Conditionals II: no truth conditions?

Transcription

1 Conditionals II: no truth conditions? UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Arguments for the material conditional analysis As Edgington [1] notes, there are some powerful reasons for thinking that the material conditional analysis is right. The or-to-if inference seems good: or-to-if inference And so does the not-and inference: A B if A,B (A B) if A, B not-and inference If either of these are valid entailments, they show that A B entails if A,B. And we ve already seen that if A,B has to entail A B, if Modus Ponens is to be valid for the indicative conditional. If all of these are valid entailments, the indicative conditional is equivalent to the material conditional. Can you think of a case where you d be reluctant to make any of these inferences? In the section on relevance, we learned to be skeptical about drawing conclusions about entailment from intuitions about the goodness of inferences. So there is room for maneuver here: we could try to explain why the or-to-if and not-and inferences are good modes of reasoning without taking them to be valid. We ll see a couple examples of this a bit later. 2 Arguments against the material conditional analysis We ve seen how Thomson (and Grice) try to explain away the counterintuitive consequences of the material conditional analysis. Edgington points out that their story can at best explain why we refrain from asserting conditionals when we only know that their antecedents are false or their consequents true. So she proposes to focus on other things: when do we accept them? (believe them) when do we reject them? (disbelieve them) when do we think them likely/unlikely? 2.1 Questionnaire The questionnaire [1, p. 33] is a way of filtering out conversational norms (you re not making an assertion, just indicating which you think is true). April 27,

2 2.2 Partial acceptance 1. The Republicans will win (R). No 2. The Republicans won t win ( R). Yes 3. Either the Republicans won t win or it will be sunny tomorrow. ( R S) Yes 4. If the Republicans win, the Obama Health care plan will be expanded. (if R, E) No Here we see a difference between disjunctions and conditionals: In the case of disjunctions, the predicted Gricean contrast between what is reasonable to believe and what it is reasonable to say, given one s grounds, is discernible. In the case of conditionals, it is not. [1, p. 33] 2.2 Partial acceptance If you think about states of partial acceptance, conditionals don t behave as the material analysis suggests. Example: (1) If I flip this coin, it will land heads. Edgington says: you should have about 50% confidence in this, if you think it s a fair coin. And this confidence should not depend on how confident you are that I will flip this coin. But on the material analysis, it will. On the material analysis, you should get more confident that the conditional is true as you get less confident that I will flip the coin. (How many agree with Edgington that it is right to have 50% confidence in the conditional? Does anyone think one should just say it s false? Edgington says: if someone is told the probability is 0 that if you toss it it will land heads, he will think it is a doubletailed or otherwise peculiar coin. ) Edgington proposes the following principle [1, p. 34]: If A entails B, it is irrational to be more confident of A than of B. Her argument against the validity of the or-to-if inference is that it is perfectly rational to be more confident of A B than of if A,B. This case against the truth-functional account cannot be made in tersm of beliefs of which one is certain. Someone who is 100 percent certain that the Labour Party won t win has (on my account of the matter) no obvious use for an indicative conditional beginning If they win. But someone who is, say, 90 percent certain that they won t win can have beliefs about what will be the case if they do. The truth-functional account has the immensely implausible consequence that such a person, if rational, is at least 90 per cent certain of any conditional with that antecedent. [1, p. 34] She makes a similar argument against the claim that B entails if A,B: 1. The Democrats will win (D). Yes April 27,

3 2.3 Rejection 2. Either it will be sunny tomorrow or the Democrats will win. (S D) Yes 3. If the economy falls into recession in August, the Democrats will win. (if R, D) No Try drawing your allocation of credence to D, R, D, and R. The more credence you give, the bigger the area. 2.3 Rejection Material conditional account says that rejecting a conditional requires accepting that its antecedent is true. This seems crazy, and can t be explained by Gricean means. 2.4 Bizarre validities Edgington points out that the material implication account gives bizarre predictions about the validity of inferences: William Hart s new proof of the existence of God [1, p. 37]: 1. If God does not exist, then it is not the case that if I pray my prayers will be answered. (if G, if P,A) 2. I do not pray. ( P) 3. Therefore (by the material conditional analysis), it is the case that if I pray my prayers will be answered. (if P,A) 4. So (modus tollens) God exists. (G) Also, the material conditional analysis predicts that this is a tautology: (if A,B) (if A,B) But intuitively it seems possible to reject both disjuncts. If I go to the store, I will see Jack. 3 Edgington s positive view Edgington proposes that indicative conditionals don t have any truth conditions at all. Conditionals not part of fact-stating discourse. Instead of explaining their meanings by saying under what conditions they are true, she proposes to say what mental states they express. When we judge that if A, B, she says, we are not judging that some proposition, that if A, B, is true. We are, rather, judging that B under the supposition that A. Similarly, when we judge it 60% likely that if A, B, we are not judging that some proposition (whose truth conditions we might try to articulate) is 60% likely to be true. Rather, we are judging that B is 60% likely to be true, under the supposition that A. In more detail, she holds [1, p. 38]: April 27,

4 4. Argument that a truth-conditional account should be truth-functional Conditional Likelihood: X believes that (judges it likely that) if A, B to the extent that he judges that A&B is nearly as likely as A, or, roughly, equivalently, to the extent that he judges A&B to be more likely than A& B. If you re happy with numerical assignments of credence, then this amounts to [1, p. 39]: A person s degree of confidence in a conditional, if A, B, is the conditional probability he assigns to B given A. Standardly the conditional probability of B given A is defined using the ratio: conditional probability Pr(A B) Pr(B A) = Pr(A) David Lewis [2] showed that there s no way to assign truth conditions to sentences of the form if A,B that will validate the Equation: Pr(if A, B) = Pr(B A) (Of course, his proof uses some assumptions, which you might reject.) So if Edgington is right that the degree to which you should believe if A,B is your subjective probability of A given B, then Lewis s triviality proof is an argument for the no-truth-conditions view. But Edgington doesn t want to assume precise values, so she doesn t rely on this. Instead she relies on intuitive cases like the coin case (discussed above), and the following argument. 4 Argument that a truth-conditional account should be truth-functional On Edgington s view, there is no way to assign truth conditions to an indicative conditional: there is no proposition such that asserting it to be the case is equivalent to asserting that B is the case given the supposition that A is the case [1, p. 30]. We ve seen why she rejects the material conditional account, which is the only plausible truth-functional account of the conditional. So now we need to see why she thinks that no non-truthfunctional (e.g. modal) truth conditions for the conditional can be given. She does this by arguing that if indicative conditionals have truth conditions, then they must be truthfunctional: I shall now show that wherever truth-functionality is assumed to fail, there are consequences incompatible with the positive thesis about the acceptance of a conditional. [1, 42ff]. Since she has already argued against truth-functional accounts, this gives her a general argument against truth-conditional accounts. The main premise of her argument is the Conditional Likelihood principle stated above. The argument takes the form of a tetralemma. Suppose truth-functionality fails. Then we must have at least one of the following cases for if A,B. April 27,

5 4. Argument that a truth-conditional account should be truth-functional 1. if T, F can be either T or F. 2. if T, T can be either T or F. 3. if F, T can be either T or F. 4. if F, F can be either T or F. Case 1 can be easily eliminated, since we take if A,B to entail A B. That leaves three interesting possibilities. Edgington is going to argue that none of them is possible. That will show that truth functionality can t fail. Case 2 If this case can obtain, C 1. Someone may be sure that A is true and sure that B is true, yet not have enough information to decide whether If A, B is true; one may consistently be agnostic about the conditional while being sure that its components are true (as for A before B ). However [1, p. 44]: C 1 is incompatible with our positive account. Being certain that A and that B, a person must think A&B is just as likely as A. He is certain that B on the assumption that A is true. So this possibility must be rejected. Establishing that the antecedent and consequent are true is surely one incontrovertible way of verifying a conditional [1, p. 44]. Think about whether anyone who accepts truth conditions for if that sometimes make a conditional with true antecedent and consequent true, and sometimes false, must accept C1. Would it be possible to give an account on which certainty that the antecedent and consequent were true would suffice for certainty in the conditional, but the mere truth of the antecedent and consequent would not suffice for the truth of the conditional? Case 3 Now suppose someone is sure that B but is uncertain whether A. On our positive account, he knows enough to be sure that if A, B: If B is certain, A&B is just as probable as A. This also accords with common sense. [1, p. 45] Case 4 Now consider someone who is sure that A and B have the same truth value, but is uncertain which. For example he knows that John and Mary spent yesterday evening together, but doesn t know whether they went to the party. According to our positive account and according to common sense, he knows enough to be sure that if John went to the party, Mary did. (J&M is as likely as J... [1, p. 45] April 27,

6 5. Edgington on the or-to-if inference REFERENCES Conclusion If conditionals have truth conditions, they are truth functional. But they are not truth functional. So they don t have truth conditions. The mistake is to think of conditionals as part of fact-stating discourse. Further upshot: the criterion for the validity of deductive arguments needs to be restated in the light of this thesis. Not truth preservation, since conditional sentences do not have truth conditions. Something to do with high subjective probability (Adams). 5 Edgington on the or-to-if inference In most normal cases where we assert A or B, Edgington shows, the or-to-if inference is reliable, in the sense that if we have high credence in the premises, we should have high credence in the conclusion. These are cases where we have intermediate credence in both A and B, and we do not accept the disjunction on the basis of one of the disjuncts alone. If I am agnostic about A, and agnostic about B, but confident that A or B, I must believe that if not-a, B. (See figure on p. 40.) However, in cases where we accept the disjunction only because we think one of the disjuncts is very likely, the or-to-if inference breaks down. Her example: (2) It is either 8 o clock or 11 o clock. Suppose you re 90% confident that it s 8 o clock, but you think there s a small chance your clock is broken. Since you re 90% confident that it s 8 o clock, you should be 90% confident that it s either 8 o clock or 11 o clock. But in this case the or-to-if inference fails. You don t accept the conditional (3) If it is not 8 o clock, it is 11 o clock. For, if it is not 8 o clock, it could many different times. (See figure on p. 41.) Edgington has explained why the or-to-if inference seems so intuitively compelling, and shown why it is nonetheless not valid. References [1] Dorothy Edgington. Do Conditionals Have Truth-Conditions. In: A Philosophical Companion to First-Order Logic. Ed. by R. I. G. Hughes. Indianapolis: Hackett, 1993, pp [2] David Lewis. Probabilities of Conditionals and Conditional Probabilities. In: Philosophical Review 85 (1976), pp April 27,