How to Think About Deductive Logic

Logical consequences are the . . . beacons of wise men.

—T. H. Huxley, Science and Culture

Insanity is often the logic of an accurate mind overtasked.

—Oliver Wendell Holmes, The Autocrat of the Breakfast Table

The logic of an argument is the reasonableness conferred on the argument’s conclusion by its premises. In an argument that is logically successful the conclusion follows from the premises—or, to put it differently, the premises support the conclusion.

Although we often use the term logical as a synonym for reasonable, we are clearly using it in a narrower way in this text, since good reasoning requires more than a certain kind of relationship between premises and conclusion. What does logic have to do with reasoning? Recall that reasoning is the thinking we do to answer questions that interest us; it is modeled by arguments—good reasoning by good arguments, bad reasoning by bad ones. Good logic is one of the merits of arguments; and good logic is important, since we need to understand how it is that beliefs of ours are supported by others that we judge to be true. But logic is only part of the story. We must also judge whether the premises are true; further, we must judge whether the argument is relevant to the conversation that gave rise to it. And the argument must be clear enough for us to be able to tell. An argument is a model of good reasoning only when it exhibits all four of these merits—not merely good logic.

When used properly, as Huxley notes, logic can serve as a beacon for the wise. But when we rely on it to the exclusion of the other merits of arguments, then, as Oliver Wendell Holmes suggests, at its very worst it can tidily sever our connection to reality.

10.1 Deduction and Induction

For any argument, the best way to think about its logic is to ask this question: If the premises were true, would that make it reasonable to believe the conclusion? This is roughly the same thing as asking any of these questions:

Is the argument’s logic good?
Does the conclusion follow from the premises?
Do the premises support the conclusion?

Logic is traditionally divided into two broad categories according to the level of support the argument aims to provide the conclusion. In deductive arguments, the premises are intended to guarantee, or make certain, the conclusion. To determine whether the logic of a deductive argument is successful, a good rule of thumb is to ask questions such as these:

Do the premises guarantee the conclusion?
If the premises were true, would that make the conclusion certain?

For example, in the deductive argument All men are mortal, Socrates is a man, so Socrates is mortal, it is easy to see that the truth of the premises would make certain the conclusion, and thus that it is logically successful.

In inductive arguments, however, the premises are intended merely to count toward, or make probable, the conclusion. To determine whether the logic of an inductive argument is successful, a good rule of thumb is to ask these questions:

Do the premises count toward the conclusion?
If the premises were true, would that make the conclusion probable?

Take, for example, the inductive argument All the men that I know are mortal; therefore, all men are mortal. The premise certainly seems to count toward the conclusion—although it is hard to say how much. It is easy to say, however, that despite its counting toward the conclusion, it cannot make the conclusion certain.

If you read more broadly on this topic, you will find that a few authors adopt different terminology. Deductive logic is sometimes referred to as demonstrative or apodictic logic (apodictic is from a Greek word for demonstrative) while inductive logic is sometimes referred to as nondemonstrative or ampliative logic (ampliative because the conclusion amplifies, or adds to, the premises). Furthermore, the boundaries are sometimes drawn in different ways. The terms deductive and inductive are, for example, sometimes strictly reserved for arguments in which the logic succeeds. We will use the terms more broadly, however, allowing for deductions and inductions that fail as well as for those that succeed.^[1] Finally, there is the common but mistaken definition of deduction as “reasoning from the general to the particular” and of induction as “reasoning from the particular to the general.” Some deductive arguments do move from the general to the particular (our familiar All men are mortal, Socrates is a man, so Socrates is mortal, for example), but here is a simple deductive argument that moves from the particular to the general:

This marble is red. That one is also red. And that one is too.
These are all the marbles. Therefore, all the marbles are red.

And many others go from the general to the general or the particular to the particular. The mark of deduction is simply the aim for a conclusion guaranteed by the premises.

Likewise, some inductive arguments do move from the particular to the general (the preceding argument, for example that All the men that I know are mortal, therefore all men are mortal). But here is a simple one that moves from the general to the particular:

Most men are mortal. Socrates is a man. So, Socrates is mortal.

All that is required for induction is simply the aim for a conclusion merely made probable by the premises.

You should be aware of this lack of unanimity so that you are not puzzled if you find variant accounts when you read other sources. The account in this text aims to provide the best mix of accuracy, practicability, and common usage.

10.2 Deductive Validity

Since a deductive argument is one in which the premises are intended to guarantee the conclusion, a logically successful deductive argument is one in which this guarantee is achieved. In looking at a particular argument, does it seem as though the argument’s conclusion would be made certain if the premises were assumed to be true? Then, chances are, you are looking at a deductive argument that is logically successful. That is the case with this argument:

A logically successful deductive argument such as this is valid. We will call an argument valid if and only if it is impossible for an argument with such a form to have true premises and a false conclusion. Conversely, it is invalid if and only if it is possible for an argument with such a form to have true premises and a false conclusion. Validity, therefore, is a perfect preserver of truth. If you want to be sure of true conclusions then find a valid form, feed in true premises (setting aside how to be sure that the premises are true!), and out will come a true conclusion.

There are two mistakes, however, that you should avoid. First, resist the temptation to think that validity also perfectly preserves falsity. It does not. A valid argument with false premises can still have a true conclusion. Note the following argument:

This is indeed a valid argument with false premises and a false conclusion. But with one small adjustment we get the following argument:

The argument is still valid and still has false premises; but now the conclusion is true. Validity does not perfectly preserve falsity.

Second, don’t jump to the conclusion that every argument with true premises and a true conclusion is valid. Suppose I more or less indiscriminately take four sentences that everyone would agree are true:

1. George Washington was the first president of the United States.
2. Triangles have three sides.
3. Three plus one equals four.
4. Dogs normally have four legs.

All it takes to have an argument  is for at least one statement to be offered as reason to believe another statement. The reason does not have to be a good one. All I need to do is argue as follows:

You wonder how many legs dogs usually have? Well, surely you know that George Washington was president number one. Combine that with the fact that triangles have three sides. Since one plus three is four, it follows that dogs have four legs.

Silly (though I’ve seen equally silly arguments offered by believers in numerology), but it is an argument. Furthermore, the premises and the conclusion are all true. But it clearly is not valid.

The lesson is this: in an invalid argument, you can find any combination of truth-values in the premises and conclusion. And in a valid argument you can likewise find any combination of truth-values in the premises and conclusion—but with one major exception. A valid argument—by definition—cannot have true premises and a false conclusion.

10.3 Validity Counterexamples

If you know, then, that an argument has true premises but a false conclusion, you know that the argument is invalid. Similarly, if you know that the materials in a building are good but the building collapses anyway, you know the problem must be in the way the materials were put together.

But it is seldom so easy. More often than not you must decide about validity when the conclusion is true, or when a premise is false, or when you are unsure about whether they are true or false. In many cases you will recognize the form as one that has been introduced and named in this text; if so, provide the name of the valid or invalid form as part of your defense of your judgment. But for any deductive argument that is invalid, even if you can provide the name of the invalid form, you should also provide a validity counterexample. This is a simple two-step method for checking any argument for validity. The first step is to extract the form that the argument is depending on for logical success (using the principles described in Chapter 6). The second is to attempt to construct a new argument by appropriately substituting new sentences, predicates, or names in a way that produces obviously true premises and an obviously false conclusion. If you can thus use the argument’s form to create a new argument with true premises and a false conclusion, then you have shown that it is possible for an argument with this form to have true premises and false conclusion. You in this way show the argument to be invalid. But if you cannot do this, you have a case for the argument’s validity.

Let us return to Socrates:

The first step in constructing a validity counterexample, extracting its form, yields this:

The second step is to produce obviously true premises and an obviously false conclusion by substituting a new property for F, a new property for G, and a different name for A. Alas, it cannot be done. This is a good reason to conclude that the original argument is deductively valid. But try the same thing on this variation:

The first step yields this:

In this case, the second step is easy. Try these assignments for the variables:

F: ponds
G: bodies of water
A: Atlantic Ocean

This yields the following argument:

Since this argument uses the form depended on by the original argument, yet has obviously true premises and an obviously false conclusion, it shows that it is possible for the form to have true premises and a false conclusion. Thus, it shows the original argument to be invalid.

Consider an argument from the philosopher Descartes; the question is whether your mind is nothing more than a part of your body:

If mind and body are one and the same, then mind (like body) is divisible. However, the mind cannot be divided into parts. Consequently, mind and body are not the same.

If we attempt to provide a validity counterexample, we first extract the form that it seems to depend on, namely, this:

We then look for sentences to substitute for P and Q that will produce obviously true premises and an obviously false conclusion. It cannot be done, for the argument is valid.

For the sake of example, suppose that Descartes had been concerned with a different question, and had argued thus:

If mind and body are one and the same, then mind (like body) is divisible. But mind and body are not the same thing. Consequently, mind cannot be divided into parts.

Is this new argument valid? The answer is not immediately apparent. Let’s try to produce a validity counterexample. First, the form it seems to depend on is this:

Can we substitute for P and Q in a way that produces obviously true premises and an obviously false conclusion? Easily. Consider the following argument:

The premises are clearly true, the conclusion clearly false. The argument in question is shown to be invalid, since it has been shown that it is possible for an argument with its form to have true premises and a false conclusion.^[2]

10.3.1 The Value and the Limitations of Validity Counterexamples

Validity counterexamples can be a powerful tool. In this book you will be introduced only to the most common deductive forms. With this tool in hand, you will not only be able to see vividly the invalidity of the invalid ones in the book, but you will also be in the position to evaluate the logic of any deductive argument not included in the book.

Here, for example, is one such argument. There is an interesting passage in Descartes’ Meditations in which he points out that our senses sometimes deceive us; note, for example, mirages and hallucinations.^[3] Therefore, he says, it just could be that our senses always deceive us. Here is one attempt to clarify that argument.

The argument appears to be deductive—it looks as though the premise is offered as a guarantee of the conclusion—but it is an uncommon sort of argument and surely does not depend on any deductive form that is covered in this book. To test it by the method of validity counterexample, let us first extract what seems to be the logical form the argument is depending on:

It is possible that seems here to roughly mean, there is a way of imagining the world so that.

Having taken the first step, we now see if we can take the second. The first few things we try may fail. Try, for example, trees for F and evergreen for G. The premise Some trees are evergreen would be true. But it seems that the conclusion, It is possible that all trees are evergreen, would also be true. (They aren’t all evergreen, but there is a way of imagining the world such that they are.) But let’s try paintings for F and forgeries for G. That gives us the following argument.

The premise is clearly true; and the conclusion is false, since there cannot be a forgery unless there is at some time an original to be forged.

This is by no means the last word on Descartes’ argument—we may, for example, be somehow misunderstanding the form that Descartes is depending on, or he may not intend it to be a deductive argument. But the forgery argument provides good reason to think that if it is deductive, it is invalid. Note that even though I have provided a validity counterexample, I am not counting it as absolutely conclusive for showing that the argument is invalid. All we really show by a counterexample is that the form we have extracted from the argument is an invalid form. The possibility may remain that we have extracted the wrong form—that there is some other form, as yet undetected, that the argument is really depending on for its logical success. Take yet again our Socrates argument:

It is correct to say that this argument takes the following form:

That is, it is true that the argument is made up of three different sentences. But anyone can easily produce a counterexample to that form, substituting for P and Q any two obviously true sentences and for R any obviously false sentence.

The problem is that the form I have identified is not the form that the argument depends on for its logical success. For it is not the relationship among its complete sentences that the argument depends on for logical success, but the relationship among its various predicates and names. That is to say, referring back to Chapter 6, it does not depend on sentential logic but on predicate logic.

Just as in the case of truth, you are stuck with epistemic probabilities when it comes to your judgments about logic. It may be that the best anyone can do is to judge arguments as almost certainly valid or invalid. Although this is worth keeping in mind, since it is always worth being reminded that we might be making a mistake, there is ordinarily no need to hedge your judgments about logic in this way. Once you become comfortable with making judgments about validity, the level of probability will ordinarily be so high that it will make good practical sense to express your judgments simply as valid or invalid.

Note, on the other hand, that if you cannot find a counterexample, then that is good reason to judge an argument valid. But it is, at most, a good reason; it is not an absolutely conclusive reason, since your inability to find one may be better explained by your lack of imagination in thinking up ways to produce true premises and a false conclusion.

10.4 Some Valid Deductive Forms

A handful of deductive forms are so obviously valid that they almost never occur in ordinary arguments. They tend to be taken for granted. On the rare occasions that they are explicitly invoked, it is either for rhetorical purposes or because there is a special need for care in spelling out an argument.

One such form is repetition, in which the structure is this:

This occurs when a premise is simply repeated—perhaps disguised in different terminology—as the conclusion of an argument, as in the following:

Walking is healthy since it is good for you.

It is the most obvious case of validity we can find, since, for an argument with this form to have a true premise and false conclusion, it would have to violate the law of noncontradiction. Such arguments are typically bad ones—not because of any logical problem but because they usually commit the fallacy of begging the question.

Two obviously valid forms are found in both–and arguments. These are arguments that include a premise of the form P and Q, which we will term a both–and statement (sometimes known as a conjunction, though we will reserve that term for a valid form of argument). The statements that fit into the variables P and Q we will simply refer to as the both–and statement’s parts. (They are more formally called conjuncts.) Simplification and conjunction are valid both–and forms that are closely allied to repetition. Simplification takes this form:

The both–and premise asserts the truth of its two parts; the argument concludes that one of the two parts is true. Once clarified, a remark such as the following might be seen as taking this form:

It’s going to be rainy and cold tomorrow, so of course it’s going to be rainy.

Conjunction goes in the other direction. Its form is this:

As you can see, it conjoins two statements. An argument such as the following, once clarified, might be seen as depending on this form:

He’s 6’4″. His hair is black. So, there you have it—he’s tall and dark.

In each case, it should be obvious that true premises would make a false conclusion impossible.

Bear in mind that common stylistic variants for and may need to be translated, according to the guidelines of Chapter 6, into the standard constant for purposes of clarification. These include the following:

Translating the stylistic variant does not necessarily preserve all the meaning of the translated expression; it merely translates what matters from a strictly logical point of view. Suppose the conclusion above had been He’s tall but dark, expressed that way because I know you are looking for someone who is tall and blond. Translating but into and makes more vivid its logical role of conjoining He is tall with He is dark; but it loses the conversational role of signaling your likely disappointment.

Finally, there is double negation. To say it is not the case that the statement is false—where it is not the case is one negation and false doubles it—is ordinarily a complicated way of saying the statement is true. Suppose I assert, “It is false that exercise is good for you,” to which you may reply, “It is not false that exercise is good for you.” You might just as well have replied, “Exercise is good for you,” but you have communicated the same thing by doubly negating it. This can go in either direction:

Neither case lends itself to a validity counterexample; if the premise is true, so is the conclusion.

As already mentioned, these forms are so obvious that they seldom occur explicitly. And if they do occur, they are seldom interesting enough to warrant the trouble it takes to clarify them separately. So, although they are worth knowing about, it will usually make the best practical sense to eliminate them in the streamlining phase of the clarification process.

10.4.1 The Fallacies of Composition and of Division.

Two famous fallacies that date back to antiquity^[5] can typically best be seen as misguided applications of the valid forms of conjunction and simplification.

The fallacy of composition is the mistake of concluding that a property applies to the whole of something because it applies to all of its parts. My team is the best team because it has the best players might at first look like a good argument, although My book is a good book because it is made up of good words does not. But both commit the fallacy of composition. The words, regardless of how good they are, obviously have to work together in the right way to make the book a good one; so, likewise, must the players to make the best team. This is reminiscent of conjunction, but importantly different. A valid conjunction would go something like this:

Player A is the best shooting guard, player B is the best point guard, player C is the best center, player D is the best power forward, and player E is the best small forward, therefore Players A, B, C, D, and E are each the best.

It differs from the fallacy because there is no shifting of the property best from the players to the team; it applies only to players throughout the argument.

Economists, trying to avoid a similar trap in their field, have formulated this maxim:

The sum of all locally optimal decisions is not always globally optimal.

That is to say, even if each person is making decisions that are in that person’s best interest (they are “locally optimal”), it doesn’t add up to what is best for society (what is “globally optimal”). We must sometimes sacrifice our own best interest if we are to serve the larger interest. Those who miss this point commit the fallacy of composition.

The fallacy of division is the reverse—it is the mistake of concluding that a property applies to one or more of the parts because it applies to the whole. My team is the best so my center is the best is an example. It does seem a great deal like simplification. But the closest valid simplification might look like this: Each of my players is the best so my center is the best. In the valid version the property best does not shift from team to player; it is applied to the same thing in both premise and conclusion.

10.5 Evaluating the Truth of Premises with Not or And

We have covered two logical constants in this chapter: not and and. We will briefly consider whether there is anything special to think about when evaluating the truth of premises that include them.

10.5.1 Negative Premises

Negative premises and both–and premises are, for the most part, uncomplicated. Negation is typically a simple on-off switch. Add it is not the case to the front of a statement and its truth-value is reversed. Dolphins are mammals is true. So, It is not the case that dolphins are mammals is false.

There are traps, however, that you should avoid. You may, for reasons of style, choose to put not somewhere inside the sentence rather than tacking It is not the case to the front of it. Attitude contexts, which report someone’s attitude—what someone believes, feels, or wants—present one such trap. It is not the case that she believes you are guilty means one thing, while She believes that it is not the case that you are guilty means something else. The second version does not allow for the possibility that she has no view on the question of your guilt. Modal contexts, which state modes such as probability, possibility, and necessity, provide another opportunity for caution. It is not possible that you are guilty means one thing, while It is possible that you are not guilty has quite a different meaning.

10.5.2 Both–and Premises

Both–and statements are also usually straightforward. If you are almost certain of each part that it is true, then you should judge the both–and statement as almost certainly true. If even one part is almost certainly false, then the both–and statement is almost certainly false.

It is not so straightforward, however, when you can say of the parts merely that they are probably true or false. Usually you can arrive at the probability of the both–and statement by applying this simple rule: multiply the probabilities of the parts. Suppose your plans for tomorrow depend on two things: good weather and your ability to get off work. You are interested in whether the following both–and statement is true:

Tomorrow’s weather will be good and I can get off work tomorrow.

You believe that each part is probably true; the TV forecaster said there was roughly a 35 percent chance of showers, and your boss lets people off approximately 3 out of every 4 times they ask. This means there is about a .65 probability for Tomorrow’s weather will be good and about a .75 probability for I can get off work tomorrow. Multiplying the probabilities of the two parts, you find that the probability of the both–and statement is in the neighborhood of a mere .49. This is in the same neighborhood as .50, so you can’t decide whether the both–and statement is true. Take special note of this: even if you judge the parts to be fairly probable, you might find that the probability of the both–and statement is .50 or below.

The simple rule of multiplying the probabilities of the parts, however, doesn’t work when the truth of one part would affect the probability of the other part.^[6] Suppose you work for a company that lays cement. There is more work when the weather is good. So even though the boss generally lets people off about 3 in every 4 times, the chances of getting a day off in good weather drop to about 1 in 2. There is a broader rule that applies here (and it encompasses the simpler situation as well): when you multiply the probabilities of the parts, for the affected part, use the probability that the part would have if the other part were true. So, in this case, multiply .65 (the probability for Tomorrow’s weather will be good) by .50 (the 1-in-2 probability for I will be able to get off tomorrow when I assume that tomorrow’s weather will be good). It may be time to start thinking about changing your plans.

You will usually have to make an educated guess about probability assignments. You might not have specific information about the frequency probabilities of the parts or, even if you do, you might have additional information that bears on the probabilities of the parts. It can still be helpful to convert these judgments temporarily into numbers so that you can be guided by the rules of probability. Suppose that after you heard the weather forecast you saw some clouds rolling in, making it less probable than the predicted .65 that tomorrow’s weather will be good. The most you can now say is that you can’t decide; but tentatively call it .50. And suppose you know the boss is in a bad mood this week, meaning that his general practice of letting people off about 1 in every 2 times in good weather is overoptimistic. You’re not sure how overoptimistic, but tentatively call it a .30 probability that he will let you off on the assumption that the weather is indeed good. Multiplying these two numbers produces a .15 probability for the both–and statement. This is misleadingly precise; but it does vividly show that you have strong grounds for saying that the both–and statement is very probably false.

10.6 Summary of Chapter Ten

Good logic, which is one of several criteria for good reasoning, is present when an argument’s premises (whether true or not) support its conclusions, or, alternatively, when its conclusion follows from its premises. Deductive logic has to do with those arguments that aim to make certain, or guarantee, their conclusions; inductive logic has to do with those arguments that aim merely to make probable, or count toward, their conclusions. Later chapters will introduce various forms of each sort, making it easy to keep them straight.

A successful deductive argument is valid, meaning that it depends on a form such that it is impossible for an argument with that form to have true premises and a false conclusion. A validity counterexample can provide a useful—though not perfect—test for validity. It first extracts the form the argument depends on and, second, makes substitutions for all variables in a way that produces an argument with obviously true premises and an obviously false conclusion. When you evaluate an argument in this text, you should provide a validity counterexample for every deductive argument that is invalid. Also, if there is a name for the invalid form, you should state the name.

The most obviously valid deductive forms—which clearly do not lend themselves to attack by validity counterexample—include repetition, double negation, simplification, and conjunction. Their logic is simple, but evaluating the truth of their premises—especially in the case of both–and statements—can be helped by special rules regarding the epistemic probabilities of the parts.

10.7 Guidelines for Chapter Ten

• An argument with true premises and a false conclusion should be judged invalid. Every other combination of truth-values in the premises and conclusion can occur in either a valid or invalid argument.
• Demonstrate invalidity by creating a validity counterexample, which illustrates that it is possible for an argument with such a form to have true premises and a false conclusion.
• Although the method of validity counterexample is very useful, it isn’t perfect. Failure to come up with a counterexample could be due to lack of imagination. Success in coming up with a counterexample could be due to overlooking the actual form depended on by the argument.
• The most obviously valid forms of deductive logic—such as repetition, simplification, conjunction, and double negation—can normally simply be paraphrased away when clarifying an argument.
• Beware of the fallacies of composition and of division, which are patterned closely after the valid forms of conjunction and simplification. They are invalid, because the property shifts in application from the part to the whole (in composition) or from the whole to the part (in
  division).
• Negating a statement reverses its truth-value; be very careful, however, about placing the negation inside the statement, especially in attitude and modal contexts.
• When the two parts of a both–and statement are merely probable, tentatively assign them a probability (even if the result is misleadingly precise) so that you can apply the rules of probability. Convert the numbers back into everyday language for your final evaluation.

10.8 Glossary for Chapter Ten

Both–and argument—one of a loosely defined group of deductive arguments that have a both–and statement as a premise.

Both–and statement—a statement of the form P and Q. Also called a conjunction, though we are reserving this term for a valid deductive form.

Conjunction—valid deductive form, as follows:

The term is also sometimes used for a both–and statement.

Deductive argument—an argument in which the premises are intended to guarantee, or make certain, the conclusion. To determine whether the logic of a deductive argument is successful, a good rule of thumb is to ask questions such as these:

Do the premises guarantee the conclusion?
If the premises were true, would that make the conclusion certain?

Alternatively termed apodictic or demonstrative argument.

Double negation—valid deductive form, as follows:

Fallacy of composition—the mistake of concluding that a property applies to the whole of something because it applies to each of its parts.

Fallacy of division—the mistake of concluding that a property applies to one or more of the parts because it applies to the whole.

Inductive argument—an argument in which the premises are intended merely to count toward, or make probable, the conclusion. To determine whether the logic of an inductive argument is successful, a good rule of thumb is to ask these questions:

Do the premises count toward the conclusion?
If the premises were true, would that make the conclusion probable?

Alternatively termed probabilistic, nondemonstrative, or ampliative argument.

Invalid—a deductive argument that is not logically successful. An argument is invalid if and only if it is possible for an argument with such a form to have true premises and a false conclusion.

Logic—the reasonableness conferred on an argument’s conclusion by its premises. In an argument that is logically successful the conclusion follows from the premises—or, to put it differently, the premises support the conclusion. In deductive arguments, this is strictly a matter of the fit of the conclusion to the premises. In inductive arguments, it is also a matter of the fit of the conclusion to the total available evidence.

Part—a statement connected to another by and. Also known as conjunct.

Repetition—valid deductive form, as follows:

Simplification—valid deductive form, as follows:

Valid—a logically successful deductive argument. An argument is valid if and only if it is impossible for an argument with such a form to have true premises and a false conclusion.

Validity counterexample—a two-step method for checking any argument for validity. The first step is to extract the form that the argument is depending on for logical success. The second step is to attempt to construct a new argument by appropriately substituting new sentences, predicates, or names in a way that produces obviously true premises and an obviously false conclusion.