by Antony Flew
2.4 Arguments of the same form as that of the soldier at Stalingrad are employed wherever any hypothesis is tested, both in science proper and everywhere else. Sherlock Holmes, for instance, might argue: Since If the thief entered this way, then there would be footprints in the flowerbed and since There are no footprints, therefore, I deduce that The thief did not enter this way. The form of the argument, and it is a valid form, is: If This is so, then That is so; but That is not so; therefore, This is not so either. Since This is so and That is so are propositions and not subjects or predicates, it is sensible, if you want to introduce symbolism, to make your symbolism appropriately different from that suggested earlier for subjects and predicates (paragraph 1.48). The established convention is to use the lower case Latin letters p, q, and r. Since This is not so and That is not so are simply the denials or negations of, respectively, This is so and That is so, the further convention is to symbolize the negations of p, q, and r as, respectively, ~p, ~q, and ~r (read as not p, not q, and not r). Putting this new symbolism to work at once, we can now express the present valid form of argument as follows: If p then q, but ~q, therefore ~p.
2.5 Let us now distinguish this important and valid form from two other forms that are invalid. Let the two conditionals again be granted: If there is a God, then the soldier will return; and If the thief entered this way, then there will be footprints in the flowerbed. Now suppose: The soldier has returned; and There are footprints in the flowerbed. It will not do to infer from these pairs of premises the conclusions: There is a God and The thief entered this way. In such very simple cases, in which none of our public or private concerns are engaged, it is easy to see why it will not do. The crux is that the premises so far provided do not preclude the possibility of some alternative explanation. Even in a Godless world some soldiers would presumably return. So why not this one? And certainly had the thief entered this way, then the thief would have left footprints in the flowerbed. But nothing has so far been said to rule out the possibility that any footprints found there might have been made by someone other than the thief.
2.6 The second of the two invalid forms approaches from the opposite direction. Let the two conditionals be granted yet again: If there is a God, then the soldier will return; and If the thief entered this way, then there will be footprints in the flowerbed. But now suppose: There is no God and The thief did not enter this way. It will not do to infer from these fresh pairs of premises the conclusions: The soldier will not return and There will be no footprints in the flowerbed. The crux this time is that such conditionals say nothing about the situation on the alternative assumption that their antecedents are not true. It may well be true that If one becomes a schoolteacher, then one will not succeed in amassing a fortune. But, depressingly, this gives a person no positive reason to believe that he or she will be more successful in amassing a fortune even if choosing not to become a teacher.
2.7 Four possible moves can be made with such conditional propositions: two valid and two invalid. Some readers may find it helpful to run through the quadruplet systematically. But if for you the emphasis in this sort of foursome reel is too much on the word “reel,” then skip to the next paragraph.
2.8 The valid move of arguing If p then q, but ~q, therefore ~p involves denying the consequent (the “then” bit) of the conditional to disprove the antecedent (the “if” bit): hence, Denying the Consequent. The first invalid move, illustrated in paragraph 2.5, consists in arguing If p then q, but q, therefore p. This involves affirming the consequent in the baseless hope of thereby proving the antecedent: hence, Affirming the Consequent. A second valid move is too obvious and too obviously sound to require illustration. This is to argue: If p then q, and p, therefore q. In this we affirm the antecedent in order to prove the consequent: hence, for short, Affirming the Antecedent. The second invalid move completes the ring of changes. This invalid move, illustrated in paragraph 2.6, is to argue: If p then q, but ~p, therefore ~q. Here we deny the antecedent in the baseless hope of thereby disproving the consequent: hence, for short, Denying the Antecedent.
2.9 Important and fundamental questions about scientific method turn on elementary logical points expounded in the previous eight paragraphs. The heart of the matter lies in an asymmetry with regard to verification and falsification. Hypotheses are tested by deducing consequences which would follow if the hypothesis were true. When a hypothesis is tested and even one of its consequences is found in fact not to obtain, then that hypothesis, at least as originally formulated, is decisively demonstrated to have been false. Thus, the original hypothesis entertained by the soldier’s wife (the hypothesis tested in the inferno of Stalingrad) was decisively disproved by her husband’s death there. Holmes’s hypothesis that the thief entered the house by a certain way was similarly falsified—shown, that is, to be false—when no footprints were found in the flowerbed. But when a hypothesis is tested and one of its consequences is in fact found to obtain, it is still not thereby demonstrated that the hypothesis must be true. Suppose that that German soldier had returned safe and sound. This would not have proved the existence of his wife’s God. Nor would the discovery of footprints in the flowerbed have ruled out the possibility that these were made by someone else and that the thief used a different route to enter the house.
2.10 This asymmetry with respect to verification and falsification also characterizes such open universal propositions as All swans are white or All living creatures are mortal. These propositions are called “universal” because they refer to all the members of a class and “open” because the members of that class cannot be exhaustively listed. A single genuine counterexample is sufficient to disprove any such open universal proposition. But no matter how many confirming examples you produce, it is impossible to prove the open, universal proposition with any correspondingly decisive finality. It was this logical observation which led Francis Bacon (1561–1626) to make his often-quoted remark that “the force of the negative instance is greater.” It was the same logical observation which provoked T. H. Huxley (1826–1895), a friend and defender of Charles Darwin (1809–1882), to speak of “The great tragedy of science—the slaying of a beautiful hypothesis by an ugly fact.”
2.11 It was the observation of these logical asymmetries that served as one foundation of Sir Karl Popper’s enormously influential and salutary philosophy of science. No open universal proposition can ever be confirmed beyond all conceivable future correction, although some can be decisively and finally disproved. Yet the crucial elements in every imaginable scientific theory, and any acceptable candidates for the status of law of nature, must be propositions of this kind. So Popper drew the exciting moral that science is and can only be a matter of endless striving and of endless inquiry. The best we have or ever could have can and could only be the best so far (Magee 1973 and Popper 1959 and 1963).
2.12 This is all very well if it is taken to mean only that there will always be more for scientists to discover. But Popper himself, by failing to make and employ the crucial distinction between two senses of the word “possible,” became trapped in the intolerably contradictory position both of maintaining that there is no such thing as real, certain knowledge and of glorying in the enormous and wide-ranging advances in scientific knowledge. Thus, in The Logic of Scientific Discovery (1959) he asserted that “The old scientific idea of episteme [the Greek word for knowledge]—of absolutely certain, demonstrable knowledge—has proved to be an idol. . . . Every scientific statement must remain tentative for ever. . . .” (pp. 280–81). He nevertheless later published a book entitled Objective Knowledge: An Evolutionary Approach (Popper 1972).
2.13 Possibilities may be either logical or practical. To say that some occurrence is logically possible, that it is conceivable, is to say only that it is possible to provide a coherent and intelligible description of what would be happening if that logically possible occurrence were actually to occur. That somebody should jump over the Statue of Liberty is conceivable, that is, logically pos
sible. But everyone knows that it is practically impossible. Therefore, to show that no open universal proposition can ever be confirmed beyond all logical possibility of future correction—which is what in effect Popper did show—is not to show that such scientific propositions cannot, in the ordinary understanding of the word “know,” be known to be true. For in that understanding, to be known a proposition does not have to be demonstrated. That is, it does not have to be shown that to deny that proposition would be to contradict yourself. It is sufficient that it should in fact be true and that the person claiming to know it either possesses sufficient evidencing reason or is otherwise in a position justifiably to claim to know it.
2.14 As the ancient Greeks used to say of their greatest poet: “Even Homer sometimes nods.” So we should be distressed but not surprised to discover that Popper once supplied us with an interesting example of the fallacy of Denying the Antecedent. Yet the right lesson to learn from this embarrassing lapse is utterly Popperian. It is the need for unresting critical alertness—perhaps especially in our attention to those with whom we most agree and whom we most admire.
2.15 The lapse occurs when Popper is reconsidering that medieval favorite, All men are mortal. He remarks that this “mortal” ought to be construed as “liable to die.” This is certainly true. It was because they were in fact construing it as (naturally) “liable to die,” that those who believed in the (miraculous) translation both of the prophet Elijah and of Mary the mother of Jesus rightly refused to accept these as counterexamples. The claim that these two exceptional people were by a miraculous overriding of the laws of nature taken straight up into heaven does not deny, but takes it for granted, that they, too, like all other humans, were naturally liable to death. But for those miracles they, too, would in due course have died ordinary, unprivileged, natural deaths. Popper next proceeds to report that it was “part of Aristotle’s theory that every generated creature is bound to decay and die. . . . But this theory was refuted by the discovery that bacteria are not bound to die, since multiplication by fission is not death . . .” (Popper 1972, p. 10). Certainly from the hypothesis that All living things are bound to decay and die we can deduce: first, All bacteria are bound to decay and die; second, All humans are bound to decay and die; and so on indefinitely. And certainly the falsity of the first of these consequents is sufficient to show the falsity of that antecedent. But this is fallacious to argue from the falsity of that antecedent to the falsity of any of those consequents.
2.16 The facts, first, that All men are mortal is indeed a consequence of All living things (including men) are mortal, and, second, that because of the splitting bacteria the latter proposition is now known to be false, are not sufficient reasons for saying that the former proposition must be false also. Suppose they were sufficient. Then to refute any theory would be simultaneously to refute all the propositions deducible from that theory, and hence all the propositions it was capable of explaining. This would carry a curious and a catastrophic consequence. Any established theory which is refuted is refuted through its failure to explain recalcitrant facts. And it would surely be unusual and certainly scandalous if a theory were to have become established notwithstanding that it could not explain any even of the facts already known in its heyday. On the present supposition, however, by refuting any established theory we should be left with nothing for any new theory to explain except the fresh facts deployed in that very refutation—which is absurd.
2.17 The four moves listed systematically in paragraph 2.8 can be further illuminated with the aid of distinctions between necessary and sufficient conditions, distinctions far and away more valuable than any of the eminently forgettable technicalities of that paragraph. The purpose of the next four paragraphs, therefore, is to explain the logical structure of those four moves and the logical relations between them by first explaining and then employing the notions of logically necessary and logically sufficient conditions. Those in no mind to mind their p’s and q’s should skip at once to paragraph 2.22.
2.18 When given the truth of one proposition p the truth of another proposition q follows necessarily, then the truth of p is, by definition, a logically sufficient condition of the truth of q. That Bertie is an Englishman is thus a logically sufficient condition of the truth that Bertie is a man. When some proposition r necessarily cannot be true unless some other proposition s is also true, then the truth of s is, again by definition, a logically necessary condition of the truth of r. That Cynthia has been married is a logically necessary condition of the truth of the proposition that Cynthia is divorced. You could not even begin to get divorced unless you have first been married.
2.19 It should be obvious that to say that p is a logically sufficient condition of q is not to say either that p is the only logically sufficient condition of q or that it is a logically necessary condition of q. Since Frenchmen are also men, that Bertie is a Frenchman would be an alternative, logically sufficient condition of the truth that Bertie is a man. For the same good and sufficient reason it is not a logically necessary condition of the truth that Bertie is a man that Bertie is an Englishman.
2.20 Again, to say that s is a logically necessary condition of r is not to say either that s is the only logically necessary condition of r or that it is a logically sufficient condition of r. Since it is not the case—even for countries with high divorce rates, such as Sweden and the United States—that everyone who gets married gets divorced, either forthwith or eventually, it may well be true that Cynthia has married and has not been divorced. And even if everyone getting married proceeded at once to divorce, to say that someone has married would be to say less than to say he or she is now divorced. Thus, that Cynthia was married is a logically necessary but not the only logically necessary nor the logically sufficient condition of her now being divorced. (Many of us, however, must have known marriages of which it would have been fair, though unseemly, to comment: “Well, I suppose marriage is a logically necessary precondition of divorce.”)
2.21 Putting the distinction between necessary and sufficient conditions immediately to work, we can say that the two fallacies represent misunderstandings of what it is for one proposition to be any sort of sufficient condition of another. For to say If p then q is to say simply that p is some sort of sufficient condition of q. The second invalid move, to argue If p then q, but ~p, therefore ~q confounds this initial statement that p is a sufficient condition of q with the claim that p is a necessary condition of q. Were p indeed a necessary condition of q then we could infer ~q from ~p. The first invalid move, to argue If p then q, but q, therefore p, confounds the same initial statement that p is a sufficient condition of q with the claim that p is the only sufficient and necessary condition of q. Were p indeed the only sufficient and necessary condition of q then we could infer p from q. Or, in other words, this second fallacy consists in the confusion of If p then q with If and only if p then q. Since English—like, so far as I know, all other natural languages—lacks a single word for “if and only if,” logicians have invented the artificial and seemingly misspelled vocable “iff.” In these terms this second fallacy of Denying the Antecedent can be characterized as the mistake of deducing from If p then q what would indeed follow from Iff p then q.
2.22 If things seem to have been moving either too fast or too abstractedly in the five previous paragraphs, and especially if you think of yourself as a student rather than as a general reader, then you would be well advised to settle down with a pencil and paper to run through, in your own way and at your own pace, the various logical points about conditionals treated in all the first sixteen paragraphs of this chapter. Half the point of having and using the simple symbolism introduced in paragraph 2.4 and the symbols introduced earlier in paragraph 1.48 is to make this sort of checking easier and quicker. (The other half is to escape the often misleading distractions of individual interests in, and beliefs about, some of the particular propositions which may have served as premises or conclusions in some argument offered as
an example for examination.) Certainly no one can reasonably expect to acquire any facility even with this very modest minimum of symbolism without undertaking a little private practice of this sort. In general, any argument which is either difficult or contentious should be examined closely and in writing. It is altogether too easy—even when the intentions of all concerned are impeccable, which is not always the case—for the swiftness and fleetingness of the spoken word to deceive the mind. And just because what is not recorded is not recorded, everyone’s ground may shift without anyone noticing it.
2.23 Suppose someone says, as someone often does: If I (Jones) do not obtain some much-desired good then there is no justice in this world. And suppose, as sometimes happens, Jones does obtain that much-desired good. Are we entitled from these two premises alone to infer the conclusion which someone else is sure now to point out to Jones: that There is some justice in the world? Certainly this conclusion is true, and perhaps justice did indeed demand that Jones should obtain that much-desired good. But this argument as an argument—and that is what we are talking about—is invalid. It is an instance of (attempting to disprove the consequent by) Denying the Antecedent.
2.24 There are three reasons why it is perhaps not immediately obvious that the particular argument about Jones is fallacious. First, the proposed conclusion is true; second, we get hazed by the negatives in both antecedent and consequent; and, third, the fact that justice was done on this occasion—assuming it was—is by itself and without recourse to any other argument a sufficient reason for saying that There is some justice in the world. We remove all three difficulties if we get that pencil and paper, and write down: p = I (Jones) do not obtain this; and q = There is no justice in this world. Given these interpretations of p and q we can now symbolize the argument: If p then q, but ~p, therefore ~q. And this is manifestly a form of argument the invalidity of which we came fully to appreciate earlier; especially in paragraphs 2.6, 2.8, 2.14–2.16, and 2.18–2.19. Of course, we also have to appreciate that with these interpretations of p and q, the negations ~p and ~q become, respectively, Jones obtains this and There is some justice in the world. The rule is that double negation, “not not,” cancels out, leaving no negation at all.