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§6: DEDUCTIVE LOGIC

Chapter 17: Deductive validity and invalidity

Ben Bayer Drafted April 25, 2010

Revised August 23, 2010 Deduction vs. induction reviewed In chapter 14, we spent a fair amount of time introducing the distinction between inductive and deductive reasoning. To explain the distinction, we made an analogy to the difference between the packing and unpacking of luggage. Induction is the process by which we “pack” together observations in a retainable cognitive form that is usable in the future. Deduction is the process of “unpacking” that knowledge, of using it in application to specific situations. Previously, we contrasted induction from deduction in order to focus on induction, the source of generalization and scientific knowledge. Now that we have already dwelled on that contrast, we will briefly dwell on some very basic principles of deductive reasoning in this, the last section of our book. We will discuss only the most basic principles of deduction, because many logicians have written a great deal explaining these principles elsewhere. From ancient Greece, where geometers perfected the first formal deductive systems and where Aristotle first reflected on the principles that governed these systems and human reasoning more generally—to the Middle Ages, when philosophers extended and perfected Aristotle’s logic—to the late 19th and early 20th century, when mathematicians and philosophers developed a new system that supplemented, and in some cases, may have highlighted limitations of Aristotle’s system—we can see that logicians understand the rules of deductive logic with far greater systematicity and precision than they do the rules of inductive logic. If you want to learn more about the rules of deductive logic, there are whole other texts and courses available on this subject alone.

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Recall that induction is defined as the form of inference that generates conclusions taken to be wider (more universal) in content than the observations contained in the premises, whereas deduction is the form of inference in which the conclusion states no more content than that stated in the premises (and is necessitated by them). Part of the reason deduction is understood so well is that it is confined to “unpacking” the content of the stated premises. Deduction works with a finite number of premises which contain a finite amount of information; as a consequence, there are only so many implications one can derive from them. The rules of deductive reasoning help to delimit precisely which implications one can derive, and how. This is in contrast to inductive inference, which works with an entire body of observations and background knowledge as its starting point, not all of which can easily be summarized in the form of stated premises. This makes it much harder to know what conclusions follow from the starting points of induction: it is, after all, very hard even to specify these starting points. None of this should be taken as disparaging the usefulness of deduction. The fact that its rules are well-understood does not mean that they are always easy to apply or that the conclusions of deductive arguments are obvious. Neither is true. Recall, for example, the deduction we used to solve the riddle in chapter 14 (“Brothers and sisters have I none, but this man’s father is my father’s son”). And though we cannot possible study the content or even all of the method used to perform them, we should mention just a few examples of sophisticated deductive argumentation that have helped scientists uncover surprising new facts about the world. Induction may be the indispensible fuel of human reasoning power, but deduction is where the rubber hits the road. Way back in chapter 1 when we were discussing the ancient evidence for the conclusion that the Earth is a sphere, we mentioned that the Greeks were even able to infer the size of the Earth from (literally) mundane observations to within a small margin of error of the correct conclusion. Their argument was deductive, beginning with premises taken from

trigonometry and observations of measured quantities. Here is Eratosthenes proof of the Earth’s circumference:

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1. The angle of the shadow at noon at Alexandria is 7.2˚ 2. Alternating interior angles are equal. 3. Therefore, the angle between Alexandria and Syene is 7.2˚ 4. The distance from Alexandria to Syene is 500 miles. 5. But (Circumference of earth/500 miles)=360˚/7.2˚ 6. Therefore Circumference of earth = 25,000 miles. 7. But Circumference/π = Diameter, 25,000/3.14 = 8,000 miles. 8. Therefore, the diameter of earth is 8,000 miles.

Using knowledge of the size of the Earth, and a related argument for the size of the moon, the Greeks were even able to determine the distance to the sun:

1. The angle between the sun and the moon during a half moon is 87˚

2. Cos 87˚ = moon distance/sun distance 3. Sun distance/moon distance = 1/cos 87˚ =

19 4. Therefore, the sun’s distance from the earth

is 19 times its distance to the moon. Through mere connections of deductive logic, the Greeks were able to travel millions of miles with their minds. A final example of the amazing power of deductive reasoning is so complicated that we cannot even represent the argument in symbolic terms here that the reader is likely to understand. Using fundamental laws of electricity and magnetism that he himself had formulated, James Clerk Maxwell, famously constructed a proof in 1864 showing that a change in a electrical field would induce a corresponding change in the magnetic field. But it was already known that the reverse was true, that a changing magnetic field would cause a change in the electrical field. It followed logically that the first would cause the second, which would in turn cause the first, and then the second, etc. Maxwell was predicting the existence of what we now call electromagnetic waves. He even predicted their speed, which corresponded closely to the known speed of light: Maxwell had shown that light was an electromagnetic wave. His discovery was the basis of countless innovations in 20th century electronics.

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We cannot hope even to begin to teach you how to be an Eratosthenes or a Maxwell. The deductions you will learn to perform

will be comparable simple. But you have to walk with simple syllogisms before you can run with whole chains of sophisticated deduction. Two requirements of reasoning in general, and of deduction in particular Before we describe some basic rules of deductive reasoning and methods of evaluating the quality of deductive argument, it is worth reviewing the most general requirements of reasoning of any kind. Recall our three basic rules:

1. The premises of the argument must be known and known better than the conclusion (they must be well-evidenced).

2. The premises must be relevant to the conclusion (they must be likely to establish the conclusion as true).

3. The argument’s premises must contain all of the known relevant evidence.

The first requirement applies to deductive arguments no differently than it applies to inductive ones. Premises can be shown to be well-evidenced if they are either derived from some form of direct awareness, or from some further premises that are themselves derived from a form of direct awareness. Either way, the premises must be ultimately derived from some form of direct knowledge, like sensory observation. We discussed this requirement in more detail in chapter 3. Regarding the second requirement, in good non-deductive arguments premises are relevant to their conclusion when they are such that if they are true, the conclusion is at the very least highly likely to be true. (Some might suggest that inductive arguments can still establish conclusions that are made necessary not only by stated premises, but by the totality of unstated background knowledge.) The relevance relationship between the premises and conclusion of a good deductive argument, on the other hand, is of such a special type of relevance that if these premises are true, the conclusion absolutely must be true, and this follows from nothing more than its premises. We will explore this special type of relevance in the next section: it is called validity.

Picture credit 1: http://commons.wikimedia.org/wiki/File:Onde_electromagnetique.svg

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Back in chapter 7, we also discussed how the third requirement applies differently to inductive and deductive reasoning. The whole function of deductive inference is just to unpack the implications of some stated premises. If a given conclusion is not an implication of these premises, the argument containing the conclusion is simply not valid. If the conclusion is an implication of these premises, the argument containing it is valid, and no assessment of other relevant evidence is needed to see this: those premises are all the relevant evidence there is. This is not the case in inductive inference, which can establish conclusions as at least somewhat probable independently of other considerations, and the probability of an inductive conclusion can change for better or worse as more evidence is brought in. All of this, however, is dependent on understanding the nature of the difference between deductive and inductive relevance. We will now focus exclusively on the topic of deductive relevance, i.e., validity. As a preliminary introduction to the uniqueness of deductive relevance, consider the following two arguments. Which one is inductive, and which is deductive? Is there any difference between the kinds of relevance you see in each?

All rain is from condensed cloud vapor. Rain is falling in New Orleans. Therefore, cloud vapor is condensing in New Orleans.

We know that water evaporates when it is hot. We know that water condenses on cool surfaces. We see clouds present whenever it rains. All rain is from condensed cloud vapor.

Deductive validity, soundness, and invalidity The name given to deductive relevance is “validity.” An argument is said to be deductively valid if its conclusion cannot be false if its premises are assumed to be true; i.e., it is such that if its premises are true, its conclusion must be true. A key part of this definition is that validity is a fact about arguments conditioned on the assumption that their premises are true. It is purely an assessment of the relevance of these premises, independent of whether or not they are really true. Judging validity involves a kind of pretense: we pretend that the premises are true, and see what follows—even if we know they are not. Validity, then, is distinct from soundness: an argument is said to be deductively sound just in case its conclusion follows validly from premises,

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when all these premises are known to be true. Every sound argument is valid, but not every valid argument is sound. An argument can be valid even if its premises are known to be false: what makes it valid is that it is such that if the premises were true, they conclusion would be true. We need examples to bring out this distinction. Our favorite stock example of a deductively valid argument is also an example of a deductively sound argument:

All men are mortal. Socrates was a man. Therefore, Socrates was mortal

As we demonstrated in chapter 14 through the use of simple circle diagrams, if these premises are true, the conclusion has to be true. What makes this argument sound in addition to valid is that the relationship between the premises and conclusion is not merely hypothetical: we do accept that these premises are true, and hence that the conclusion is, as well. Contrast the Socrates argument with the following pair of valid but unsound arguments:

All men are immortal. Fido is a man. Therefore, Fido is immortal.

All dogs are philosophers. Socrates is a dog. Therefore, Socrates is a philosopher.

Suppose that Fido, per his name, really is a dog. In that case, the first argument scores a trifecta of unsoundness: each and every statement, both premises and the conclusion, is false. But the argument is valid. If all men were immortal and Fido were a man, he would be immortal. The same can be said for the second example, though in this case, we have two false premises in effect cancelling each other out and yielding a true conclusion. You should think of deductive validity as a testing a purely hypothetical relationship between premises and conclusion. When we evaluate validity, we are, in a way, performing a test in our imagination. But it is a test in the imagination that is constrained rigorously by our premises

And surely this instinct of the dog is very charming;—your dog is a true philosopher. Why? Why, because he distinguishes the face of a friend and of an enemy only by the criterion of knowing and not knowing. And must not an animal be a lover of learning who determines what he likes and dislikes by the test of knowledge and ignorance? Most assuredly. And is not the love of learning the love of wisdom, which is philosophy? --Plato, Republic, Book II

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and our understanding of the meaning of any statements of the form “S is P”: if All S is P, and Fido is an S, then Fido must also be a P. Deductive validity involves a special kind of relevance because its relevance is based strictly on the form of its premises and conclusion. We will say more about what this means shortly. If you really understand the concept of deductive relevance, i.e. of validity, you will appreciate why the following is a fairly clear example of an invalid argument:

All men are mortal. All mice are mortal. Therefore, all mice are men.

We know both of the premises to be true. But the conclusion is quite obviously false. Recall: a valid argument is one whose conclusion cannot be false if its premises are assumed to be true. In this case, not only can we assume the premises to be true, we know they are! So we know that the truth of this conclusion could not follow from assuming the truth of these premises, because we know the conclusion is false, even as we know that the premises are true. Examples like this, in which we know the premises are true but the conclusion false, are case studies of invalid arguments. Knowing that these are the most obvious case studies proves to be useful for evaluating arguments as invalid even when we do not know whether their premises and conclusion are true or false. If you really, really understand the concept of validity, you will even be able to discern why the following are examples of invalid arguments:

All people are animals. All animals move. Therefore, all philosophers are people.

This example is tricky to someone who is new to the rules of validity, because each premise is true—and so is the conclusion. Yet it is invalid. The premises “All people are animals” and “All animals move” says something about animals, and we might think this is a common factor that could yield some new conclusion. But the conclusion, “All philosophers are people,” is one which, Picture credit 2:

http://picasaweb.google.com/lh/photo/zXenfB_2m42RKpV2ORj-Zwhttp://picasaweb.google.com/lh/photo/zXenfB_2m42RKpV2ORj-Zw

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though we know to be true, seems to have nothing at all to do with the premises. The premises, after all, said nothing about philosophers. If you understand the concept of deductive validity, you should now clearly see why validity is entirely a matter of the relationship of relevance between the premises and the conclusion. In this argument, all of the premises are true, as is the conclusion, but the premises do not make the conclusion true. Nothing links the premises to the conclusion. As a final test of your understanding of validity, consider a second example:

All human beings are mortal All philosophers are mortal. Therefore, all philosophers are human beings

Unlike the previous example, there is no new concept that appears in the conclusion that makes it obviously irrelevant to draw. The conclusion mentions something about philosophers and their relationship to people, and both were mentioned in the premises. Making things trickier still is the fact that both premises are true, as is the conclusion. But if you remember that not every set of true premises is related to every true conclusion, you should remember that these true premises do not necessarily make this true conclusion true. We could imagine a situation in which, though the premises are true, the conclusion is false. Suppose that all people are mortal, as are all philosophers. But it turns out that some philosophers are mortal Martians. In this case, it would not follow that all philosophers are people. Even though the present conclusion is true given the facts we know about philosophers, we would not be averse to changing it if we discovered new kinds of philosophers in the universe, even if we continued to accept the premises as true. If, in order for an argument to be sound, it must be both valid and in possession of true premises, it follows that every invalid argument is also unsound (though some unsound arguments are valid). All of the possible combinations are summarized on the following table:

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As you can see, no special combination of premise types (true or false) will make an argument valid. Any of the four possible combinations in the first column can be invalid. Of the valid arguments, only one possibility allows the argument to be sound: the case in which the argument has true premises and a true conclusion. Of course there are examples of valid arguments where this does not hold: valid arguments can have false premises with a true conclusion, or false premises with a false conclusion. What makes them valid is that if their premises were true, their conclusion would be as well. And all of the invalid arguments are of course also unsound: to be sound an argument must be valid as well. With these distinctions in mind, can you tell which if any of the following arguments are valid? And which of the valid ones are also sound?

All U.S. senators are federal representatives. All federal representatives work in Washington, DC. Therefore, all U.S. senators work in Washington, DC

Some politicians are promise-makers. Some promise-makers are sincere. Therefore, some politicians are ambitious. Some politicians are liars. Some liars are killers. Some politicians are killers. Some politicians are truth-tellers. Some truth-tellers are fools. Therefore, some politicians are fools.

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The counterexample method of testing validity How easy was it for you to tell that any of the above arguments were invalid? It is particularly hard to tell in cases where you think that the premises and the conclusion are true. How can you tell whether or not the true premises make the conclusion true? In the last section, we mentioned a way of evaluating this by trying to think of a situation in which the premises could be true but the conclusion false. If we think of the way in which true premises make conclusions true as analogous to a kind of cause and effect relationship, then when are looking for an example in which the premises might be true but the conclusion false, we are doing something like applying Mill’s negative method of difference for refuting causal claims. We’re looking for a case in which the alleged cause (the truth of the premises) is present, but the effect (the truth of the conclusion) is absent. Any time we find an argument with true premises and a false conclusion, we know it cannot be valid for this reason. This is reflected by the one block missing examples in our table of different possible argument types:

What if we simply can’t think of a situation in which the given premises are true, but the conclusion false? What if we don’t even know when the premises of a given argument are true and its conclusion false? It can be difficult to know how to do either, and our failure to do either doesn’t show that the argument is valid; we might simply lack imagination. We need a more systematic method of finding counterexamples, so that if we fail even

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after applying this method, it is at least more likely that what we have in the end is a valid argument. We need a method that helps us transform arguments into equivalents for which counterexamples are easier to find. There is such a method that works with some understanding of what it is it about a set of premises that makes a conclusion true. Consider a case where true premises obviously do make the conclusion true.

All men are mortal All mortals are living All men are living.

What makes these premises guarantee the truth of the conclusion? It is something about the abstract pattern of the argument. Notice in this case that there are terms in the premises that also appear in the conclusion (“men . . . . living”). A third term (“mortal”) is common to both premises, but missing from the conclusion. And there is something about the order in which these terms appear that is important. After all, the following argument with the exact same terms would clearly not be valid:

All men are mortal All living things are mortal. All living things are men.

The only difference between this argument and the previous is that the terms in the second premise and the conclusion have been reversed. The second premise is still true, but the conclusion is false.

So there is something about the abstract pattern of the terms in the premises and conclusion that makes one argument valid and the other invalid. So what happens when we turn to an entirely different argument, composed of different terms, but possessing the same abstract pattern, the same form? An argument of exactly the same form would also have the same deductive validity—and, we presume, because it has the same form. For example:

All vampires are immortal. All immortals are undead. All vampires are undead.

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What exactly do we mean by the “form” of an argument? The form of a statue, for example, is its shape, as opposed to its matter or stuff. We can make copies of a statue using many different materials—marble, bronze, plaster, etc.—but they are copies of that statue as long as they are of the same shape. When we speak of the form of an argument, we mean the “shape” of the statements composing it, the order in which various terms appear in premises, and the order of the same terms in the conclusion. If it is the form of an argument that allows its premises to determine the truth of its conclusion of a

deductive argument, perhaps having the wrong form could account for the invalidity of deductive arguments. Let’s look at an example of an argument that we can agree is obviously invalid, with premises that are obviously true and a conclusion that is obviously false: All men are mortal. All mice are mortal. All mice are men. If it is an argument’s form that allows the truth of the premises

to guarantee the truth of its conclusion, what is the form of the argument above? We represent it this way: All A are B. All C are B. All C are A. We know that an argument of this form can include true premises and a false conclusion, because the “All mice or men” argument is of this form. So if any other argument has this same form, we know that nothing about its form guarantees that true premises must be accompanied by true conclusions. And since form is the source of deductive validity, missing the right form means the argument cannot be valid.

Picture credit 4: http://www.flickr.com/photos/su-lin/266340072/

Picture credit 5: http://www.flickr.com/photos/jerry7171/15315121/

Picture credit 3: http://www.flickr.com/photos/blondavenger/2455815329/sizes/m/in/photostream/

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Suppose, for example, that we were asked to evaluate the validity of these arguments:

All whales are cetaceans. All dolphins are cetaceans. All dolphins are whales. All derivatives are options. All futures are options. All futures are derivatives. All naiads are fairies. All dryads are fairies. All dryads are naiads.

Notice that each of these arguments has the same “bad form” as the “All mice are men” argument. If the abstract form of that argument is consistent with having true premises and a false conclusion, we know that nothing about the form of these arguments will guarantee a true conclusion. So even if we don’t know what a cetacean is, what the difference between a derivative and an option is, or the meaning of any of the terms in the last argument, we can still know—without knowing whether any of these statements are true or false—that these arguments simply lack the necessary form to be valid. They lack the necessary form, because they are all instances of the form “All A are B, All C are B; Therefore all C are A.” These observations help give us a recipe for finding counterexamples that prove the invalidity of many arguments. We know that if an argument has true premises and a false conclusion, it cannot be valid. But if the form of another argument would permit the substitution of the obviously invalid argument, we know that that the other argument itself does not have the form that it needs to be valid. Here, then, are the steps by which this counterexample method works.

1. First, we break the argument down into premises and conclusions. Suppose we want to test the following argument for validity:

Some traders are not brokers. All buyers are traders. Some buyers are not brokers.

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2. Next, we find the abstract form of the argument, as follows:

Some A are not B. All C are A. Some C are not B.

3. Finally, we find substitution instances for each of these abstract variables that amount to an argument in which the premises are obviously true, and the conclusion obviously false. Here is one such example:

Some animals are not mammals. All dogs are animals. Some dogs are not mammals.

It is useful, for the purposes of the counterexamples involved in this method, to work with concepts like animals, mammals, and dogs, for which we already have clear knowledge of the category relationships in question. We know that it is obviously true that animals are not mammals if we know of fish and birds. We know it is true that all dogs are animals (none are plants). And yet, we know that the conclusion here is false: it’s false that some dogs are not mammals. Does someone think they are reptiles or amphibians? Since the argument about traders, brokers, and buyers has the form that permits this obviously invalid argument, nothing about the form of the traders/brokers/buyers argument supports its validity. We can conclude it is invalid.

This three-step method helps clarify the concept of validity for us, especially insofar as it may have been difficult to detect when arguments containing actually true premises and actually true conclusions could still fail to be valid. Here we return to the example of this type of argument mentioned before:

Some politicians are truth-tellers. Some truth-tellers are fools. Therefore, some politicians are fools.

The abstract form of this argument is as follows:

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Some As are Bs. Some Bs are Cs. Therefore, some As are Cs.

But there are fairly straightforward counterexamples (and probably many more) to any argument utilizing this pattern: Some friends are unhappy people. Some unhappy people are enemies. Therefore, some friends are enemies. Consider a final analogy to help explain how an argument’s form helps determine its validity. Arguments are not the only things that have a form. Anything with a physical shape also has a form (as our earlier example of statues brought out). Sometimes in matters outside of logic, the physical form or shape of a thing can also determine whether or not it has a relevant property or power to get something done. To take a somewhat curious example, the shape of a thing can determine what kinds of spaces it can fit through. Suppose we have a triangular hole of a specific size cut into a wall. Non-triangular objects greater than a specific surface area could not fit through the hole, because of its shape. Suppose we use a bundt cake mold to create circular objects of a larger size. Well then the circular form of the cake mold does not make objects of the right size to fit through this triangular hole:

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But suppose that we can fill that cake mold with many different substances. We can fill it with water and freeze it, or with concrete mix, or with Jello. Many of these substances will yield “cakes” that will be too solid to fit through the hole. Using Jello, the size and shape will still be wrong to fit easily through the hole, but perhaps the right kind of squeezing can still accomplish the task. The Jello example is analogous to the examples above in which argument’s form is invalid even if the premises and conclusion are true. In these cases, there is nothing about the form of the argument that transmits truth from the premises to the conclusion. Still, the conclusion is true, but it’s true as a matter of luck. Likewise, in this example, it’s not the form of the cake as supplied by the bundt that allows it to pass through the triangular hole. All things being equal it wouldn’t have so passed. But by chance, the material in this case is squishy enough that it can pass through after all, just like a conclusion might have been true by chance. Sadly, this doesn’t mean that testing validity is always a piece of cake…. Here, then, for final practice, is another example of an argument that may be tested for validity using this counterexample method: Some philosophers are not poets, for some singers are not poets (P) and some poets (P) that are not singers (S) are philosophers. Which of the following best captures the form of this argument?

1. Some S are not P. Some P are F. Some F are not P.

2. Some P that are not S are F. Some P are not F. Some S are not P.

3. Some S are not P. Some P that are not S are F. Some F are not P.

4. Some F are not P. Some S are not P. Some P that are not S are F.

5. Some F are not P. Some S are not P. All P that are not S are F.

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If you look carefully, you will see that it has to be 3. So which of the following substitutions gives us a counterexample, a case in which we have two true premises and a false conclusion which shows that the argument as presented is not valid?:

1. P = mammals, S = animals, F = dogs. 2. P = cats, S = trees, F = animals. 3. P = mammals, S = cats, F = animals. 4. P = animals, S = trees, F = cats. 5. P = fish, S = dogs, F = mammals.