Chapter Eleven: If–Then Arguments

If–then arguments, also known as conditional arguments or hypothetical syllogisms, are the workhorses of deductive logic. They make up a loosely defined family of deductive arguments that have an if–then statement—that is, a conditional—as a premise. The conditional has the standard form If P then Q. The if portion, since it typically comes first, is called the antecedent; the then portion is called the consequent.

These arguments—often with implicit premises or conclusions—are pressed into service again and again in everyday communication. In The De-Valuing of America, for example, William Bennett gives this brief if–then argument:

If we believe that good art, good music, and good books will elevate taste and improve the sensibilities of the young—which they certainly do—then we must also believe that bad music, bad art, and bad books will degrade.

The if–then premise—lightly paraphrased—is this:

If good art, good music, and good books elevate taste and improve the sensibilities of the young, then bad music, bad art, and bad books degrade taste and degrade the sensibilities of the young.

The second premise—set off in the original by dashes—is:

Good art, good music, and good books elevate taste and improve the sensibilities of the young.

And the implicit conclusion is this:

Bad music, bad art, and bad books degrade taste and degrade the sensibilities of the young.

Whether the argument is sound depends on whether the logic of the argument is successful and whether the premises are true. We now look at each of these two categories of evaluation.

11.1 Forms of If–Then Arguments

The arguments of this chapter are deductive, so the success of their logic is entirely a matter of form. The form of Bennett’s argument in the preceding paragraph is the most common and the most obviously valid. It is normally termed affirming the antecedent; a common Latin term for this form is modus ponens, which means “the method (or mode, from modus) of affirming (or propounding, from ponens).”

Almost as common is the valid form denying the consequent; the Latin term for this is modus tollens, which means “the method of denying.”

This is the form of my argument if I say to you, “If you decide to adopt that puppy, then you’re going to be stuck at home for a long time. But you could never accept that—you live for your trips. This pup’s adorable, but it’s not for you.”

Each of these two valid forms may be contrasted with an invalid form that unsuccessfully mimics it. The invalid form that is tempting due to its similarity to affirming the antecedent is the fallacy of affirming the consequent; its structure is this:

I’ve committed this fallacy if I argue, “If you decide to adopt that puppy, then you’re going to be stuck at home for a long time. Fortunately you hate to sleep in any bed other than your own. So, this pup’s for you!” After all, you may love staying at home but also have a severe allergy to dog hair; the conclusion surely does not follow.

And deceptively similar to denying the consequent is the fallacy of denying the antecedent; this invalid form is as follows:

I made this mistake in the following argument: “If you decide to adopt that puppy, then you’re going to be stuck at home for a long time. But, knowing you, of course you’re not going to decide to adopt the puppy. So, it follows that you’re not going to be such a homebody anymore.” If you pass on the puppy because of your asthma, that has no bearing on your travel plans one way or the other. Again, the conclusion does not follow.

Recall that when you find these fallacious forms, there is normally no need to apply the principle of charity in your paraphrase. The ease with which such mistakes are made (thus earning each fallacy a name of its own) is usually reason to think that the arguer might have been truly mistaken in his or her thinking, and thus is reason to clarify the argument in the invalid form. [1]

Another form of argument, a valid one, that belongs to the if–then family is often termed transitivity of implication. This form of argument links if–then statements into a chain, as follows:

I’ve given an argument of this form if I contend, “If you decide to adopt that puppy, then you’re going to be stuck at home for a long time. But if you’re stuck at home for a long time, you’d better fix your toaster oven. So, if you decide to adopt that puppy, you’d better fix your toaster oven.” There is no limit to the number of if–then links that this chain could contain and still be valid. [2]

Incidentally, Lewis Carroll’s argument at the chapter’s opening presents some interesting evaluative possibilities. Here is one reasonable paraphrase:

  1. If it is, then it is.
  2. It is not.
  3. ∴ It is not.

On the one hand, it has the form of the fallacy of denying the antecedent, which is invalid; on the other hand, it has the form of denying the consequent, which is valid. Further, it also has the form of repetition—looking only at 2 and C—which is also valid. The solution to the puzzle is that it is valid—not because the two valid forms outnumber the one invalid one, but because we should charitably suppose that the valid form is the one that was intended. Charity, unfortunately, cannot prevent us from noting that whatever the form, this argument probably commits the fallacy of begging the question. And if it does, then it does.

If–Then Arguments

Valid Invalid
Affirming the antecedent Affirming the consequent
Denying the consequent Denying the antecedent
Transitivity of implication

EXERCISES Chapter 11, set (a)

Create a brief argument that takes the specified form.

Sample exercise. Transitivity of implication.

Sample answer. If I run out of gas I’ll be late. And if I’m late I’ll get fired. So, if I run out of gas I’ll get fired.

  1. Affirming the antecedent
  2. Affirming the consequent
  3. Denying the consequent
  4. Denying the antecedent
  5. Transitivity of implication

11.1.1 Stylistic Variants for If–Then

If–then arguments, as we have seen, make crucial use of statements of the form If P then Q as premises. Using the terminology of Chapter 6, the expression if–then is the logical constant of such statements, while P and Q are the variables—sentential variables, you will recall—replaceable by declarative sentences as the content of the argument.

These constants are anything but constant in ordinary language; a wide variety of everyday English expressions are used to express if–then. In the structuring phase of the clarifying process, it is important that you translate them into standard constants. This helps bring the structure of the argument close to the surface and makes it much easier to tell whether the argument is logically successful.

All of the expressions listed below—and many more—can be used as stylistic variants for if–then. More precisely, each of the expressions can be translated, for logical purposes, into If P then Q.

Stylistic Variants for if P then Q

Q if P.
P only if Q.
Only if Q, P.
Assuming P, Q.
Q assuming P.
Supposing P, Q.
Q supposing P.
Given P, Q.
Q given P.
That P is a sufficient condition for that Q.
That Q is a necessary condition for that P.

This list includes some of the most obvious variants, but it is not comprehensive. Unexpected variants for if–then statements show up with regularity. A politician says, for example, “Vote for my bill and I’ll vote for yours.” This can be taken as a stylistic variant for, “If you vote for my bill, then I’ll vote for yours.” A story about new television shows says, “With good summer ratings, the series will end up on the fall schedule of NBC.” This translates into, “If the series gets good summer ratings, then it will end up on the fall schedule of NBC.” And language watcher Thomas Middleton, complaining in the Los Angeles Times about a tendency he has noticed among teens to use expressions like “and then my friend goes so-and-so” instead of “and then my friend said so-and-so,” presses his point thus:

The ability to say things . . . is consummately precious, and to describe “saying” as “going” is to debase this glorious gift. It is to treat speech as though it were no more than, as Random House says, making a certain sound—like a cat’s purr.

This passage, it seems, translates into something like the following:

If someone describes “saying” as “going,” then that person debases the gift of speech and treats it as though it were no more than making a sound.

Be very careful, however, with words like with, and, and to; they are rarely stylistic variants for if–then. It is only when they are used in these distinctive kinds of contexts that they should be taken this way.

Guideline. Translate the stylistic variants for the if–then premise into the standard constant.

EXERCISES Chapter 11, set (b)

Translate the stylistic variant in each of the following if–then statements.

Sample exercise. As long as history textbooks make white racism invisible in the 19th century, students will never be able to analyze racism intelligently in the present.—James Loewen, Lies My Teacher Told Me

Sample answer. If history textbooks make white racism invisible in the 19th century, then students will never be able to analyze racism intelligently in the present.

  1. Oxygen is necessary for combustion.
  2. The governor agreed Tuesday to a legislative compromise for ending the community colleges’ financial troubles—but only if lawmakers can find another $121 million.
  3. Teachers should assign passages and require students to summarize the passages in their own words. Do that consistently, and students will not only learn to write a lot better, they will also learn to analyze, evaluate, sort out, and synthesize information.
  4. Those who are not willing to give to everyone else the same intellectual rights they claim for themselves are dishonest, selfish, and brutal. —from Robert Ingersoll, Ingersoll: The Immortal Infidel
  5. “To address kids in masses, you have to be an entertainer, which I’m not,” Dr. Seuss said, sounding a little like the Grinch.—Los Angeles Times
  6. “Strip a woman’s body of its breasts and hips, of all of its nurturing curves, and replace it with enough stringy, sinewy muscle, and a lot of people will simply not know what to make of what you have left.”
    —Pumping Iron II: The Women

EXERCISES Chapter 11, set (c)

Clarify each of the if–then arguments. Then state whether the argument is valid and provide the name of the valid or invalid form.

Sample exercise. “I submit that the author is thoroughly wrong to criticize analogical argumentation, that if argument by analogy were really as weak as he allows we would not use it as extensively as we do.”—book review in Teaching Philosophy

  1. If argument by analogy is as weak as the author allows, then we do not use argument by analogy as extensively as we do.
  2. [We do use argument by analogy as extensively as we do.]
  3. ∴ Argument by analogy is not as weak as the author allows.
    Valid, denying the consequent.
  1. Universal mandatory screening for AIDS can be justified on the basis of beneficence when a therapeutic intervention is available or when an infectious state puts others at risk merely by casual contact. However, neither is the case with AIDS. Thus, there is no demonstrable public health benefit that justifies universal mandatory screening.—N. F. McKenzie, ed., The AIDS Reader (Even though an extremely charitable reading of this argument might suggest otherwise, go ahead and clarify it as a fallacy.)
  2. “The prolonged study of ethics does not by itself make you a better person. If it did, philosophy professors would in general be better people than average. But they aren’t.”—William Bennett, The De-Valuing of America
  3. “If the North Koreans are smart—and we know they are smart—they will move in the direction of reform.”—Daryl Plank, Korea expert and visiting fellow at Washington’s Heritage Foundation
  4. “If, instead of offering the occasional high-profile prize of $35 million, New York awarded 350 prizes of $100,000, making not one multimillionaire, but a great number of $100,000 winners, it would create an environment where far more people would know, or know of, large prize winners. More people will buy tickets if they know large prize winners. So experimenting with such a format should reverse the current negative trend.” —letter to the editor, New York Times
  5. “Ladies and Gentlemen, I’ll be brief. The issue here is not whether we broke a few rules or took a few liberties with our female party guests. We did. But you can’t hold a whole fraternity responsible for the behavior of a few sick perverted individuals. For if you do, then shouldn’t we blame the whole fraternity system? And if the whole fraternity system is guilty, then isn’t this an indictment of our educational institutions in general? I put it to you, Greg: Isn’t this an indictment of our entire American society? Well, you can do what you want to us, but we’re not going to sit here and listen to you badmouth the entire United States of America.”—Eric “Otter” Stratton, in the film Animal House (This is a complex argument with the subconclusion If the fraternity is guilty, then the entire American society is guilty.)
  6. “And if, for example, antiabortionism required the perverting of natural reason and normal sensibilities by a system of superstitions, then the liberal could discredit it—but it doesn’t, so he can’t.”—Roger Wertheimer, Philosophy and Public Affairs

11.1.2 Singular Inferences

One common variation on the preceding forms is worth our attention. Note the remark made by legendary heavyweight boxing champ Joe Frazier to the short-lived and less legendary title holder, Jimmy Ellis:

You ain’t no champ. You won’t fight anybody. A champ’s got to fight everybody.

This provides several opportunities for following the rules of paraphrasing arguments—a stylistic variant for if–then, the need to follow the principle of charity (because of the rather extreme words everybody, anybody, and even what Frazier means by being a champ), wording to be matched, and emptiness to be avoided (because of the word you). The result of clarifying it is something like this:

  1. If any person deserves to be the heavyweight boxing champion, then that person will fight all worthy contenders.
  2. Jimmy Ellis will not fight all worthy contenders.
  3. ∴ Jimmy Ellis does not deserve to be the heavyweight boxing champion.

This looks very much like denying the consequent—that is, it seems to depend on this form:

But the Q of premise 1 and the Q of premise 2 do not really match, nor do the P of premise 1 and the P of C. For there is no mention of Jimmy Ellis anywhere in premise 1, yet Jimmy Ellis is the subject of premise 2 and of the conclusion. This certainly does not harm the argument’s logic, however, since Jimmy Ellis is included—as a single person—among those encompassed by the term any person in the first premise. So, for practical purposes, we can continue to call this form denying the consequent, but with a slight difference. It will be identified as singular denying the consequent

The same modification is permitted for every form of sentential logic that we cover, assuming two things hold. First, there must be a universal statement as a premise—that is, a premise with a term like all, none, anything, or nothing, to mention a few examples. If any person deserves to be the heavyweight boxing champion, then that person will fight all worthy contenders is universal, since it applies to any person. Second, there must be a conclusion in which a single instance is specified that is encompassed by the universal term. Jimmy Ellis will not fight all worthy contenders provides an example, since Jimmy Ellis is encompassed by any person. All the if–then forms mentioned above can be modified in this way. Singular affirming the antecedent and singular transitivity of implication are also valid forms, while the fallacy of singular affirming the consequent and the fallacy of singular denying the antecedent are invalid ones.

Guideline. When an argument has both a universal premise and a conclusion about a single thing that is encompassed by the universal premise, consider whether it is the singular version of a sentential logical form.

Singular If–Then Arguments

Valid Invalid
Singular affirming the antecedent Singular affirming the consequent
Singular denying the consequent Singular denying the antecedent
Singular transitivity of implication

EXERCISES Chapter 11, set (d)

Clarify the following arguments as examples of singular if–then arguments. Then state whether the argument is valid and provide the name of the valid or invalid form.

Sample exercise. “Q: You mentioned that Bundy was mentally ill?
A: Sane people do not go round killing dozens of women, and the person that the state of Florida strapped in the electric chair was a man who was severely mentally ill.”—I. Gray and M. Stanley, eds., A Punishment in Search of a Crime: Americans Speak Out Against the Death Penalty

  1. If anyone is sane, that person does not kill dozens of women.
  2. [Bundy killed dozens of women.]
  3. ∴ Bundy was not sane.
    Valid, singular denying the consequent.
  1. If someone is in Birmingham, then that person is in Alabama. And if someone is in Alabama, then that person is in America. So, if I am in Birmingham, then I am in America.
  2. I conceded that speeding is sufficient reason for getting pulled over by a police officer. But I wasn’t speeding. So I should not have been pulled over.
  3. One argument for the immorality of adultery might go something like this: it involves the breaking of a promise, and it is immoral to break a promise.—from a lecture by philosopher Richard Wasserstrom
  4. I refer you to the verdict in the English Court sustaining Whistler’s contention that a man did not wholly own a picture by simply buying it. So, although I may have sold my painting to him, I have a right to protect my picture from the vandalism of his cleaning it.—from American artist Albert Pinkham Ryder

11.2 Evaluating the Truth of If–Then Premises

If–then statements usually propose a special connection between the if-clause and then-clause. Identifying the specific nature of the connection is usually the key to judging the truth of such a statement and to successfully defending that judgment. [3]

Sometimes the proposed connection is causal, as in the case of the statement If you push the ignition button, then the car will start. Pushing the button would cause the car to start. But in other cases the proposed connection is broadly logical; the if-clause does not cause the then-clause but is offered as counting toward or even guaranteeing its truth. [4] Consider the statement that became a book and movie title—If it’s Tuesday, this must be Belgium. It’s being Tuesday cannot cause this to be Belgium, but could presumably (combined with other statements about the itinerary) count in favor of the belief that this is Belgium. Or consider the statement If there is intelligent life on other planets, then we are not alone in the universe. That there is intelligent life elsewhere in the universe is just what we mean by not being alone in the universe. So the connection here is also a logical one—and in this case it is such a tight connection that we can safely call it self-evidently true.

Whether the proposed connection is causal or logical, it is helpful to think of the if-clause as not being offered as alone sufficient for the then-clause. When we use if–then statements we are typically allowing for other relevant factors as well. We have simply picked out the if-clause for special mention because it is the one factor that happens to be most important in the context. These implicit assumptions about other relevant factors are termed secondary assumptions (or auxiliary hypotheses).

Return to the if–then statement If you push the ignition button, then the car will start. Behind such a statement there usually are implicit secondary assumptions about many other factors that contribute to the starting of the car—but that are presumed to be already in place, and thus do not merit mention. They may include assumptions about the specific situation, such as these:

There is a functioning engine in the car.
There is gas in the tank and the starter battery is not dead (if it has an internal combustion engine).
The battery pack is charged (if it has an electric engine).
The key fob is nearby.
The ignition system is not defective.

They may also include more general assumptions about the relation between the if-clause and then-clause, such as this:

Ignition systems are designed to start properly functioning cars.

And they may include even broader principles that guide much of our reasoning, such as this:

The laws of nature will not suddenly change.

When you judge an if–then statement to be true, a good way to defend your judgment is to identify the secondary assumptions that are most likely to be in question, given the circumstances, and to point out their truth. You might say, for example,

I judge this premise to be very probably true because this is what ignition systems are designed to do, and there is no reason to think that this car is out of fuel or is defective in some other way.

Thus a connection between if-clause and then-clause is affirmed.

Alternatively, if you judge the if–then statement to be false, a good defense is to point out that a secondary assumption is false; for example,

This premise is probably false, since the headlights were left on all day and the battery is dead by now.

Thus you have denied one of the secondary assumptions and shown that the connection between if-clause and then-clause is severed.

The same strategy works well for if–then statements in which the connection is broadly logical rather than causal. Consider If you are reading this book, then you understand English. One important secondary assumption is This book is written in English. Another is Reading something just means that you understand it. (You might wonder whether this, or the earlier life on other planets example, should count as a secondary assumption, since it is part of the very meaning of the terms used—what we have in preceding chapters called self-evidently true. It will nevertheless make good practical sense in this text for us to count it so.) So here is an exemplary defense of the statement:

This premise is certainly true, since the book is written in English, and part of what it means to read something is to understand it.

Again, its truth is defended by pointing out the cords that connect if-clause to then-clause.

Consider, finally, If New York City were in Quebec, then it would still be in the United States. There is, unfortunately, no way of knowing what secondary assumptions are supposed to connect this if-clause and then-clause. Is New York City to be located further north, or Quebec further south? And, on either scenario, what historical events would have caused such a difference—and would they, perhaps, have resulted in Quebec’s being included within the United States? There is simply not enough information to decide. The best evaluation of this premise, then, would be something like this:

I can’t decide whether this premise is true or false. There is no way of knowing whether New York City is to be located further north, Quebec further south, or what relevant historical events might have led to it.

The daughter of Rudolf Carnap, one of the great philosophers and logicians of the 20th century, tells of asking her father, when she was a young child, “If you were offered a million dollars, would you be willing to have your right arm amputated?” “I don’t know,” he replied. “Would I be given an anesthetic?” Lack of information about relevant secondary assumptions can sometimes make it impossible for any of us, even Carnap, to say any more than “can’t decide” in evaluating if–then statements.

Guideline. Defend your judgment that an if–then statement is true by affirming the truth of the most questionable secondary assumptions. Defend your judgment that it is false by showing that a secondary assumption is false.

EXERCISES Chapter 11, set (e)

For each of the following if–then statements, list the most plausible and relevant secondary assumptions (or explain why you cannot do so). Then provide a judgment of the premise’s truth by reference to your list. (They are not provided with any context, so you will have to use your imagination.)

Sample exercise. If there were more solar panels available, then air pollution would decrease.

Sample answer. Secondary assumptions: Consumers will buy and install more solar panels if they become available. Solar panels produce less air pollution than the more conventional forms of energy production.
Probably false, since, at least currently, in much of the world consumers do not have enough economic incentive to convert to solar energy.

  1. If there were more classical music on the radio, then there would be more appreciation of classical music among the public.
  2. If George W. Bush was the 45th president, then Barack Obama was the 46th.
  3. If it rains tomorrow, then you should take your umbrella to work.
  4. If people recycled more, the environment would be in better shape.
  5. If I were two feet taller, I would have played in the NBA.
  6. If it’s Tuesday, this must be Belgium.
  7. If the stock market rises next year, then Alphabet stock will rise in value.
  8. If all private ownership of guns were made illegal, then violence in our country would drop dramatically.
  9. If you can do 20 pushups, then you are in good shape.
  10. If you are planning to go to medical school, then you can expect to take several science courses.

11.2.1 The Retranslation Mistake

Consider the statement If New York City is in the state of New York, then it is in the United States. It is certainly true, but it is tempting to defend that judgment by more or less repeating the if–then statement in slightly different words, as follows:

My view is that the premise is certainly true, since New York City has to be in the United States, given that it is in the state of New York.

You have said nothing that goes beyond the premise itself, thus nothing that would be enlightening to the reasonable objector over your shoulder. You have merely retranslated the if–then constant back into one of its stylistic variants! Be careful to avoid this sort of defense. (I’ll leave it as an exercise for you to identify the simple secondary assumption that provides the crucial connection for this if–then statement.)

Guideline. Do not defend your judgment of an if–then statement by simply rewording the statement (or, if false, by rewording the denial of the statement).

11.2.2 Truth Counterexamples

It can be especially tempting to ignore mention of secondary assumptions when the if-clause is clearly true and the then-clause is clearly false. These are the most straightforward cases, for if you know that the if-clause is true and the then-clause is false, you know that the if–then statement is false. The if–then statement has vividly failed to deliver on its promise.

But even here it is better, if possible, to show the severed connection between the two by identifying the false secondary assumption. Take, for example, If New York City is in the state of New York, then it is in Canada. You might defend your judgment as follows:

I consider the premise to be certainly false since, based on my experience in my own travels and based on the testimony of every authority I’ve ever encountered, New York City is in the state of New York and it is not in Canada (but in the United States).

But this makes no mention of any connection between the if-clause and then-clause. If there is supposed to be one, it is the assumption that the state of New York is itself in Canada. And your defense is stronger if you include the rejection of this assumption, as follows:

Further, the state of New York is wholly located within the United States, not Canada.

There can be, however, exceptions to this rule. One exception applies when it is a universal if–then statement that is false. Universal if–then statements, recall, are if–then statements with a universal term like anything, anyone, nothing, or nobody in the if-clause. An example we have already seen is If any person deserves to be the heavyweight boxing champion, then that person will fight all worthy contenders. A property—such as deserving to be champ—is applied universally—to any person—rather than to a single instance. When such statements are false, the method of truth counterexample can be a simple and effective way of defending that judgment. This method identifies a single instance in which the if-clause is obviously true and the then-clause is obviously false.

A newspaper story on the homeless, for example, contains the line, “No one is poor by choice.” This is a stylistic variant of the universal if–then statement, “If anyone is poor, then it is not by choice.” Yet the same newspaper, on the facing page, has a story about Mother Theresa’s religious order, stating, “These nuns have voluntarily taken an oath of poverty.” Here we have a ready-made truth counterexample. The nuns are instances of the if-clause’s truth—they are poor—and at the same time are instances of the then-clause’s falsity—their poverty is by choice. Thus armed, your defense of your judgment of the universal if–then statement can be stated simply as follows: “The premise is certainly false, since certain orders of nuns are poor by choice.”

Guideline. When a universal if–then statement is false, try to defend that judgment by providing a truth counterexample.

EXERCISES Chapter 11, set (f)

Provide a truth counterexample for each of these false universal if–then statements. If necessary, first translate stylistic variants into the standard constant.

Sample exercise. Only animals that can fly are endowed with wings.

Sample answer. If any animal has wings, then it can fly. Certainly false; the ostrich has wings but cannot fly.

  1. No major-party presidential or vice presidential candidate has been a female.
  2. If any substance is made of metal, then it is attracted to a magnet.
  3. Everything reported in the newspaper is true.
  4. Nice guys finish last.
  5. What goes up must come down.

11.2.3 The Educated Ignorance Defense

Another occasion for ignoring secondary assumptions—also occurring under the true if-clause/false then-clause scenario—is when the evidence for the truth of the if-clause and the evidence for the falsity of the then-clause are each stronger than the evidence for the truth of the if–then statement. In these cases, even though you may not know which secondary assumption is at fault, it can be reasonable to say that the premise is false “because some secondary assumption—not yet identified—is mistaken.” This we will term the educated ignorance defense (“Ignorance” because you admit ignorance regarding which secondary assumption is faulty; “educated” because you nevertheless have good evidence that the if-clause is true and the then-clause is false.)

Return, for example, to our car-starting example. Imagine that when you pick up your car after extensive repairs your mechanic says to you, “If you push the ignition button, then the car will start.” He has extremely good reasons to believe this is true. He has checked out all of the systems—in short, his experience and expert judgment support the truth of any secondary assumption that might be reasonably questioned. You push the ignition button. But the car does not start.

Something has to give. There are three statements for which you apparently have very good evidence:

If you push the ignition button, then the car will start.
You push the ignition button. (The if-clause is true.)
The car will not start. (The then-clause is false.)

They cannot all be true at the same time. You will probably quickly give up on the truth of the if–then statement, not knowing what went wrong but knowing quite well that you pushed the button and the car didn’t start. But the mechanic, who has especially good reasons to believe the if–then statement—he did the work, and he has his reputation to think about—will probably start off by doubting the if-clause, asking you suspiciously, “Are you sure you pushed the ignition button?” “I’m sure,” you reply, anxiously pushing it again and again. “Let me see,” he says with a hint of disdain and gets in to push the button himself. Only when it does not start for him does he say, “Well, OK, I was mistaken, but I just can’t figure out what’s wrong with it.”

The mechanic’s initial reluctance to give up the truth of the if–then statement is because he cannot imagine which secondary assumption is mistaken. And he only concedes that the if–then statement is false when he sees that evidence in favor of the if-clause—that the button has been pushed—is conclusive. (The evidence that the then-clause is false—that the car did not start—is already conclusive.) He is still ignorant of which secondary assumption to blame, but now that he is duly educated—about the truth of the if-clause and falsity of the then-clause— he can reasonably resort to the educated ignorance defense. Eventually something better than educated ignorance will be required if the car is to be driven away.

Science provides many examples of this defense. In the 18th century, for example, astronomers used the new Newtonian mechanics to accurately predict the orbits of many of the planets in our solar system. The following if–then statement describes the general shape of these predictions:

If Newtonian mechanics is true, then the orbit of planet A will be observed to be F.

(In this case, A is the name of the planet and F is a mathematical description of the predicted observed orbit of the planet around Earth.) After many successes, the astronomers did their work on the orbit of Uranus and discovered, to everyone’s surprise, that the predicted orbit did not accord with their observations. They thus found themselves with good evidence for the following three statements, not all of which could be true:

If Newtonian mechanics is true, then the orbit of Uranus will be observed to be F.
Newtonian mechanics is true. (The if-clause is true.)
It is not the case that the orbit of Uranus is observed to be F. (The then-clause is false).

They checked and rechecked their equipment to be sure of their evidence that the then-clause was false, but they found their surprising observations to be accurate. They reminded themselves of the mountains of other evidence in favor of the if-clause. And they checked and rechecked their calculations, in the futile hope of finding some faulty secondary assumption that would falsify the if–then statement. In the end, the only reasonable thing to do was to reject the if–then statement with a defense something like this:

This premise is probably false; the support for Newtonian theory is so strong, and the quality of this observation so good, that it is most likely that some not-yet-identified faulty secondary assumption lies behind its falsity.

Incidentally, that is where things stood until the 19th century, when the Englishman John Adams and the Frenchman Urbain Leverier, working independently, realized that the mistake had been in assuming Uranus is the outermost planet. Due to this secondary assumption, the earlier Newtonians had not factored into their calculations any gravitational pull from the other side of Uranus. They each reworked the calculations and predicted where they should be able to observe an outer planet exercising gravitational attraction on Uranus. In 1846 they independently observed this planet, later named Neptune, in the predicted location.

The strategy of saying, “There is some unidentified secondary assumption that is mistaken” should be employed with great care. Again, it works only when there is independent strong evidence in favor of the truth of the if-clause and against the truth of the then-clause. These lines are from the final letter written to his wife by one of the doomed soldiers of the German Sixth Army outside Stalingrad:

“If there is a God,” you wrote me in your last letter, “then he will bring you back to me soon and healthy.” But, dearest, if your words are weighed now you will have to make a difficult and great decision.

Her own words, quoted by her husband, committed her to the statement If God exists, then the soldier will return to his wife soon and healthy. The report of his death that she later received supported this statement: The soldier will not return to his wife soon and healthy. But by a valid denying the consequent argument, these two premises entail God does not exist. This, then, presented his wife with the difficult and great decision that the soldier foretold—she must stop believing in God, or she must go back on her own words.

Let’s set this up in the same way as we did with the auto mechanic and the Newtonians. There are three statements before her, at least one of which must be false:

If God exists, then the soldier will return to his wife soon and healthy.
God exists. (The if-clause is true.)
The soldier will not return to his wife soon and healthy. (The then-clause is false.)

Let’s suppose that instead of giving up her belief in God, she chose the option of going back on her words and rejecting her if–then statement. Her most reasonable defense, as we have seen, would be for her to sever the connection between the if-clause and the then-clause by identifying and rejecting the false secondary assumption. Candidates might include:

God cares about human suffering.
God cares about the suffering of this particular soldier and his wife.
God is able to prevent this suffering.
God knows about this suffering.

But let’s further suppose that she insisted on continuing to embrace all these secondary assumptions, on the grounds that to do otherwise would be to unsuitably diminish God. Instead, she took the step that many believers in God take—the step of saying, “God’s ways are beyond the understanding of man. When I get to heaven he will reveal to me his reasons. Until then, I will continue to believe in him.” This is an attempt to use the educated ignorance defense. We give up on the if–then statement in the expectation that we will eventually discover the car’s mechanical defect, the flaw in our astronomical calculations, or the hidden mysteries of God.

Whether this is a reasonable move for the soldier’s wife depends on one condition: it is educated ignorance—and thus a reasonable defense—only if the wife has independent strong evidence that God exists (evidence for the if-clause). If she does not—if she accepts by faith alone not only God’s mysterious ways but also his very existence—then she cannot reasonably defend her rejection of the if–then statement unless she identifies and rejects the false secondary assumption.

Guideline. It is reasonable to judge an if–then premise false “because some secondary assumption must be mistaken, though I don’t know which one” only if there is very powerful evidence both that the if-clause is true and that the then-clause is false.

EXERCISES Chapter 11, set (g)

In each problem there are three statements, at least one of which must be false. Provide an “educated ignorance” defense for the claim that the if–then statement is false; you’ll need to state evidence for the if-clause and against the then-clause in your defense.

Sample exercise.
If the instructor is fair, then he will not give higher grades to males than to females.
The instructor is fair.
The instructor gave higher grades to males than females.

Sample answer. The if–then statement is probably false, although I can’t say exactly what the mistaken assumption is. Even though the record shows that in this class the males did much better than the females, he has a widespread reputation for bending over backward to treat everyone fairly. It seems likely that his reputation is deserved and that in this case there is an explanation that will eventually emerge.

  1. If you patch that hole, then the roof will stop leaking. You patch the hole. The roof does not stop leaking.
  2. If your boyfriend loves you, then he will be on time tonight. Your boyfriend loves you. He is not on time tonight.
  3. If you are the most talented, then you will win the talent show. You are the most talented. You do not win the talent show.
  4. If it is impossible to move physical objects by only thinking about it, then when Uri Geller concentrates on bending the spoon it will not bend. It is impossible to move physical objects by only thinking about it. When Uri Geller concentrates on bending the spoon it does bend.

Strategies for Evaluating the Truth of If–Then Statements

What you know How to evaluate the if-then statement
1. You can identify secondary assumptions.
a. You can show a secondary assumption is probably false. Probably false.
b. You can show the most questionable secondary assumptions are probably true. Probably true.
c. Otherwise Can’t decide.
2. You cannot identify secondary assumptions.
a. You can provide a truth counterexample. Almost certainly false
b. You can provide the educated ignorance defense. Probably false.
c. Otherwise. Can’t decide.

11.2.4 Secondary Assumptions and Indirect Arguments

Secondary assumptions can also play an important role in the evaluation of indirect arguments (which we have also called reductios). Introduced in Chapter 10, such arguments, in their simplest form, exhibit the structure of denying the consequent. They begin with a statement that may seem quite innocuous and attempt to show that it is false by pointing out, in what amounts to an if–then premise, an absurd consequence that it forces on you. You accept the absurdity of the consequence by accepting a premise that says the then-clause is false. You must then conclude, by the valid form of denying the consequent, that the seemingly innocuous if-clause must be rejected. [5] An example is found in these remarks by David Wilson (no known relation to the author), adapted from a newspaper report:

Melina Mercouri, Greece’s minister of culture, swept into the staid old British Museum to examine what she called the soul of the Greek people—the Elgin Marbles. Lord Elgin took them from the Parthenon in Athens in the early 19th century. Mercouri is expected to make a formal request soon for the marbles’ return. But Dr. David Wilson, director of the British Museum, opposes the idea. “If we start dismantling our collection,” Wilson said, “it will be the beginning of the end of the museum as an international cultural institution. The logical conclusion of the forced return of the Elgin Marbles would be the utter stripping of the great museums of the world.”

Wilson’s argument can be clarified thus:

  1. If it is acceptable to force the British to return the Elgin Marbles to Greece, then it is acceptable to strip the great museums of the world.
  2. [It is not acceptable to strip the great museums of the world.]
  3. ∴ It is not acceptable to force the British to return the Elgin Marbles to Greece.

Melina Mercouri must avoid the conclusion without rejecting premise 2, so her only recourse is to reject the if–then premise. But when she does reject it, she is no position to respond with the educated ignorance defense; the evidence for the if-clause is exactly what is in question, so for her to simply say the if-clause is obviously true would be to beg the question. In short, her only reasonable strategy is to reject the if–then premise by identifying a faulty secondary assumption that it depends on. Here is a strong candidate for the role of faulty secondary assumption:

The only principle for returning the Elgin Marbles would be that any item, great or small, removed from its original culture, whether by consent or by force, must be returned to that culture.

This secondary assumption is clearly false. So Mercouri might defend her rejection of premise 1 as follows:

Premise 1 is almost certainly false, since it assumes that all items must be returned to their original culture; but the return of the Elgin Marbles only depends on a principle calling for the return of great national treasures that have been forcibly removed.

What Mercouri would be doing is accusing Wilson of committing the fallacy of non causa pro causa (introduced in Chapter 10). This is the fallacy of blaming the absurd consequence (It is acceptable to strip the great museums of the world) on what is set forth as its cause (It is acceptable to force the British to return the Elgin Marbles to Greece) instead of blaming the unnoticed assumption that is the real cause of the absurdity (All items must be returned to their original culture).

Because indirect arguments are typically offered in support of controversial conclusions, only rarely can the educated ignorance approach be used in evaluating them without begging the question. Be especially watchful for faulty secondary assumptions behind the if–then premise of indirect arguments; when there is such an assumption, the indirect argument can be criticized for committing the fallacy of non causa pro causa.

Guideline. In indirect arguments, be alert for faulty secondary assumptions behind the if–then premise.

EXERCISES Chapter 11 set (h)

Clarify each of these simple indirect arguments; then evaluate only the if–then premise, on the grounds that it commits the fallacy of non causa pro causa. (Use the Elgin Marbles case as your sample.)

  1. If you are right in your claim that income taxes should be eliminated, then you must accept the consequence that the government will be left with no money to do even its most important business. But we would all agree that government cannot be done away with. So income taxes must remain.
  2. If children who misbehave are not immediately and severely punished, they will grow up with the belief that misbehavior has no negative consequences. We all agree, of course, that our children cannot be allowed to grow up with that belief. So don’t spare the rod with your children.

EXERCISES Chapter 11, set (i)

Each of the passages below indicates what could be seen as a misuse of secondary assumptions. In the Kelvin case, clarify the denying the consequent argument and identify the secondary assumption that, perhaps, Kelvin should have questioned. In the Azande case, clarify the affirming the antecedent argument that the Azande are trying to avoid, and identify the secondary assumption that they, perhaps, too readily reject to show that the if–then premise is false.

  1. Lord Kelvin, the leading British physicist of his day, dismissed Darwin’s work on the ground that it violated the principles of thermodynamics. The sun could be no more than 100 million years old; evolution demanded a much longer period in which to operate; therefore evolution must be rejected. Kelvin wasted no time pursuing the minutiae of the geological and palaeontological evidence on which evolution was based. Physics in the guise of thermodynamics had spoken clearly and whatever failed to fit into its scheme had to be rejected. Kelvin’s thermodynamics were later shown to be wrong. Unaware of radioactivity, he had inevitably failed to allow for its effect in his calculations.—Derek Gjertsen, Science and Philosophy
  2. According to the Azande, witchcraft is inherited unilinearly, from father to son, and mother to daughter. How, therefore, can I accept that my brother is a witch and yet deny that I am also infected? To prevent this absurdity arising, the Azande adopt further “elaborations of belief.” They argue, for example, that if a man is proven a witch beyond all doubt, his kin, to establish their innocence, deny that he is a member of their clan. They say he was a bastard, for among Azande a man is always of the clan of his genitor [natural father] and not of his pater [mother’s legal husband]. In this and other ways, Evans-Pritchard concluded, the Azande freed themselves from “the logical consequences of belief in biological transmission of witchcraft.”—Derek Gjertsen, Science and Philosophy

11.3 If–Then Arguments With Implicit Statements

If–then arguments, like any other sort of arguments, frequently have implicit premises or conclusions. To use a term from earlier in the book, they are frequently enthymemes. In extreme cases, only the if–then premise is explicit. Suppose, for example, that you’ve complained for the 10th time that the party across the hall is too loud, and I say to you, “Hey, if you can’t beat ‘em, join ‘em.” What I’ve actually given is an affirming the antecedent argument. I’ve explicitly provided the if–then premise; the implicit premise, obviously, is You can’t beat them; and the implicit conclusion is You should join them.

Consider the following, more sophisticated example, from a New York Review of Books review of a book of film criticism by Stanley Cavell:

When Katharine Hepburn in The Philadelphia Story brightly says, “I think men are wonderful,” Cavell hears an “allusion” to The Tempest that amounts “almost to an echo” of Miranda’s saying, “How beauteous mankind is!” If this is an echo, then Irene Dunne’s saying of her marriage, “It was pretty swell while it lasted” is a reminiscence of Gibbon’s Decline and Fall.

This argument is an example of denying the consequent. But only one statement of the argument is explicit. The full clarification proceeds thus:

  1. If Hepburn’s remark “I think men are wonderful” in ThePhiladelphia Story is an echo of Miranda’s “How beauteous mankind is” in The Tempest, then Irene Dunne’s saying of her marriage, “It was pretty swell while it lasted” is a reminiscence of Gibbon’s Decline and Fall.
  2. [Irene Dunne’s saying of her marriage, “It was pretty swell while
    it lasted” is not a reminiscence of Gibbon’s Decline and Fall.]
  3. ∴ [Hepburn’s remark “I think men are wonderful” in ThePhiladelphia Story is not an echo of Miranda’s “How beauteous mankind is” in The Tempest.]

This is the reviewer’s sideways—but effective—way of saying that perhaps Cavell takes himself a bit too seriously.

11.3.1 If–Then Bridges

In the preceding examples, only the if–then premise was explicit. But in other cases, only the if–then premise is implicit. Note, for example, this episode recorded by Jean Piaget in his book, The Child’s Conception of the World:

A little girl of nine asked: “Daddy, is there really God?” The father answered that it wasn’t very certain, to which the child retorted: “There must be really, because he has a name!”

This does not look, on the face of it, like an if–then argument. But there must be an implicit premise connecting the two parts of her retort. A good clarification, it seems, is this:

  1. [If any name exists, then the thing it names exists.]
  2. God has a name.
  3. ∴ God exists.

Premise 1 serves as a universal if–then bridge. It is a universal if–then statement (note the term any) and serves as a bridge of sorts between 2 and C. We might have proposed a more specific sort of bridge, as follows:

1*. [If God has a name, then God exists.]

Either bridge produces a valid argument—the first one by singular affirming the antecedent, the second one by affirming the antecedent. But the second doesn’t produce an argument that will convince us—after all, you can add a premise to any argument that says, “If the premises are true, then the conclusion is true,” and thereby say something that the arguer surely intended, without saying anything illuminating. (There is a specialized term for such an if–then statement, namely, the corresponding conditional of an argument.) When the conditional is expressed in its universal form, on the other hand, we get some idea of the general principle being assumed by the arguer.

Guideline. Consider providing a universal if–then bridge when an explicit link between premise and conclusion has not been provided by the arguer.

EXERCISES Chapter 11, set (j)

Clarify each of these arguments, proposing for each a universal if–then bridge.

Sample exercise. “An idealist is one who, on noticing that a rose smells better than a cabbage, concludes that it will also make better soup.”—H. L. Mencken

  1. [If anything smells better than another thing, then it also tastes better.]
  2. A rose smells better than a cabbage.
  3. ∴ A rose tastes better than a cabbage.
  1. Twitter, as another high-tech innovation, will inevitably fragment community rather than enhance it.
  2. “Spirituality turns out to be central to cognitive psychology, and therefore to artificial intelligence, and therefore to computer science, and therefore to the whole history of science and technology.”
    —David Gelernter, The Muse in the Machine
  3. Most striking of their claims is that among the 7,000 people executed in this century, at least 23 people, and possibly many more, have been innocent. This alone is reason, the authors conclude, that the death penalty should be abolished.—review of Hugo Bedau and Michael Radelet, In Spite of Innocence
  4. “There will always be books. You can show them off on the shelf behind you during a Zoom meeting, and you still can’t do that with your iPhone.”—Robert Maxwell
  5. “Although these textbooks purport to be a universal guide to learning of great worth and importance, there is a single clue that points to another direction. In the six years I taught in city and country schools, no one ever stole a textbook.”—W. Ron Jones, Changing Education

11.4 Bringing It All Together

After learning a wide array of distinct skills, you now have the opportunity to use all of them together. If–then arguments provide us with our first of six groupings of arguments that can be substantial and interesting. And you are now equipped to fully clarify and evaluate them.

There is nothing new to be said, but a few things bear repeating. In your evaluation, separately evaluate the truth of the premises (considering each premise individually), the logic of the argument (naming the form if it has a name, and providing a validity counterexample if it is invalid), the soundness of the argument (which depends entirely on truth and logic), and, if necessary, the conversational relevance of the argument. Always provide a defense of your judgment, and do so as if there were a reasonable objector over your shoulder whom you were trying to persuade.

I’ll provide a sample clarification and evaluation of this brief argument found in Gilbert Harman’s The Nature of Morality:

Total pacifism might be a good principle if everyone were to follow it. But not everyone is, so it isn’t.

    1. If everyone followed total pacifism, then total pacifism would be a good principle.
    2. Not everyone follows total pacifism.
    3. ∴ Total pacifism is not a good principle.

    Premise 1 is probably true, since the main objection to total pacifism is that it leaves you with no defense against those who are not pacifists. But if everyone were a pacifist, that would be no problem. (This seems to be the main secondary assumption of the premise.)

    Premise 2 is certainly true. We all have firsthand experience with violent people, not to mention the experience of them that we have via the media.

    Invalid, fallacy of denying the antecedent. Here is a validity counterexample:

      1. If anything is a chihuahua, then it is a dog.
      2. Uga, the Georgia Bulldogs mascot, is not a chihuahua.
      3. ∴ Uga, the Georgia Bulldogs mascot, is not a dog

      Unsound, due to invalidity.

      EXERCISES Chapter 11, set (k)

      Clarify and evaluate. Where appropriate, provide implicit statements in the clarification (including universal if–then bridges) and original validity counterexamples in the evaluations.

      1. “With the layout of the San Francisco-Oakland area, a rail line there had a better chance than most,” said rail critic Peter Gordon, a regional planner at USC. “If it doesn’t work there, and I assert it doesn’t, there is no way rail transit will work in a place like Los Angeles.” —Los Angeles Times
      2. “But if evolution proceeded as a lockstep, then the fossil record should display a pattern of gradual and sequential advance in organization. It does not, and I regard this failure as the most telling argument against an evolutionary ratchet.”—Stephen Gould, Panda’s Thumb (It might help you to know that Gould is a noted professor of paleontology at Harvard. By a “ratchet” and a “lockstep” he means a “gradual, uniform progression.”)
      3. “If each man had a definite set of rules of conduct by which he regulated his life he would be no better than a machine. But there are no such rules, so men cannot be machines.”—A. M. Turing, Mind (Turing is an expert in artificial intelligence, i.e., what it would take to make a machine think.)
      4. “Q: Even so, don’t you think that the use of computers reinforces a child’s problem-solving ability? A: If that were true, then computer professionals would lead better lives than the rest of the population. We know very well that isn’t the case.”—interview with computer expert Joseph Weizenbaum, Le Nouvel Observateur (The argument is in the answer.)
      5. Pjeter Ivezaj, a U.S. citizen, was sentenced Wednesday to seven years in prison by a panel of judges in Yugoslavia for participating in peaceful anti-Yugoslav activities in this country. Said his brother Frano, 28, “I intend to take this matter to the court of world opinion. If U.S. citizenship has any value, which I believe it does, now is the time for the U.S. government to make a move.”—Detroit Free Press
      6. Scientists believe that the most likely location of our country’s next severe earthquake is a fault zone that centers on New Madrid, MO. But residents here, where the quake probably will be centered, are unimpressed. “I’m just a non-believer,” said L. H. Rector, publisher of the New Madrid Weekly Record, whose motto is “The only paper in the world that cares about New Madrid.” “It hasn’t been proved that we’re going to have one,” Rector said. “Now, out in California, they can see the fault. It’s been proved. But no one here in New Madrid has seen one fault.”—Los Angeles Times (Provide Rector’s argument with a universal if–then bridge.)
      7. “The church never wanted disease to be under the control of man. Timothy Dwight, president of Yale College, preached a sermon against vaccination. His idea was that if God had decreed from all eternity that a certain man should die with the smallpox, it was a frightful sin to avoid and annul that decree by the trick of vaccination. Smallpox being regarded as one of the heaviest guns in the arsenal of heaven, to spike it was the height of presumption.”—Robert Ingersoll, Ingersoll, The Immortal Infidel
      8. The key premise is that a human fetus is a full-fledged, actualized human life. Supposing human embryos are human beings, their innocence is beyond question. So nothing could justify our destroying them except, perhaps, the necessity of saving some other innocent human life.—Roger Wertheimer, “Understanding the Abortion Argument” (Clarify as a complex argument with an if–then bridge in the second inference.)
      9. “If there be righteousness in the heart, there will be beauty in the character. If there be beauty in the character, there will be harmony in the home. If there be harmony in the home, there will be order in the nation. If there be order in the nation, there will be peace in the world.”—Confucius (Suppose that this is a transitivity of implication argument with an implicit conclusion.)
      10. “If a man could not have done otherwise than he in fact did, then he is not responsible for his action. But if determinism is true, it is true of every action that the agent could not have done otherwise. Therefore, if determinism is true, no one is ever responsible for what he does.”—Winston Nesbitt and Stewart Candlish, Mind
      11. Their children, they soon saw, were being presented by EPIC, an organization to help children learn to think about values, with hypothetical situations that called for the students to make choices and decisions. On a spaceship survival trip, an EPIC question asks, “Determine what to take with you. Pretend the ship develops trouble and the load must be lightened. What could you discard?” There was, to a number of parents, something very troubling about these types of questions. The questions seemed to carry with them the presumption that the children were free to reason through to their own answers. If they could do that, it meant that there were no moral absolutes, and nothing was clearly right or wrong, good or bad. This was not the worldview of fundamentalists who believe in the literal word of the Bible. “Once you tell a child that he has to decide upon his own values system, that’s like saying that values are not real, and you can just make them up as you go along,” said Marjorie McNabb, a former Episcopalian who now attends a Baptist church. “Children would be better raised by a street gang than EPIC. At least, they’d learn two values, courage and loyalty. That’s better than no values.”—Los Angeles Times (Look for the actual argument to begin in about the middle, with the words “If they could do that. . . .”)
      12. In spite of the popularity of the finite-world picture, however, it is open to a devastating objection. In being finite the world must have a limiting boundary, such as Aristotle’s outermost sphere. That is impossible. This objection was put forward by the Greeks, reappeared in the scientific skepticism of the early Renaissance and probably occurs to any schoolchild who thinks about it today. On the basis of the objection, one must conclude that the universe is infinite.—J. J. Callahan, Scientific American (The stylistic variant for the if–then statement is unusual—it is “in being . . . must have . . .”)
      13. “There is one insuperable obstacle to a belief in ghosts. A ghost never comes naked: he appears either in a winding-sheet or ‘in his habit as he lived.’ To believe in him, then, is to believe that not only have the dead the power to make themselves visible after there is nothing left of them, but that the same power inheres in textile fabrics. Supposing the products of the loom to have this ability, what object would they have in exercising it? And why does not the apparition of a suit of clothes sometimes walk abroad without a ghost in it? These be riddles of significance. They reach away down and get a convulsive grasp on the very taproot of this flourishing faith.”—Ambrose Bierce, Devil’s Dictionary (Treat this as an indirect argument in which the only explicit premise is really the one starting “To believe in him . . .”)
      14. There was little doubt, then, that if the earth moved through an immovable sea of ether, there would be an ether wind, and if there were an ether wind, the Michelson-Morley apparatus would detect it. In fact, both scientists were confident that they would not only find such a wind, but that they could also determine (by rotating the slab until there was a maximum difference in the time it took light to make the two journeys) the exact direction, at any given moment, of the earth’s path through the ether. Michelson was astounded and disappointed. This time the astonishment was felt by physicists all over the world. Regardless of how Michelson and Morley turned their apparatus, they found no sign of an ether wind! Michelson never dreamed that this ‘failure’ would make the experiment one of the most successful, revolutionary experiments in the history of science. The reason Michelson and Morley were unable to detect an ether wind, Einstein said, is simple: there is no ether wind.—Martin Gardner, Relativity for the Million

      11.5 Summary of Chapter Eleven

      There are three common valid forms of if–then arguments: affirming the antecedent, denying the consequent, and transitivity of implication. There are two common invalid forms: the fallacy of affirming the consequent and the fallacy of denying the antecedent. When an argument takes one of these forms but has both a universal if–then premise and a conclusion about a single instance to which the universal applies, describe it in the same terms but for the addition of the phrase singular. . . .

      When paraphrasing, translate variants such as P only if Q and Q assuming P into the standard constant If P then Q.

      When judging the truth of if–then premises, concentrate chiefly on the proposed connection—whether causal or broadly logical—between the if-clause and the then-clause. Typically the if-clause is not alone presumed to be sufficient for the then-clause, but to be sufficient only in combination with secondary assumptions that are themselves not in question. The if-clause is the only one mentioned because it is presumed, in this particular context, to be the only factor in doubt. In defending your judgment that an if-then premise is true, point out the truth of the most doubtful secondary assumptions. In defending your judgment that the if–then premise is false, point out the falsity of a secondary assumption. In this way you either reinforce or sever the connection between if-clause and then-clause.

      When the if-clause is clearly true and the then-clause is clearly false, you may have the opportunity to effectively show the falsity of the if–then premise without reference to secondary assumptions in two different ways. First, you may provide a truth counterexample, assuming the if–then premise is universal. And second, you may provide an “educated ignorance” defense, which requires that the evidence for the truth of the if-clause and the falsity of the then-clause is strong—much stronger than the evidence for the if–then premise itself.

      If–then arguments are frequently enthymematic. When the if–then premise is the implicit statement, be especially attuned to the likely need for a universal if–then bridge.

      11.6 Guidelines for Chapter Eleven

      • Translate the stylistic variants for the if–then premise into the standard constant.
      • When an argument has both a universal premise and a conclusion about a single thing that is encompassed by the universal premise, consider whether it is the singular version of a sentential logical form.
      • Defend your judgment that an if–then statement is true by affirming the truth of the most questionable secondary assumptions. Defend your judgment that it is false by showing that a secondary assumption is false.
      • Do not defend your judgment of an if–then statement by simply rewording the statement (or, if false, by rewording the denial of the statement).
      • When a universal if–then statement is false, try to defend that judgment by providing a truth counterexample.
      • It is reasonable to judge an if–then premise false “because some secondary assumption must be mistaken, though I don’t know which one” only if there is very powerful evidence both that the if-clause is true and that the then-clause is false.
      • In indirect arguments, be alert for faulty secondary assumptions behind the if–then premise.
      • Consider providing a universal if–then bridge when an explicit link between premise and conclusion has not been provided by the arguer.

      11.7 Glossary for Chapter Eleven

      Affirming the antecedent—valid deductive form, as follows:

      Also known as modus ponens, which is Latin for “the method (or mode, from modus) of affirming (or propounding, from ponens).”

      Antecedent—the if-clause of an if–then statement.

      Consequent—the then-clause of an if–then statement.

      Denying the consequent—valid deductive form, as follows:

      Also known as modus tollens, which is Latin for “the method of denying.”

      Educated ignorance defense—defense of your judgment that an if–then premise is false even though you cannot tell which secondary assumption is at fault (thus, it reflects ignorance); it can be a reasonable defense only if your evidence for the truth of the if-clause and for the falsity of the then-clause is especially strong (thus, the defense is educated).

      Fallacy of affirming the consequent—invalid deductive form, as follows:

      Fallacy of denying the antecedent—invalid deductive form, as follows:

      Fallacy of singular affirming the consequent—invalid affirming the consequent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

      Fallacy of singular denying the antecedent—invalid denying the antecedent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

      If–then argument—one of a loosely defined group of deductive arguments that have an if–then statement as a premise. Also known as a conditional argument or hypothetical–syllogism.

      If–then statement—a statement in the form of If P then Q. Also known as a conditional.

      Secondary assumption—when an if–then statement is asserted, this is an assumption made, often implicit because it is not in doubt, about another factor besides the if-clause that contributes to the truth of the then-clause. Also known as auxiliary hypothesis.

      Singular affirming the antecedent—valid affirming the antecedent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

      Singular denying the consequent—valid denying the consequent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

      Singular transitivity of implication—valid transitivity of implication in which the if–then premises are universal and the conclusion is about a single instance that is encompassed by the universal term.

      Transitivity of implication—valid deductive form, as follows:

      It can have any number of if–then premises. It can also have a negative conclusion, as follows:

      Truth counterexample—strategy for defending your judgment that a universal if–then premise is false, by identifying a single instance in which the if-clause is obviously true and the then-clause is obviously false.

      Universal statement—a premise with a term like all, none, anything, or nothing.

      1. Note, incidentally, that If P then Q and If not Q then not P, which are called the contrapositives of one another, are logically equivalent; each is true—or false—under exactly the same circumstances. Consider, for example, these contrapositives: If construction begins before July 4, then the building will be ready to be occupied before snow falls. If the building is not ready to be occupied before snow falls, then construction did not begin before July 4. This equivalence produces an interesting implication for the four if–then forms that have just been introduced. There is always a logically equivalent affirming the antecedent argument for every denying the consequent argument (and vice versa). Likewise for each fallacy of denying the antecedent/affirming the consequent. Consider, for example, the following denying the consequent argument.
      1. If construction begins before July 4, then the building will be ready to be occupied before snow falls.
      2. The building will not be ready to be occupied before snow falls.
      3. ∴ Construction did not begin before July 4.

      This is equivalent to the following affirming the antecedent argument—note that the only change is to substitute for premise 1 its contrapositive:

      1. If the building is not ready to be occupied before snow falls, then construction did not begin before July 4.
      2. The building will not be ready to be occupied before snow falls.
      3. ∴ Construction did not begin before July 4.
      definition

      One of a loosely defined group of deductive arguments that have an if–then statement as a premise. Also known as a conditional argument or hypothetical–syllogism.

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      A statement in the form of If P then Q. Also known as a conditional.

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      The if-clause of an if–then statement.

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      The then-clause of an if–then statement.

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      Valid deductive form, as follows:

      Also known as modus ponens, which is Latin for “the method (or mode, from modus) of affirming (or propounding, from ponens).”

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      Valid deductive form, as follows:

      Also known as modus tollens, which is Latin for “the method of denying.”

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      Invalid deductive form, as follows:

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      Invalid deductive form, as follows:

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      Valid deductive form, as follows:

      It can have any number of if–then premises. It can also have a negative conclusion, as follows:

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      Valid denying the consequent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

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      A premise with a term like all, none, anything, or nothing.

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      Valid affirming the antecedent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

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      Valid transitivity of implication in which the if–then premises are universal and the conclusion is about a single instance that is encompassed by the universal term.

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      Invalid affirming the consequent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

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      Invalid denying the antecedent in which the if–then premise is universal and the conclusion is about a single instance that is encompassed by the universal term.

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      When an if–then statement is asserted, this is an assumption made, often implicit because it is not in doubt, about another factor besides the if-clause that contributes to the truth of the then-clause. Also known as auxiliary hypothesis.

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      Strategy for defending your judgment that a universal if–then premise is false, by identifying a single instance in which the if-clause is obviously true and the then-clause is obviously false.

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      Defense of your judgment that an if–then premise is false even though you cannot tell which secondary assumption is at fault (thus, it reflects ignorance); it can be a reasonable defense only if your evidence for the truth of the if-clause and for the falsity of the then-clause is especially strong (thus, the defense is educated).