Part Five: Evaluating Deductive Logic

Chapter Twelve: Either–Or Arguments and More

Chrysippus considers the actions of the dog, who comes to a path that leads three ways in search of his master, whom he has lost, and sniffs first one way and then another. Having assured himself of two, because he does not find his master’s track, without more ado the dog furiously follows the third path. Such a dog, Chrysippus says, must necessarily discourse thus with himself: “I have followed my master’s scent this far, he must of necessity pass by one of these three ways; it is neither this nor that, then consequently he is gone this other.” And by this conclusion of discourse assuring himself, coming to the third path, the dog sniffs no more, but by the power of reason suffers himself to be carried violently through it.

—Michel de Montaigne, Essays

TOPICS

  • Either–Or Arguments
  • Evaluating the Truth of Either–Or Premises
  • Dilemmas
  • Categorical Arguments

12.1 Either–Or Arguments

, sometimes more formally called disjunctive syllogisms, are so common and intuitive that—if Chrysippus, cited above, is right—even our pets are capable of using them. Either–or arguments prominently feature a statement of the form P or Q, which is called an .(sometimes known as a disjunction, though we will reserve that term for a valid form of argument). P and Q are the , known by logicians as disjuncts.

Here is a valid either–or argument form:

  1. P or Q.
  2. Not P.
  3. Q

You are following this form, for example, if you argue, “He’s either lying or he’s crazy. He’s not lying. He must be crazy.”

Another valid form is:

  1. P or Q.
  2. Not Q.
  3. P

For example: “He’s either lying or he’s crazy. He’s not crazy. He must be lying.”

Valid either–or arguments can include more than one alternative, as in this case:

  1. P or Q or R.
  2. Not P.
  3. Q or R.

For example: “He’s either lying or he’s crazy or, well, I’m crazy. I don’t think he’s lying. So one of us must be crazy.”

And, finally, here’s another example of a valid either–or form with more than two alternatives:

  1. P or Q or R.
  2. Not P.
  3. Not Q.
  4. R

For example: “He’s either lying or he’s crazy or, well, I’m crazy. I can’t see how he would be lying. And he’s the sanest person I know. So I must be crazy.”

An ordinary English term covers all of these valid forms: the . There is also a more formal Latin term, modus tollendo ponens, which means “the method of denying in order to affirm.”[1] In principle, there is no limit to the number of alternatives in the either–or premise. All that matters, to ensure that it is a valid process of elimination argument, is that the conclusion includes all the alternatives that have not been eliminated by a premise.

When clarifying either–or arguments that eliminate more than one alternative, eliminate each alternative in a separate premise. Don’t, for example, represent He is neither lying nor crazy as a single premise; rather, separate it into two premises, He is not lying and He is not crazy. This puts you in the position to evaluate the truth of each claim independently.

Chrysippus’s cerebral dog seems to be using the fourth valid form of the process of elimination. A clarification of the argument might look like this:

  1. The master took the first path or the master took the second path or the master took the third path.
  2. The master did not take the first path.
  3. The master did not take the second path.
  4. The master took the third path.

As always, be prepared to translate stylistic variants for or into the standard constant. Some of the expressions that can be variants for or are in the following list.

Stylistic Variants for P or Q

Q or P.
Either P or Q.
P unless Q.
P or else Q.
If not P, then Q.

Also, when clarifying be alert for implicit statements. Often the either–or premise will be the implicit one. Suppose I said, “He’s not crazy. And he’s not lying. So I must be crazy.” It is evident that I’m relying on the implicit premise He’s lying or he’s crazy or I’m crazy.

Guideline.  Translate stylistic variants for the either–or premise into the standard constant. Also, be alert for implicit statements, including the either–or premise.

EXERCISES Chapter 12, set (a)

Paraphrase the either–or statement in each of these passages, translating the stylistic variant into the standard constant.

Sample exercise. “One could not be a successful scientist without realizing that, in contrast to the popular conception supported by newspapers and mothers of scientists, a goodly number of scientists are not only narrow-minded and dull, but also just stupid.” —James Watson, co-discoverer of the structure of DNA, in his memoir The Double Helix

Sample answer. One is not a successful scientist or one realizes that many scientists are stupid.

  1. Either Berkeley or Stanford would have been a good university for you to attend.
  2. Unless you have salami today, I’ll have the tuna sandwich.
  3. Mr. Duesberg charges that the HIV virus is incapable of causing AIDS. He contends that a virus that later immobilizes the immune system cannot exist in a person’s body for several years unless it causes serious ill effects. —Chronicle of Higher Education
  4. I wasn’t drunk. I wasn’t impulsive. I damned well wanted a tattoo. Unless you’re convinced you want it, you shouldn’t do it. —John McPhee, Looking for a Ship

EXERCISES Chapter 12, set (b)

Clarify each of the either–or arguments below.

Sample exercise. “His ‘Nothing’ doesn’t suggest the debates of existentialism these days so much as it does the plight of the homeless, bedding down on the sidewalk for another night. Today we know what Beckett is talking about; he has become a realist. And yet he didn’t change one inch over the last 30 years. It must have been us.” —obituary for writer Samuel Beckett

Sample answer.

  1. [Beckett changed over the last 30 years of his career or Beckett’s audience changed over the last 30 years of his career.]
  2. Beckett didn’t change over the last 30 years of his career.
  3. Beckett’s audience changed over the last 30 years of his career.
  1. In a departmental meeting, one professor commented, “Either this is the best bunch of graduate students we have produced or I’ve lost my mind.” He paused, then concluded, “This bunch is the best.”
  2. The Atlanta baseball team accused the Dodgers of ratting on them to the National League office for allowing players to work out ahead of the date agreed to with the players’ association. “The league told us the complaint came from Vero Beach,” said an Atlanta official in the their camp at West Palm Beach. “That means it either came from the Dodgers or the New Orleans Saints (who train at Dodgertown in July).” “And we’ve eliminated the Saints,” said another Atlanta official. “It’s just a petty thing to do, but that’s the Dodgers for you.” —from a February newspaper story
  3. (A priest is apparently unconscious on the floor of a bar.) Man 1: “He got bushwhacked.” Man 2: “There ain’t but two reasons why a man gets bushwhacked: love or money.” (We hear the sound of the two men going through the unconscious priest’s pockets and then the jingling of small coins.) Man 2: “Well it ain’t money. Musta been love.” —Powder River Policy, a play by E. Jack Newman
  4. “It goes without saying that my work couldn’t be bogus or I wouldn’t be doing it.” (Second premise and conclusion are implicit.) —Chronicle of Higher Education, quoting response by Professor John Bockris of Texas A&M to criticisms of his attempts to convert mercury to gold
  5. “But the power of Marxism cannot be explained solely by his theories; for these were at least partially limited by his nineteenth-century experience, and they have been superseded by the considerable development of the social science. The power of Marxism must therefore be located to a considerable degree in its religious impulse and its moral protest.”—Paul Kurtz, Free Inquiry

12.1.1 Other Related Forms

A handful of other valid forms use either–or statements. A fairly rare yet very simple form is called , as follows:

  1. P
  2. P or Q.

It is rare because (like the forms repetition, simplification, and conjunction from Chapter 10) there is seldom any reason to use it. Perhaps I might argue in this way: “Why do I agree with you that the assignment for tomorrow was either postponed or even canceled? Because—I happen to know—the fact is, it was canceled.” Not a very illuminating argument, but a valid one all the same. There is a commonsense way of thinking about this form: if one of the alternatives is true, then either that alternative or something else is true.

Two other related forms that are occasionally useful are together referred to as (after the logician who formulated them). They combine either–or statements with negative statements and both–and statements, as follows:

  1. Not (P or Q).
  2. Not P and not Q.
  1. Not (P and Q)
  2. Not P or not Q.

There is likewise a commonsense way of thinking about these. In the case of the first law, the argument says that if an either–or statement is false, then neither of the alternatives is true. From It is false that either Hamilton or Burr would’ve been a good president, it follows validly that Hamilton would not have been a good president, and Burr would not have been a good president.

The second law tells us that if a conjunction is false, then at least one of the conjuncts is false. So, from It is false that Hamilton and Burr would both have been good presidents, it follows validly that Either Hamilton would not have been a good president, or Burr would not have been a good president.

EXERCISES Chapter 12, set (c)

State, in standard form, a conclusion that validly follows from each premise according to the specified rule. (We are concerned here only with logical validity, so it does not matter whether the premise is true.)

Sample exercise. By the applicable DeMorgan’s law. Neither the rain nor the snow kept the mail carrier from delivering the mail.

Sample answer. The rain did not keep the mail carrier from delivering the mail and the snow did not keep the mail carrier from delivering the mail.

  1. By disjunction. Halley’s comet will return in 2061.
  2. By the applicable DeMorgan’s law. To say that her work is both good and original is profoundly misleading.
  3. By the process of elimination. You guaranteed that we would either hit the jackpot or lose everything. Let’s put it this way—we didn’t hit the jackpot.
  4. By the applicable DeMorgan’s law. I expected the championship to be won by either the Bulls or the Lakers. I was wrong.
  5. By the process of elimination. As I expected, the championship was won by either the Bulls or the Lakers. And it wasn’t the Bulls.
  6. By disjunction. Abraham Lincoln was the first president of the United States.

12.1.2 The Exclusivity Premise

So far, we have been concerned with either–or statements that communicate the core idea that at least one of these alternatives is true.[2] But there are also exclusive either–or statements, which are used to communicate, in addition to the core idea, the idea that only one of these alternatives is true. This addition of exclusivity—of the truth of only one of the alternatives—can have implications for the evaluations of both truth and logic.

Sometimes the alternatives happen to be exclusive, but the exclusivity is not important to the argument. Suppose you and I agree that the greatest American writer is either Mark Twain or Herman Melville. They cannot both be the greatest, so the either–or statement is exclusive. You then state that you do not think highly of Melville, and thus you consider Twain to be the greatest. You have offered a valid process of elimination argument that does not depend on the exclusivity of the two alternatives; it depends solely on the core idea that at least one of the alternatives is true, and it should be evaluated solely on that basis.

But suppose the discussion had taken a different turn; suppose after we agreed that the greatest was either Twain or Melville, you instead had remarked, “Surely Twain is the greatest, so Melville is not.” It would be tempting, though highly uncharitable, to paraphrase this argument according to the following form:

  1. P or Q.
  2. P
  3. Not Q.

This form is in fact invalid; the name of the invalid form is the . Its invalidity can easily be demonstrated by a validity counterexample. Suppose a student who is having a hard time in class comes to me for advice, and I discover that, out of nothing but sheer laziness, he seldom attends and that he never does the exercises. So I say to him, “Well, you can come to class more often or you can do the exercises.” Someone might then reason thus:

  1. The student can come to class more often or the student can do the exercises.
  2. The student can come to class more often.
  3. The student cannot do the exercises.

The premises are obviously true and conclusion obviously false, according to my story. The form, then, is invalid.

A more charitable paraphrase of the Twain-or-Melville argument would revise premise 1 so that it conveys not only the core idea of the either–or statement, but also the additional idea of exclusivity. Called the , it has the following form: P or Q and only one.[3] The overall argument, then, would take this form:

  1. P or Q and only one.
  2. P
  3. Not Q.

You will not be able to find a counterexample for this form. It is valid, and we will call it . There was a time when this form was known as modus ponendo tollens—that is, the method of affirming in order to deny. This Latin term, however, has fallen out of use, and you should probably avoid it even if you love Latin. Those who do not know quite as much as you do will suspect that you have doubly erred: that you have not only misidentified the argument as modus tollendo ponens (the formal Latin term, still in use, for the process of elimination), but also that you have garbled the term.

Some Arguments with Either–Or Premises

Valid Invalid
Process of elimination Fallacy of affirming an alternative
Disjunction
DeMorgan’s laws
Affirming an exclusive alternative
Guideline.  When the context and the logic of the argument call for it, paraphrase exclusive alternatives by including and only one as part of the standard constant.

EXERCISES Chapter 12, set (d)

Clarify each of these either–or arguments, then evaluate its logic. (Some of them require an exclusivity premise, others do not.)

Sample exercise. You’re right that either the Missouri or the Mississippi River is the longest in the United States. In fact, it’s the Missouri—I remember how surprised I was to find that out in school. So, it isn’t the Mississippi!

Sample answer.

  1. The Missouri River is the longest river in the United States or the Mississippi River is the longest river in the United States, and only one.
  2. The Missouri River is the longest river in the United States.
  3. The Mississippi River is not the longest river in the United States.
    Valid, affirming an exclusive alternative.
  1. You should major in something useful, like business, unless you’re really committed to studying ideas for the sake of studying ideas. But I don’t get the impression that you’re an idea person. So stick with business.
  2. Sure, I’ve been to Abilene before. Wait a minute—it was either Abilene or Amarillo. Oh, yeah, now I remember, that was Amarillo. So, no, I guess I haven’t been to Abilene.
  3. The department says that either Smith or Jones will be teaching the course. But I talked to Smith—he’s going to be out of the country. So apparently it’s going to be Jones.
  4. Her letters of reference say that she’s qualified to work as either a manager or a sales representative. I can see from the interview that she’s qualified as a sales rep. So we can conclude, based on the letters, that she’s not qualified as a manager.

12.2 Evaluating the Truth of Either–Or Premises

There is an obvious—but not always very helpful—thing to be said about evaluating the truth of either–or statements. All it takes to judge an either–or statement to be certainly true is to find that one of the alternatives is certainly true. And to find the either–or statement to be certainly false, both alternatives must be certainly false. This, for example, would be the simple basis for judging it certainly true that Either two plus two equals four or the moon is made of green cheese, and certainly false that Either two plus two equals five or the moon is made of green cheese.

This strategy serves us well in cases where we have overwhelming evidence for the truth or falsity of the alternatives. But there are many cases in which there is less certainty—cases in which we might be able to make a judgment about the probability of each alternative but are not sure what to say about the either–or statement. Here is a simple, two-step rule that can guide you in such cases: first, add the individual probabilities for each of the alternatives; and second, subtract the probability that both alternatives are true.

Suppose I want to offer a research job to a student using funding that can be used only to support an upperclassman. I do not know what class she is in, and I need to evaluate the truth of this statement:

She is a junior or she is a senior.

Until I have the chance to ask her or to check her file, all I have to go on is my knowledge that enrollments drop by 2 percent with each class. This would mean that it has 28 percent freshmen, 26 percent sophomores, 24 percent juniors, and 22 percent seniors. I would thus assign a .24 probability to the alternative She is a junior and a .22 probability to the alternative She is a senior. Step one, adding the probabilities, yields a probability of .46. Since there is no probability that she is both a junior and a senior, step two requires me to subtract nothing. Based on the limited frequency information I have, the probability of the either–or statement is .46—that is, there is a very slight probability that the statement is false, and that she is ineligible.

Suppose, however, I have a different source of funding, one that can be spent only on graduates of California high schools, whatever their class, or on seniors, whatever state they are from. Now I must judge the following either–or statement:

She is a senior or she is a California high school graduate.

Again, assume that I haven’t had the chance to collect information specifically about her, but only have information about frequencies. I again assign a .22 probability to the first alternative and, knowing that 90 percent of the undergraduates are graduates of California high schools, I assign a .90 probability to the second alternative. After taking the first step of adding them together, I have a 1.11 probability. But it would be a glaring mistake to stop here. Saying that the probability of a statement’s truth is greater than 1.00 is as incongruous as an athlete’s saying, “I always give 110 percent.” By definition, 100 percent is everything that the athlete can give and 1.00 is as high as a probability can be.

The problem is that students who are both seniors and California high school graduates are being counted twice—once in each category. Using the Chapter 10 guidelines for both–and statements,[4] I conclude that 20 percent of the students are both seniors and California high school graduates. I subtract this .20 probability from the 1.11 of the first step (that is, I subtract the students that were counted twice), and I conclude that there is a .91 probability that the either–or statement is true. Based on the very limited information that I now have, it is highly probable that the statement is true, and that she is eligible for the funding.

As we saw in the discussion of both–and sentences in Chapter 10, we often don’t have such tidy information about frequency probabilities at our fingertips. Further, even when we do, we usually have other relevant evidence as well—suppose the student asked me to write her a letter of reference to law school, and that I’ve heard her talk about missing snow in the wintertime. This evidence must be considered in evaluating the overall epistemic probability of the either–or statement. In light of this additional information, I may now think that She is a senior is somewhat probable, but may now be undecided about She is a graduate of a California high school.

Strategies for Evaluating the Truth of Either–Or Statements

What you know about the alternatives How to evaluate the either-or statement
Both alternatives are almost certainly false. Almost certainly false.
At least one alternative is almost certainly true. Almost certainly true.
Alternatives are merely probable. Add the probabilities of the alternatives, then subtract the probability that both are true.

Despite the vagueness of our probability judgments, it can be helpful to assign tentative numbers to the alternatives so that arithmetic can provide a helping hand. Let’s take “somewhat probable” to be .60 and “can’t decide” to be .50; these add up to 1.10. I now have to subtract the probability that she is both, which is .30; this yields a .80 probability for the either–or statement. Although this number is misleadingly precise, it is clear that the new information still allows me to judge the either–or statement as probably true—though somewhat less probable than it was.[5]

Guideline.  When the alternatives of an either–or 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.

EXERCISES Chapter 12, set (e)

Explain your calculations and then state your evaluation of the truth of the statement based on the information provided.

Sample exercise. The Republican candidate for president will be elected or the Democratic candidate for president will be elected. (The form is P or Q. P is .55; Q is .43; the probability of both is zero, since there can’t be two presidents—or, to put it in the terminology of Chapter 10, if we assumed the Republican candidate were president, then the probability of the Democratic candidate’s also being president would be zero; and .55 times zero is still zero.)

Sample answer. Add .55 and .43, then subtract 0, for .98. It is almost certainly true.

  1. His father will buy a Lexus next year, or his father will buy a Cadillac next year. (P is .30; Q is .05; probability of both is .01.)
  2. My heart was beating last year, or I was breathing last year. (P is 1.00; Q is 1.00; both is 1.00)
  3. My new love will either break my heart or make me the happiest person alive. (Assign plausible probabilities yourself.)
  4. My library book was due today, or it was due yesterday. (P is .5; Q is .4; both is zero.)

12.3 Dilemmas

In everyday life, a dilemma is a problem. Not so in logic, where a points out the consequences, whether good or bad, of two inevitable alternatives. (The word comes from the Greek words di, for two, and lemma, for proposition.) They come in two varieties: either–or dilemmas and, less common, if–then dilemmas.

12.3.1 Either–Or Dilemmas

usually take one of these four forms:

First,

  1. P or Q.
  2. If P then R.
  3. If Q then R.
  4. R

For example: “I’m definitely going to take either Psychology 101 or Biology 101. If I take Psychology 101, I’ll satisfy a requirement for my major. If I take Biology 101, I’ll satisfy a requirement for my major. So either way I’ll satisfy a requirement for my major.”

Second,

  1. P or Q.
  2. If P then R.
  3. If Q then S.
  4. R or S.

For example: “I’m definitely going to take either Psychology 101 or Biology 101. If I take Psychology 101, then I’ll have to get up at 6 a.m. If I take Biology 101, I’ll have to park on the other end of campus. So I’ll either have to get up at 6 a.m. or park on the other end of campus.”

Third,

  1. P or Q.
  2. If R then not P.
  3. If R then not Q.
  4. Not R.

For example: “I’m definitely going to take either Psychology 101 or Biology 101. Now, if I also take Sociology 101, then I won’t be able to take Psychology 101. Furthermore, if I take Sociology 101, I won’t be able to take Biology 101. So, there’s no way I’ll take Sociology 101.”

Fourth,

  1. P or Q.
  2. If R then not P.
  3. If S then not Q.
  4. Not R or not S.

For example: “I’m definitely going to take either Psychology 101 or Biology 101. If I take Sociology 101, then I won’t be able to take Psychology 101. And if I take Philosophy 101, then I won’t be able to take Biology 101. So, I’ll have to do without either Sociology 101 or Philosophy 101.”

The first two are by far the most common. As you can see, they are modeled after the valid form of affirming the antecedent; in effect, they say, “You must affirm the antecedent of either the first or the second if–then statement.” The next two, similarly, are modeled after denying the consequent. There are no widely accepted names for these forms; so, when evaluating a valid dilemma, identify it simply as correct form for a dilemma.

In popular jargon—which presumes that dilemmas are problems—the two alternatives expressed in the first premise are often referred to as the horns of the dilemma. Showing that the either–or premise is false is termed escaping between the horns of the dilemma, since it provides a way out of the forced choice. And showing that one of the if–then premises is false is termed attacking a horn of the dilemma, since this renders the choice harmless by removing the threat of the purported consequence.

Dilemmas are usually enthymematic. Sometimes both the either–or premise and the conclusion are implicit; sometimes both of the if–then premises are implicit. As an example for both clarifying and evaluating, here is a dilemma expressed in a political cartoon from the early 1970s, shortly after Republican operatives were caught breaking into the Democratic Party offices in the Watergate office complex:

President Nixon to his press secretary, Ronald Zeigler: “OK, Mr. Press Secretary, give me some answers! If I knew about the Watergate Caper, what am I doing in the White House? And if I didn’t know anything about the affair what am I doing in the White House?”

CLARIFICATION

    1. [Nixon knew about Watergate or Nixon didn’t know about Watergate.]
    2. If Nixon knew about Watergate, then Nixon did not deserve to be president.
    3. If Nixon did not know about Watergate, then Nixon did not deserve to
      be president.
    4. [Nixon did not deserve to be president.]

EVALUATION

(The anti-Nixon view, for illustration’s sake.)

TRUTH

Premise 1. Certainly true, since the form P or not P is the law of the excluded middle.

Premise 2. Probably true; if he knew about it then he lied repeatedly about it, and we should hold our presidents to higher moral standards than this. An objector might protest that we have come to expect presidents to lie to us whenever it is politically expedient, even that they need to do so due to the peculiar demands of the office. But political leadership is strongest when it incorporates moral leadership, and only the strongest political leaders deserve to be president.

Premise 3. Probably true; a president should have enough influence over his inferiors either to insure that they are in line with his policies or to be quickly informed when they are seriously out of line. An objector might argue that a president surely cannot be expected to be fully informed about every unauthorized activity of his associates. This is true, but the Watergate break-in involved people at a high enough level that this does not excuse him.

LOGIC

The argument is valid, since it has the correct form for an either–or dilemma.

SOUNDNESS

Probably deductively sound.

Note that dilemmas don’t require us to learn much that is new. Evaluating the logic is simple, and evaluating the truth of the premises—given that they combine if–then and either–or premises—is already a familiar process.

EXERCISES Chapter 12, set (f)

Clarify and evaluate (to the extent that context allows) each of these dilemmas.

  1. “If one worries a lot, one is obviously unhappy, since worry itself is one of the most painful things in life. If one fails to worry enough, then (at least so I have been told) one may be even worse off because one may fail to take the precautions necessary to ward off even greater catastrophes than worry.”—Raymond Smullyan, This Book Needs No Title
  2. “The strapping rodeo bull rider grabbed Jerry Jeff Walker’s arm in a viselike grip and stared angrily at the singer, who bore a beatific, faraway expression on his face. “Didn’t you hear me, boy?” he growled. “I told you to play that song about red-necks. Now play it. Fast.” Stoned and drunk and uncertain if he was in a honky-tonk in Austin or in Oklahoma City, Walker struggled to concentrate on his dilemma. If he played the song, which he knew the cowboy hated, he would probably be beaten up. If he refused the request, he would also be beaten up. Finally, he began to play. The cowboy hit him three times, smashed his guitar, and left him bloody.”—Newsweek
  3. “ ‘There’s so much pressure,’ Tarkanian says. ‘But what else would I do? I have no other skills. Two years ago I was losing and I was going nuts, and I was thinking of getting out. I have a very good friend who said, “What would you do if you quit? You’d go nuts.”’ It’s refreshing to meet a man who knows where he’s going.”—Scott Ostler, Los Angeles Times, on basketball coach Jerry Tarkanian
  4. “At this point an annoying, though obvious, question intrudes. If Skinner’s thesis is false, then there is no point in his having written the book or in our reading it. But if his thesis is true, then there is also no point in his having written the book or our reading it.”—Noam Chomsky, “The Case Against B.F. Skinner,” in the New York Review of Books, criticizing Skinner’s view that humans have no free will, but act only in ways that are wholly determined by the physical world
  5. I described to him an impudent fellow from Scotland, who maintained that there was no distinction between virtue and vice. Johnson: “Why, Sir, if the fellow does not think as he speaks, he is lying; and I see not what honor he can propose to himself from having the character of a liar. But if he does really think that there is no distinction between virtue and vice, why, Sir, when he leaves our houses let us count our spoons.” —James Boswell, Life of Samuel Johnson, referring to philosopher David Hume
  6. Either God cannot eliminate the evil that is in the world, or he does not want to eliminate it. If he cannot eliminate evil, then he is not omnipotent. And if he does not want to eliminate it, he is not perfectly good. Therefore, either God is not omnipotent or he is not perfectly good. —classical form of the argument from evil against the existence of God

12.3.2 If–Then Dilemmas

are like either–or dilemmas, but for one exception. Whether you take one of the alternatives presented in the either–or premise depends on something else; whether you are stuck with the outcome in the conclusion hangs on the same condition. For any either–or dilemma, prefix If T, then . . . to the either–or premise and to the conclusion (and leave alone the two if–then premises); that makes it an if–then dilemma. Below is the scheme for the first, most common, form of if–then dilemma:

  1. If T, then P or Q.
  2. If P then R.
  3. If Q then R.
  4. If T, then R.

For example: “If I decide on psychobiology as my major, then I’m definitely going to take either Psychology 101 or Biology 101 next term. If I take Psychology 101, I’ll satisfy a requirement for my major. If I take Biology 101, I’ll satisfy a requirement for my major. So, if I decide on psychobiology as my major, then I’ll satisfy a requirement for my major next term.”

The other forms of if–then dilemmas follow the same pattern.

Types of Valid Dilemmas

  1. Either–or dilemmas
  2. If–then dilemmas

EXERCISES Chapter 12, set (g)

Clarify and evaluate (to the extent that context allows) each of these dilemmas. If–then and either–or dilemmas are mixed together.

Sample exercise. From Redd Foxx’s classic “Sanford and Son” TV series: Fred to Lamont, afraid he has shot his neighbor, “If I go and find him dead, I’ll have a heart attack. If I go and find him alive, I’ll jump around for joy and have a heart attack. So, either way, if I go, I’ll go.”

Sample answer.

CLARIFICATION

  1. [If Fred visits next door, then Fred finds his neighbor alive or Fred finds his neighbor dead.]
  2. If Fred finds his neighbor dead, then Fred will have a heart attack.
  3. If Fred finds his neighbor alive, then Fred will have a heart attack.
  4. [If Fred visits next door, then Fred will have a heart attack.]

EVALUATION

TRUTH

Premise 1 is probably true (in the fictional world). It assumes that the neighbor is there and that Fred will find the neighbor if he is there. Those assumptions are, I suppose, likely; and if they turn out to be true, there’s no doubt that he’ll be either dead or alive.

Premises 2 and 3 are probably false (in the fictional world); I don’t know much about Fred’s health in the fictional world of “Sanford and Son,” but this seems to assume that Fred’s health is extremely fragile, and also that in a situation comedy they would allow a leading character to have a heart attack. These assumptions seem unlikely to me—probably the premises are exaggerated for the humor.

LOGIC

This argument is valid—it has correct form for an if–then dilemma.

SOUNDNESS

Probably unsound, due to probably false premises 2 and 3.

  1. If I get a job next summer, then I’ll either get a better car or move into a nicer apartment. If I get a better car, the money I earn will be gone. If I move into a nicer apartment, then the money I earn will be gone. So, if I get a job next summer, the money I earn will be gone. (Evaluate this on the assumption that it applies to you.)
  2. In ad 642, as legend has it, the Caliph Omar commanded that all the books in the Great Library of Alexandria be burned as fuel to heat the city. The city fathers begged him to spare what was one of the Seven Wonders of the World. With diabolical logic, Omar refused. “The Koran is the source of all wisdom,” said the Caliph. “So if these books all agree with the Koran, they are redundant and thus can be burned. If they disagree, then they are heretical and thus should be burned.”
  3. Suppose you wish to get “enlightened” and a person asks you: “Why do you wish to get enlightened?” Now, the amazing thing is that however you answer, you will find yourself trapped! Suppose you answer, “For my sake.” Then people will descend upon you with the fury of hell and say: “You selfish egotist! How will that help the problems of the world? It is purely a selfish enterprise!” On the other hand, suppose you answer, “I primarily wish to help others, but I must first get enlightened myself before I can spread enlightenment to others.” Well, if you give that answer, people will descend upon you with the fury of hell and say: “You arrogant, conceited egotist! So it is up to you to enlighten others, eh? You have to be in the limelight. This whole ‘enlightenment’ business is just to feed people’s vanities!” —Raymond Smullyan, This Book Needs No Title
  4. “Either my piece is a work of the highest rank, or it is not a work of the highest rank. In the latter (and more probable) case I myself am in favour of it not being printed. And in the former case it’s a matter of indifference whether it’s printed twenty or a hundred years sooner or later. After all, who asks whether the Critique of Pure Reason, for example, was written in 17x or y.” —letter to Bertrand Russell from Ludwig Wittgenstein regarding his troubles finding a publisher for the Tractatus

12.4 Categorical Arguments

12.4.1 Categorical Syllogisms

Until the last century or so, the study of logic was the study of the categorical syllogism. We will not, however, need to cover this style of argument in any detail. An important goal of this book is to equip you with the tools for handling arguments that you are likely to encounter in your own experience, but you will probably never meet a categorical syllogism anywhere outside of a textbook. This section is included primarily for the historical context it provides and for the opportunity to introduce a common argument form that is related to the categorical syllogism.

The categorical syllogism, as codified by Aristotle in the fourth century bc, is made up of two premises and a conclusion, each of which takes one of these four forms:

A:   All F are G.
I:   Some F are G.
E:   No F are G.
O:   Some F are not G.

The statements identified as A and I, which are affirmative, are named for the first two vowels of the Latin affirmo; E and O, the negative statements, are named for the first two vowels of the Latin nego.

Since these statements have to do entirely with whether things that are in one category also belong to some other category (whether things that are F are also G), they are . And an argument, or syllogism, made up entirely of categorical statements is a . (Recall from Chapter 6 that the variables here are predicates, not sentences; so categorical syllogisms fall under the heading of predicate logic rather than sentential logic.)

12.4.2 Early Methods of Evaluating Logic

Logic students in the Middle Ages were asked to evaluate categorical syllogisms like this one:

  1. All prophets are men.
  2. Some prophets are not mortal.
  3. Some men are not mortal.

They would have determined that the argument was valid based on the following verse—made up of personal names of the era—which identifies all valid categorical syllogisms:

Barbara, Celerant, Darii, Ferioque, prioris;
Cesare, Camestres, Festino, Baroco, secundai;
Tertia, Darapti, Disami, Datisi, Felapton,
Bocardo, Ferison habet. Quarta insuper addit
Bramantip, Camenes, Dimaris, Gesapo, Fresison.

The first three vowels of each name pick out which three categorical statements make up the syllogism. Our argument is AOO, so it matches the name Baroco. Does the verse deem it valid? Yes. There are four different ways in which the Fs and Gs can be arranged in any categorical syllogism. The terms prioris, secundai, tertia, and quarta in the doggerel indicate whether the Fs and Gs are in the first, second, third, or fourth arrangement. Our argument is in the second arrangement, and Bacoro is indeed grouped with the names labeled secundai. The student thus ascertains that the argument is valid by solving this puzzle.

This system, unfortunately, did little to inspire a deep understanding of logic within those who memorized it. A better approach was to have students memorize a series of rules followed by any valid categorical syllogism. Our sample argument, for example, conforms to rules such as these:

There must be exactly three terms. (Our example has prophet, mortal,
and men.)
At least one premise must be of the all or none variety. (Premise 1 is.)
It cannot have two negative premises. (Ours has only one, premise 2.)
If the conclusion is particular (I or O), it cannot have two universal premises (A or E). (Ours, appropriately, does not.)
If one premise is negative, then the conclusion must be negative.
(Ours is.)

Because it follows these rules, it can be evaluated as valid.

12.4.3 The Venn Diagram Method

The best system—because it is simple and because it promotes understanding of the underlying logic of any categorical syllogism—was introduced by mathematician John Venn in the 19th century. Alas, he made this contribution when Aristotelian logic was about to be eclipsed by the vastly superior system of logic founded by Gottlob Frege.

Venn’s system, as it is normally implemented, requires the drawing of three overlapping circles, one circle representing each of the three categories. (See the accompanying figure.) For our sample argument, we can represent the first premise by shading the entire area of the prophet circle that does not overlap with the man circle (showing that there is no such thing as a prophet that is not a man, that is, All prophets are men). We then represent the second premise by putting an x in the part of the prophet circle that does not overlap the mortal circle (showing that there do exist some prophets that are outside the category of mortality, that is, Some prophets are not mortal); we avoid the shaded area, of course, since premise 1 has precluded our putting anything there. We can then see that the argument is valid, since it can be read off the diagram; we find an x in the man circle that is outside the mortal circle, that is, Some men are not mortal.

12.4.4 Singular Categorical Arguments

There is, however, one near relative of the categorical syllogism that does occur with some frequency and thus does merit a more careful look. Recall this familiar example:

  1. All men are mortal.
  2. Socrates is a man.
  3. Socrates is mortal.

Strictly speaking, this does not count as a categorical syllogism because neither the second premise nor the conclusion are in the form of A, E, I, or O.[6] The insertion of Socrates into the second premise and conclusion moves the argument from the universal to the particular. Following the pattern of the preceding chapter, we might term this form a . Not all singular categorical arguments are valid. The preceding argument is valid and can be defended on the grounds that it has correct form for a singular categorical argument. The following argument, however, would not be valid:

  1. All men are mortal.
  2. Socrates is mortal.
  3. Socrates is a man.

The evaluation of the logic of this argument should say that it has incorrect form for a singular categorical syllogism and should provide a validity counterexample.

For the sake of simplicity, however, I suggest that such arguments ordinarily be paraphrased so that they become examples of singular if–then arguments. As noted in Chapter 11, All F are G can be a stylistic variant for If anything (or anyone or anytime) is an F, then it is a G. The example above, then, is the exact logical equivalent of this:

  1. If anything is a man, then it is mortal.
  2. Socrates is a man.
  3. Socrates is mortal.

And it is valid by singular affirming the antecedent.

If the first premise had been No men are mortal, it would have validly followed that Socrates was immortal; the clarification would have looked like this:

  1. If anything is a man, then it is not mortal.
  2. Socrates is a man.
  3. Socrates is not mortal.

There are a few easy-to-spot variants of categorical statements that also can be translated into if–then statements. Only G is F, for example, is logically equivalent to All F are G and can typically be paraphrased as If anything is F, then it is G. Here is an example from ordinary English:

Now they know that she was a real princess, since she had felt the pea that was lying on the bedstead through twenty mattresses and twenty eiderdown quilts. Only a real princess could be so sensitive. —Hans Christian Andersen, “The Princess and the Pea”

The clarification would look like this:

  1. If anyone is sensitive enough to feel a pea through 20 mattresses and 20 quilts, then she is a princess.
  2. She is sensitive enough to feel a pea through 20 mattresses
    and 20 quilts.
  3. She is a princess.

It is valid by singular affirming the antecedent.

There are some rare cases in which it is better to leave a categorical statement as it is rather than paraphrase it as an if–then statement. Standard format requires, in certain cases, that you represent a conclusion in the form All F are G. In Chapter 14, for example, we will look at arguments like this: “All computer programs I ever used have had bugs in them. I conclude, then, that all computer programs have bugs.” As we will see, clarifying the argument in standard format gives us this:

  1. All sampled computer programs have bugs in them.
  2. All computer programs have bugs in them.

What makes these arguments distinctive is that they move from a sample to the general population; this move is made clear when the conclusion looks exactly like the premise but for the omission of the word sampled. This clarity of structure would be lost if we paraphrased the conclusion’s categorical statement as an if–then statement, as you can see:

  1. All sampled computer programs have bugs in them.
  2. If anything is a computer program, then it has bugs in it.

Keep the categorical statement in such cases. This means that you will still have the occasional singular categorical argument to evaluate. Suppose, for example, I further argued, “So, the word processing program I’m using right now has a bug in it.” The clarified complex argument would look like this:

  1. All sampled computer programs have bugs in them.
  2. All computer programs have bugs in them.
  3. [This word processing program is a computer program.]
  4. This word processing program has a bug in it.

The inference from 1 to 2 will be dealt with in Chapter 14; the inference from 2 and 3 to C is valid on the grounds that it has correct form for a singular categorical argument.

Guideline.  Convert categorical statements, in most cases, into if–then statements. Retain the categorical statement, however, when standard format for the argument calls for it, as with the conclusion in some inductive generalizations.

EXERCISES Chapter 12, set (h)

Clarify the following arguments, treating the categorical statements as stylistic variants for if–then statements. Then evaluate the logic. (Not all are valid.)

Sample exercise. Only those with incomes above $100,000 per year are eligible. I know what you make. No need to apply.

Sample answer.

  1. If any person is eligible, then the person’s income is above $100,000.
  2. [Your income is not above $100,000.]
  3. You are not eligible.
    Valid, singular denying the consequent.
  1. Nobody with an income below $100,000 per year is eligible. I know what you make. No need to apply.
  2. Only the lonely come to this nightclub. I was lonely. So, of course, I had to come.
  3. Everything he cooks is delectable. He’s the chef tonight. So, as you might expect, it will be delectable.
  4. Only those actions that can be demonstrated by empirical evidence to warrant criminal sanctions should be punished. On this basis, prostitution should not be punishable in our legal system.
  5. Yon Cassius has a lean and hungry look; such men are dangerous. —Shakespeare, Julius Caeser

12.5 Summary of Chapter Twelve

Two valid forms of either–or arguments are the process of elimination and disjunction. An invalid form is the fallacy of affirming an alternative; if, however, an argument with this form has an exclusive either–or premise (which can be paraphrased as P or Q and only one) it is a valid example of affirming an exclusive alternative.

Several other valid forms use either–or premises in combination with both–and premises (DeMorgan’s laws) or if–then premises (dilemmas).

All either–or statements express the core idea that at least one of the alternatives is true. Thus, if all alternatives are false then the either–or statement is false; otherwise, it is true. When thinking in terms of probabilities, a helpful guideline is to add the probabilities of both alternatives (but then to subtract the probability that both are true) to arrive at the probability for the entire either–or statement.

Some arguments require an either–or statement that, in addition to the core idea, expresses the idea that only one of the alternatives is true—that is, that the alternatives are exclusive. In such cases, paraphrase as P or Q and only one.

Historically, logic has mainly been concerned with categorical syllogisms. These arguments occur rarely in ordinary language, and logicians in the last century have abandoned the study of the categorical syllogism for much more sophisticated approaches that are taught in formal logic courses. But the singular categorical argument is related to the categorical syllogism and does occur in ordinary language from time to time. It is usually best, however, to treat categorical statements like All F are G and No F is G as stylistic variants of if–then statements and to paraphrase—and evaluate—them accordingly.

12.6 Guidelines for Chapter Twelve

  • Translate stylistic variants for the either–or premise into the standard constant. Also, be alert for implicit statements, including the either–or premise.
  • When the context and the logic of the argument call for it, paraphrase exclusive alternatives by including and only one as part of the standard constant.
  • When the alternatives of an either–or 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.
  • Convert categorical statements, in most cases, into if–then statements. Retain the categorical statement, however, when standard format for the argument calls for it, as with the conclusion in some inductive generalizations.

12.7 Glossary for Chapter Twelve

Affirming an exclusive alternative—valid argument form, as follows:

  1. P or Q and only one.
  2. P
  3. Not Q.

Historically known as modus ponendo tollens, Latin for “the method of
affirming in order to deny.”

Alternative—a statement connected to another by or. Also known as disjunct.

Categorical statement—statement of the form All F are G, No F are G, Some F are G, or Some F are not G.

Categorical syllogism—one of a family of deductive arguments, some valid and others invalid, each with three categorical statements—two as premises, one as conclusion.

DeMorgan’s laws—valid deductive forms, as follows:

  1. Not (P or Q).
  2. Not P and not Q.
  1. Not (P and Q).
  2. Not P or not Q.

Dilemma—a valid argument form that points out the consequences, whether good or bad, of two inevitable alternatives. The word comes from the Greek words di, for two, and lemma, for proposition. Dilemmas come in two varieties: either–or dilemmas and if–then dilemmas.

Disjunction—valid deductive form, as follows:

  1. P
  2. P or Q.

The term is sometimes also used for an either–or statement.

Either–or argument—one of a loosely defined group of arguments that has an either–or premise. Also called disjunctive syllogism.

Either–or dilemma—a valid dilemma that begins with an either–or premise. The most common forms are these:

  1. P or Q.
  2. If P then R.
  3. If Q then R.
  4. R
  1. P or Q.
  2. If P then R.
  3. If Q then S.
  4. R or S.
  1. P or Q.
  2. If R then not P.
  3. If R then not Q.
  4. Not R.
  1. P or Q.
  2. If R then not P.
  3. If S then not Q.
  4. Not R or not S.

Either–or statement—statement of the form P or Q. Also sometimes called a disjunction.

Exclusivity premise—either–or premise that includes the notion that only one of the alternatives is true; it has the form P or Q and only one.

Fallacy of affirming an alternative—an invalid argument form, as follows:

  1. P or Q.
  2. P
  3. Not Q.

If–then dilemmas—valid dilemmas that are constructed like any either–or dilemmas, except for an if–clause prefixed both to the either–or premise and to the conclusion. The most common form is as follows:

  1. If T, then P or Q.
  2. If P then R.
  3. If Q then R.
  4. If T, then R.

Process of elimination—a valid form of either–or argument in which premises eliminate alternatives, while the conclusion includes all the alternatives that have not been eliminated by a premise. Examples include:

  1. P or Q.
  2. Not P.
  3. Q
  1. P or Q.
  2. Not Q.
  3. P
  1. P or Q or R.
  2. Not P.
  3. Q or R.
  1. P or Q or R.
  2. Not P.
  3. Not Q.
  4. R

Also called modus tollendo ponens, Latin for “the method of denying in order to affirm.”

Singular categorical argument—a deductive argument, either valid or invalid, with a universal categorical statement as a premise, but with a conclusion about a single instance that is included in the universal category. For example, a valid form is:

  1. All F are G.
  2. A is F.
  3. A is G.

  1. Some logicians reserve the term disjunctive syllogism for these valid forms alone.
  2. When only this core idea is intended, the or is called the non-exclusive or. (It is sometimes, alternatively, misleadingly called the inclusive or.)
  3. Some writers use the premise P or Q and not both. This does not serve the purpose as well as a generic form; if there are three exclusive alternatives, it would miss the point to say P or Q or K and not all three. That would still allow two to be true, which is not exclusive.
  4. I multiply .22 times .9, with no need to subtract anything since the truth of neither part would affect the probability of the other part.
  5. For an exclusive either–or statement, there is an extra step in judging its truth. It takes the form P or Q and only one. Only one is short for Only one of the alternatives is true. This is itself a statement—call it R—that must be evaluated. For our purposes, the form to be evaluated is (P or Q) and R. Evaluate P or Q first. Then, using the rule for the probability of both–and sentences, evaluate (P or Q) and R. Suppose you conclude that the probability of P or Q is .70 and that the probability that only one of them is true (that is, the probability of R) is .99. The probability of the exclusive either–or statement, then, is just under .70—that is, it is somewhat probable.
  6. Aristotelian logic does have its awkward techniques for converting such arguments to categorical syllogisms.

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