Read the case State v. Ransom (pp. 411-425). Consider your verdict. Prepare a document that expresses your deliberation that justifies your vote of guilty or not guilty. The goal is not simply a w

14 Necessary and Sufficient Conditions

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Whether you are reading an editorial, watching a commercial, or serving on a jury, detecting the exact conclusion is of the first importance. But that is often a complex task. If you are serving on the jury during a breaking or entering case, it is easy enough to state the conclusion that the prosecution is attempting to prove: The defendant is guilty of breaking or entering. That’s right as far as it goes, but it doesn’t go very far.

Necessary Conditions

The judge’s instructions to the jury will describe exactly what must be proved in order to convict of breaking or entering. If you don’t know exactly what counts as breaking or entering, then your careful attention to the evidence and your finely honed logical capacities will be useless. This is true for every argument you examine: If you aren’t clear about the conclusion of the argument, it will be impossible to effectively evaluate the argument.

If you are serving as a jury member (in North Carolina) in a breaking and entering trial, the judge’s instructions will probably include the following:

Now I [the judge] charge that for you [the jury] to find the defendant guilty of felonious breaking or entering, the State [the prosecution] must prove four things beyond a reasonable doubt.

First, that there was either a breaking (which simply means the opening or removal of anything blocking entry) or an entry (walking or reaching in would be an entry) by the defendant.

Second, the State must prove that it was a building that was broken or entered.

Third, that the owner or tenant did not consent to the breaking or entering.

And fourth, that at the time of the breaking or entering, the defendant intended to commit some specific felony.1

So the judge instructs, along with some other points. And the jury retires to consider its verdict. Now we come down to brass tacks, and probably to rampant confusion. For Chauncey (one of your fellow jurors) the verdict is perfectly clear: The defendant was caught in the house, so obviously the defendant is guilty as charged. Now suppose that you agree with Chauncey that the evidence shows the defendant was caught in the house; does that prove the defendant guilty of breaking or entering?

No, of course it doesn’t. Chauncey is so fixed on the fact that two of the conditions for breaking or entering have been met (the defendant did enter, and entered a building) that he has neglected the other two conditions. That is, Chauncey has confused necessary conditions with sufficient conditions. One necessary condition for finding the defendant guilty of breaking or entering is that you be convinced that the defendant either broke or entered. (Thus if you think that perhaps the defendant neither broke nor entered, you would have to vote not guilty.) But while that is a necessary condition for guilt, it is not sufficient.

In many cases in which the defendant is charged with breaking or entering, there is no doubt that the defendant entered the building. In such a case the defendant may admit that she entered the building, but claim that the owner had given permission. Or perhaps the defendant will admit entering, but claim that he had no intention of committing any felony. (“I lost the key to my apartment,” he claims, “and when I went around to the back to crawl in a window, I got confused and crawled in the window to the neighboring apartment thinking it was my own; they all look alike, there are 12 identical apartments running along the back of the building, the light was bad, and I didn’t realize my mistake until I was in the apartment and heard someone yelling for the police. It was an innocent mistake: certainly I wasn’t intending to steal anything.”) This is a case in which the defense admits that the first three conditions for breaking or entering are met; but the defense case rests on its claim that the fourth necessary condition (that at the time of the entering the defendant intended to commit a felony such as larceny) was not met. And if you—as a juror—decide that there is a reasonable doubt that the defendant intended to commit a felony, then you must vote not guilty.

Each of the four conditions is a necessary condition for finding the defendant guilty of breaking or entering. Taken altogether, they are sufficient for finding the defendant guilty of breaking or entering. That is, if you are certain that all four conditions are met (one, the defendant did either break or enter; two, it was a building into which the defendant either broke or entered; three, the owner or tenant had not given permission; and four, the defendant did intend to commit a specific felony), then you should conclude that the defendant did violate the law.

We have been considering the charge of breaking or entering: It has four conditions, each of which is necessary and jointly they are sufficient. But for other charges the necessary and sufficient conditions may be quite different. Consider a case in which a defendant in North Carolina (the law and conditions will obviously

Argue Your Case

You are defense counsel for Corena, a teenager charged with breaking or entering. Corena was picked up by the police in a high school late at night: There’s no question that she was there, and that she had broken into the locked building (she has admitted prying a window open in order to get inside). What questions would you now ask Corena—this is your first interview with her—in order to decide what sort of defense to present (or perhaps whether to plead guilty and ask for mercy)? In other words, what sort of defense could you make against the breaking or entering charges?

differ from state to state) is charged with illegal gambling. The judge instructs the jury as follows:

Now I charge that for you to find the defendant guilty of gambling, the state must prove one of two things beyond a reasonable doubt:

First, that the defendant bet money on a game of chance.

Or second, that the defendant played at a game of chance because money was being bet on it.

So I charge that if you find from the evidence beyond a reasonable doubt that on or about August 15, 2010, the defendant either bet money on a game of chance or played at a game of chance because money was being bet on it, it would be your duty to return a verdict of guilty as charged. However, if you do not so find or have a reasonable doubt as to both these things, it would be your duty to return a verdict of not guilty.2

Having heard all the evidence, the closing arguments, and the judge’s charge to the jury, you retire to the jury room to begin deliberation. Beatrice (one of the jurors) argues that the verdict must be not guilty: “I’m convinced that the defendant was in the alley where the dice game was taking place, and he was betting money on it. But the game was broken up by the police before he ever had his turn at rolling the dice. There’s really no evidence that he ever held the dice in his hand, and certainly no evidence that he ever rolled them. So it has not been proved that he ever played the game at all. And since there is at least a reasonable doubt about whether he played the game, we can’t find him guilty of gambling.” What is wrong with Beatrice’s argument for acquittal?

Distinguishing Necessary from Sufficient Conditions

Beatrice has confused sufficient conditions with necessary conditions. If the defendant either played at a game of chance because money was being bet on it or bet money on a game of chance, then the defendant is guilty of gambling. Each of those conditions is sufficient; neither of those conditions is necessary. If the law were that in order to be guilty of gambling one must both bet money on and play a game of chance, then Beatrice’s point would strongly support a verdict of not guilty; but since to be guilty of gambling (in contrast to felonious breaking or entering) each condition is sufficient, Beatrice’s argument does not offer grounds for acquittal.

Argue Your Case

You are a defense attorney, and your client has been charged with felonious gambling. Apparently there is no evidence that the defendant himself placed any bets. However, there are several eyewitnesses who are willing to testify that he was a major participant in a darts match at the Dew Drop Inn, in which other people bet large sums of money. Thus the prosecution is arguing that the defendant meets a sufficient condition for being guilty of felonious gambling: He played at a game of chance because money was being bet on it. What line would your defense take? On what grounds might you argue that the defendant is not guilty (or at least that there is reasonable doubt of the defendant’s guilt) of felonious gambling?

Sufficient Conditions

In sum, when considering exactly what conclusions the prosecution and the defense are trying to prove, it is crucial to understand the necessary and sufficient conditions for verdicts of guilty and of not guilty. In order to be quite certain that you are comfortable with this, try thinking about the breaking or entering case from another angle. We considered what the prosecution was trying to prove in order to prove the defendant guilty of breaking or entering, and certainly that is the best way to think about it, since the prosecution bears the burden of proof, and the defense does not have to prove innocence. However, as an exercise think about what would be sufficient for a verdict of not guilty in a case of felonious breaking or entering. That is, when the defense presents its case (in answer to the prosecution’s arguments), exactly what is the defense trying to prove? Exactly what would be sufficient to support a verdict of not guilty? Or in other words, give one sufficient condition for a verdict of not guilty.

If you answered that a sufficient condition for a not guilty verdict is that the defendant neither broke nor entered, then you are correct: That is sufficient for a verdict of not guilty. But in fact it is more than sufficient. Is there a weaker claim that would still be a sufficient condition for a verdict of not guilty? (Remember: The burden of proof is on the prosecution.)

A weaker claim that would still be a sufficient condition for a not guilty verdict is: There is a reasonable doubt that the defendant either broke or entered. That is, it is a sufficient condition for a not guilty verdict if one can show that there is a reasonable doubt about any one of the necessary conditions for a guilty verdict. If one of the necessary conditions is not proved beyond a reasonable doubt, that is sufficient for a verdict of not guilty. Of course, if the defense can prove that one of the necessary conditions for guilt is not met (not that it is merely doubtful, but that it actually is false), that would be more than sufficient for a verdict of not guilty.

When we talk about the judge’s instructions to the jury, then necessary and sufficient conditions can sound rather formidable. Actually, you think in terms of necessary and sufficient conditions all the time—though you may not use those terms. Sitting in the coffee shop this morning, one of your friends asserted: “The Maple Leafs have to have a good goalkeeper to win the Stanley Cup.” That is, a good goalkeeper is a necessary (not a sufficient) condition of the Maple Leafs winning the Stanley Cup. And before your history class, you told a friend: “If I make a B on this exam, then I’ll pass the course.” Which is to say, making a B is a sufficient condition for passing the course. Consider a few more examples. If the Cardinals had one more good starting pitcher, they would win the pennant; if I finish my term paper tonight, I’ll go to the beach this weekend; if you hit the ball to Sarah’s backhand, you will beat her; if the inflation rate drops, the Democrats will win the election. All those cases make claims about sufficient conditions. One additional good starting pitcher is a sufficient condition for the Cards to win the pennant; finishing your term paper is a sufficient condition for going to the beach; and so on. You have also made many statements about necessary conditions, although again you probably did not use that phrase. You said, for example, that the Celtics would win the title only if they had a healthy point guard, that you would have another drink only if someone would drive you home, that you would shop at Walmart only if it were unionized, that you would go out with Ralph only if he were magically turned into a handsome prince. In those cases, you are claiming that a healthy point guard is a necessary condition for winning the title, that having someone drive you home is a necessary condition for taking another drink. So don’t let all this talk about necessary and sufficient conditions intimidate you: You have been thinking in terms of necessary and sufficient conditions for years.

One more illustration. Suppose we are driving rather aimlessly around North America. After a few days of wandering, we find ourselves facing an enormous body of water. We’re uncertain of our location, but we know (somehow) that this is either Lake Ontario or the Pacific Ocean, but we don’t know which. Suddenly, off in the distance, you spot a whale surging up through the waves. “Hark,” you say, “this must be the Pacific Ocean!” So, how did you reach that conclusion?

There are two possibilities. Probably you thought that since there are whales, this must be an ocean. That is, whales are a sufficient condition for determining that the water is an ocean. But you might also have reasoned from the other direction: An ocean is a necessary condition for whales. Both ways of thinking are equally correct: Whales are a sufficient condition for the existence of an ocean, and an ocean is a necessary condition for whales. In sum, when A is a sufficient condition for B, then B is a necessary condition for A.

Here are some rather pedestrian examples. Drinking a quart of gin is a sufficient condition for drunkenness. So if you know that Alisdair drank a quart of gin, you can safely conclude that Alisdair is drunk; and if Alisdair is not drunk, you are justified in concluding that Alisdair did not drink a quart of gin. But you are not justified in concluding, from the fact that Alisdair is snookered, that Alisdair drank a quart of gin (he might have drunk a quart of vodka, which also suffices for drunkenness). And from the fact that Alisdair has sworn off quarts of gin, you cannot conclude that Alisdair is not drunk; he may have substituted a quart of bourbon. In short, a quart of gin is sufficient for drunkenness, but not necessary.

Having four legs is a necessary condition for winning the Kentucky Derby. Thus if we know that Secretariat won the Kentucky Derby, we are justified in concluding that Secretariat has four legs. And if we know that Raunchy Runner has only three legs, we know that Raunchy Runner cannot win the Kentucky Derby. But from the fact that Run Dusty Run did not win the Kentucky Derby, we certainly cannot conclude that Run Dusty Run does not have four legs. That is, four legs is a necessary—but not a sufficient—condition for winning the Kentucky Derby.

Conditional Statements

A few terms will make it easier to discuss necessary and sufficient conditions. An “if-then” statement—such as “If the sun shines, we’ll go on a picnic”—is called a conditional statement. The first part of the statement (“the sun shines”) is called the antecedent. (That’s easy to remember: “ante” means prior to.) The antecedent is the first, or prior, part of the conditional statement. When you play poker, each player makes an ante prior to dealing the cards; antebellum mansions are mansions built before the “bellum,” or before the war; antediluvian means that something is so old and outdated that it existed before the Biblical flood. (Of course, that model doesn’t always fit: An antelope is not an animal that precedes a lope.) The last half of the conditional statement (“we’ll go on a picnic”) is called the consequent. (Please notice that that’s consequent, and not consequence.) So a conditional statement is simply one in which the antecedent is a condition of the consequent. The consequent is true on the condition that the antecedent is true; in other words, the consequent is asserted to be conditionally true.

It is sometimes useful to write conditional statements in shortened symbolic form. To do so, we need a symbol for “if-then.” There are two standard ones: → and ⊃. (Not very imaginatively, logicians call the first symbol an arrow, and the second a horseshoe.) You can use either symbol you prefer; I’ll use the arrow. So if we use S to stand for “the sun shines” and P to stand for “we’ll go on a picnic,” then “If the sun shines, we’ll go on a picnic” would be symbolized as:

S→PS→P

What this states is that S is a sufficient condition for P. It claims that if S is true, then P must also be true.

In actual fact, the antecedent is rarely a completely sufficient condition for the consequent. What the conditional statement should be understood to say is that if the antecedent is true (and there are no weird extraordinary circumstances), then the consequent will also be true. If the sun shines, we’ll go on a picnic; the sun shining is a sufficient condition for going on a picnic. But there are a lot of other conditions that are assumed and that are so numerous and so obvious that it would be useless to try to specify them. If I tell you that “If the sun shines, we’ll go on a picnic,” you take for granted that that means something like “If everything remains normal and the sun shines, then we’ll go on a picnic.” Obviously, the sun shining by itself is not entirely sufficient, for if the sun shines but I am severely injured by a meteorite or there is a major earthquake or there is a meltdown at a nearby nuclear power plant, then we certainly cannot go on a picnic. I can’t state all those conditions in my conditional statement (“If the sun shines and there is no nuclear meltdown and no meteorite falls on me and there is no earthquake and the air continues to contain sufficient oxygen to sustain life and I am not arrested for conspiracy and the car does not turn into a pumpkin, then we’ll go on a picnic”). When I say that “If the Cardinals get another good starting pitcher, they will win the pennant,” I obviously mean if all conditions remain the same, then the additional pitcher will be sufficient for the Cards to win the pennant. (I am not claiming that the Cards would win with an additional pitcher even if all the other players suddenly decided to become Buddhist monks and quit the game of baseball; but since that would be a great change from current conditions, it is not necessary to add that qualification.) Even in court it is rightfully assumed that jurors will understand that a statement of sufficient conditions does not state every condition, but instead assumes that the situation is normal. For example, betting money on a game of chance is a sufficient condition for being guilty of violating the North Carolina gaming statutes. But of course there are other conditions that are assumed: No one is holding a gun to the head of the bettor, the bettor is not working as an undercover agent for the police, and so on.

Alternative Ways of Stating Necessary and Sufficient Conditions

Consider this statement: “If the defendant’s fingerprints are on the pistol, then the defendant is guilty.” (Don’t worry for the moment whether that conditional statement is true; just focus on exactly what it means.) How would you express that statement in terms of necessary conditions? In terms of sufficient conditions?

“The defendant’s fingerprints are a sufficient condition for the defendant’s guilt.” Or alternatively, and a little less naturally—but still correctly—it could be stated: “The defendant’s guilt is a necessary condition of the defendant’s fingerprints being on the pistol.”

The sentence—“If the defendant’s fingerprints are on the pistol, then the defendant is guilty”—might be phrased differently without changing the meaning: “The defendant is guilty if the defendant’s fingerprints are on the pistol.” (The sentence still means the same: “The defendant’s fingerprints are a sufficient condition for the defendant’s guilt.”) In this case, the antecedent (“the defendant’s fingerprints are on the pistol”) occurs last in the sentence, while the consequent (“the defendant is guilty”) occurs first. The moral is this: Don’t assume that the first part of every conditional statement is the antecedent; you must read the sentence carefully and determine what it actually means. You must determine what is being presented as the condition, the if, the antecedent, and what is said to follow from that condition (the consequent).

Suppose that instead the sentence is this: “The defendant is guilty only if the defendant’s fingerprints are on the pistol.” That “only” seems an innocent addition, but it changes the claim entirely. How would you express that sentence in terms of necessary and sufficient conditions? (Think carefully about it. If you can answer that question, then you have a good grasp of necessary and sufficient conditions. Of course if you cannot answer it, that does not imply that you do not have a good grasp of necessary and sufficient conditions. It is a sufficient—rather than necessary—condition of understanding.)

How can “The defendant is guilty only if the defendant’s fingerprints are on the pistol” be stated in terms of necessary and sufficient conditions? That sentence asserts that the defendant’s fingerprints are a necessary condition for the defendant’s guilt. (If there are no fingerprints, the defendant is not guilty.) Or another way of putting it: The defendant’s guilt is a sufficient condition for finding the defendant’s fingerprints. (That is, if the defendant is guilty, you will find the defendant’s fingerprints on the pistol.) This is a common and useful way of focusing on necessary conditions. For example, a defense attorney who wants to focus the attention of the jury on the essential role played by a rather weak prosecution witness might say in her summation to the jury, “You can find the defendant guilty only if you believe the testimony of the prosecution’s key witness—that scurrilous character who has been convicted three times for perjury and twice for fraud. The prosecution’s entire case rests on that sleazy foundation.”

Necessary and sufficient conditions can be a trifle tricky. Here’s a hint that might help you keep track of them. Consider a statement like this: If the economy improves, then the Democrats will win the election. We could represent that as a conditional statement, thusly:

That is,

E→WE→W

Once we have it in the form of a conditional statement, it’s easy to read off statements of sufficient and necessary conditions. Just remember this golden rule of conditional statements: The antecedent is a sufficient condition for the consequent, and the consequent is a necessary condition for the antecedent. Thus, the economy improving is a sufficient condition for the Democrats’ winning; and the Democrats’ winning is a necessary condition of the economy improving. So, of course, once you know necessary conditions you can easily state sufficient conditions (and vice versa), just by plugging in that handy formula. When you know, for example, that oxygen is a necessary condition of life, then you also know that life is a sufficient condition for oxygen (i.e., where you find life you know there must be oxygen); and that could also be written as:

life→Oxygenlife→Oxygen

Both Necessary and Sufficient

One more thing. Occasionally we want to assert a particularly strong relation. For example, some television evangelists claim that “You will go to Heaven if and only if you follow my religion.” That is, you will go to Heaven if you follow my religion, and you will go to Heaven only if you follow my religion. Or in other words, following my religion is both a necessary and a sufficient condition for going to Heaven. Another example: If the only way you can develop collywobbles disease is by being bitten by the collywobble bug, and if the bite of the collywobble bug always causes collywobbles disease, then the bite of the collywobble bug is a necessary and sufficient condition for collywobble disease; one develops collywobble disease if and only if one is bitten by the collywobble bug. But note that that claim is stronger than—and quite different from—the claim that if one is bitten by the collywobble bug, one will develop collywobble disease. And the “if and only if” claim is also stronger than the claim that if one develops collywobble disease, then one was bitten by the collywobble bug. The statement that “One develops collywobble disease if and only if one is bitten by the collywobble bug” is stronger than either of those claims because it combines both of them. (If you know that it’s true that “If one is bitten by the collywobble bug, one develops collywobble disease,” then from the fact that you have collywobble disease, what can you conclude about what bit you? Nothing. Perhaps you were bitten by the collywobble bug, or perhaps you were bitten by some other bug that carries collywobble disease; perhaps you developed collywobble disease from some other source. But, if you know that it’s true that “One develops collywobble disease if and only if one is bitten by the collywobble bug,” then from the fact that you have collywobble disease you could rightfully conclude that you must have been bitten by the dreaded collywobble bug.)

Suppose the judge instructs you that “A person is not criminally responsible if and only if his unlawful act was the product of mental disease or mental defect.” In that case, if you determine that the defendant’s assault was the product of mental disease, then you know that the person is not criminally responsible. And, if you determine that the assault was not the product of mental disease or mental defect, then you know that the defendant is criminally responsible (according to the judge’s instructions).

Valid Inferences from Necessary and Sufficient Conditions

Suppose we are deliberating our verdict in a case of first-degree burglary. A witness for the defense has claimed that she was with the defendant during the entire night on which the burglary occurred and that the defendant never left her house. This testimony, if believed, establishes a strong alibi for the defendant. One juror comments, “If the alibi witness was telling the truth, then the defendant is not guilty. And certainly she was telling the truth: I believed every word she said. Therefore, the defendant is not guilty.” That juror’s argument might be presented thus:

The truthfulness of the alibi witness’s testimony is a sufficient condition for finding the defendant not guilty. The alibi witness was truthful. Therefore, the defendant is not guilty.

The argument might be presented in symbolic form as follows. (A stands for “the alibi witness is telling the truth”; D stands for “the defendant is not guilty”; the three dots in a triangle shape mean “it follows that”—they mark the conclusion; and, as you remember, the arrow stands for “if-then.”)

A→DAModus Ponens−Valid∴DA→DA    Modus Ponens-Valid∴D

Modus Ponens

That is a deductively valid argument. I don’t know whether the premises are true or are not true (perhaps the alibi witness was in fact lying), but certainly the conclusion follows from the premises. If the premises are true, then the conclusion must be true. A is a sufficient condition for D; A is the case; therefore, D must follow. This form of argument is so common that it has been christened with a special name: modus ponens. Sometimes it is also called affirming the antecedent. That’s a good name for this argument form, since that’s exactly what it does. It makes a conditional statement, and then affirms the antecedent of that conditional statement, and draws as a conclusion the consequent of that conditional statement.

So far in this chapter we have talked mainly about sufficient conditions, but conditional statements also make claims about necessary conditions. Back to the jury room. One juror says, “If the alibi witness was telling the truth, then the defendant is innocent.” Another juror replies, “Well, the defendant must be innocent, because certainly that angelic alibi witness was telling the truth.” In this case, the juror is focusing on the innocence of the defendant as a necessary condition of the truthfulness of the witness. The point to note is that any conditional statement can be expressed either in terms of necessary conditions or in terms of sufficient conditions. Thus, a conditional statement such as “If the witness is telling the truth, then the defendant is innocent” can be correctly expressed in terms of sufficient conditions (“The truthfulness of the witness is a sufficient condition for the innocence of the defendant”), and it can also be correctly expressed in terms of necessary conditions (“The innocence of the defendant is a necessary condition for the truthfulness of the witness”). Sometimes it will seem more natural to express a conditional statement in terms of sufficient conditions; at other times it will seem more natural to express a conditional statement in terms of necessary conditions (which way seems more natural probably depends on what our particular interest is). For example, the statement “If there are whales, then that is saltwater” could be stated as “Whales are a sufficient condition for saltwater” (we would find that the natural way of stating it if we were trying to decide whether a body of water was saltwater or freshwater, and we saw a whale cavorting in it); or it could be stated as “Saltwater is a necessary condition for whales” (that would be the natural way of stating it if we were interested in knowing where to look for whales or what sort of conditions we would have to maintain in order to keep whales in our fishbowl).

With the whale–saltwater example in hand, it is easy to see why it is essential to keep the antecedent and consequent in the proper order. It is one thing to say: “If there are whales, then there is saltwater.” It is something quite different to say: “If there is saltwater, then there are whales.” The former statement is true; the latter statement is false (there was saltwater for millions of years before whales evolved). If whales are a sufficient condition for saltwater, then saltwater is a necessary condition for whales; but whales being sufficient for saltwater does not make saltwater sufficient for whales, and saltwater being necessary for whales does not make saltwater sufficient for whales. If I make a perfect score on every exam, then I’ll pass logic; that is not the same as saying that if I pass logic, then I’ll make a perfect score on every exam. It is true that if tuition, room, and board are provided free at State U., then many students will attend State U.; it is false that if many students attend State U., then tuition, room, and board will be free.

Now go back to the juror who focused on necessary conditions: “The defendant must be innocent, because certainly the alibi witness was telling the truth.” That is, the innocence of the defendant is a necessary condition of the truthfulness of the alibi witness. In this case, the juror is again reasoning by modus ponens: If the witness is telling the truth, then the defendant is innocent; the witness is telling the truth; therefore, the defendant is innocent. But imagine another juror, who is absolutely convinced that the defendant is not innocent, but is instead guilty as charged. (We need not worry here about why the juror is so certain of that—perhaps because the juror believes, as many jurors do, that anyone charged with a crime must be guilty, or perhaps because the juror took one look at the defendant and decided that the defendant looked like the “criminal type.” This juror accepts the truth of the conditional statement (“If the witness is telling the truth, then the defendant is innocent”) but also believes that the defendant is not innocent. What conclusion must that juror draw concerning the truthfulness of the alibi witness?

That juror would have to conclude that the alibi witness is not telling the truth. Since the innocence of the defendant is a necessary condition of the veracity of the witness, if the juror believes that the defendant is not innocent then the juror believes that a necessary condition of the witness being truthful is not met; therefore, the witness cannot be telling the truth. The same symbols will be used: A stands for “the alibi witness is telling the truth”; D stands for “the defendant is innocent”; → is the symbol for “if-then”; and ˜ will be used as a negation sign, simply meaning “it is not the case that.” Using those symbols, the juror’s deduction can be symbolized as follows:

A→D~DModus Tollens−Valid∴~AA→D~D    Modus Tollens-Valid∴~A

This form of argument is very common. You use it all the time. Your friend Joe is a notorious procrastinator who never works on anything until the very last moment. Thus you know that “If Joe is in the library, then Joe must have a term paper due tomorrow morning.” But you also know that “Joe does not have a term paper due tomorrow morning.” Thus you conclude—as the conclusion of a valid deductive argument—that Joe is not in the library. Or take another example: “If Louise drank that entire fifth of gin last night, then Louise will have a terrible hangover this morning. But Louise obviously does not have a hangover. Therefore, Louise did not drink the entire fifth of gin.”

Modus Tollens

This is such a common form of valid deductive argument that it has been given a special name: modus tollens. You might prefer to call it denying the consequent, since that is exactly what it does: One premise is a conditional, another premise denies the consequent of that conditional, and the conclusion is the denial of the antecedent of the conditional. (Another way of thinking of it is that a modus tollens argument denies a necessary condition of X, and on the basis of that denial concludes that X is not the case.)

We have looked at two very common valid argument forms that use conditional statements: modus ponens and modus tollens. Unfortunately, there are also some common invalid, fallacious argument forms that look and sound a lot like the valid ones. It is important to reason correctly from necessary and sufficient conditions, but it is equally important to detect and avoid some very common errors. Back again to the burglary case and the alibi witness. All the jurors agree that “If the alibi witness is telling the truth, then the defendant is innocent.” Suppose a juror now says, “It’s true that if the alibi witness is telling the truth, then the defendant is innocent, but I just did not believe that alibi witness—her story didn’t ring true to me. Therefore, the defendant is not innocent: The defendant is guilty as charged.” That sounds like a good argument; it sounds a lot like the valid argument form modus tollens. But it is not. It is an invalid argument. Think carefully about that argument, and try to state exactly how it differs from modus tollens. Also, try to state, in terms of necessary and sufficient conditions, exactly what error that argument commits.

The valid argument form modus tollens goes like this:

A→D~DModus Tollens−Valid∴~AA→D~D    Modus Tollens-Valid∴~A

In modus tollens, the second premise denies the consequent. But in the invalid argument under consideration (“If the alibi witness is telling the truth, then the defendant is innocent; the alibi witness is not telling the truth; therefore, the defendant is not innocent”), the second premise does not deny the consequent; instead, the premise denies the antecedent:

A→D~ADenying the Antecendent−Invalid∴~DA→D~A    Denying the Antecendent-Invalid∴~D

That may appear to be an insignificant difference, but in fact it makes all the difference in the world: Modus tollens is a valid argument, but this argument—which denies the antecedent—is an invalid, fallacious argument. (That’s exactly what this invalid argument form is called: the fallacy of denying the antecedent.) The difference is that in modus tollens the premise denies a necessary condition for A and then deduces—correctly—that A is not the case. In the fallacious argument the premise denies a sufficient (not a necessary) condition for D, and from that, one cannot conclude that D is not the case. Perhaps that sounds a bit tricky. It’s not. Think of it this way: If LeBron James and Kevin Durant play basketball for Western High School, then Western High will have a winning season. (That’s true, right? I mean, no matter who the other players are, if LeBron and Kevin play for Western High, then Western High is going to win more than its share of basketball games.) But LeBron James and Kevin Durant will not play for Western High (that’s also true; both LeBron and Kevin have other plans for the basketball season). Does it follow that Western High will not have a winning season? Certainly not. LeBron and Kevin are a sufficient condition for a winning season, but they are not a necessary condition. To conclude that because LeBron and Kevin will not play for Western, therefore Western will not have a winning season is to mistake a sufficient condition for a necessary condition. Take another example, one that is perhaps more immediately relevant. If you study logic 8 hours every day for the rest of the semester, then you will pass logic (I personally guarantee it). But, you say, I can’t possibly study logic 8 hours every day—it would take all my time from my other courses, not to mention cutting into my social life. So what follows? That you will not pass logic? Of course not. Studying logic 8 hours every day is a sufficient condition for passing logic, but it is by no means a necessary condition.

• If I study logic 8 hours every day, then I’ll pass logic.

• I shall not study logic 8 hours every day.

• Therefore, I shall not pass logic.

That argument commits the fallacy of denying the antecedent. It is an invalid argument. It mistakenly treats a sufficient condition as if it were a necessary condition.

There is another fallacious argument form that sounds and looks a lot like a valid argument form but is invalid. Once again to the jury room. The jurors agree that if the alibi witness is telling the truth, then the defendant is innocent. Now suppose that there is a dispute about the truthfulness of the alibi witness: Some jurors think she is telling the truth, others insist she is lying. In the middle of this dispute, word comes from the judge that the jurors can go home! Another person has just confessed to committing the burglary, and all charges against the defendant have now been dropped. The trial is over; the defendant is innocent. At that point one of the jurors who had been insisting on the truthfulness of the alibi witness turns to one of the doubters and says, “There, now you see, she was telling the truth, just as I claimed.” Is the truthfulness of the witness in fact proved by the innocence of the defendant?

The juror has made a mistake: The fact that the defendant is innocent (together with the conditional that if the witness is telling the truth, then the defendant is innocent) does not prove that the witness is telling the truth. The juror has confused a sufficient condition with a necessary condition. The truthfulness of the witness is a sufficient condition—but is not necessary—for the innocence of the defendant. If Sabrina makes a perfect score on every exam, then she’ll pass the course. So suppose that Sabrina passes the course. Can we conclude that Sabrina made a perfect score on every exam? Of course not. Making a perfect score on every exam is sufficient for passing the course: a perfect score on every exam would guarantee a passing grade. But that is not necessary for passing the course. Or if you prefer, we can look at this from another angle. Passing the course is a necessary result of perfect scores on every exam; passing the course is a necessary condition of making perfect scores on all exams. But a passing grade in the course is not a sufficient condition of making perfect scores on every exam. Thus Sabrina may pass the course even though she did not achieve perfection on every exam. In the same way, the innocence of the defendant is a necessary condition of the truthfulness of the alibi witness, but it is not a sufficient condition. The defendant may be innocent even though the alibi witness is not telling the truth.

Look closely at the form of the argument. The form of the juror’s argument is as follows:

A→DDAffirming the consequent−Invalid∴AA→DD    Affirming the consequent-Invalid∴A

That looks and sounds a lot like a valid modus ponens type of argument. But there is a very important difference. In the valid argument form modus ponens, the second premise affirms the antecedent of the conditional statement, and then draws the consequent as a conclusion. But in the invalid argument above, the juror affirms the consequent (not the antecedent) in the second premise, and then draws the antecedent as a conclusion. And that invalid, fallacious argument form is called by exactly that name: the fallacy of affirming the consequent.

Like denying the antecedent, affirming the consequent is a very common fallacy. It sounds like a perfectly good argument—and usually both the person giving the argument and the persons hearing the argument are deceived by it. For example, a senator supporting a tax decrease argues, “If taxes are too high, then the economy slows down; and the economy certainly has slowed down; obviously, then, taxes must be too high.” And it does sound obvious, until one examines the argument closely and recognizes the fallacy of affirming the consequent. “If grading becomes easier at Home State U., then the grade point averages of students will increase. And indeed students are getting higher grade point averages. So grading at Home State U. must be getting easier.” Sounds plausible and maybe the conclusion is true, but certainly that argument does not prove that the conclusion is true, since it is invalid: It commits the fallacy of affirming the consequent.

Detecting Argument Forms

Recognizing valid modus ponens and modus tollens arguments (and distinguishing them from the look-alike fallacies of affirming the consequent and denying the antecedent) requires careful attention to the forms of the arguments. But those forms can be tricky to detect. For one thing, the premises can be switched around (the conditional premise isn’t always first), and the conclusion isn’t always stated last. For example:

• If Jill drove the getaway car, then Jill is guilty.

• Jill drove the getaway car. Therefore, Jill is guilty.

That’s the valid argument form modus ponens. But suppose the argument is stated like this:

• Jill is guilty, because Jill drove the getaway car.

• And if Jill drove the getaway car, then Jill is guilty.

That’s exactly the same argument, with exactly the same argument form: modus ponens. The premises and conclusion are in a different order, but that doesn’t change the form of the argument.

Another source of potential confusion is negation. Look again at the modus ponens argument just considered:

• If Jill drove the getaway car, then Jill is guilty.

• Jill drove the getaway car. Therefore, Jill is guilty.

That argument could be diagrammed like this:

D→GD∴GD→GD∴G

Like all modus ponens arguments, one premise affirms the antecedent of the other premise. But what about this argument:

If Joe did not intend to steal the jacket, then Joe is not guilty of shoplifting. Joe did not intend to steal the jacket. Therefore, Joe is not guilty of shoplifting.

That argument could be represented in symbols like this:

~I→~G~I∴~G~I→~G~I∴~G

At first glance the form of that argument may look very different from the modus ponens argument of the previous example. But in fact the form is the same: The argument is just dressed up with a few negations. The antecedent of the conditional is this: Joe did not intend to steal. The other premise affirms that antecedent. Since the antecedent is a negation, the affirmation of that antecedent is also a negation. If I say “Nixon was not a good president,” you affirm what I say by also stating a negation: “True, Nixon was not a good president.” To deny my negative statement, you would say: “Nixon was a good president.” Or you could say: “It is not the case that Nixon was not a good president,” which—through double negation—is the equivalent of saying simply “Nixon was a good president.”

The same points apply to argument forms like the fallacy of denying the antecedent. The standard form of that fallacy is this:

If the Maple Leafs score five goals, then the Maple Leafs will win. But the Maple Leafs will not score five goals. Therefore, the Maple Leafs will not win.

In symbolic form, this is stated as:

S→G~S∴~WS→G~S∴~W

Compare that with this argument:

If Joe did not know the car was stolen, then Joe is not guilty. But Joe did know the car was stolen. Therefore, Joe is guilty.

In symbolic form, that’s:

~K→~GK∴G~K→~GK∴G

Again, that may look rather different from the prior example of denying the antecedent; but it’s exactly the same form. The antecedent of the conditional is a negation: Joe did not know. The negative antecedent is denied by saying: Joe did know.

If that seems confusing, think more about negations and denials of negations. It’s a snowy, overcast day. Since I love the snow, I greet you cheerily: “It’s a beautiful day!” Your taste in weather runs to balmier temperatures, and you deny my statement with a negation: “It is not a beautiful day.” If I am in the mood to argue, I may deny your negation, in either of two ways: “It is a beautiful day” or alternatively (by negating your negation), “It is not the case that it is not a beautiful day.” So don’t let negations obscure things: Just look at the forms of the arguments.

When we add negations to our arguments, and then try to figure out what form the argument is, sometimes it looks a bit scary. For example,

L→G~G∴~LL→G~G∴~L

You immediately (or almost immediately) recognize that as modus tollens, right? But suppose it looks like this:

~L→~GG∴L~L→~GG∴L

That’s just the same thing: the second premise denies the consequent of the first premise (the consequent is not G, and the second premise denies that by asserting G). When we put it in that form, it may look a bit intimidating. But it’s not. You use that argument form all the time, and you immediately understand it when you hear it. There’s a great old blues song, and the main line of the song goes like this: “If I don’t love you, baby, then grits ain’t groceries, eggs ain’t poultry, and Mona Lisa was a man.” If someone says that to you—“If I don’t love you, baby, then grits ain’t groceries”—you immediately know that he or she is saying that “I do love you, baby.” Right? But how did you reach that conclusion? By modus tollens. The argument goes like this:

• If I do not love you, then grits are not groceries.

• Grits are groceries.

• Therefore, I love you.

Four Argument Forms

These are the four argument forms discussed in this chapter. Modus ponens and modus tollens are legitimate, nonfallacious arguments; denying the antecedent and affirming the consequent are fallacious.

P→QP∴QP→QP∴Q

P→Q~Q∴~PP→Q~Q∴~P

P→Q~P∴~QP→Q~P∴~Q

P→QQ∴PP→QQ∴P

Loses something in translation, doesn’t it? Still, it’s just the modus tollens form of argument; symbolized, it looks like this:

~L→~GG∴L~L→~GG∴L

Actually, as originally stated, the argument is an enthymeme. One premise (“Grits are groceries”) is regarded as so obvious that it need not be stated. But you have to fill it in to get the whole argument.

Now, that’s as hard as these get; and if you don’t get frightened by all the squiggles and arrows, that’s really not so hard, right? One last point, but an important one: when diagramming arguments, you will find it much easier if you start with the conclusion.