24.221 Metaphysics Counterfactuals When the truth functional material conditional ‘→ ’ (or ‘⊃’) is introduced, it is normally glossed with the English expression ‘If ..., then ...’. However, if this is the correct gloss there are a number of surprising features. Firstly, a sentence of the form ‘p → q’ will always be true when the antecedent, p, is false; and, secondly, it will always be true if the consequent, q, is true. But there are certainly some uses of ‘If ..., then ...’ which do not have these features. First consider (1) If Bush had not won the last election, then Nader would have won it. The antecedent of this sentence is false: Bush did win the last election. But we still don’t want to say that the sentence is true. If Bush hadn’t won the last election, Gore would almost certainly have done so. There was virtually no chance of Nader winning. So we don’t want to read the ‘If ..., then ...’ as a material conditional. Now consider (2) If Bush had polled only twenty votes across the whole country, then he would have won the last election. This time the consequent is true. But again we don’t want to say that the conditional is true: if Bush had polled only twenty votes they would not have won the election (at least, one hopes that’s true). So once again we don’t want to read the ‘If ..., then ...’ as a material conditional. What should we conclude? One possibility would be to say that the material conditional is just the wrong reading for the ‘If..., then...’ construction in English. But there are plenty of cases in which it seems to get it right. More plausible is the idea that there are two different English constructions that make use of ‘If ..., then ...’; and indeed, the syntax of English bears this out. Consider the two sentences (3) If Oswald didn’t shoot Kennedy, then someone else did. (4) If Oswald hadn’t shot Kennedy, then someone else would have Clearly these don’t mean the same thing. The first is not implausibly read as the material conditional. All that is ruled out is the possibility that the antecedent is true (i.e. Oswald didn’t shoot Kennedy) and the consequent is false (i.e. nobody else shot him either). But the second sentence cannot be read as a material conditional. The fact that the antecedent is false (since, let us suppose, Oswald did shoot Kennedy) doesn’t, by itself, make the sentence true. So it looks as though there are two quite different ‘If..., then’ constructions in English, marked by the different mood of the verbs involved. In (3) the verbs are in the simple indicative mood; in (4) they are subjunctive, as indeed they are in (1) and (2) (‘had shot’, ‘would have shot’, ‘had won’, ‘would have won’ etc.). 1Following the standard practice of grammarians, we’ll call such conditionals ‘counterfactuals’, and symbolize them: (P 䖪→ Q) Truth Conditions for Counterfactuals In developing truth conditiions for counterfactuals we follow the account given by David Lewis, who says (roughly): (P 䖪→ Q) is true iff the closest possible world (i.e. closest to the actual world) in which the antecedent, P, is true, is a world in which the consequent, Q, is also true (or, in other words, (P 䖪→ Q) is true iff the closest P-world is a Q-world). What do we mean here by ‘closest’? This is a measure of similarity. The closest P-world to the actual world is the world in which P is true which is most similar to the actual world. So the account of counterfactuals amounts to this: a counterfactual (P 䖪→ Q) is true just in case the world most similar to the actual world in which P is true is a world in which Q is true. This means in order to assess the truth value of a counterfactual we have to make an assessment about similarities between worlds; and that is going to be a rather vague business. But we shouldn’t let that put us off the account. The truth value of counterfactuals is itself vague; the account should mirror that vagueness. (Note: we said that this account was roughly that given by Lewis; in fact we have simplified his account in a number of ways. The most significant concerns our talk of the closest P-world. There are two ways in which there might fail to be such a world, and yet the counterfactual still be true. First, there might two or more P-worlds that are equally close; provided that these worlds are all Q-worlds, that shouldn’t make the counterfactual come out false. Second, there might be an infinite series of P-worlds, each one of which is closer to the actual world than the one before— compare the infinite series of fractions 1/2, 1/4, 1/8, 1/16 ... each of which is closer to zero that the one that comes before; again, provided that these are all Q-world, the counterfactual Lewis avoids these problems by saying that (P 䖪→ Q) will be true iff there is a possible world, w, which is both a P-world and a Q-world, and that any P-world which is as close or closer to the actual world than w is also a Q-world. But it’s not so easy to get one’s mind around this formulation; so we’ll stick with our simpler approximation.) No other world can be as similar to a world as that world is to itself. Identity is the limit case of similarity. But if that is so, then, if the actual world is a P-world, (P 䖪→ Q) will be true just in case the actual world is a Q-world. That might seem to be wrong: surely we would never say ‘If Oswald hadn’t shot Kennedy, someone else would have’ if we knew that in fact Oswald hadn’t shot him. But, as ever in providing a semantics for natural language, we need to distinguish that which is false from that which is pragmatically unacceptable on other grounds. It is true that we would normally not utter a counterfactual if we knew that its antecedent was true; but that could be because, in such circumstances, we would be in a position to assert the consequent itself, and so it would be misleading to assert something weaker. You wouldn’t say ‘If they were to