A short introduction to formal logic

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1 A short introduction to formal logic Dan Hicks v0.3.2, July 20, 2012 Thanks to Tim Pawl and my Fall 2011 Intro to Philosophy students for feedback on earlier versions. My approach to teaching logic has benefited a great deal from numerous exchanges with John Milanese. Contents 1 The idea of logic Arguments and argument patterns Truth, validity, and soundness Deductive validity Testing validity Some common argument patterns Conditionals Categories Other sentence logic arguments Induction Practical reasoning Conversational logic Modal logics Working with arguments in the wild Formulating an argument Reconstructing (and evaluating) an argument Enthymemes Improving arguments Terminology Additional resources 35 Note for instructors 37 Fig. 1: Categorical syllogisms 38 Fig. 2: Other argument patterns 39 1 The idea of logic Logic is the branch of philosophy that deals with arguments. We often think of arguments in connection with disagreements, debates, and even shouting. Call this way of thinking about arguments argument as disagreement. For example, we might talk about the argument (meaning the disagreement) between Republicans and Democrats over taxes and the budget. But there s another way of thinking about arguments. This is less common than argument as disagreement, but it s still common. For example, suppose the Republicans say that we should balance the budget because it would help the economy. Here we would say that the Republicans are giving an argument to support their position on the budget. Call this way of thinking about arguments argument as reason. An argument, on this way of thinking, is a reason why someone does (or should) believe something: the Republicans, in giving their argument, are giving reasons why you (or 1

2 whoever they re talking to) should believe that we should balance the budget. It may seem to you that philosophy is mostly about argument as disagreement. You re probably not wrong to think this. But logic deals only with argument as reason. Logicians philosophers who study logic are interested in the connection between reasons and the claims they support. Throughout this pamphlet, we re going to be concerned entirely with arguments as reasons (even when those reasons show up in the context of a disagreement). Our goal will be to equip you with some basic concepts a logical toolkit for use in your philosophy class. 1.1 Arguments and argument patterns It s not the case that all logic textbooks are obligated to discuss this argument. But it s such a standard example that I can t help but start with it. (1) All humans are mortal. (2) Socrates is a human. (3) Socrates is mortal. (1,2) (Socrates was a philosopher in ancient Athens, and is often considered the father of Western philosophy. He famously died by drinking poison.) This argument is in premise-conclusion form. In premise-conclusion form, each distinct claim of the argument is written on its own line, and the lines are numbered to make them easy to refer to. Line (3) is called the conclusion of the argument. (See what I did there, with the line number?) The conclusion is the claim that the argument is meant to support. If I were giving you this argument, I would be trying to convince you that Socrates is mortal. (It s not a very interesting or surprising conclusion. But simple examples like this are easier to follow. You ll encounter many arguments with more interesting conclusions in your philosophy class.) Lines (1) and (2) of this argument are its premises. The premises are the reasons given to support the conclusion: if you believe that all humans are mortal and that Socrates is a human, then you have reason to believe that Socrates is mortal. Both premises and conclusions are steps of the argument. Premise (1) is the first step; the last step, (3), is the conclusion. This is a good time to mention two basic rules for writing arguments in premise-conclusion form. First, every step is a declarative sentence. You can think of a declarative sentence as a sentence that can be true or false. Here are some things that aren t declarative sentences, and so can t be steps of an argument: Is a human. This one isn t grammatically a sentence it doesn t have a subject. Make tacos! This one is a sentence, but it s an imperative sentence a command. There s a special logic for imperative sentences, but it s not one we ll discuss in this pamphlet. Is it raining? Another sentence, but this one is a question. I like corn. This one is actually tricky. It can be read as a declarative sentence: a report (true or false) of how I feel about corn. But it can also be read as an optative, which is an expression of a wish, desire, or preference. On this reading, this sentence is like saying Hooray for corn! which can t be true or false. 2

3 Our second rule is that premises come first, conclusions come last. This rule is just about order: the conclusion of the argument should be the very last step, and the first step (at least) should be a premise. (There are some exceptions, but we won t cover them here). Besides the steps and their numbers, the example up above had some other useful symbols. (3) started off with this symbol: This symbol is read hence or therefore, and it signals a conclusion. We know that (3) is the conclusion because of the symbol. This might seem obvious now it s the last step, so of course it s the conclusion. But we ll see more complicated arguments below, and will be very important. (If you can t figure out how to make on your computer, just start the conclusion with the word hence instead.) There were also the numbers in parentheses after (3): (1,2). These numbers indicate logical support. They tell us that step (3) is supported by steps (1) and (2), taken together. Again, this is obvious in this simple example so I won t be using them regularly in this pamphlet but it ll be very useful when we get to more complicated arguments. For now, compare the earlier argument with this one: (4) All cats are mammals. (5) Whiskers is a cat. (6) Whiskers is a mammal. The two arguments look quite similar. We could even borrow from algebra and replace the nouns and names with letters: (7) All As are Bs. (8) x is a A. (9) x is a B. And we could go the other direction, replacing the generic letters with nouns and names. (Actually, that s how I thought up the example about Whiskers.) When we have two arguments that are similar in this way, we say that they have the same logical form or pattern. Here s another pair of examples in English, along with the generic-letter version: (10) If Obama is President then Biden is VP. (11) Obama is President. (12) Biden is VP. (10 ) If p then q. (11 ) p (12 ) q (10 ) If it s raining then the sidewalks will be slippery. (11 ) It s raining. (12 ) The sidewalks will be slippery. (The is a prime, so for example (10 ) is ten-prime and (11 ) is eleven double-prime.) These arguments all have the same pattern, though of course it s different from the pattern we saw with Socrates and Whiskers. The next thing to notice is that, in arguments with the same pattern, the premises are related to the conclusion in the same way. The premises (10) and (11), taken together, support (12) in the same way that (10 ) and (11 ), taken together, support (12 ). And that, in turn, means that we can ask and answer some interesting questions about an argument even if all we know is the logical form. By studying the generic version of the argument (10 12 ), and without even thinking about President Obama are whether the sidewalks really are slippery, we can learn something about how the premises, in each case, support the conclusion. And we 3

4 can go on to apply that to other arguments with the same pattern, whether they re about President Obama or the planet Mars or whatever. The move from English-language arguments (or, more generally, arguments in natural languages) to their patterns is like the move from drawing shapes on a piece of paper to studying geometry. Indeed, the study of argument-patterns goes by two names: formal logic because it studies logical form and mathematical logic because it s studied mathematically. If you re reading this pamphlet for a logic class, you ll go on to learn some of that mathematics. (If you re math-phobic, don t worry: you can be excellent at logic without knowing anything at all about algebra or calculus or any of that.) But, if you re not, you can still use the tools we develop here to analyze arguments. In fact, that s why I wrote this: so students who haven t learned any mathematical logic at all can still use it. So far, we ve only seen arguments with a single conclusion. These arguments are called simple arguments. We can build compound arguments by putting together simple arguments. Here s an example: (13) Either Obama is President or McCain is President. (14) McCain isn t President. (15) Obama is President. (13, 14) (16) If Obama is President then Biden is VP. (17) Biden is VP. (15, 16) In compound arguments, just like simple arguments, we arrange the premises first and the conclusion last. But, unlike simple arguments, compound arguments will have steps in the middle, that are neither independent premises nor the final conclusion. In this example, (15) isn t an independent premise it follows from (13) and (14) but it s also just a step on the way to (5). These intermediate steps are called subconclusions. Like conclusions, we mark subconclusions with and numbers indicating logical support. Notice that, after (17), we list only (15) and (16). This is because these only two steps, all by themselves, give support directly to the conclusion. The other steps (13) and (14) support the conclusion indirectly, by directly supporting (15). 1.2 Truth, validity, and soundness So far, we ve developed some important concepts to understand the anatomy of an argument: its premises, conclusions, and pattern. Next, we want to evaluate arguments: do the premises, taken together, give us a reason to believe the conclusion? The most familiar concept here is truth: we say that it s true that Obama is President, that it s false that it s raining, and so on. Defining truth is a huge philosophical problem; many philosophers think of truth as something like correspondence to reality, but there are plenty of philosophers who think this definition won t do. I ll assume that truth is something like correspondence. Notice that, with this conception of truth, truth isn t the same as belief and opinion. Contrast these three sentences: Juanita believes that humans are responsible for global warming. Juan believes that humans aren t responsible for global warming. Humans are responsible for global warming. The first two sentences are about what people believe. These sen- 4

5 tences are true if, and only if, these people actually believe these things. (If, and only if, the sentences reporting their beliefs correspond to what they believe.) The third sentence has nothing to do with what anyone believes. It s true if, and only if, humans actually are responsible for global warming. (If, and only if, the sentence blaming humans corresponds to the effects of our actions.) That means whether the third sentence is true has nothing to do with whether people agree on it: Juanita s belief doesn t make the third sentence true, and Juan s belief doesn t make the third sentence false. Their disagreement might make it hard for us to tell whether or not the third sentence is true, but that s a matter of our knowledge of the world, not the way the world is. Disagreement, by itself, doesn t make anyone s beliefs false. The last paragraph shouldn t seem outrageous or surprising. But now I m going to make a very controversial move. Contrast these three sentences: Juanita believes that we shouldn t eat meat. Juan believes that we should eat meat. We should eat meat. Notice, first, that the relationship between these three sentences is the same as the relationship between the last three sentences: the first is about what Juanita believes, the second is about what Juan believes, and the third doesn t seem to have anything to do with what anyone believes. The controversial move is to assume that the third really doesn t have anything to do with what anyone believes. Like the claim that humans are responsible for global warming, we should eat meat is true if, and only if, it corresponds to the way the world actually is. This probably strikes you as much more surprising and outrageous than the global warming stuff. You ve probably been taught that there s a difference between fact and opinion, and you ve used this distinction to say things like there s a fact about whether humans are responsible for global warming and whether we should eat meat is a matter of individual opinion. But we often act as though there are some facts about what everyone should do: not commit murder or rape; not be racist or sexist; pay taxes; protect small children. We believe that it s true that you should pay your taxes, regardless of what anyone thinks; similarly, we believe that someone who believes that murder is acceptable is wrong, and has a false belief. Philosophers call the facts corresponding to claims about what we should do (if there are any such facts) moral facts or moral truths. And philosophers disagree about whether there are moral facts. I m going to assume that there are moral facts, but only because they make thinking about the logic of what we should do easier. If it s simply true or false that we should eat meat, it s easier to think about the logic of arguments for and against eating meat. Arguments cannot be true or false. The individual steps of an argument can be true or false, and usually the person giving an argument would like the conclusion to be true, but the argument itself is neither true nor false. Only declarative sentences can be true or false. Instead of talking about the truth of arguments, we talk about their validity. The two ideas come together in the idea of soundness; these are illustrated in the diagram below. truth of individual sentences soundness of whole argument validity of whole argument All of the arguments we ve seen so far have been valid. An ar- 5

6 gument that s not valid is called invalid. Here s an example of an invalid argument: (1) All cats are mammals. (2) Whiskers is a mammal. (3) Whiskers is a cat. (Since we ve started a new subsection, and so that the numbers don t get too big, I ve started over the numbering of the steps.) Validity is the core concept of logic: everything we do as logicians is about trying to determine whether arguments are valid. Just like arguments cannot be true or false, conclusions cannot be valid or invalid. That goes for any sentence, but students are often tempted to say that about conclusions when they really mean the whole argument is valid or invalid. Unlike truth, philosophers all basically agree on the official definition of validity. The problem is that this definition is complicated. 1 Here s a simpler definition: validity An argument is valid if, and only if, its premises give good reason to believe the conclusion. This definition is useful for some cases, but doesn t work with others. It works with (1-3): Even if Whiskers is a mammal and all cats are mammals, Whiskers might be some other kind of mammal, like a dog or an aardvark. Since the premises don t rule out this possibility, they don t give us good reason to believe the conclusion. So (1-3) is invalid. Next, consider these two arguments: (4) If Obama is President then Biden is VP. (5) Obama is President. (6) Biden is VP. (7) If McCain is President then Palin is VP. (8) McCain is President. (9) Palin is VP. It should seem reasonable to think that (4-6) is valid: these premises do indeed give good reason to believe this conclusion. What about (7-9)? This argument has a false premise, (8). You might think that a false premise doesn t give good reason to believe the conclusion; so this argument would be invalid. But someone else might say that a premise if you accept it for the sake of argument or suppose that it were true can give good reason to believe the conclusion, whether it s true or false. If someone really, sincerely believes that McCain is President, then that person has good reason to believe that Palin is VP. They re out of touch with reality, but they re not reasoning incorrectly. Things get even more difficult when we move to formal logic dealing just with patterns, not particular arguments. In formal logic, we don t know whether the premises are true or false. All we have to work with are the generic versions: (10) if p then q (11) p (12) q If we re going to say anything at all about validity in formal logic, it will have to be independent of the truth of the premises. It will have to depend only on the pattern itself. And it turns out that, for many arguments, their validity does depend only on the pattern itself: 1 How complicated? Just learning basic tests of validity for the simplest kinds of logic requires an entire semester-long course. Some mathematicians and philosophers devote their entire careers to studying logic. 6

7 formal validity property A pattern has the formal validity property (FVP) if, and only if, every argument with that pattern has the same validity. The FVP isn t based on the definition of validity up above. It s based on the much more complicated, actual definition of validity. Fortunately, you don t need to know this definition to use the FVP. The pattern (10-12) has the FVP: every argument with this pattern has the same validity. Now, (4-6) has this pattern, and it s valid. So every argument with this pattern is also valid. In particular, (7-9) is valid. Here s the pattern for (1-3): (13) All F s are Gs. (14) x is a G. (15) x is a F. x is a variable for an individual (like Whiskers); F and G are classes or sets of individuals. This argument has the FVP. (1-3) has this pattern and is invalid. So this argument, which has the same pattern, is also invalid: (16) All ethicists are philosophers. (17) Peter Singer is a philosopher. (18) Peter Singer is an ethicist. (Peter Singer is an ethicist who has written several famous papers and books on poverty, vegetarianism, and medical ethics.) The examples of Whiskers (1-3) and Singer (16-18) illustrate how validity isn t a matter of the premises being true. Every step in each of these arguments is true. The problem is that the (true) premises don t support the (true) conclusion. For arguments with a FVP pattern, the question of validity is settled purely by the logical relationship between its premises and its conclusion. It has nothing to do with whether any of the steps are true or false or even whether we have any idea what they re talking about. Both of these arguments, because they have the same pattern as (10-12), are valid: (19) If McCain is President than Obama is VP. (20) McCain is President. (21) Obama is VP. (22) If bandersnatches are frumious then mome-raths outgrabe. (23) Bandersnatches are frumious. (24) Mome-raths outgrabe. (The nonsense words in (22-24) come from a famous poem, Jabberwocky, by Lewis Carroll, who wrote Alice in Wonderland and was also a logician.) Truth (of the steps) and validity (of the argument as a whole) are independent. We ve seen examples of an invalid argument with all true steps, and examples of a valid argument with all false steps. (This would be a good time to stop and review. Go back and find these examples, and explain to yourself why they re valid or invalid.) My students usually start talking about flow or the argument hanging together and making sense at this point. I think all of this language is trying to say that validity depends on the relation between the steps of the argument, not features of the steps taken individually. In other words, when we re trying to decide whether or not an argument is valid, we don t look at the individual steps. Instead, we look at the argument as a whole. The definition we started with suggests that, but it could 7

8 be more explicit. If we have the FVP and plenty of examples of valid and invalid patterns, we have that explicitness. Not all patterns have FVP. Here are some examples of an important exception: (25) Michelle Obama is the First Lady. (26) The First Lady lives in the White House. (27) Michelle Obama lives in the White House. (28) The robber is the guard s daughter. (29) The guard believes his daughter was at home during the robbery. (30) The guard believes the robber was at home during the robbery. (25-27) is valid, but (28-30) is not. In many cases, when we have two labels for the same thing, we can replace one label with the other and get a valid argument. However, in some cases especially dealing with what people believe this doesn t work. Suppose the guard doesn t know that his daughter is the robber the guard doesn t know (28). He believes that his daughter and the robber are completely different people. He also believes (falsely) that his daughter was at home during the robbery. Then (29) is true it s a true report of what the guard believes. (Note the difference between the guard s belief and the sentence reporting the belief. The first is false and the second is true!) And (30) is false: the guard doesn t believe the robber was at home; he believes the robber was out committing the robbery! So we can t swap one label ( the guard s daughter ) for the other ( the robber ) in this case. Patterns without FVP are called questionable. They live between valid and invalid. Questionable arguments aren t flawed; it s more accurate to think of them as fragile. A questionable argument can be a very good argument its premises can provide very good reason to believe the conclusion. (As in (25-27).) But it s easy to mistake a very good questionable argument for a very bad questionable argument. So we have to be careful with questionable arguments, and think carefully about whether the premises support the conclusion on a case-by-case basis. Distinguishing patterns with and without FVP is difficult. And, for ones with FVP, it s often difficult to tell whether each is valid or invalid. You won t be expected to do any of that in this pamphlet. Instead, when we get to the list of common patterns ( 2), I ll tell you whether they re valid, questionable, or invalid. Most of the argument patterns we ll meet have FVP. That means we ll be able to say, precisely and for all cases, whether they re valid or invalid. Still, some common patterns are questionable, so it s important to keep the distinction in mind. We bring the concepts of truth and validity back together with the concept of soundness. soundness An argument is sound if, and only if, it is valid and all of its premises are true. By contrast, an argument is unsound if either it is invalid or at least one of its premises is false. Arguments (1-3) and (16-18) are unsound because they are invalid. While arguments (4-6), (7-9), and (19-21) are all valid, only (4-6) is sound; the others are unsound because they have false premises. 1.3 Deductive validity Philosophers often like to work with a version of validity that s much stronger than the way I ve defined it here, which is called 8

9 deductive validity: deductive validity An argument is deductively valid if, and only if, it is impossible for the premises to be true and the conclusion false. Consider these two arguments: (1) Dan is a reliable authority on logic. (2) Dan says that all sound arguments are valid. (3) All sound arguments are valid. (4) All sound arguments have true premises. (5) This argument is sound. (6) This argument has true premises. Both arguments are valid (and sound). But (4-6) guarantees its conclusion in a way that (1-3) doesn t. The fact that I m a reliable authority on logic doesn t make it impossible for me to get things wrong sometimes. So there s a chance that (3) is false even when (1) and (2) are true. On the other hand, given (4) and (5), it s completely impossible for (6) to be false. These premises, together, guarantee that the conclusion is true. If you go on to study formal logic, you ll probably focus on deductive validity. But often deductive validity is too demanding for practical purposes. We don t usually need absolute guarantees for the conclusions of our arguments; we just need good, even if imperfect, reasons. Since this pamphlet is all about practical formal logic, I m not going to say much about deductive validity. 1.4 Testing validity Logic, as such, is only concerned with validity. The question of whether a given premise is actually true is usually turned over to someone else, or at least to a different branch of knowledge. For most of the arguments in the last subsection, a logician can tell us which ones are valid and which ones are invalid, but we ll have to ask someone who knows something about politics to determine whether the premises of these arguments are true. In other words, logic only gives us tools for testing the validity of arguments. If you take a formal logic class, you ll learn some sophisticated tools for testing arguments. However, like with math, it takes quite a bit of explanation and practice to understand just how and why these tools work. We won t go into that detail here. So, instead of explaining all the nuts and bolts to you, I m going to introduce you to a couple of simple tests that come out of the more sophisticated tools. This is like using a graphing calculator to graph the function y = 2x + 4. The inner workings of the calculator are much, much more complicated than this simple function. But you don t have to understand why the calculator does what it does in order to use it; you just have to know how to tell the calculator to graph the function The pattern test Our first test is called the pattern test. We ve already used the pattern test in the last subsection, when we compared an argument to a known pattern to determine whether it was valid. Here s an example with a pattern we haven t seen before: 9

10 (1) Either McCain is President or Obama is President. (2) McCain isn t President. (3) Obama is President. (4) either p or q (5) not-p (6) q This pattern has the formal validity property, so every argument with it has the same validity. And, in particular, every argument with this pattern is valid. So (1-3) is valid. To use the pattern test, first make sure the argument is in premise-conclusion form. Second, compare it to some known patterns. Finally, third, read the validity of the argument off the pattern. Let s look at another example: (7) Some Republicans believe that humans are responsible for global warming. (8) Everyone who believes that humans are responsible for global warming thinks we should reduce our CO 2 admissions. (9) Some Republicans think we should reduce our CO 2 admissions. This argument has this pattern, though it might not be obvious at first: (10) Some F s are Gs. (11) All Gs are Hs. (12) Some F s are Hs. F, G, and H all stand for sets of things. Here are their values for (7-9): F: Republicans G: people who believe that humans are responsible for global warming H: people who think we should reduce our CO 2 admissions This pattern is valid, and so (7-9) is valid. Here s one more example, this time with an invalid argument: (13) If Obama is President than Biden is VP. (14) If Obama isn t President then Biden isn t VP. And this pattern is invalid. So (13-14) is invalid. (15) if p then q (16) if not-p then not-q The pattern test has an important advantage and an important disadvantage, and these are closely related. The advantage of the pattern test is that you don t have to think about why an argument is valid or invalid; you just have to be able to check it against a list of examples. The disadvantage is that using the pattern test doesn t do much to help you understand why some arguments are valid and some are invalid; you re just matching up patterns. Maybe it s clear to you why (1-3) and (7-9) are valid, and (13-15) isn t. But, if it s not clear to you, the pattern test isn t liable to seem arbitrary. So, if you use the pattern test a lot, it s helpful to occasionally use the other test The counterexample test The second test is the counterexample test. To use this test, first make sure the argument is in premise-conclusion form. Second, try to imagine a situation in which the premises are all true and the conclusion is false. Then, third, if you can imagine such a 10

11 situation, the argument invalid. Or, fourth, if you can t imagine such a situation, the argument is valid. A situation real or fictional in which the premises of an argument are true and the conclusion is false is called a counterexample hence the name of the test. Counterexamples are especially useful for understanding why questionable and invalid arguments are question or invalid. Consider our most recent example: (13) If Obama is President then Biden is VP. (14) If Obama isn t President then Biden isn t VP. In the first few decades of the US, the VP wasn t picked by the Presidential candidate before the election as part of the ticket; instead, he or she was the person who got second in the election. Suppose we still had this system. Since the election was very controversial, we might also suppose that there were two possible outcomes: McCain wins, and Obama only gets a small share of the vote; or Obama wins, and McCain only gets a small share of the vote. In either case, Biden, who was a much less controversial politician, would have gotten second. Then, in this situation, (13) is true: If Obama is President then Biden (who got second, after Obama) would be VP. But (14) is false: If Obama wasn t President then Biden (who got second, after McCain) would be VP. Just in case that was too complicated, here s a simpler argument with the same pattern: (15) If the window is hit by a baseball then it ll break. (16) If the window isn t hit by a baseball then it won t break. There are lots of different ways to break a window; hitting it with a baseball is just one. For example, if it s hit with a rock, it ll break. So (19) is true but (20) is false, and the argument is invalid. We saw another use of the counterexample test in the last subsection. This was (28-30): (17) The robber is the guard s daughter. (18) The guard believes his daughter was at home during the robbery. (19) The guard believes the robber was at home during the robbery. Suppose the guard doesn t know that his daughter is the robber the guard doesn t know (21). He believes that his daughter and the robber are completely different people. He also believes (falsely) that his daughter was at home during the robbery. Then (22) is true it s a true report of what the guard believes. And (23) is false: the guard doesn t believe the robber was at home; he believes the robber was out committing the robbery! Here we have a counterexample: a situation in which the premises are true and the conclusion is false. So this argument is invalid. Unlike the pattern test, the counterexample test can help you understand why an argument is valid or invalid. It can also help you identify logical gaps in an argument premises that the author has assumed, or that can be added to make the argument valid. For example, you might repair argument (17-18) by adding a premise in this way: (20) If Obama is President then Biden is VP. (21) If Biden is VP then Obama is President. (22) If Obama isn t President then Biden isn t VP. Another advantage of the counterexample test is that you can use it when you can t identify the pattern of the argument. Even if you didn t realize that (7-9) could be rewritten in line with the pattern (10-12), you could still use the counterexample test to con- 11

12 firm that it s valid. Similarly, if you encounter an argument with very complex premises, the counterexample test might be easier to use than the pattern test. The counterexample test has a couple of important weakness, and they both come from the way the test relies on your imagination. First, even if you can t think of a counterexample, this might be due to a lack of creativity rather than the validity of the argument. If you didn t know about the way we used to elect the VP, for example, you might not think of the counterexample to argument (17-18). Second, philosophers often disagree about the outcome of thought experiments. A situation that you think is quite plausible and shows that the argument is invalid might strike someone else as completely implausible and so show nothing at all about the validity of the argument. 2 Some common argument patterns To use the pattern test, we need a stockpile of patterns, both valid and invalid. This section gives you exactly that. The field of biology can be subdivided based on the kinds of organisms that are studied: microbiology (which studies bacteria and other microscopic organisms), botany (which studies plants), and zoology (which studies animals), for example. Similarly, the field of logic can be subdivided based on the kinds of arguments that are studied. There are two common ways to make these divisions: by the strength of the arguments and by the logical concepts that the argument involves. I m going to take the second approach, since this will make it easier for you to find the pattern for an argument you re analyzing. But I ll also tell you how strong each argument is along the way. Example: your argument involves a conditional, so take a look at 2.1; you see that it s an instance of modus ponens, and so it s valid. All of these arguments are listed, in brief, in figs. 1 and Conditionals One of the most frequently used patterns is modus ponens. (The Latin name means the method of affirmation.) (1) if p then q (2) p (3) q We ve seen several examples of modus ponens already. Modus ponens is valid. Premise (1) is also an example of a conditional statement a statement that says one thing (p, called the antecedent) is a condition for another thing (q, called the consequent). (1) says that, if the antecedent is true, then the consequent is also true. (2) says that the antecedent is true. And so (3) says that the consequent is true. Another pattern that uses a conditional premise is modus tollens ( the method of denial ): (4) if p then q (5) not-q (6) not-p Modus tollens is like modus ponens in reverse: (1) and (4) are the same, but (5) says that the consequent is not true. So, in (6), the antecedent cannot be true either. Modus tollens is also valid. In formal logic, it s very important to make all of the connections between our ideas explicit. Consider this argument: 12

13 (7) Obama received the most votes in the last Presidential election. (8) Obama is President. This argument assumes that the reader understands the relationship between getting the most votes and being President. But, when we analyze the logical form, that relationship can t be assumed: (7 ) o is R (8 ) o is P This argument, as it stands, is invalid. To fix it, we need to bridge the logical gap between R (the person who received the most votes) and P (the President). But that s easy to do, using a conditional and modus ponens: (7 ) o is R (8 ) if o is R then o is P (9 ) o is P The added premise, in English, says that if Obama received the most votes then he is President. The conditional connects the premise we already had to the conclusion, makes the argument an instance of modus ponens, and so makes the argument formally valid. When you are reconstructing English arguments in premiseconclusion form, this is an extremely useful technique. Some conditionals in English are stronger than others. Compare these three: If Maria throws a rock at the window then it will break. If Obama gets the most votes in 2012 then he ll be re-elected. If this number is an even prime then it is the number 2. The first conditional is rather weak: if Maria throws a rock at the window but it s just a pebble, or Jorge catches it, or something else gets in the way, then the window won t break after all. The second conditional is stronger, but there are still ways in which Obama can lose the election even if he gets the most votes. (In 2000, Al Gore got the most votes nationally, but didn t win the election because of the way the electoral college works. This outcome has only happened a few times.) And the third conditional is the strongest you can get: it s mathematically impossible for any even number besides 2 to be prime. The validity of arguments with conditionals depends on the strength of the conditionals. This instance of modus ponens is valid, but not deductively valid, because of its weak conditional: (10) If Maria throws a rock at the window then (in most cases) it will break. (11) Maria will throw a rock at the window. (12) The window will break. By contrast, this instance of modus tollens is deductively valid, because of its strong conditional: (13) If Whiskers is a cat then (by definition) she is a mammal. (14) Whiskers is not a mammal. (15) Whiskers is not a cat. In (10), it s possible for the antecedent to be true and the consequent to be false. So it s possible for the premises (10, 11) to be true and the conclusion (12) false these premises don t guarantee the conclusion in the way deductive validity requires. In (13), it s impossible for the antecedent to be true and the consequent to be false. So it s impossible for the premises (13, 14) to be true and the conclusion (14) false these premises guarantee the conclusion. 13

14 Weak conditionals are often used to relate cause and effect: If Thao throws Juan a birthday party then Juan will be happy. If Juanita gives Jaime a ride home then Jaime won t have to walk. If Obama gets the most votes in 2012 then he ll be re-elected. If I don t refill my gas tank then my car won t start. These causal conditionals are almost always weak: they describe one way in which the effect (consequent) could be caused, but not the only way. So causal arguments using modus ponens and modus tollens are almost always valid but not deductively valid. What about reasoning from effects to causes? It s tempting to make an argument like this: (16) If Thao throws Juan a birthday party then Juan will be happy. (17) Juan is happy. (18) Thao threw Juan a birthday party. This argument has this pattern, called affirming the consequent: (19) if p then q (20) q (21) p (Notice that, compared to modus ponens, we ve swapped the second premise and the conclusion.) (16) just tells us one way in which Juan could be made happy. Why think, in this particular case, that Thao was responsible? (17) just tells us that the consequent of (16) is true; that doesn t tell us anything at all about the antecedent! This pattern is invalid. For similar reasons, denying the antecedent is also invalid: (22) if p then q (23) not-p (24) not-q And so is the fallacy of the single cause: (25) Making teachers accountable causes students to learn more. (26) Providing more funding for schools doesn t cause students to learn more. This version is a little easier to read than the more formal version: (27) If teachers are made more accountable then students will learn more. (28) It s not the case that if we provide more funding to schools then students will learn more. (29) if p then q (30) not-(if r then q) Here, the premise says that something (p) is the cause of some effect (q). The conclusion says that something else (r) cannot be the cause of that same effect. This argument is invalid because there is usually more than one way to bring about some effect: we could probably get students to learn more by making teachers more accountable, providing more funding to schools, or both. The fallacy of the single cause often shows up in criticisms of other people s arguments. Suppose Xi wrote a letter to the editor of a newspaper, arguing that, if we provide more funding to 14

15 schools, then students will learn more. Xin writes a response to the letter, attempting to criticize Xi: in fact, Xin says, if teachers are made more accountable then students will learn more. But Xin s response doesn t actually criticize Xi. If both making teachers more accountable and providing more funding to schools would cause students to learn more, then both Xi and Xin are right. Xin s response is only a criticism if we assume that there s one unique cause of student learning. But this response seems false. Xin has committed the fallacy of the single cause. We can fix up some of these arguments by adding some additional information. Suppose we know that Thao is the only person liable to throw Juan a birthday party, and that Juan is liable to be unhappy on his birthday if he doesn t get a party. Then the fact that Juan is happy does give us reason to believe that Thao threw him a party. Here s how we can state this argument more formally and generally: (31) Thao throwing Juan a birthday party is the best available explanation of Juan being happy. (32) Juan is happy. (33) Thao threw Juan a birthday party. This pattern is called inference to the best explanation or abduction. (34) p is the best available explanation of q (35) q (36) p Abduction lets us infer from effects backwards from observed effects to unobserved causes. It is valid. When checking an abductive argument for soundness or trying to convince someone that your abductive argument is sound it s best to test the proposed explanation against some other possibilities. For instance, perhaps Maria threw Juan a party. But Thao is Juan s best friend, and Maria doesn t even like Juan very much. So Thao throwing the party is a better explanation than Maria throwing the party. Comparing several rival explanations in this way amounts to checking to see how many different causes there are for the effect. The historian and philosopher of science Thomas Kuhn gave the following five criteria for a good explanation: consistency: The explanation should be consistent with both itself and other accept ideas about cause and effect. simplicity: The explanation avoids unnecessary complications. accuracy: The explanation makes accurate predictions about observable events. fruitfulness: The explanation suggests possibilities for further exploration and development. scope: A general explanation can be applied to many different phenomena. For example, the theory of evolution by natural selection is a very good explanation by these criteria: it is consistent with other ideas about biology; it provides a simple mechanism for evolution over time, without assuming, for example, that evolution is directed towards some ultimate purpose; many of its predictions have turned out to be accurate; it has suggested many further lines of research; and it applies to many, many features of living organisms. Here s one more argument involving conditionals: (37) If Whiskers purrs then Whiskers is a cat. (38) If Whiskers is a cat then Whiskers has fur. (39) If Whiskers purrs then Whiskers has fur. 15

16 This argument doesn t say that Whiskers does purr, or does have fur, or is a cat. Instead, it links two conditionals together. It s called a hypothetical syllogism: (40) if p then q cats mammals things with fur mammals (41) if q then r (42) if p then r As with modus ponens, a hypothetical syllogism is valid. 2.2 Categories Categorical logic is the oldest kind of logic, going back to the Ancient Greek philosopher Aristotle. It deals with the relationship between sets, classes, or categories of things. In this argument, the classes are cats, mammals, and things with fur : (1) All mammals have fur. (2) All cats are mammals. (3) All cats have fur. We symbolize this argument using letters for each of the categories: (1 ) all M is F (2 ) all C is M (3 ) all C is F Sometimes categorical logic is studied in high school and junior high school math classes, using Venn diagrams and Euler diagrams. Here are the Venn and Euler diagrams for the last argument: things with fur cats (Venn diagrams were developed by the nineteenth-century British logician John Venn. Euler diagrams were developed by the eighteenth-century Swiss mathematician Leonhard Euler.) We also use categorical logic to talk about the individual members of categories: (4) All mammals have fur. (5) Mowser is a mammal. (6) Mowser has fur. When we symbolize these arguments, we use a lower-case letter for the individual: (4 ) all M is F (5 ) m is M (6 ) m is F When we evaluate validity, we can treat the lower-case letter as the name of a single-member class the class of all the things that are m. Until the late nineteenth century, much of the study of logic focused on categorical logic involving three classes. These arguments are called categorical syllogisms. Logicians developed terminology to precisely describe the patterns of categorical syllogisms and 16

17 determine whether they were valid and invalid. (These arguments, when valid, are all deductively valid, as in (1-3).) To help memorize the valid patterns, they gave them names. All 15 deductively valid categorical syllogisms are listed, with their names, in fig. 1. However, categorical syllogisms have a number of limitations. For one thing, it s often quite difficult to make these arguments sound, because they involve very strict relations among very broad categories. (2), for example, is false, because some mammals (like whales, dolphins, hippos, and shaved cats) don t have fur. For another, the terminology and rules don t work as soon as we add in a fourth class, as in this example: (7) All cats are either mammals or birds. (8) All mammals are warm-blooded. (9) All birds are warm-blooded. (10) All cats are warm-blooded. This argument is deductively valid, but the rules for categorical syllogisms can t tell us that. Driven by these limitations, in the nineteenth century logicians began to develop a much more powerful, flexible, and interesting logical system called quantifier logic or predicate logic. Quantifier logic can do everything that categorical syllogisms can do, and much, much more. But this also means that quantifier logic is much more complicated far too complicated for me to explain in this short introduction. 2.3 Other sentence logic arguments Over the last two subsections, you may have noticed that we were using two different kinds of variables. When we were discussing conditionals, we used variables p and q that stood in for sentences. And when we were discussing categorical logic, we used variables S, P, and M that stood in for categories of things. For example, we might fill in p with Whiskers is a cat and P with is a cat or the property of catness. The logic of arguments that use sentence variables is called sentence logic or propositional logic. In this subsection, we ll take a look at a few other common argument patterns from sentence logic. One fairly common pattern from sentence logic is disjunctive syllogism: (1) either p or q (2) not-p (3) q (4) Either Obama is President or McCain is President. (5) McCain isn t President. (6) Obama is President. (You can replace (2) with not-q to conclude p compare the pattern to the example below.) This pattern is deductively valid. A related pattern is called affirming a disjunct: (7) Either I ll have Mexican for dinner or I ll have sushi for dinner. (8) I ll have sushi for dinner. (9) I won t have Mexican for dinner. (10) either p or q (11) p (12) not-q (Again, the order doesn t matter.) As with the conditional, there are several ways to understand the disjunction (either-or). English typically uses an exclusive disjunction: either p or q but not both. The example above makes the most sense with an exclusive 17

18 disjunction. Since p (having Mexican) excludes q (having sushi), if p is true then q must be false. With an exclusive disjunction, this pattern is deductively valid. But sometimes we use an inclusive disjunction: either p or q but maybe also both. We might have some of fusion Mexican sushi. Here s another example: (13) Either I ll study hard or I ll fail the test (and maybe also both, oh no!). (14) I ll study hard. (15) I won t fail the test. It would be nice if studying hard could guarantee passing, but unfortunately sometimes you end up with both. Since p (studying) doesn t exclude q (failing), we can t conclude that q is false just because p is true. With an inclusive disjunction, this pattern is invalid. So, in general, affirming a disjunct is questionable: you need to figure out which kind of disjunction the author is using before you evaluate the argument. Our last pattern from sentence logic combines conditionals and disjunctions. (16) Whiskers is either a cat or a bird. (17) If Whiskers is a cat then Whiskers is warm-blooded. (18) If Whiskers is a bird then Whiskers is warm-blooded. (19) Whiskers is warm-blooded. This pattern is called dilemma or proof by cases. It is valid. As a proof by cases, this pattern is used to show that something is true whatever the case: we re in case A or case B or.... And in each of these cases, p is true. So, whatever the case, p is true. The example above is a proof by cases. As a dilemma, this pattern is often used to show that, whatever we do, something bad will happen, as in this example: (20) Either we balance the budget or we let the debt continue to grow. (21) If we balance the budget then we damage the economy. (22) If we let the debt continue to grow then we damage the economy. (23) We damage the economy (whatever we do). Here s the pattern for a dilemma: (24) either p or q (25) if p then r (26) if q then r (27) r 2.4 Induction Induction is a mode of reasoning where we infer generalizations from particular cases. Arguments using induction are called inductive arguments. Often the particular cases are actual observations of some category of objects, and the generalization is meant to cover all objects of that type; this is called enumerative induction or simple induction. For example: (1) Robin #1 was observed eating worms. (2) Robin #2 was observed eating worms. (3) Robin #3 was observed eating worms. (4) All robins eat worms. 18

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