injured in an accident if you are not wearing one.” Is this a claim that can be a support for the conclusion? In other words, if we put the conclusion in the form of a question again as we did before, would this sentence be an adequate reason in response? Let us see.
Question: Why must you wear a seat belt whenever you drive?
Answer: Because statistics show that you are 10 times more likely to be injured in an accident if you are not wearing one.
The answer provides a reason in support of the conclusion, and thus, we have another prem- ise. Now we have most of the puzzle completed, as follows:
Premise 1: Driving without a seat belt is dangerous. Premise 2: Statistics show that you are 10 times more likely to be injured in an accident if you are not wearing one. Premise 3: ? Therefore, you ought to wear a seat belt whenever you drive.
We have one more sentence left in the argument, which reads: “Besides, in our state you can get fined $100 if you are caught not wearing one.” Is this a premise? Well, it is uncer- tain, since the sentence is not presented in the form of a claim. So let us paraphrase it as a claim as follows: “Not wearing a seat belt can result in a $100 fine.” This is now a claim, and the paraphrasing has not altered the meaning, so we can proceed to our question: Is this a
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Section 2.2 Putting Arguments in the Standard Form
premise for the argument that we are examining? Once again, let us put the conclusion into a question:
Question: Why must you wear a seat belt whenever you drive?
Answer: Because not wearing a seat belt can result in a $100 fine.
This is a claim that can be a support for the conclusion, and thus, we have another premise. We can now see the argument presented more formally as follows:
Driving without a seat belt is dangerous. Statistics show that you are 10 times more likely to be injured in an accident if you are not wearing one. Not wearing a seat belt can result in a $100 fine. Therefore, you ought to wear a seat belt whenever you drive.
The Necessity of Paraphrasing As we have discussed, extracting the fundamental claims from a written or a spoken argument often involves paraphrasing. Paraphrasing is not merely an option but rather a necessity in order to uncover the intended argument in the best way possible. Most other arguments presented to you (especially those in the media) will not consist of only premises and the conclusion in clearly identifiable language. Furthermore, many arguments will be much longer and compli- cated than the seat belt argument example. Often, arguments are presented with many other sentences that do not serve the purposes of an argument, such as empty rhetorical devices, filler sentences that aim to manipulate your emotions, and so on. So your task in extracting an argument from such sources is akin to that of a surgeon—removing all those linguistic tumors that obscure the argument in order to reveal the basic claims presented and their supporting evidence. In other words, you should expect to do paraphrasing as a necessary task when you attempt to draw an argument in the standard form from almost any source.
It is important to recognize that not everyone who advances an argument does so clearly or even coherently. This is precisely why the structure of the standard argument form is such a powerful tool to command. It offers you the machinery to distinguish arguments from what are not arguments. It also helps you unearth the elements of an argument that are buried under complicated prose and rhetoric. And it helps you evaluate the worthiness of the argument presented once it has been fully clarified. You should paraphrase all claims when presenting them in the standard argument form, whether the claims are implied in a long argumentative essay or speech or in shorter arguments that may be ambiguous or unclear. (To understand the added benefits, see Everyday Logic: Modesty and Charity.)
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Section 2.2 Putting Arguments in the Standard Form
Thinking Analytically Identifying an argument’s components as we have just done is an example of analytical think- ing. When we analyze something, we examine its architectural structure—that is, the relation of the whole to its parts—to identify its parts and to see how the parts fit together as a whole.
Let us examine an excerpt from President Barack Obama’s (2014) speech on Ebola as a way of bringing the new skills from this section all together:
Everyday Logic: Modesty and Charity
The goal of paraphrasing is to find the best presentation of the premises and conclusions intended. By presenting the argument offered in its best possible light, this will help you see not only how far off the argument is from an optimal defense, but also how good it is despite its bad presentation. Why should you be so charitable?
First we must keep in mind that ideas are important, even if the ideas are not ours. So we must always give our utmost due diligence to the examination of ideas. Sometimes even the roughest presentation of ideas can contain the most impressive pearls of insight. If we are not charitable to the ideas of others, then we might miss out on hidden wisdom.
Second, modesty is a good intellectual habit to develop. It is very easy to fall into the trap of thinking that our thoughts are the best ones around. This is generally far from the truth. The most fruitful innovations of mankind have been quite unexpected, often as the result of someone paying attention to others’ ideas and coming up with a new way of putting them to use. This applies to all sorts of things, including everything from the ways in which cooking methods turned into regional cuisines, to scientific discoveries, product innovations, and the emergence of the Internet.
That modesty has advantages is not a new idea. In the 1980s Peter Drucker wrote the book Inno- vation and Entrepreneurship, in which he recounts, among many other stories, the story of how Ray Kroc founded the burger chain McDonald’s®. As the well-known story goes, Kroc bought a hamburger stand from the McDonald brothers, along with their invention of a milkshake machine. Although Kroc never invented anything, his entrepreneurial genius was in seeing the potential of a hamburger, fries, and milkshake business that catered to mothers with little chil- dren and turning this vision into a billion-dollar standardized operation (Drucker, 1985/2007).
Even if you dislike McDonald’s, the point is that Kroc noticed the potential for something that many, including the McDonald brothers themselves, had overlooked. Gems are everywhere in the world of ideas, but we often have to dust them off, remove all the excess baggage, and extract what is good in them. Intellectual modesty allows us to do this; we don’t blind our- selves by assuming our own ideas are best. Once we seek to fully understand others’ ideas and allow them to challenge our own, we can do all sorts of good things: understand an idea more clearly, understand someone better, and understand ourselves (our values, what we find important, and so on) better as well.
Given that our human social world is characterized by diversity of ideas, modesty also marks the path of cooperation, harmony, and respect among human beings. This is one of the many small ways in which the application of logical reasoning can help us all have better lives and better relations with other people. If we could all use logical reasoning on a regular basis, per- haps we would not have as many wars and atrocities as we have today.
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Section 2.2 Putting Arguments in the Standard Form
In West Africa, Ebola is now an epidemic of the likes that we have not seen before. It’s spiraling out of control. It is getting worse. It’s spreading faster and exponentially. Today, thousands of people in West Africa are infected. That num- ber could rapidly grow to tens of thousands. And if the outbreak is not stopped now, we could be looking at hundreds of thousands of people infected, with pro- found political and economic and security implications for all of us. So this is an epidemic that is not just a threat to regional security—it’s a potential threat to global security if these countries break down, if their economies break down, if people panic. That has profound effects on all of us, even if we are not directly contracting the disease. (para. 8)
We have identified “The West African Ebola epidemic is a potential threat to global security” as the conclusion. What are the premises? Read the passage a few times while asking yourself, “Why should I think the epidemic is a global threat?” Obama says that the epidemic is not like others, that it is growing faster and exponentially. He moves from there being thousands of people infected, to tens of thousands, to the possibility of hundreds of thousands. So far, everything is about how fast the epidemic is growing.
In the middle of the seventh sentence, the president switches from talking about the growth of the epidemic to claiming that it has profound economic and security implications. What is the basis for the claim that the growth will have these effects? Notice that it is not in the seventh sentence, at least not explicitly. However, the last part of the eighth sentence does address this. In that sentence, Obama suggests three conditions that might lead to a global security threat: “if these countries break down, if their economies break down, if people panic.” So the extreme growth of the epidemic may lead to the breakdown of economies or countries, or it may lead to widespread panic. If any of these things happen, there are “profound effects on all of us.” Therefore, the epidemic is a potential threat to global security. We can now list the premises, and indeed the entire argument, in standard form as follows:
The West African Ebola epidemic is growing extremely fast. If the growth isn’t stopped, the countries may break down. If the growth isn’t stopped, the economies may break down. If the growth isn’t stopped, people may panic. Any of these things would have profound effects on people outside of the region. Therefore, the West African Ebola epidemic is a potential threat to global security.
Notice that putting the argument in standard form may lose some of the fluidity of the origi- nal, but it more than makes up for it in increased clarity.
Practice Problems 2.2
Identify the premises and conclusions in the following arguments.
1. Every time I turn on the radio, all I hear is vulgar language about sex, violence, and drugs. Whether it’s rock and roll or rap, it’s all the same. The trend toward vulgarity has to change. If it doesn’t, younger children will begin speaking in these ways, and this will spoil their innocence.
2. Letting your kids play around on the Internet all day is like dropping them off in downtown Chicago to spend the day by themselves. They will find something that gets them into trouble.
(continued)
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Section 2.2 Putting Arguments in the Standard Form
3. Too many intravenous drug users continue to risk their lives by sharing dirty nee- dles. This situation could be changed if we were to supply drug addicts with a way to get clean needles. This would lower the rate of AIDS in this high-risk population as well as allow for the opportunity to educate and attempt to aid those who are addicted to heroin and other intravenous drugs.
4. I know that Stephen has a lot of money. His parents drive a Mercedes. His dogs wear cashmere sweaters, and he paid cash for his Hummer.
5. Dogs are better than cats, since they always listen to what their masters say. They also are more fun and energetic.
6. All dogs are warm-blooded. All warm-blooded creatures are mammals. Hence, all dogs are mammals.
7. Chances are that I will not be able to get in to see Slipknot since it is an over-21 show, and Jeffrey, James, and Sloan were all carded when they tried to get in to the club.
8. This is not the best of all possible worlds, because the best of all possible worlds would not contain suffering, and this world contains much suffering.
9. Some apples are not bananas. Some bananas are things that are yellow. Therefore, some things that are yellow are not apples.
10. Since all philosophers are seekers of truth, it follows that no evil human is a seeker after truth, since no philosophers are evil humans.
11. All squares are triangles, and all triangles are rectangles. So all squares are rectangles.
12. Deciduous trees are trees that shed their leaves. Maple trees are deciduous trees. Thus, maple trees will shed their leaves at some point during the growing season.
13. Joe must make a lot of money teaching philosophy, since most philosophy professors are rich.
14. Since all mammals are cold-blooded, and all cold-blooded creatures are aquatic, all mammals must be aquatic.
15. If you drive too fast, you will get into an accident. If you get into an accident, your insurance premiums will increase. Therefore, if you drive too fast, your insurance premiums will increase.
16. The economy continues to descend into chaos. The stock market still moves down after it makes progress forward, and unemployment still hovers around 10%. It is going to be a while before things get better in the United States.
17. Football is the best sport. The athletes are amazing, and it is extremely complex.
18. We should go to see Avatar tonight. I hear that it has amazing special effects.
19. All doctors are people who are committed to enhancing the health of their patients. No people who purposely harm others can consider themselves to be doctors. It fol- lows that some people who harm others do not enhance the health of their patients.
20. Guns are necessary. Guns protect people. They give people confidence that they can defend themselves. Guns also ensure that the government will not be able to take over its citizenry.
Practice Problems 2.2 (continued)
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There’s snow on the ground.
It’s cold outside.
Section 2.3 Representing Arguments Graphically
2.3 Representing Arguments Graphically In the preceding section, we discussed the component parts of an argument and how we can identify each when we encounter them in writing. Although the standard argument form is useful and will be used throughout this book, you may find it easier to display the structure of an argument by drawing the connections between the parts of an argument. We will start by learning some simple techniques for diagramming arguments. An argument diagram (also called an argument map) is just a drawing that shows how the various pieces of an argument are related to each other.
Representing Reasons That Support a Conclusion The simplest argument consists of two claims, one of which supports the other—which means that one is the premise and the other is the conclusion. For example:
There is snow on the ground, so it must be cold outside.
To represent this argument, we put each claim in a box and draw an arrow to show which one supports the other. We can diagram this argument in the following way:
Notice that the claims are represented by simple, complete sentences. The premise is at the start of the arrow, and the conclusion is at the end. The arrow represents the process of inferring the conclusion from the premise. Seeing snow on the ground is indeed a reason for believing that it is cold.
But arguments can be more complex. First, consider that an argument may have more than one line of support. For example:
There’s snow on the ground.
It’s cold outside.
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There’s snow on the ground.
It’s February in Idaho.
It’s cold outside.
It’s February in Idaho.
It’s a very cold year.
There’s snow on the ground.
It’s cold outside.
Section 2.3 Representing Arguments Graphically
The important thing here is that the two lines of support are independent of each other. Knowing that it is February in Idaho is a reason for thinking that it is cold outside, even if you do not see snow. Similarly, seeing snow outside is a reason for thinking it is cold regardless of when or where you see it.
Second, it can also be the case that a single line of support contains multiple premises that work together. For example, although February in Idaho offers good grounds for thinking it is cold outside, this reason is strengthened if it also happens to be a particularly cold year. A year being particularly cold is not by itself much of a reason to think it is cold outside. Even a cold year will be warm in the summer. But a February day in a cold year is even more likely to be cold than a February day in a warm one. We represent this by starting the arrow at a group of premises (bottom):
It’s February in Idaho.
It’s a very cold year.
There’s snow on the ground.
It’s cold outside.
There’s snow on the ground.
It’s February in Idaho.
It’s cold outside.
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It’s February in Idaho.
It’s a very cold year.
There’s snow on the ground.
John came in with snow on his boots.
It’s cold outside.
Section 2.3 Representing Arguments Graphically
Although arrows can sometimes start at a group of claims, they always end at a single claim. This is because every simple argument or inference has only one conclusion, no matter how many premises it may have.
Finally, arguments can form chains with some claims being used as a conclusion for one infer- ence and a premise for another. For example, if your reason for thinking that there is snow on the ground is that your friend John just came in with snow on his boots, this can be indicated in a diagram as follows:
Notice that the claim “There is snow on the ground” is a conclusion for one inference and a premise for another. From these basic patterns we can build extremely complicated arguments.
It’s February in Idaho.
It’s a very cold year.
There’s snow on the ground.
John came in with snow on his boots.
It’s cold outside.
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It’s February in Idaho.
It’s a very cold year.
There’s snow on the ground.
Most people outside aren’t wearing coats.
John came in with snow on his boots.
It’s cold outside.
Section 2.3 Representing Arguments Graphically
Representing Counterarguments We will discuss one more refinement, and then we will have all of the basic tools we need for constructing argument maps. Sometimes lines of reasoning count against a conclusion rather than support it. If we look out the window and notice that most of the students outside are not wearing coats, that might lead us to believe that it is not very cold even though it is Febru- ary and we see snow. We will represent this sort of contrary argument by using a red arrow with a slash through it:
Just as with supporting lines of reasoning, opposing lines may have multiple premises or chains. From the point of view of logic, these lines of opposing reasoning are not really part of the argument. However, such reasoning is often included when presenting an argument, so it is useful to have a way to represent it. This is especially true when you are trying to under- stand an argument in order to write an essay about it. It is good practice to note what objec- tions an author has already considered so that you do not just repeat them.
It’s February in Idaho.
It’s a very cold year.
There’s snow on the ground.
Most people outside aren’t wearing coats.
John came in with snow on his boots.
It’s cold outside.
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2 1
4
3
Section 2.3 Representing Arguments Graphically
With that, you have all the basic tools you need to create argument diagrams. In principle, arguments of any complexity can be represented with diagrams of this sort. In practice, as arguments get more complex, there are many interpretational choices about how to repre- sent them.
Diagramming Efficiently One issue that arises when creating argument diagrams is that including each premise and conclusion can make diagrams large and cumbersome. A common practice is to number each statement in an argument and make the diagram with circled numbers representing each premise and conclusion. See Figure 2.1 for an illustration of the seat belt example from the previous section.
The seat belt example is not a complex argument, but the diagram in Figure 2.1 is able to show how the hidden assertion in the first question is supported by the second statement and how, together with the third assertion, the conclusion is supported. Sketching diagrams that show the relationship among the premises and their connections to the conclusion is very helpful in understanding complex arguments. Yet you must keep in mind that the diagramming is the second stage of the process, since you will have to first identify the ele- ments of the argument.
Figure 2.1: Diagramming the structure of an argument
This diagram shows the relationship between each of the sentences in the seat belt example. Here are the claims: 1. Don’t you know that driving without a seat belt is dangerous? 2. Statistics show that you are 10 times more likely to be injured in an accident if you are not wearing one. 3. Besides, in our state you can get fined $100 if you are caught not wearing one. 4. You ought to wear one even if you are driving a short distance. Notice how numbering the individual components of each argument and diagramming them will help you see the relationship among the pieces and how the pieces work together to support the conclusion.
2 1
4
3
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Section 2.4 Classifying Arguments
2.4 Classifying Arguments There are many ways of classifying arguments. In logic, the broadest division is between deductive and inductive arguments. Recall that Section 2.1 introduced the notion of inference, the process of drawing a judgment from the reasons offered in the premises. The distinction between deductive and inductive arguments is based on the strength of that inference. A con- clusion can follow from the premises very tightly or very loosely, and there is a wide range in between. For deductive arguments, the expectation is that the conclusion will follow from the premises necessarily. For inductive arguments, the expectation is that the conclusion will follow from the premises probably but not necessarily. We shall explore these two kinds of arguments in greater depth in subsequent chapters. In this section our goal is to achieve a basic grasp of their respective definitions and understand how the two types differ from one another. Finally, we will improve our understanding of the concept of an argument by com- paring arguments to explanations, which are often mistaken for arguments.
Practice Problems 2.3
Draw an argument map of each of the following arguments, using the described method of numbering each statement and making a diagram with circled numbers representing each premise and conclusion.
1. (1) I know that Stephen has a lot of money. (2) His parents drive a Mercedes. (3) His dogs wear cashmere sweaters, and (4) he paid cash for his Hummer.
2. (1) Guns are necessary. (2) Guns protect people, because (3) they give people con- fidence that they can defend themselves. (4) Guns also ensure that the government will not be able to take over its citizenry.
3. (1) If you drive too fast, you will get into an accident. (2) If you get into an accident, your insurance premiums will increase. Therefore, (3) if you drive too fast, your insurance premiums will increase.
4. Since (1) all philosophers are seekers of truth, it follows that (2) no evil human is a seeker after truth, since (3) no philosophers are evil humans.
5. (1) This cat can experience pain. So (2) it has the right to not suffer. (3) Since we shouldn’t cause suffering, (4) we should not harm the cat.
6. (1) If we change the construction of the conveyer belt, then the timing of the line will change. (2) Thus, if the timing of the line doesn’t change, then we didn’t change the construction of the conveyor belt. (3) In fact, the timing of the line hasn’t changed. (4) So that means we didn’t change the conveyer belt.
7. (1) The affordable health care act is becoming less popular. (2) Cultural sentiment is increasingly negative, and (3) the Senate and House are progressively moving toward opposition to it. (4) Just last week five Democratic senators joined their Republican counterparts to attempt to block certain aspects of the act.
8. (1) Everyone should have to study logic. (2) It is becoming more important to be able to adapt to changes and (3) to evaluate information in today’s workplace. (4) Logic enhances these abilities. (5) Plus, logic helps protect us against manipulators who try to pawn off their fallacious arguments as truth.
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Section 2.4 Classifying Arguments
Deductive Arguments In logic the terms deductive and inductive are used in a technical sense that is somewhat different than the way the terms may be used in other contexts. For example, Sherlock Holmes, the protagonist in Sir Arthur Conan Doyle’s detective novels, often referred to his own style of reasoning as deductive. In fact, the popularity of Sherlock Holmes introduced deductive reasoning into ordinary speech and made it a com- monplace term. Unfortunately, deductive reasoning is often misunderstood, and in the case of Sherlock Holmes, his clever style of reasoning is actually more inductive than deductive. For example, in The Adven- ture of the Cardboard Box, he says:
Let me run over the principal steps. We approached the case, you remember, with an absolutely blank mind, which is always an advantage. We had formed no theories. We were simply there to observe and to draw inferences from our observations. (Doyle, 1892/2008, para. 114)
The foregoing does not describe deductive reasoning as it is employed in logic. In fact, Sher- lock Holmes mostly uses inductive rather than deductive reasoning. For now, the simplest way to present deductive arguments is to say that deductive reasoning is the sort of reasoning that we normally encounter in mathematical proofs. In a mathematical proof, as long as you do not make a mistake, you can count on the conclusion being true. If the conclusion is not true, you have either made an error in the proof or assumed something that was false. The same is true of deductive reasoning, because good deductive arguments are characterized by their truth-preserving nature—if the premises are true, then the conclusion is guaranteed to be true also. Consider the following deductive argument:
All married men are husbands. Jacob is a married man. Therefore, Jacob is a husband.
In this example, the conclusion necessarily follows from the given premises. In other words, if it is true that all married men are husbands and, moreover, that Jacob is a married man, then it must be necessarily true that Jacob is a husband.
But suppose that Jacob is a 3-year-old boy, so he is not a married man. Would the argument still be a good deductive argument and, thereby, truth preserving? The answer is yes, because deductive reasoning reflects the relations between premises and the conclusion such that if it were to be the case that the premises were true, then it would be impossible for the conclu- sion to be false. If it so happens that Jacob is a 3-year-old boy, then the second premise would not be true, and thus, the necessity for the conclusion to be true is broken.
Wiley Miller/Cartoonstock
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Section 2.4 Classifying Arguments
However, this does not mean that all we need are true premises and a true conclusion. Good deductive arguments are not free form; rather, they use specific patterns that must be followed strictly in the inferential operation. Although this might sound rigid, the greatest advantage of good deductive arguments is that their precise structure guides us into grasping a truth that we might not otherwise have recognized with the same certainty. The use of deductive reasoning is quite broad—in science, mathematics, and the examination of moral problems, to name a few examples. Subsequent chapters will demonstrate more about the powerful machinery of deductive arguments.
Inductive Arguments In contrast to deductive arguments, good inductive arguments do not need to be truth pre- serving. Even those that have true premises do not guarantee the truth of their conclusion. At best, true premises in inductive arguments make the conclusion highly probable. The prem- ises of good inductive arguments offer good grounds for accepting the conclusion, but they do not guarantee its truth. Consider the following example:
The produce at my corner store is stocked by local farmers every day. They have a bakery, too, and they refill their shelves with fresh-baked bread twice a day. I have been shopping at my corner store continuously for 5 years, and every day is the same. Therefore, my corner store will have fresh produce and baked goods every day of the week.
Let us suppose that all the premises are true. After 5 years of going to the corner store and getting to know its practices and the quality of its daily offerings, the conclusion would seem to be highly probable. But is it necessarily true? At some point the store may change hands, close, or experience something else that interrupts its normal operations. Such cases show that even though the reasoning is good, the conclusion is not guaranteed to be true just because the premises are true.
Another way to think of what is going on here is to address a likely familiar fact of the human condition: Past experience does not guarantee that the future will be the same. Think of that great car you loved that did not require any expensive maintenance—and then suddenly one day it started to break down bit by bit with age. Time changes the performance of things. Or think of the great quality of a clothing brand you counted on year after year that one day was no longer as good. Products also change with time as the leaders of the manufacturing company change or the standards become somewhat relaxed. Things change. Sometimes the changes are for the better, sometimes for the worse. But our observation of how things are now and have been in the past does not guarantee that things will remain the same in the future. Accordingly, even if the conclusion in our corner store example seems sufficiently jus- tified for us to venture saying that it is true, the fact is that at some point it could change. At best, we can say that the premises give us good grounds to assert that it is probably true that the store will have good produce and baked goods this coming week.
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Section 2.4 Classifying Arguments
Despite having a weaker connection between premises and conclusion, inductive arguments are more widely used than deductive arguments. In fact, you have likely been using inductive reasoning your entire life without knowing it. Think about the expectation you have that your car, house, or other object will be in the location you last left it. This expectation is based on good inductive reasoning. You have good reasons for expecting your car to be sitting in the parking space where you left it. We can represent your reasoning as follows:
I left my car in that spot. I have always found my car in the same parking spot I left it in. Therefore, my car will be in that spot when I return.
Of course, having good reason is not the same as having a guarantee, as anyone who has expe- rienced having their vehicle stolen can attest. This is the difference between deductive and inductive arguments. Because inductive arguments only establish that their conclusions are probable, the conclusions can turn out to be false even when the premises are all true. The chance may be small, but there is always a chance. By contrast, a good deductive argument is airtight; it is absolutely impossible for the conclusion to be false when the premises are true. Of course, if one of the premises is false, then neither kind of argument can establish its con- clusion. If you misremember which spot you parked in, then you are not likely to find your car immediately, even if it is right where you left it.
Arguments Versus Explanations Mastering logical reasoning requires not only understanding what arguments are, but also being able to distinguish arguments from their closest conceptual neighbors. Although it might be clear by now why news articles, debates, and commands are not considered argu- ments, we should take a closer look at explanations, because they are commonly mistaken for arguments and present a similar framework. Arguments provide a methodical defense of a position, presenting evidence by means of premises in support of a conclusion that is dis- puted. Explanations, in contrast, tell why or how something is the case.
Suppose that we have the following claim:
We have to travel by train instead of by plane.
If you disagree with this decision, then you might question this claim, thus presenting a request for evidence. Accordingly, an argument would be the appropriate response. We could then have the following:
The total cost for plane tickets is $2,000. The total cost for train tickets is $1,000. We have a budget of $1,200 for this trip. Therefore, we have to travel by train instead of by plane.
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Section 2.4 Classifying Arguments
Now, suppose that you do not question the claim, but you still want to know why we have to travel by train. This is not a request for evidence for the conclusion. Rather, this is a request for the cause that leads to the conclusion. This is thus a request for an explanation, which may be as simple as this:
Because we do not have enough money for plane tickets.
The point of an argument is to establish its main claim as true. The point of an explanation is to say how or why its main claim is true. In arguments, the premises will likely be less controver- sial than the conclusion. It is difficult to convince someone that your conclusion is true if they are even less likely to agree with your premises. In explanations, the thing being explained is likely to be less controversial than the explanation given. There is little reason to explain why or how something is true if the listener does not already accept that it is true. Unlike argu- ments, then, explanations do not involve contested conclusions but, instead, accepted ones. Their point is to say why or how the primary claim is true, not to provide reasons for believing that it is true. This explanation might be fairly straightforward, but distinguishing between arguments and explanations in real life may seem a bit more blurry.
As an example, suppose you try to start your car one morning and it will not start. You recall that your son drove the car last night and know that he has a bad habit of leaving the lights on. You see the light switch is on. You now understand why the car will not start. In our scenario, you found out your car would not start and then looked around for the reason. After noticing that the light switch was on, you came up with the following explanation:
Your son left the lights on. Leaving the lights on will drain the battery. A drained battery will prevent the car from starting. That’s why your car won’t start.
It is an explanation because you already know that your car will not start; you just want to know why.
On the other hand, suppose that after your son got home last night, you noticed that he left the lights on. Rather than turn them off or tell him to do it, you decide to teach him a lesson by let- ting the battery go dead. In the morning you have the following conversation with your son:
You: I hope you don’t need to go anywhere with the car this morning.
Son: Why?
You: You left the car’s lights on last night.
Son: So?
You: The lights will have completely drained the battery. The car won’t start with a dead battery, so it’s not going to start this morning.
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Section 2.4 Classifying Arguments
In this case the thing you are most sure of is that your son left the lights on. You reason from that to the conclusion that the car will not start. In this scenario, knowing that the lights were left on is a reason for believing that the car will not start. You are trying to convince your son that the car will not start, and the fact that he left the lights on last night is the starting point for doing so. We can show the structure of your argument as follows:
Your son left the lights on. Leaving the lights one will drain the battery. A drained battery will prevent the car from starting. Therefore, your car won’t start.
Notice that the structure of this argument is the same as the structure of the explanation example. The only difference is whether you are trying to show that the car will not start or to understand why it will not start after already realizing that it will not. Finding the structure will help you understand the details of the argument or explanation, but it will not, by itself, help you determine which one you are dealing with. For that, you have to determine what the author is trying to accomplish and what the author sees as common ground with the reader. Understanding the structure of what is said can help you become clearer about what the author is doing, so it is a good thing to look for, but understanding the structure is not enough.
Determining whether a passage is an argument or an explanation is thus often a matter of inter- preting the intention of the speaker or writer of the claim. A good first step is to identify the main point or central focus of the passage. What you are looking for is the sentence that will be either the conclusion to the argument or the claim being explained. If the author has not done so, paraphrase the main claim as a single, simple sentence. Try to avoid including words like because or therefore in your paraphrase. Ask yourself, if this is an argument, what is its conclu- sion? Once you have identified the potential conclusion, try to determine whether the author is attempting to convince you that that sentence is true, or whether the author assumes you agree with the sentence and is trying to help you understand why or how the sentence is true. If the author is trying to convince you, then the author is advancing an argument. If the author is try- ing to help you get a deeper understanding, the author is providing an explanation.
It is important to be able to tell the difference between arguments and explanations both when listening to others and when crafting our own arguments and explanations. This is because arguments and explanations are trying to accomplish different goals; what makes an effective argument may not make an effective explanation.
Moral of the Story: Arguments Versus Explanations If the main claim is accepted as true from the beginning, then the speaker or writer may be advancing an explanation, not an argument. If the point of a passage is to convince the reader that the main claim is true, then it is most likely an argument. Of course, you may question an explanation, thus requesting an argument that the explanation is correct.
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Summary and Resources
Summary and Resources
Chapter Summary This chapter introduced the standard argument form, which is the principal tool that we will employ in the ensuing chapters. We examined the elements of an argument in standard form, starting from the fundamental notion of claim to an argument’s proper parts—premises and conclusion—and the relationship between these, or what we call inference. Although the standard argument form is simple, the relationship between those claims we call premises and those we call conclusions is crucial to distinguishing between different kinds of arguments. Diagramming these relationships is merely one way we can analyze arguments more fully.
In this chapter we also briefly discussed two kinds of arguments—deductive and induc- tive. However, each one of these will be addressed individually in subsequent chapters as we employ them in more sophisticated applications. Additionally, we explored how to identify arguments in the sources we encounter, as well as how to extract what we find and paraphrase it so that it can be presented in the standard form. Finally, we discussed how to distinguish arguments from explanations and presented a simple method for making such a distinction.
As you continue to read this book, remember that logic is not learned by reading alone. Learning logic demands taking notes of structures and terminology, and it requires practice. Accordingly, practice the exercises provided in each chapter. Once you gain mastery of the standard argument form, you will be able to recognize good arguments from bad arguments, and you will be able to present good arguments in defense of your views. This is a powerful skill to have, and it is now in your hands.
Critical Thinking Questions
1. Try to find a political commercial, and outline the argument that is presented in the commercial. Is it easy or difficult to find premises and conclusions in the content of the commercial? Does the argument relate to politics or to something outside of politics? Are there components of the ad that you think attempt to manipulate the viewer? Why or why not?
2. How can you utilize what you have learned in this chapter about arguments in your own life? At work? At home? How does an understanding of being able to outline and structure arguments translate into your everyday activities?
3. Now that you understand the components of an argument, think back to a time that someone you know attempted to provide an argument but failed to do so in a con- vincing fashion. What were the mistakes that this person made in his or her reason- ing? What were the structural or content errors that weakened the argument?
4. Suppose that your child refuses to go to bed. You want to convince your child that he or she needs to get to sleep. You feel the urge to say, “You have to go to bed because I said so.” However, you are now trying to use what you are learning in this course. What argument would you present to your child to try to convince him or her to go to sleep? Do you think that a strong argument would be effective in convincing your child? Why or why not?
5. Suppose you have a coworker who refuses to help you with a mandatory project. You want to convince him that he needs to help you. What premises would you use to support the conclusion that he ought to help you with the project? Assuming that he fails to find your argument convincing, what would you do next? Why?
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Summary and Resources
Web Resources http://austhink.com/critical/pages/argument_mapping.html The group Austhink provides a number of resources on argument mapping, including tutori- als on how to diagram arguments.
http://www.manyworldsof logic.com/index.html The Many Worlds of Logic website discusses many of the topics that will be covered in this book.
Key Terms
argument The methodical defense of a position advanced in relation to a disputed issue; a set of claims in which some, called premises, serve as support for another claim, called the conclusion.
claim A sentence that presents an assertion that something is the case. In logic, claims are often referred to as propositions in order to recognize that these may be true or false.
conclusion The main claim of an argument; the claim that is supported by the premises but does not itself support any other claims in the argument.
conclusion indicators The words that signal the appearance of a conclusion in an argument.
explanations Statements that tell why or how something is the case. Unlike argu- ments, explanations do not involve contested conclusions but, instead, accepted ones.
inference The process of drawing the nec- essary judgment or, at least, the judgment that would follow from the reasons offered in the premises.
premise indicators The words that signal the appearance of a premise in an argument.
premises Claims in an argument that serve as support for the conclusion.
standard argument form The structure of an argument that consists of premises and a conclusion. This structure displays each premise of an argument on a separate line, with the conclusion on a line following all the premises.
Answers to Practice Problems Practice Problems 2.1
1. claim 2. claim 3. claim 4. nonclaim 5. nonclaim 6. claim 7. claim 8. nonclaim 9. claim
10. claim 11. nonclaim 12. claim 13. nonclaim 14. claim 15. nonclaim 16. claim 17. claim 18. nonclaim
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Summary and Resources
19. claim 20. nonclaim 21. nonclaim 22. claim 23. claim 24. nonclaim 25. claim 26. nonclaim 27. claim 28. claim 29. nonclaim
30. claim 31. nonclaim 32. claim 33. claim 34. nonclaim 35. nonclaim 36. claim 37. claim 38. nonclaim 39. claim 40. nonclaim
Practice Problems 2.2
1. There are three premises: (1) “Every time I turn on the radio, all I hear is vulgar lan- guage about sex, violence, and drugs,” (2) “Whether it’s rock and roll or rap, it’s all the same,” and (3) “The trend toward vulgarity has to change.” The conclusion is “If it doesn’t, younger children will begin speaking in these ways, and this will spoil their innocence.” Notice that the final sentence is an “If . . . , then . . .” statement. Remem- ber that these forms of sentences are single statements. The entire final sentence is the conclusion of this argument.
2. The premise is “Letting your kids play around on the Internet all day is like dropping them off in downtown Chicago to spend the day by themselves.” The conclusion is “They will find something that gets them into trouble.”
3. There are three premises: (1) “Too many intravenous drug users continue to risk their lives by sharing dirty needles,” (2) “This would lower the rate of AIDS in this high-risk population,” and (3) “allow for the opportunity to educate and attempt to aid those who are addicted to heroin and other intravenous drugs.” The conclusion is “This situation could be changed if we were to supply drug addicts with a way to get clean needles.”
4. There are three premises: (1) “His parents drive a Mercedes,” (2) “His dogs wear cashmere sweaters,” and (3) “he paid cash for his Hummer.” The conclusion is “I know that Stephen has a lot of money.” Notice that there are two premises in the final sentence. Remember that words like and and as well as usually indicate that there are multiple statements being made in a single sentence.
5. There are two premises: (1) “they always listen to what their masters say” and (2) “They also are more fun and energetic.” The conclusion is “Dogs are better than cats.” Remember that the word since is often a premise indicator. That means that the state- ment that follows the word since is often a premise.
6. There are two premises: (1) “All dogs are warm-blooded” and (2) “All warm-blooded creatures are mammals.” The conclusion is “all dogs are mammals.” Remember that the word hence is a conclusion indicator. It often comes before the conclusion of an argument.
7. There are two premises: (1) “it is an over-21 show” and (2) “Jeffrey, James, and Sloan were all carded when they tried to get in to the club.” The conclusion is “Chances are that I will not be able to get in to see Slipknot.” Remember that the word since is
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Summary and Resources
often a premise indicator. That means that the statement that follows the word since is often a premise.
8. There are two premises: (1) “the best of all possible worlds would not contain suf- fering” and (2) “this world contains much suffering.” The conclusion is “This is not the best of all possible worlds.” Remember that the word because is often a premise indicator. That means that the statement that follows the word because is often a premise.
9. There are two premises: (1) “Some apples are not bananas” and (2) “Some bananas are things that are yellow.” The conclusion is “some things that are yellow are not apples.” Remember that the word therefore is a conclusion indicator. It often comes before the conclusion of an argument.
10. There are two premises: (1) “all philosophers are seekers of truth” and (2) “no philosophers are evil humans.” The conclusion is “no evil human is a seeker after truth.” Remember that the words it follows that are conclusion indicators. They often come before the conclusion of an argument. Also, remember that the word since is often a premise indicator.
11. The premise is “All squares are triangles and all triangles are rectangles.” The con- clusion is “all squares are rectangles.” Remember that the word so is a conclusion indicator. It often comes before the conclusion of an argument.
12. There are two premises: (1) “Deciduous trees are trees that shed their leaves” and (2) “Maple trees are deciduous trees.” The conclusion is “maple trees will shed their leaves at some point during the growing season.” Remember that the word thus is a conclusion indicator. It often comes before the conclusion of an argument.
13. The premise is “most philosophy professors are rich.” The conclusion is “Joe must make a lot of money teaching philosophy.” Remember that the word since is often a premise indicator. That means that the statement that follows the word since is often a premise.
14. There are two premises: (1) “all mammals are cold-blooded” and (2) “all cold-blooded creatures are aquatic.” The conclusion is “all mammals must be aquatic.” Notice that there are two premises in the first sentence. Remember that words like and and as well as usually indicate that there are multiple statements being made in a single sentence.
15. There are two premises: (1) “If you drive too fast, you will get into an accident” and (2) “If you get into an accident your insurance premiums will increase.” The conclu- sion is “if you drive too fast, your insurance premiums will increase.” Remember that the word therefore is a conclusion indicator. It often comes before the conclusion of an argument.
16. There are three premises: (1) “The economy continues to descend into chaos,” (2) “The stock market still moves down after it makes progress forward,” and (3) “unemployment still hovers around 10%.” The conclusion is “It is going to be a while before things get better in the United States.” Notice that there are two premises in the second sentence. Remember that words like and and as well as usually indicate that there are multiple statements being made in a single sentence.
17. There are two premises: (1) “The athletes are amazing” and (2) “it is extremely com- plex.” The conclusion is “Football is the best sport.” Notice that there are two premises in the second sentence. Remember that words like and and as well as usually indicate that there are multiple statements being made in a single sentence.
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2
1
43
2
1
4
3
1
3
2+
1
2
3+
Summary and Resources
18. The premise is “I hear that it has amazing special effects.” The conclusion is “We should go to see Avatar tonight.”
19. There are two premises: (1) “All doctors are people who are committed to enhancing the health of their patients” and (2) “No people who purposely harm others can con- sider themselves to be doctors.” The conclusion is “some people who harm others do not enhance the health of their patients.” Remember that the words it follows that are a conclusion indicator. When you see these words, think “a conclusion is coming.”
20. There are three premises: (1) “Guns protect people,” (2) “They give people confi- dence that they can defend themselves,” and (3) “Guns also ensure that the gov- ernment will not be able to take over its citizenry.” The conclusion is “Guns are necessary.”
Practice Problems 2.3
1. 2
1
43
2.
2
1
4
3
3. 1
3
2+
4. 1
2
3+
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2
4
3+
1
2
4
3+
1
2
1
3
4
1
2 3+
5
4+
Summary and Resources
5.
2
4
3+
1
6.
2
4
3+
1
7.
2
1
3
4
8.
1
2 3+
5
4+
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1
1An Introduction to Critical Thinking and Logic
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Learning Objectives After reading this chapter, you should be able to:
1. Explain the importance of critical thinking and logic.
2. Describe the relationship between critical thinking and logic.
3. Explain why logical reasoning is a natural human attribute that we all have to develop as a skill.
4. Identify logic as a subject matter applicable to many other disciplines and everyday life.
5. Distinguish the various uses of the word argument that do not pertain to logic.
6. Articulate the importance of language in logical reasoning.
7. Describe the connection between logic and philosophy.
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Section 1.1 What Is Critical Thinking?
This book will introduce you to the tools and practices of critical thinking. Since the main tool for critical thinking is logical reasoning, the better part of this book will be devoted to discuss- ing logic and how to use it effectively to become a critical thinker.
We will start by examining the practical importance of critical thinking and the virtues it requires us to nurture. Then we will explore what logic is and how the tools of logic can help us lead easier and happier lives. We will also briefly review a critical concept in logic—the argument—and discuss the importance of language in making good judgments. We will con- clude with a snapshot of the historical roots of logic in philosophy.
1.1 What Is Critical Thinking? What is critical thinking? What is a critical thinker? Why do you need a guide to think criti- cally? These are good questions, but ones that are seldom asked. Sometimes people are afraid to ask questions because they think that doing so will make them seem ignorant to others. But admitting you do not know something is actually the only way to learn new things and better understand what others are trying to tell you.
There are differing views about what critical thinking is. For the most part, people take bits and pieces of these views and carry on with their often imprecise—and sometimes conflicting— assumptions of what critical thinking may be. However, one of the ideas we will discuss in this book is the fundamental importance of seeking truth. To this end, let us unpack the term critical thinking to better understand its meaning.
First, the word thinking can describe any number of cognitive activities, and there is certainly more than one way to think. We can think analytically, creatively, strategically, and so on (Sousa, 2011). When we think analytically, we take the whole that we are examining—this could be a term, a situation, a scientific phenomenon—and attempt to identify its components. The next step is to examine each component individually and understand how it fits with the other com- ponents. For example, we are currently examining the meaning of each of the words in the term critical thinking so we can have a better understanding of what they mean together as a whole.
Analytical thinking is the kind of thinking mostly used in academia, science, and law (includ- ing crime scene investigation). In ordinary life, however, you engage in analytical thinking more often than you imagine. For example, think of a time when you felt puzzled by some- one else’s comment. You might have tried to recall the original situation and then parsed out the language employed, the context, the mood of the speaker, and the subject of the com- ment. Identifying the different parts and looking at how each is related to the other, and how together they contribute to the whole, is an act of analytical thinking.
When we think creatively, we are not focused on relationships between parts and their wholes, as we are when we think analytically. Rather, we try to free our minds from any boundaries such as rules or conventions. Instead, our tools are imagination and innovation. Suppose you are cooking, and you do not have all the ingredients called for in your recipe. If you start thinking creatively, you will begin to look for things in your refrigerator and pantry that can substitute for the missing ingredients. But in order to do this, you must let go of the recipe’s expected outcome and conceive of a new direction.
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Section 1.1 What Is Critical Thinking?
When we think strategically, our focus is to first lay out a master plan of action and then break it down into smaller goals that are organized in such a way as to support our outcomes. For exam- ple, undertaking a job search involves strategic planning. You must identify due dates for applications, request let- ters of recommendations, prepare your résumé and cover letters, and so on. Thinking strategically likely extends to many activities in your life, whether you are going grocery shopping or planning a wedding.
What, then, does it mean to think criti- cally? In this case the word critical has nothing to do with criticizing others in a negative way or being surly or cynical.
Rather, it refers to the habit of carefully evaluating ideas and beliefs, both those we hear from others and those we formulate on our own, and only accepting those that meet certain stan- dards. While critical thinking can be viewed from a number of different perspectives, we will define critical thinking as the activity of careful assessment and self-assessment in the process of forming judgments. This means that when we think critically, we become the vigilant guard- ians of the quality of our thinking.
Simply put, the “critical” in critical thinking refers to a healthy dose of suspicion. This means that critical thinkers do not simply accept what they read or hear from others—even if the information comes from loved ones or is accompanied by plausible-sounding statistics. Instead, critical thinkers check the sources of information. If none are given or the sources are weak or unreliable, they research the information for themselves. Perhaps most importantly, critical thinkers are guided by logical reasoning.
As a critical thinker, always ask yourself what is unclear, not understood, or unknown. This is the first step in critical thinking because you cannot make good judgments about things that you do not understand or know.
The Importance of Critical Thinking Why should you care about critical thinking? What can it offer you? Suppose you must make an important decision—about your future career, the person with whom you might want to spend the rest of your life, your financial investments, or some other critical matter. What considerations might come to mind? Perhaps you would wonder whether you need to think about it at all or whether you should just, as the old saying goes, “follow your heart.” In doing so, you are already clarifying the nature of your decision: purely rational, purely emotional, or a combination of both.
Ferlistockphoto/iStock/Thinkstock
Critical thinking involves carefully assessing information and its sources.
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Section 1.1 What Is Critical Thinking?
In following this process you are already starting to think critically. First you started by asking questions. Once you examine the answers, you would then assess whether this information is sufficient, and perhaps proceed to research further information from reli- able sources. Note that in all of these steps, you are making distinctions: You would distinguish between relevant and irrelevant questions, and from the relevant questions, you would distin- guish the clear and precise ones from the others. You also would distinguish the answers that are helpful from those that are not. And finally, you would separate out the good sources for your research, leaving aside the weak and biased ones.
Making distinctions also determines the path that your examination will follow, and herein lies the connection between critical thinking and logic. If you decide you should examine the best reasons that support each of the possible options available, then this choice takes you in the direction of logic. One part of logical reasoning is the weighing of evidence. When making an important decision, you will need to identify which factors you consider favorable and which you consider unfavorable. You can then see which option has the strongest evidence in its favor (see Everyday Logic: Evidence, Beliefs, and Good Thinking for a discussion of the importance of evidence).
Consider the following scenario. You are 1 year away from graduating with a degree in busi- ness. However, you have a nagging feeling that you are not cut out for business. Based on your research, a business major is practical and can lead to many possibilities for well-paid employment. But you have discovered that you do not enjoy the application or the analysis of quantitative methods—something that seems to be central to most jobs in business. What should you do?
Many would seek advice from trusted people in their lives—people who know them well and thus theoretically might suggest the best option for them. But even those closest to us can offer conflicting advice. A practical parent may point out that it would be wasteful and possibly risky to switch to another major with only 1 more year to go. A reflective friend may point out that the years spent studying business could be considered simply part of a journey of self-discovery, an investment of time that warded off years of unhappiness after gradua- tion. In these types of situations, critical thinking and logical reasoning can help you sort out competing considerations and avoid making a haphazard decision.
We all find ourselves at a crossroads at various times in our lives, and whatever path we choose will determine the direction our lives will take. Some rely on their emotions to help them make their decisions. Granted, it is difficult to deny the power of emotions. We recall more vividly those moments or things in our lives that have had the strongest emotional
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Can you recall a time when you acted or made a decision while you were experiencing strong emotions? Relying on our emotions to make decisions undermines our ability to develop confidence in our rational judgments. Moreover, emotional decisions cannot typically be justified and often lead to regret.
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Section 1.1 What Is Critical Thinking?
impact: a favorite toy, a first love, a painful loss. Many interpret gut feelings as revelations of what they need to do. It is thus easy to assume that emotions can lead us to truth. Indeed, emotions can reveal phenomena that may be otherwise inaccessible. Empathy, for example, permits us to share or recognize the emotions that others are experiencing (Stein, 1989).
The problem is that, on their own, emotions are not reliable sources of information. Emotions can lead you only toward what feels right or what feels wrong—but cannot guarantee that what feels right or wrong is indeed the right or wrong thing to do. For example, acting self- ishly, stealing, and lying are all actions that can bring about good feelings because they satisfy our self-serving interests. By contrast, asking for forgiveness or forgiving someone can feel wrong because these actions can unleash feelings of embarrassment, humiliation, and vulner- ability. Sometimes emotions can work against our best interests. For example, we are often fooled by false displays of goodwill and even affection, and we often fall for the emotional appeal of a politician’s rhetoric.
The best alternative is the route marked by logical reasoning, the principal tool for developing critical thinking. The purpose of this book is to help you learn this valuable tool. You may be wondering, “What’s in it for me?” For starters, you are bound to gain the peace of mind that comes from knowing that your decisions are not based solely on a whim or a feeling but have the support of the firmer ground of reason. Despite the compelling nature of your own emo- tional barometer, you may always wonder whether you made the right choice, and you may not find out until it is too late. Moreover, the emotional route for decision making will not help you develop confidence in your own judgments in the face of uncertainty.
In contrast, armed with the skill of logical reasoning, you can lead a life that you choose and not a life that just happens to you. This power alone can make the difference between a happy and an unhappy life. Mastering critical thinking results in practical gains—such as the ability to defend your views without feeling intimidated or inadequate and to protect yourself from manipulation or deception. This is what’s in it for you, and this is only the beginning.
Everyday Logic: Evidence, Beliefs, and Good Thinking
It is wrong always, everywhere, and for anyone to believe anything on insufficient evidence. —W. K. Clifford (1879, p. 186)
British philosopher and mathematician W. K. Clifford’s claim—that it is unethical to believe anything if you do not have sufficient evidence for it—elicited a pronounced response from the philosophical community. Many argued that Clifford’s claim was too strong and that it is acceptable to believe things for which we lack the requisite evidence. Whether or not one absolutely agrees with Clifford, he raises a good point. Every day, millions of people make deci- sions based on insufficient evidence. They claim that things are true or false without putting in the time, effort, and research necessary to make those claims with justification.
You have probably witnessed an argument in which people continue to make the same claims until they either begin to become upset or merely continue to restate their positions without adding anything new to the discussion. These situations often devolve and end with state- ments such as, “Well, I guess we will just agree to disagree” or “You are entitled to your opin- ion, and I am entitled to mine, and we will just have to leave it at that.” However, upon further reflection we have to ask ourselves, “Are people really entitled to have any opinion they want?”
(continued)
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Section 1.1 What Is Critical Thinking?
From the perspective of critical thinking, the answer is no. Although people are legally entitled to their beliefs and opinions, it would be intellectually irresponsible of them to feel entitled to an opinion that is unsupported by logical reasoning and evidence; people making this claim are conflating freedom of speech with freedom of opinion. A simple example will illustrate this point. Suppose someone believes that the moon is composed of green cheese. Although he is legally entitled to his belief that the moon is made of green cheese, he is not rationally entitled to that belief, since there are many reasons to believe and much evidence to show that the moon is not composed of green cheese.
Good thinkers constantly question their beliefs and examine multiple sources of evidence to ensure their beliefs are true. Of course, people often hold beliefs that seem warranted but are later found not to be true, such as that the earth is flat, that it is acceptable to paint baby cribs with lead paint, and so on. However, a good thinker is one who is willing to change his or her views when those views are proved to be false. There are certain criteria that must be met for us to claim that someone is entitled to a specific opinion or position on an issue.
There are other examples where the distinction is not so clear. For instance, some people believe that women should be subservient to men. They hold this belief for many reasons, but the pre- dominant one is because specific religions claim this is the case. Does the fact that a religious text claims that women should serve men provide sufficient evidence for one to believe this claim? Many people believe it does not. However, many who interpret their religious texts in this man- ner would claim that these texts do provide sufficient evidence for such claims.
It is here that we see the danger and difficulty of providing hard-and-fast definitions of what constitutes sufficient evidence. If we believe that written words in books came directly from divine sources, then we would be prone to give those words the highest credibility in terms of the strength of their evidence. However, if we view written words as arguments presented by their authors, then we would analyze the text based on the evidence and reasoning presented. In the latter case we would find that these people are wrong and that they are merely making claims based on their cultural, male-dominated environments.
Of course, all people have the freedom to believe what they want. However, if we think of entitlement as justification, then we cannot say that all people are entitled to their opinions and beliefs. As you read this book, think about what you believe and why. If you do not have reasons or supporting evidence for your beliefs and opinions, you should attempt to find it. Try not to get sucked into arguments without having evidence. Most important, as a good thinker, you should be willing and able to admit the strengths and weaknesses of various posi- tions on issues, especially your own. At the same time, if in your search for evidence you find that the opposing position is the stronger one, you should be willing to change your position. It is also a sign of good thinking to suspend judgment when you suspect that the arguments of others are not supported by evidence or logical reasoning. Suspending judgment can protect you from error and making rash decisions that lead to negative outcomes.
Everyday Logic: Evidence, Beliefs, and Good Thinking (continued)
Becoming a Critical Thinker By now it should be clear that critical thinking is an important life skill, one that will have a decisive impact on our lives. It does not take luck or a genetic disposition to be a critical thinker. Anyone can master critical thinking skills. So how do you become a critical thinker? Earlier in
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Section 1.2 Three Misconceptions About Logic
the chapter, logical reasoning was described as the main tool for critical thinking. Thus, the most fundamental step in becoming a critical thinker is to recognize the importance of reason as the filter for your beliefs and actions. Once you have done this, you will be in the right frame of mind to start learning about logic and identify what tools of logic are at your disposal.
It is also important to note that becoming a critical thinker demands intellectual modesty. We can understand intellectual modesty as the willingness to put our egos in check because we see truth seeking as a far greater and more satisfying good than seeking to be right. Critical thinkers do not care about seeking approval by trying to show that they are right. They do not assume that disagreement reflects a lack of intelligence or insight. Being intellectually modest means recognizing not only that we can make mistakes, but also that we have much to learn. If we are (a) aware that we are bound to make mistakes and that we will benefit when we recognize them; (b) willing to break old habits and embrace change; and, perhaps most importantly, (c) genuinely willing to know what others think, then we can be truly free to experience life as richly and satisfactorily as a human being can.
1.2 Three Misconceptions About Logic If logic is so important to critical thinking, we must of course examine what logic is. This task is not as easy as it sounds, and before we tackle it we must first dismantle some common misconceptions about the subject.
Logic Is for Robots The first misconception is that it is not normal for humans to display a command of logic. (In fact, some suggest that humans created, rather than discovered, these patterns of thought; see A Closer Look: Logic: A Human Invention?) Think of how popular culture and media often depict characters endowed with logical reasoning. In American slang they are the eggheads, the geeks, the nerds, the ones who can use their minds but have trouble relating to other people. Such people often lack compassion or social charisma, or they are emotionally unex- pressive. They are only logical and lack the blend of attributes that people actually have.
Consider the logically endowed characters on the Star Trek series. Vulcans, for example, are beings who suppress all emotions in favor of logic because they believe that emotions are dangerous. What appear to be heartless decisions by the Vulcans no doubt make logic seem quite unsavory to some viewers. The android Data—from The Next Generation series in the Star Trek franchise—is another example. Data’s positronic brain is devoid of any emotional capacity and thus processes all information exclusively by means of a logical calculus. Logic is thus presented as a source of alienation, as Data yearns for the affective depth that his human colleagues experience, such as humor and love.
Such presentations of logic as the polar opposite of emotion are false dichotomies because all human beings are naturally endowed with both logical and emotional faculties—not just one or the other. In other words, we have a broader range of abilities than that for which we give
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Section 1.2 Three Misconceptions About Logic
ourselves credit. So if you think that you are mostly emotional, then you simply have yet to discover your logical side.
Nonetheless, some believe emotions are the fundamental mark of human beings. It is quite likely that emotion has played a significant role in our survival as a species. Neuroscientists, for example, have discovered that our emotions have a faster pathway to the action centers of the brain than the methodical decision-making approach of our logical faculties (LeDoux, 1986, 1992). It pays, for example, to give no thought to running if we fear we are being hunted by a predator.
In most human civilizations today, however, dodging predators is not a main necessity. In fact, methodical reasoning is more advantageous in most of today’s situations. Thinking things through logically assists learning at all levels, produces better results in the job market (in seeking jobs, obtaining promotions, and procuring raises), and helps us make better choices. As noted in the previous section, we are more likely to be satisfied and experience fewer regrets if we reason carefully about our most critical choices in life. Indeed, logical reasoning can prove to be a better strategy for attaining the individual quest for personal fulfillment than any available alternative such as random choice, emotional impulse, waiting and seeing, and so on.
Moral of the Story: Emotions Versus Logic Embracing logical reasoning does not mean disregarding our emotions altogether. Instead, we should recognize that emotions and logic are both essential components of what it is to be human.
A Closer Look: Logic: A Human Invention? One objection to the use of logic—often from what is known as a postmodern perspective—is that logic is a human invention and thus inferior to emotions or intuitions. In other words, what some call the “rules of logic” cannot be seen as univer- sally applicable because logic originated in the Western world; thus, logic is relative and only a matter of perspective.
For example, the invention of chairs seems indispensable to those of us who live where chairs have become part of our cultural background. But those from different cultural back- grounds or those who lived during different time periods may not use chairs at all, or may employ alternative seating devices, such as the traditional Japanese tatami mats. To broadly apply the concept of chair as an appropriate place to sit would be ethnocentric, or applying the standards of one’s own culture to all other cultures.
In response to the foregoing objection, the authors of this text argue that logic is not a human invention, nor a conven- tion that spread in certain parts of the world. Rather, logic was
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Aristotle’s Organon is a compilation of six treatises in which Aristotle formulated principles that laid the foundation for the field of logic.
(continued)
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Section 1.2 Three Misconceptions About Logic
discovered in people’s ordinary encounters with reality, as early as antiquity. Based on avail- able historical records, the first study of the principles at work in good reasoning emerged in ancient Greece. Aristotle was the first to formulate principles of logic, and he did so in six treatises that ancient commentators grouped together under the title Organon, which means “instrument” (reflecting the view that logic is the fundamental instrument for philosophy, which will be discussed later in the chapter).
Importantly, other civilizations have developed logic independently of the Greek tradition. For example, Dignaga was an important thinker in India who lived a few hundred years after Aristotle. Dignaga’s work begins with certain practices of debate within the Nyaya school of Hinduism and transitions to a more formal approach to reasoning. Although the result of Dig- naga’s studies is not identical to Aristotle’s, there is enough similarity to strongly suggest that basic logical principles are not merely cultural artifacts.
In the Middle Ages, Aristotelian logic was brought to the West by Islamic philosophers and thus became part of the scholarship of Christian philosophers until the 14th or 15th century. The emergence of modern logic did not take place until the 19th and 20th centuries, during which new ways of analyzing propositions gave rise to new discoveries concerning the foun- dations of mathematics, as well as a new system of logical notation and a new system of logical principles that replaced the Aristotelian system.
Thus, the examination of good reasoning was fundamental in the development of human civi- lization. Logical reasoning has helped us to identify the laws that guide physical phenomena, which brought us to the state of technological advancement that we experience today. How else could we have erected pyramids and other marvels in the ancient world without having discovered a principle for checking the accuracy of the geometry employed to design them?
Logic Does Not Need to Be Learned A second misconception is that logic does not need to be learned. After all, humankind’s unique distinction among other animals is the faculty of rationality and abstract thought. Although many nonhuman animals have very high levels of intelligence, to the best of our knowledge, abstract thought seems to be the mark of humankind’s particular brand of rationality. Today the applications of logical reasoning are all around us. We are able to experience air travel and marvel at rockets in space. We are also able to enjoy cars, sky- scrapers, computers, cell phones, air-conditioning, home insulation, and even smart homes that allow users to regulate light, temperature, and other functions remotely via smart- phones and other devices. Logical reasoning has afforded us an increasingly better picture of reality, and as a result, our lives have become more comfortable.
However, if logical reasoning is a natural human trait, then why should anyone have to learn it? We certainly experience emotions without any need to be trained, so why would the case be different with our rational capacities? Consider the difference between natural capacities that are nonvoluntary or automatic, on the one hand, and natural capacities that involve our will, on the other. Swallowing, digesting, and breathing are nonvoluntary natural capacities, as are emotions. We usually do not will ourselves to feel happy, angry, or excited. Rather, we usually just find ourselves feeling happy, angry, or excited.
A Closer Look: Logic: A Human Invention? (continued)
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Section 1.2 Three Misconceptions About Logic
Now contrast these with voluntary natural capacities such as walking, running, or sitting. We usually need to will these actions in order for them to take place. We do not just find ourselves running without intending to run, as is the case with swallowing, breathing, or feeling excited or angry. If logic were akin to breathing, the world would likely look like a different place.
Logic is practiced with intention and must be learned, just like we learn to walk, sit, and run. True, almost everyone learns to run to some degree as part of the normal process of growing up. Similarly, almost everyone learns a certain amount of logical reasoning as they move from infant to adult. However, to be a good runner, you need to learn and practice specific skills. Similarly, although everyone has some ability in logic, becoming a good critical thinker requires learning and practicing a range of logical skills.
Logic Is Too Hard The final misconception is that logic is too hard or difficult to learn. If you have survived all these years without studying logic, you might wonder why you should learn it now. It is true that learning logic can be challenging and that it takes time and effort before it feels like second nature. But consider that we face the same challenge whenever we learn anything new, whether it is baking, automotive repair, or astrophysics. These are all areas of human knowledge that have a specific terminology and methodology, and you cannot expect to know how to bake a soufflé, fix a valve cap leak, or explain black holes without any investment in learning the subject matter.
Let us return to our running analogy. Just as we must intend to run in order to do it, we must intend to think methodically in order to do it. When we become adept at running, we do not have to put in as much effort or thought. A fit body can perform physical tasks more easily than an unfit one. The mind is no different. A mind accustomed to logical reasoning will find activities of the intellect easier than an unfit one. The best part is that if you wish to achieve logical fitness, all you need to do is learn and practice the necessary tools for it. The purpose of this book is to guide you toward this goal.
Without a doubt, learning logic will be challenging. But keep in mind that starting a logical fitness program is very much like starting a physical fitness program: There will be a little pain in the beginning. When out-of-shape muscles are exercised, they hurt. You might find that some lessons or concepts might give you a bit of trouble. When this happens, don’t give up! In a physical fitness program, we know that if we keep going, over time the pain goes
Moral of the Story: Logic as a Skill Having a natural capacity for something does not amount to being good at it. Even as emotions seem to come so naturally, some people have to work at being less sensitive or more empa- thetic. The same is true for logical reasoning.
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Section 1.3 What Is Logic?
away, the muscles get in shape, and movement becomes joyful. Likewise, as you keep working diligently on learning and developing your natural logical abilities, you will discover that you understand new things more easily, reading is less of a struggle for you, and logical reasoning is actually fun and rewarding. Eventually, you will begin to recognize logical connections (or the lack thereof ) that you did not previously notice, make decisions that you are less likely to regret, and develop the confidence to defend the positions you hold in a way that is less emotionally taxing.
1.3 What Is Logic? Having dispelled some common misconceptions, we can now occupy ourselves with a funda- mental question for this book: What is logic? A first attempt to define logic might be to say that it is the study of the methods and principles of good reasoning. This definition implies that there are certain principles at work in good reasoning and that certain methods have been developed to encourage it. It is important to clarify that these principles and methods are not a matter of opinion. They apply to someone in your hometown as much as to someone in the smallest village on the other side of the world. Furthermore, they are as suitable today as they were 200 or 2,000 years ago.
This definition is a good place to start, but it leaves open the questions of what we mean by “good reasoning” and what makes some reasoning good relative to others. Although it is admittedly difficult to cram answers to all possible questions into a pithy statement, defini- tions should attempt to be more specific. In this book, we shall employ the following defini- tion: Logic is the study of arguments that serve as tools for arriving at warranted judgments. Notice that this definition states how logic can be of service to you now, in your daily routine, and in whatever occupation you hold. To understand how this is the case, let us unpack this definition a bit.
The Study of Arguments This definition of logic does not explain that there are principles at work in good reasoning or that these princi- ples are not necessarily informed by experience: The meaning of the word argument in logic does the job. Argu- ment has a very technical meaning in logic, and for this reason, Chapter 2 is dedicated entirely to the definition of arguments—what they are, what they are not, what they consist of, and what makes them good. Later in this chap- ter, we will survey other meanings for the word argument outside of logic.
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In logic, an argument is the methodical presentation of one’s position on a topic, not a heated fight with another person.
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Section 1.3 What Is Logic?
For now, let us refer to an argument as a methodical defense of a position. Suppose that Diana is against a proposed increase in the tax rate. She decides to write a letter to the editor to pres- ent her reasons why a tax increase would be detrimental to all. She researches the subject, including what economists have to say about tax increases and the position of the opposition. She then writes an informed defense of her position. By advancing a methodical defense of a position, Diana has prepared an argument.
A Tool for Arriving at Warranted Judgments For our purposes, the word judgment refers simply to an informed evaluation. You examine the evidence with the goal of verifying that if it is not factual, it is at least probable or theo- retically conceivable. When you make a judgment, you are determining whether you think something is true or false, good or bad, right or wrong, beautiful or ugly, real or fake, delicious or disgusting, fun or boring, and so on. It is by means of judgments that we furnish our world of beliefs. The richer our world of beliefs, the clearer we can be about what makes us happy. Judgments are thus very important, so we need to make sure they are sound.
What about the word warrant? Why are warranted judgments preferable to unwarranted ones? What is a warrant? If you are familiar with the criminal justice system or television crime dramas, you may know that a warrant is an authoritative document that permits the search and seizure of potential evidence or the arrest of a person believed to have commit- ted a crime. Without a warrant, such search and seizure, as well as coercing an individual to submit to interrogation or imprisonment, is a violation of the protections and rights that individuals in free societies enjoy. The warrant certifies that the search or arrest of a person is justified—that there is sufficient reason or evidence to show that the search or arrest does not unduly violate the person’s rights. More generally, we say that an action is warranted if it is based on adequate reason or evidence.
Accordingly, our judgments are warranted when there is adequate reason or evidence for making them. In contrast, when we speak of something being unwarranted, we mean that it lacks adequate reason or evidence. For example, unwarranted fears are fears we have without good reason. Children may have unwarranted fears of monsters under their beds. They are afraid of the monsters, but they do not have any real evidence that the monsters are there. Our judgments are unwarranted when, like a child’s belief in lurking monsters under the bed, there is little evidence that they are actually true.
In the criminal justice system, the move from suspicion to arrest must be warranted. Simi- larly, in logic, the move from grounds to judgment must be warranted (see A Closer Look: War- rants for the Belief in God for an example). We want our judgments to be more like a properly executed search warrant than a child’s fear of monsters. If we fail to consider the grounds for our judgments, then we are risking our lives by means of blind decisions; our judgments are no more likely to give us true beliefs than false ones. It is thus essential to master the tools for arriving at warranted judgments.
It is important to recognize the urgency for obtaining such mastery. It is not merely another nice thing to add to the bucket list—something we will get around to doing, right after we trek to the Himalayas. Rather, mastering the argument—the fundamental tool for arriving at
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Section 1.3 What Is Logic?
A Closer Look: Warrants for the Belief in God Striving for warranted judgments might seem difficult when it comes to beliefs that we have accepted on faith. Note that not all that we accept on faith is necessarily related to God or religion. For example, we likely have faith that the sun will rise tomorrow, that our spouses are honest with us, and that the car we parked at the mall will still be there when we return from shopping. Many American children have faith that the tooth fairy will exchange money for baby teeth and that Santa Claus will bring toys come Christmas. Are we reasoning correctly by judging such beliefs as warranted? Whatever your answer in regard to these other issues, questions of religious belief are more likely to be held up as beyond the reach of logic. It is important to recognize this idea is far from being obviously true. Many deeply religious people have nonetheless found it advisable to offer arguments in support of their beliefs.
One such individual was Thomas Aquinas, a 13th-century Roman Catholic Dominican priest and philosopher. In his Summa Theo- logica (Aquinas, 1947), he advanced five logical arguments for God’s existence that do not depend on faith.
The 20th-century Oxford scholar and Christian apologist C. S. Lewis, perhaps best known for the popular children’s series The Chronicles of Narnia, did not embrace his Anglican religion until he was in his thirties. In his books Mere Christianity and Miracles: A Preliminary Study, he employs reason to defend Christian beliefs and the logical possibility of miracles.
There are, of course, many more examples. The important point to draw from this is that all of our judgments of faith—from the faith in the sun rising tomorrow to the faith in the exis- tence of God—should be warranted beliefs and not just beliefs that we readily accept without question. In other words, even faith should make sense in order to be able to communicate such beliefs to those who do not share those beliefs. Note that philosophers who have pre- sented arguments in defense of their religious views have helped transform the nature of reli- gious disagreement to one in which the differences are generally debated in an intellectually enlightening way.
We have not yet reached the point in which differences in religious views are no longer the cause of wars or killing. Nonetheless, the power of argument in the formation of our beliefs is that it supports social harmony despite diversity and disagreement in views, and we all gain from presenting our unique positions in debated issues.
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In his Summa Theologica, Thomas Aquinas advanced the idea that belief in the existence of God can be grounded in logical argument.
warranted judgments—is as essential as learning to read and write. Knowledge of logic is a relatively tiny morsel of information compared to all that you know thus far, but it has the capacity to change your life for the better.
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Section 1.4 Arguments Outside of Logic
Formal Versus Informal Logic Logic is a rich and complex field. Our focus here will be how logic contributes to the develop- ment and honing of critical thinking in everyday life. Primarily, the concepts we will discuss will reflect principles of informal logic. The principal aim in informal logic is to examine the reasoning we employ in the ordinary and everyday claims we make.
In contrast, formal logic is far more abstract, often involving the use of symbols and math- ematics to analyze arguments. Although this text will touch on a few formal concepts of logic in its discussions of deduction (see Chapter 3 and Chapter 4), the purpose in doing so is to develop methodology for good reasoning that is directly applicable to ordinary life.
1.4 Arguments Outside of Logic Although Chapter 2 will explore the term argument in more detail, it is important to clarify that the word is not exclusive to logic. Its meaning varies widely, and you may find that one of the descriptions in this section fits your own understanding of what is an argument. Knowing there is more than one meaning of this word, depending on context or application, will help you correctly understand what is meant in a given situation.
Arguments in Ordinary Language Often, we apply the word argument to an exchange of diverging views, sometimes in a heated, angry, or hostile setting. Suppose you have a friend named Lola, and she tells you, “I had an argument with a colleague at work.” In an ordinary setting you might be correct in under- standing Lola’s meaning of the term argument as equivalent to a verbal dispute. In logic, how- ever, an argument does not refer to a fight or an angry dispute. Moreover, in logic an argument does not involve an exchange between two people, and it does not necessarily have an emo- tional context.
Although in ordinary language an argument requires that at least two or more people be involved in an exchange, this is not the case in logic. A logical argument is typically advanced by only one person, either on his or her behalf or as the representative of a group. No exchange is required. Although an argument may be presented as an objection to another person’s point of view, there need not be an actual exchange of opposing ideas as a result.
Now, if two persons coordinate a presentation of their defenses of what can be identified as opposing points of view, then we have a debate. A debate may contain several arguments but is not itself an argument. Accordingly, only debates are exchanges of diverging views.
Even if a logical argument is both well supported and heartfelt, its emotional context is not its driving force. Rather, any emotion that may be inevitably tied in with the defense of the argu- ment’s principal claim is secondary to the reasons advanced. But let us add a little contextual reference to the matter of debates. If the arguments on each side of the debate are presented
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Section 1.4 Arguments Outside of Logic
well, then the debate may lead to the discovery of perspectives that each party had not pre- viously considered. As such, debates can be quite enlightening because every time our own perspective is broadened with ideas not previously considered and that are well supported and defended, it is very difficult for the experience to be negative. Instead, a good debate is an intellectually exhilarating experience, regardless of how attached one may be to the side one is defending.
Not even debates need to be carried out with an angry or hostile demeanor, or as a means to vent one’s frustration or other emotions toward the opposition. To surrender to one’s emo- tions in the midst of a debate can cause one to lose track of the opposition’s objections and, consequently, be able to muster only weak rebuttals.
Rhetorical Arguments Think about how politicians might try to persuade you to vote for them. They may appeal to your patriotism. They may suggest that if the other candidate wins, things will go badly. They may choose words and examples that help specific audiences feel like the politician empathizes with their situation. All of these techniques can be effective, and all are part of what someone who studies rhetoric—the art of persuasion—might include under the term argument.
Rhetoric is a field that uses the word argument almost as much as logic does. You are likely to encounter this use in English, communication, composition, or argumentation classes. From the point of view of rhetoric, an argument is an attempt to persuade—to change someone’s opinion or behavior. Because the goal of a rhetorical argument is persuasion, good arguments are those that are persuasive. In fact, any time someone attempts to persuade you to do some- thing, they can be seen as advancing an argument in this sense.
Moral of the Story: Defining the Word Argument To avoid conflating the two widely different uses of the word argument (that is, as a dis- pute in ordinary language and as a defense of a point of view in logic) is to use the word only in its classical sense. In its classical meaning, an argument does not refer to a vehicle to express emotions, complaints, insults, or provocations. For these and all other related meanings, there are a wide variety of terms that would do a better job, such as disagree- ment, quarrel, bicker, squabble, fight, brawl, altercation, having words, insult match, word combat, and so on. The more precise we are in our selection of words, the more efficient our communications.
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Section 1.4 Arguments Outside of Logic
Think about how you might have persuaded a sibling to do something for you when you were young. You might have offered money, tried to manipulate with guilt-inducing tactics, appealed to his or her sense of pride or duty, or just attempted to reason with him or her. All of these things can be motivating, and all may be part of a rhetorical approach to argu- ments. However, while getting someone to do something out of greed, guilt, pride, or pity can indeed get you what you want, this does not mean you have succeeded in achieving a justified defense of your position.
Some of the most impressive orators in history—Demosthenes, Cicero, Winston Churchill— were most likely born with a natural talent for rhetoric, yet they groomed their talent by becom- ing well educated and studying the speeches of previous great orators. Rhetoric depends not only on the mastery of a language and broad knowledge, but also on the fine-tuning of the use of phrases, metaphors, pauses, crescendos, humor, and other devices. However, a talent for rheto- ric can be easily employed by unscrupulous people to manipulate others. This characteristic is precisely what distinguishes rhetorical arguments from arguments in logic.
Whereas rhetorical arguments aim to persuade (often with the intent to manipulate), logical arguments aim to demonstrate. The distinction between persuading and demonstrating is crucial. Persuading requires only the appearance of a strong position, perhaps camouflaged by a strong dose of emotional appeal. But demonstrating requires presenting a position in a way that may be conceivable even by opponents of the position. To achieve this, the argument must be well informed, supported by facts, and free from flawed reasoning. Of course, an argument can be persuasive (meaning, emotionally appealing) in addition to being logically strong. The important thing to remember is that the fundamental end of logical arguments is not to persuade but to employ good reasoning in order to demonstrate truths.
Revisiting Arguments in Logic Suppose you and your friend watch a political debate, and she tells you that she thought one of the candidates gave a good argument about taxes. You respond that you thought the can- didate’s argument was not good. Have you disagreed with each other? You might think that you had, but you may just be speaking past each other, using the term argument in different senses. Your friend may mean that she found the argument persuasive, while you mean that the argument did not establish that the candidate’s position was true. It may turn out that you both agree on these points. Perhaps the candidate gave a rousing call to action regarding tax reform but did not spend much time spelling out the details of his position or how it would work to solve any problems. In this sort of case, the candidate may have given a good argu- ment in the rhetorical sense but a bad argument in the logical sense.
Moral of the Story: Persuasion Versus Demonstration Purely persuasive arguments are undoubtedly easier to advance, which makes them the per- fect tool for manipulation and deceit. However, only arguments that demonstrate with logic serve the end of pursuing truth; thus, they are the preferable ones to master.
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Section 1.5 The Importance of Language in Logic
To summarize:
• In contrast to ordinary arguments, logical arguments do not involve an exchange of any kind.
• In contrast to ordinary arguments and rhetorical arguments, logical arguments are not driven by emotions. In logic, only the reasons provided in defense of the conclu- sion make up the force of the argument.
• In contrast to rhetorical arguments, logical arguments are not primarily attempts to persuade, because there is no attempt to appeal to emotions. Rather, logical argu- ments attempt only to demonstrate with reasons. Of course, good logical arguments may indeed be persuasive, but persuasion is not the primary goal.
The goal of an argument in logic is to demonstrate that a position is likely to be true.
So before you go on to have a quarrel with your friend, make sure you are both using the word in the same way. Only then can you examine which sense of argument is the most crucial to the problem raised. Should we vote for a candidate who can get us excited about important issues but does not tell us how he or she proposes to solve them? Or shall we vote for a can- didate who may not get us very excited but who clearly outlines how he or she is planning to solve the nation’s problems?
In the rest of this book, you should read the word argument in the logical sense and no other. If the word is ever used in other ways, the meaning will be clearly indicated. Furthermore, outside of discussions of logic, you must clarify how the word is being used.
1.5 The Importance of Language in Logic The foregoing distinction of the different uses and meanings of the word argument show the importance of employing language precisely. In addition to creating misunderstandings, mis- used words or the lack of knowledge of distinctions in meaning also prevent us from formulat- ing clear positions about matters that pertain to our personal goals and happiness. Language affects how we think, what we experience, how we experience it, and the kind of lives we lead.
Language is our most efficient means of communicating what is in our minds. However, it is not the only means by which humans communicate. We also communicate via facial expres- sions, gestures, and emotions. However, these nonverbal cues often need clarifying words so we can clearly grasp what someone else is expressing or feeling, especially people we don’t know very well. If we see a stranger crying, for example, we might not be able to distinguish at first glance if the tears are from happiness or sadness. If we are visiting a foreign land and hear a man speaking in a loud voice and gesturing wildly, we might not know if he is quarrel- ling or just very enthusiastic unless we understand his language.
This suggests that words matter very much because they are the universal means for making ourselves clear to others. This may seem obvious, since we all use language to communi- cate and, generally speaking, seem to manage satisfactorily. What we do not often recognize,
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Section 1.5 The Importance of Language in Logic
however, is the difference we could experience if we took full advantage of clear and precise language in its optimal form. One result could be that many will no longer ignore what we say. Another could be that as our vocabulary expands, we will no longer be limited to what we can express to others or in what we can grasp from our experiences.
Suppose, for example, that you are invited to a dinner that unbeknownst to you introduces you to a spice you have never tasted before. As you savor the food on your plate, you may taste something unfamiliar, but the new flavor may be too faint for you, amidst the otherwise famil- iar flavors of the dish you are consuming. In fact, you may be cognitively unaware of the char- acter of this new flavor because you are unable to identify it by name and, thus, as a new flavor category in your experience.
According to philosopher David Hume (1757), many of us do not have a sensitive enough palate to actually recognize new or unfamiliar flavors in familiar taste experiences. For those who do, it would seem that the test of a sensitive palate lies not with strong flavors but with faint ones. However, recent neurobiological research suggests that our responses to taste are not entirely dependent on the refinement of our sensory properties but, rather, on higher levels of linguistic processing (Grabenhorst, Rolls, & Bilderbeck, 2008). In other words, if you cannot describe it, it may be quite possible you are unable to taste it; our ability to skillfully use language thus improves our experience.
Logicians and philosophers in general take lan- guage very seriously because it is the best means for expressing our thoughts, to be understood by others, and to clarify ideas that are in need of clarification. Communicating in a language, however, is more com- plex than we recognize. As renowned philosopher John Searle observed, “Speaking a language is engag- ing in a rule-governed form of behavior” (Searle, 1969, p. 22). This means that whenever we talk or write, we are performing according to specific rules. Pauses in speech are represented by punctuation marks such as commas or periods. If we do not pause, the meaning of the same string of words could change its meaning completely. The same prin- ciple applies in writing. But although we are more conscious of making such pauses in speech, sometimes we overlook their importance in writing. A clever saying on a T-shirt illustrates this point, and it reads as follows:
Let’s eat Grandma.
Let’s eat, Grandma.
Commas save lives.
Georgios Kollidas/iStock/Thinkstock
Hume’s essay Of the Standard of Taste stated that taste depends on the refinement of sensory properties, but recent neurobiological research suggests that taste may actually be dependent on language.
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Section 1.6 Logic and Philosophy
Indeed, even what may seem like a meaningless little comma can dramatically change the meaning of a sentence. If we want to make sure others understand our written meaning, we need to be mindful of relevant punctuation, grammatical correctness, and proper spelling. If something is difficult to read because the grammar is faulty, punctuation is missing, or the words are misspelled, these obstacles will betray the writer’s meaning.
1.6 Logic and Philosophy By this point, you may have noticed that logic and philosophy are often mentioned together. There is good reason for this. Logic is not only an area of philosophy but also its bread and butter. It is important to understand the connection between these two fields because understanding the pursuit of philosophy will help clarify in your mind the value of logic in your life.
First, however, let us confront the elephant in the room. Some people have no idea what phi- losophers do. Others think that philosophers simply spend time thinking about things that have little practical use. The stereotypical image of a philosopher, for instance, is a bearded man asking himself: “If a tree falls in the forest and there is no one else to hear it, is there sound?” Your response to this may be: “Why should anyone care?” The fact is that many do, and not only bearded philosophers: Such a question is also critical to those who work at the boundaries of philosophy and science, as well as scientists who investigate the nature of sound, such as physicists, researchers in medicine and therapy, and those in the industry of sound technology.
Spatial views regarding sound, for example, have given rise to three theories: (a) sound is where there is a hearer, (b) sound is in the medium between the resonating sound and the hearer, and (c) sound is at the resonating object (Casati & Dokic, 2014). Accordingly, the tree in the forest question would have the following three corresponding answers: (a) no, if sound is where there is a hearer; (b) no, if sound is in the medium between the resonating sound and a hearer; and (c) yes, if sound is located in the resonating object such as a human ear. This seemingly impractical question, as it turns out, is not only quite interesting but also bears tangible results that lead to our better understanding of acoustics, hearing impairments, and sound technology. The best part is that the results affect us all. Many modern technologies arose from a “tree in the forest” examination.
Moral of the Story: The Importance of Language in Logic Clarity, precision, and correctness in language are not only important to the practical quest of communicating your ideas to others; they are fundamental to the practice of logical reasoning.
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Section 1.6 Logic and Philosophy
The Goal of Philosophy Now that the practical nature of philo- sophical inquiry has been demonstrated, we can move to a more fruitful exami- nation of what exactly philosophy is. In one view, philosophy is the activity of clarifying ideas. It is an activity because philosophy is not fundamentally a body of knowledge (as is history or biology, for example) but rather an activity. The goal of philosophical activity is to clarify ideas in the quest for truth.
How does one clarify ideas? By asking questions—especially “why?,” “what does that mean?,” and “what do you mean?” Philosophers have observed that asking such questions may be a natural human inclination. Consider any 2-year-old. As he or she begins to com-
mand the use of language, the child’s quest seems to be an attempt to understand the world by identifying what things are called. This may be annoying to some adults, but if we understand this activity as philosophical, the child’s goal is clear: Names are associated with meanings, and this process of making distinctions and comparisons of similarity is essentially the philosophical mechanism for learning (Sokolowski, 1998).
Once we name things, we can distinguish things that are similar because names help us sepa- rate things that appear alike. To a 2-year-old, a toy car and a toy truck may appear similar—both are vehicles, for example, and have four tires—but their different names reflect that there are also differences between them. So a 2-year-old will most likely go on to ask questions such as why a car is not the same as a truck until she grasps the fundamental differences between these two things. This is the truth-seeking nature of philosophy.
Philosophy and Logical Reasoning Since children’s natural learning state is a philosophical attitude, by the time we start elemen- tary school, we already have a few years of philosophical thinking under our belt. Unfortu- nately, the philosophical attitude is not always sustained beyond this point. Over time, we stop clarifying ideas because we might get discouraged from asking or we just get tired or complacent. We then begin to accept everything that we are told or shown by those around us, including what we watch on television or learn through social media. Once we stop filter- ing what we accept by means of questions, as we did when we were very small children, we become vulnerable to manipulation and deceit.
When we stop using questions to rationally discern among alternatives or to make judgments concerning disputed social problems, we begin to rely entirely on emotions or on past experi- ence as the basis for our decisions and judgments. As discussed earlier in the chapter, although
christinagasner/iStock/Thinkstock
Children’s inquisitive nature personifies the act of being philosophical. Asking questions to clarify ideas or seek the truth is fundamental to engaging in philosophy.
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Summary and Resources
emotions are valid and worthwhile, they can also be unreliable or lead us to make rash deci- sions. This may be somewhat inconsequential if we are simply buying something on impulse at the mall. But if we make judgments based purely on fear or anger, then emotions have much more dire consequences, perhaps causing us to mistreat or discriminate against others.
Past experience can also be misleading. Consider Jay, a university student, who has done very well in his first four university courses. He has found the courses relatively easy and not very demanding, so he assumes that all university courses are easy. He is then surprised when he discovers that Introduction to Physics is a challenging course, when he should have rationally recognized that undertaking a university education is a challenging task. Asking himself questions about the past courses—subject matter, professor, and so on—may help Jay adjust his expectations.
Let us review two important points that we have discussed so far. First, philosophy is an activity of clarifying ideas. Second, the goal of philosophy is to seek truth about all phenom- ena in our experience. Logic provides us with an effective method for undertaking the task of philosophy and discovering truths. This view has thus remained mainstream in Western philosophy. When we think philosophically with regard to our mundane practical purposes, logic offers us the tools to break the habit of relying on our emotions, feelings, or our past experiences exclusively for making our decisions. Arriving at this recognition alone in your own case will be part and parcel of your journey, with this book as your guide.
Summary and Resources
Chapter Summary We have covered a lot of ground in this chapter. First we introduced the ideas of critical thinking and logic as tools that help us identify warranted judgments. In other words, if we have a belief, then logic helps us find an argument that warrants either our acceptance or rejection of this belief. By means of arguments, logic thus helps us clarify when our judg- ments are warranted and our beliefs are likely true. Second, we have presented a prelimi- nary understanding of the argument as a methodical defense of a position advanced in relation to a disputed issue. Arguments provide us with a structure that will help us discern fact from purely emotional appeal and identify sober judgment from wishful thinking. Third, we have defined philosophy as an activity of clarifying ideas. As such, it can be applied to ideas in every activity—for example, raising children, learning, tasks at work, cooking, mak- ing decisions—and to every discipline—for example, physics, mathematics, economics, biol- ogy, information systems, engineering, sociology, and so on.
Chapter 2 will introduce you to the argument, the principal tool of logic. Chapters 3 through 8 will teach you the applications of logical reasoning, and Chapter 9 will show you how the knowledge that you gained can be applied in your everyday life. Approach these chapters methodically: Do a first reading to get a general idea, then go back and focus on the details of each section of the chapters, always taking notes. Keep in mind that what you are learning is a method for thinking, so you cannot adopt it simply by reading. Practice what you are learning by doing the indicated exercises and activities.
The goal of this chapter has been to show you why logic is an indispensable tool in your life. (For some thoughts on how critical thinking and logic might apply to your life as a student,
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Summary and Resources
see Everyday Logic: Thinking Critically About Your Studies.) Over the course of this book, you will see how logical reasoning can help you make wiser choices. You will also find that the benefits extend beyond yourself, since by developing the habit of good reasoning you will also become more enlightened parents, better spouses, wiser voters, and more productive community members. There is a fundamental humanity in logical reasoning that brings people together rather than alienating them from one another. To achieve the habit of logical reasoning, this book will lead you in a methodical process in which each chapter will pro- vide you with an important element. Each component of this book is not only important but also necessary in learning the tools of logical reasoning.
Everyday Logic: Thinking Critically About Your Studies
You will likely find that there are multiple opportunities to apply and develop critical thinking skills in your life, but one of the most obvious opportunities at this juncture should be in your academic career. As you move forward in your studies, the decisions you make about partici- pation and study habits will affect your ability to succeed, so it is important that you approach them thoughtfully, carefully, and even critically. The goal of this feature box is to provide some insight into how good thinkers approach their studies and to offer some concrete methods for developing your own vision of academic success.
How have you approached school and education throughout your life? From a theoretical standpoint, all students know that the goal of college is to leave with skills that will allow them to pursue certain careers or, at the very least, help them survive and pursue their conception of a good life. Recall how interested you were in the world around you as a child or perhaps how excited you became when you acquired a new skill or discovered a new interest. These feelings and experiences are the essence of learning. Unfortunately, many people’s experience in formal education is not one of wonder and enjoyment, but one of boredom and tedium. The experience of the young child who found wonder and joy in discovering new things is often crushed in formal educational experiences.
So what can we do? How can we learn to love learning again and improve our thinking and study skills to make the most of our education? First you must identify and address your weak- nesses and bad habits. Do you aim only to pass a class, cramming for tests or doing the bare minimum on assignments, instead of steadily studying, reading, and taking notes for retention and understanding? Do you tune out when you think material is boring? Do you avoid asking questions because you are afraid of looking foolish or because it is easier to just accept ideas at face value? Do you allow certain activities to interfere with your studies?
It is impossible to change all of our bad habits instantaneously, but starting with just one or two can make a great difference. Here are some methods you can use to begin the journey toward becoming a better student and thinker:
• Avoid trying to multitask while studying, and perhaps even consider “fasting” from any media that tend to distract you or occupy inordinate amounts of your time. Tell oth- ers to turn off the TV, Xbox, computer, and so forth when they see you zoning out while engaging in these activities.
• Keep a journal and record urges that you have to fall into bad habits as well as goals you have for your intellectual and academic future. Make note of your triumphs over those negative urges. Review the journal regularly and reflect on how you are changing through what you are learning.
(continued)
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Summary and Resources
• Surround yourself with people who will push you to higher levels of thinking and social action.
• Read slowly and repeatedly. Having to read a text more than once does not mean you are a poor reader. The philosopher Friedrich Nietzsche said that to read well, a human must become a cow. What does that mean? It means we need to ruminate, to chew and chew until we can swallow the meal. The process continues until we swallow and the food stays down, becoming nourishment to our minds.
• Take notes and practice writing skills when you get some free time. Try to learn a new grammar or usage rule every week. For example, do you know exactly when you should use a semicolon? If not, look it up right now. It is a really simple rule.
• Teach what you are learning to others. One of the best ways to determine if you have knowledge of something is if you can explain it and teach it to someone else.
• Recognize that this will take years of practice and will probably be slow going at first. Remember that small positive changes will add up to a whole new way of thinking and approaching life over time.
Finally, always remember that we are privileged to have the opportunity to pursue education. There are billions of people that will never have the opportunity to go to school or to provide that opportunity to their loved ones. Reformatting our perspective from one of frustration to one of gratitude can do a lot to change the way we approach education and learning. As you move forward this week, think about the following questions and how you might make changes in your own life that will lead to positive intellectual change.
• What is my view of education, and what experiences led me to that view? • What are my greatest strengths as a student? • What are my greatest weaknesses as a student? • How do I waste my time, and what might I do to utilize that time more effectively? • What is something I can do today that will help me become a better student and thinker? • What am I learning, and how has what I have learned changed who I am?
Everyday Logic: Thinking Critically About Your Studies (continued)
Critical Thinking Questions
1. What does the word critical in critical thinking mean? How would you explain criti- cal thinking to someone you know?
2. Do you have reasons for your most strongly held beliefs? To what extent are they based on emotions? Are they based in factual evidence and fair reasoning? Would other people find them convincing?
3. Are there beliefs that others hold that make you upset or angry? Why? How might you change your perspective in order not to react negatively when you hear contra- dictory beliefs?
4. Is it important to use language clearly? Why or why not? What are some steps that one can take to use language more clearly?
5. What is a logical argument? What role do you think logical argument could play in your life?
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Summary and Resources
Web Resources http://www.criticalthinking.org The Foundation for Critical Thinking maintains an extensive website regarding critical thinking and related scholarship.
http://herebedragonsmovie.com If you like to watch videos, Brian Dunning’s Here Be Dragons provides a nice introduction to some of critical thinking’s advantages and tools.
http://philosophy.hku.hk/think/critical/ct.php Hong Kong professors Joe Lau and Jonathan Chan sponsor open courseware on critical thinking at this website. This is a great place to look up specific concepts and ideas within critical thinking.
http://plato.stanford.edu The Stanford Encyclopedia of Philosophy is an excellent resource for any topics related to philosophy.
http://www.iep.utm.edu The Internet Encyclopedia of Philosophy is a peer-reviewed online academic resource of articles on philosophy.
Key Terms
critical thinking The activity of care- ful assessment and self-assessment that employs logical reasoning as the princi- pal basis for accepting beliefs or making judgments.
formal logic The abstract study of argu- ments, often using symbolic notation for analysis.
informal logic The study and description of reasoning in everyday life.
logic The study of arguments as tools for arriving at warranted judgments.
philosophy The activity of clarifying ideas with the goal of seeking truth.
rhetoric The art of persuasion.
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Running Head: UNIVERSAL HEALTHCARE 1
UNIVERSAL HEALTHCARE 4
Universal Healthcare
Tasha Smith
Zachary Martin
04/03/2016
Universal healthcare coverage should be provided to all patients because it results in:
i) Improved healthcare to all people
Universal healthcare provides accessible health services to the population in different regions in the world. When patients are provided with free healthcare, a large number of patients who are unable to acquire expensive specialized care but are suffering from different conditions are likely to seek the medical services. According to HealthPac (2016), universal health coverage has changed the lifestyle of many people who live in areas with high poverty levels since they can also access better healthcare regardless of their financial status. Patients who suffer from diverse ailments that require specialised treatment such as cancer and brain tumour, therefore, can easily access high-quality health care.
ii) Preventative care among the population
Many people, in areas with high poverty levels and economic disparity, are less likely to seek preventative healthcare due to the high costs of the healthcare services. The adoption of universal health coverage will, therefore, ensure that individuals seek preventative care to avert the occurrence of fatal diseases such as tuberculosis, hypertension, and cancer.
iii) Equality and fairness
Universal health coverage also promotes the spirit of equality and fairness. This is because all people irrespective of their income, race or gender, can easily get healthcare services in any part of the world. Universal healthcare will remove the inequality barrier that divides the wealthy or the powerful in the society with the poor.
iv) Decrease in administrative healthcare costs
Universal healthcare coverage will have a positive impact on the healthcare organizations. This is because it will lead to a reduction of administrative cost that has been brought about by the fact that all activities such as billing and insurance pay-outs are undertaken by a single organization. All rules about different activities such as billing is similar to every patient hence the organization does not spend a lot of funds for these services.
v) Economic growth and better living standards
Universal healthcare results in free health services hence the population will not have not spend a lot of money in paying for health care services (Behzad, 2009). The funds that could have been used in seeking health services can, therefore, be put in income generating activities thus resulting in economic growth and improved standards of living.
In summary, the provision of free healthcare services to the public has a significant impact on their daily healthcare. Universal healthcare should, therefore, be adopted since it results in improved healthcare provision as well as better standards of living.
References
Behzad, M., (2009). Universal Healthcare in the United States of America
New York: Blackwell Press
HealthPac, (2016). Structure and Funding of Universal Healthcare
Retrieved from: http://www.healthpaconline.net/universal-health-care.htm
Rogers, K., (2012). Universal Healthcare: An Introduction
New York: McGraw Hill Press

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