The Logic of Conditionals1

THE LOGIC OF CONDITIONALS1 by Ernest Adams University of California, Berkeley The standard use of the propositional calculus ('P.C.') in analyzing the validity of inferences involving conditionals leads to fallacies, and the problem is to determine where P.C. may be 'safely' used. An alternative analysis of criteria of reasonableness of inferences in terms of conditions of justification r a t h e r than truth of statements is pro- posed. It is argued, under certain restrictions, that P. C. may be safely used, except in inferences whose conclusions are conditionals whose antecedents are incompatible with the premises in the sense that if the antecedent became known, some of the previously asserted premises would have to be withdrawn. 1. The following nine 'inferences' are all ones which, when symbol- ized in the ordinary way in the propositional calculus and analyzed truth functionally, are valid in the sense that no combination of truth values of the components makes the premises true but the conclusion false. Fl. John will arrive on the 10 o'clock plane. Therefore, if J o h n does not arrive on the 10 o'clock plane, he will arrive on the 11 o'clock plane. F2. John will arrive on the 10 o'clock plane. Therefore, if John misses his plane in New York,' he will arrive on the 10 o'clock plane. F3. If Brown wins the election, Smith will retire to private life. Therefore, if Smith dies before the election and Brown wins it, Smith will retire to private life. F4. If Brown wins the election, Smith will retire to private life. If Smith dies before the election, Brown will win it. Therefore, if Smith dies before the election, then he will retire to private life. F5. If Brown wins, Smith will retire. If Brown wins, Smith will not retire. Therefore, Brown will not win. 166 F6. Either Dr. A or Dr. B will attend the patient. Dr. B will not attend the patient. Therefore, if Dr. A does not attend the patient, Dr. B will. F7. It is not the case that if John passes history, he will graduate. Therefore, John will pass history. F8. If you throw both switch S and switch T, the motor will start. Therefore, either if you throw switch S the motor will start, or if you throw switch T the motor will start. F9. If John will graduate only if he passes history, then he won't graduate. Therefore, if John passes history he won't graduate. We trust that the reader's immediate reaction to these examples agrees with ours in rejecting or at least doubting the validity of these inferences — one would not ordinarily 'draw' the inferences if one were 'given' the premises. If this is so, it poses a problem for the application of formal logic: either applied formal logic is wrong (and it is only as an applied theory that formal logic can be said to be right or wrong at all) or, if it is right and these inferences are valid after all, then we have to explain why they 'appear' fallacious. This is the problem which this paper is concerned with. More exactly, we shall be concerned with criteria of validity for inferences involving ordinary conditionals, and with determining to what extent formal logical theory may safely be applied to analyzing and formalizing arguments involving conditionals. As will be seen, this objective is only partially achieved, if at all, and this partial 'solution' itself raises what are probably more profound difficulties. 2. Before passing to a detailed discussion, it may be useful to con- sider briefly the significance of these examples. If these inferences are truly fallacious, then clearly they call into question some of the principles of formal logic and its application. The use of formal logic in the analysis of the logic of conditionals rests on two principles, one general, the other specific to conditionals. These are: Principle 1 (general) An inference is deductively valid if and only if it is logically impossible for the premises to be true and the conclusion false. Principle 2 At least for purposes of formal analysis, 'if then' statements can be treated as truth functional — t r u e if either their antecedents are false or their consequents true, false only if their antecedents are true and the consequents false. 167 Thus baldly stated, of course, Principle 2 appears very dubious, and one might immediately be led to expect that fallacies would arise in applications of the formal theory which tacitly make the assumption. Examples F1-F9 might be regarded simply as confirmation of this expectation. Fl and F2 are in fact counterparts of the paradoxes of material implication. It should be noted, however, that the principles of material implication (i.e. the truth conditions for material conditionals) are not themselves arbitrary, but can be 'deduced' from still more funda- mental principles which must in turn be questioned if Principle 2 is questioned. For instance, the 'law' that a conditional is true if its con- sequent is true follows from Principle 1 together with the principle of conditionalization. The argument goes thus, / and q logically imply p (Principle 1). If p and q imply/, then/ implies 'if q then/' (principle of conditionalization), hence / implies 'if q then /'. If / implies 'if q t h e n / ' , and/ is true, then 'if q t h e n / ' is true (Principle 1), hence, if / is true, so is 'if q then /'. Therefore, if we reject Principle 2, we must reject either the principle of conditionalization or Principle 1, or both, as well. Similarly, the truth conditions for the material conditional follow from the apparent logical equivalence of 'not / or q' and 'if / then q' plus Principle 1. Therefore, rejection of Principle 2 requires either that Principle 1 be rejected or that the supposition that 'not / or q' is equivalent to 'if / then q' be re-examined. A closely related supposition is in fact at the heart of example F6. Looked at in another way, example F6 seems to bring into question a principle which follows from Principle 1 alone: namely, that if a conclusion follows logically from a single premise, then it also follows logically from two premises one of which is the same as the original premise. One might be willing to grant that 'If Dr. A does not attend the patient, Dr. B will' follows logically from 'Either Dr. A or Dr. B will attend the patient', but if one rejected the inference in F6, this would be a counterinstance to the above consequence of Principle 1. Finally, examples F3 and F4 call into question laws of conditionals not specifically associated with material implication. F4, if it is fallacious, is a counterinstance to the hypothetical syllogism (if 'if q then r' and 'if / then q' then 'if / then r'). A closely related principle is taken as a postulate in G. I. Lewis' [4, p. 125] theory of strict implication. It is unlikely, therefore, that fallacies of the kind given 168 here can be entirely avoided by going over to formal analysis in terms of strict implication or related systems. 3. Are the inferences in examples F1-F9 fallacious? We may concentrate for the present on Fl and ask whether C If John does not arrive on the 10 o'clock plane, then he will arrive on the 11 o'clock plane. is a logical consequence of P John will arrive on the 10 o'clock plane. To avoid irrelevant misunderstandings, let us specify that both statements are about the same person, and that both P and the antecedent of G are about the same event — J o h n ' s arrival on the 10 o'clock plane. If we attempt to answer this question by applying the criterion of validity of formal logic, formulated in Principle 1, we must ask: Is it logically possible for P to be true, but G false? To attempt to answer this question is to see that it has no clear answer. The reason is that the term 'true' has no clear ordinary sense as applied to conditionals, particularly to those whose antecedents prove to be false, as the antecedent of G must if P is true. This is not to say that conditional statements with false antecedents are not sometimes called 'true' and sometimes 'false', but that there are no clear criteria for the applications of those terms in such cases. This is, of course, an assertion about the ordinary usage of the terms 'true' and 'false', and it can be verified, if at all, only by examining that usage. We shall not conduct such an examination here, but leave it to the reader to verify by observation of how people dispute about the correctness of conditional statements whose antecedents prove false, that precise criteria are lacking. Formal logicians might perhaps be inclined to dismiss the vagueness of the ordinary usage of 'true', and provide a new and more precise definition which is appropriate to their needs. This is in fact what Tarski has done with very fruitful results in the Wahrheitsbegriff. One might even regard the truth conditions of material conditionals not as defining a new sentential connective, but rather as defining the term 'true' precisely as applied to ordinary conditionals. The danger in this approach lies in the likelihood of begging the question raised at the beginning of this section. If the application of 'true' is defined 169 arbitrarily for those cases in which it has no clear ordinary sense, then this arbitrariness will extend to the determination as to whether state- ment G is a logical consequence of P or not.

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