316 NOTES ON CONDITIONAL SEMANTICS" Robert Stalnaker Department of Linguistics and Philosophy MIT Cambridge, MA 02139 i. Introduction Philosophers have been puzzling about conditional sentences, and condition- al reasoning, at least since the time of the ancient Stoics. In this century, the problem was first raised by logicians trying to give a plausible account of the logic of conditionals, and by philosophers of science in the empiricist tra- dition trying to understand the relation between counterfactual conditionals and scientific laws, and the role of such conditionals in the formation of disposi- tional and theoretical concepts. Conditional sentences are problematic, first from an abstract point of view: they are, or at least seem to be, non-truth-func- tional, and so are not analyzable with the resources of extensional semantics, the only kind of semantics that is and unproblematic. One can, of course, define a truth-functional connective that has some of the properties of the conditional -the so-called material conditional - but it neither gives an intuitively plaus- ible account of the logic and semantics of the conditional sentences of natural language that we find ourselves using, nor does it have the promise to do the conceptual work that we would like to use conditionals to do. There are also more substantive philosophical reasons for finding condi- tionals problematic: On the face of it, counterfactual conditional sentences seem to be about unactualized possibilities, yet they also seem to be making con- tingent, factual claims - claims that must be true or false in virtue of actual facts about the world as it is, and that must be evaluated on the basis of empir- ical evidence gathered in the actual world. Since Hume first articulated these worries, philosophers in the empiricist tradition have been skeptical about the intelligibility of concepts of natural necessity and possibility, about claims about potentialities, propensities, and connections that seem to be statements about how things could or must be, and not just about how they are. Possibility and necessity might be all right so long as they concern purely conceptual rela- tions or derive from linguistic conventions, but the empiricist argues that there cannot be necessity in the world, and that is what counterfactual conditionals seem to commit us to. But while conditionals are problematic, they are also a central, and appar- ently ineliminable part of the conceptual resources we use to describe and find our way about in the world. We make contingency plans - what to do if certain situations arise - and we deliberate by considering what would be the best thing to do under various alternative hypotheses. And in justifying and evaluating ac- tions, we may appeal to hypothetical possibilities that we know are not realized: I went around the pond rather than across it because the ice was thin and I would have fallen through. The Fed was justified in lowering interest rates, because "This paper was the basis for a survey talk at the third TARK conference, March, 1990. Notes on Conditional Semantics 317 if it had not, there would have been a recession. In forming beliefs about the world, we prepare to receive evidence that we may or may not receive. Our dispo- sitions to change what we believe in response to potential evidence are expressed in conditional beliefs: if he doesn't call by ten, then we'll know that he didn't miss the train; if the test result is negative, then we can rule out AIDS. And we appeal to counterfactual possibilities in reasoning about how to interpret evidence: If he had missed the train, he would have called, so since he didn't call, he must have made the train; If the gardener had done it, there would have been muddy footprints in the parlor, but there were not. In theorizing about the world, we find more powerful generalizations and deeper ways of categorizing things byconsidering not only how things actually behave, but how they would be- have under normal or ideal conditions, or when subjected to certain conditions that they may or not in fact be subjected to. As predicted byAristotelian phys- ics, bodies in motion in the world as it is slow down and come to a stop unless a force continues to be applied to them. But contrary to what Aristotelian phys- ics implies, they would continue in a straight line at constant velocity if acted on by no force at all. Things that never dissolve in water form a heterogeneous class, difficult to generalize about. But if we group the things that dissolved with those that didn't, but would have if they had been put to the test then we can get somewhere. 2. Conditional semantics Empiricists were skeptical about counterfactual possibilities, but commit- ted to making sense of scientific theory, and of practical and inductive reason- ing. So they tried to analyze counterfactuals away - to find a way to understand them as only apparently about unrealized possibilities. But these attempts were unsuccessful, and were acknowledged to be unsuccessful by empiricists such as Nelson Goodman, whose penetrating criticisms of his own project (in 1955) showed why it was doomed to fail. I The developers of conditional semantics (starting about a decade later) adopted a different, more modest strategy: don't try to analyze away the problematic references to alternative possibilities; instead, take conditionals at face value and try to give a clear, precise and rigorous statement of what they seem, on the surface, to be saying. Ask what conceptual resources we need to postulate in order to give a straightforward interpretation of conditionals, and then help yourself to those resources, and use them to clar- ify the abstract structure of conditionals, and the relation between conditionals and the other problematic concepts that they are so deeply interconnected with. Perhaps this way we can get clearer about the role of conditionals in describing the world and reasoning about it even if we can't explain them away. On the face of it, a statement such as "If Cuomo had run in 1988 he would have won" seems to be making a claim about a counterfactual possible situation in which Mario Cuomo runs for president in 1988. The claim it is making about this counterfactual situation is that in it, Cuomo is elected. To have a model theory to give a precise statement of this straightforward account, we need a set of possible situations or possible worlds, and a selection function that picks the situation in which the proposition expressed in the consequent of the condi- tional is said to be true. This selection function must have two arguments: first, the proposition expressed in the antecedent, and second the possible sit- IGoodman, 1954. This is the classic statement of the problem of counter- factuals. 318 Stalnaker uation relative to which the conditional as a whole is being evaluated. The se- lection is dependent on the first argument the antecedent - since the world selected represents the way things would have been if the antecedent were true. The selection is dependent on the second argument - the possible world from which the conditional is being evaluated - since conditionals maybe contingent - true in some possible worlds and false in others - which implies that the possible world selected from one possible world may be different from the possible world selected from another. So suppose we have a set of possible worlds and a function that takes a possible world i and a proposition P into a possible world j = f(i,P). If i is the actual world, then j is the world that would be actual if P were true. Now in terms of these resources, we can state a rule that gives the semantic value ~- the proposition expressed - by a conditional as a function of the propositions expressed by the antecedent and consequent, (For present purposes propositions may be identified with sets of possible worlds - intuitively, the worlds in which the proposition is true.) The rule is this: "If P, the Q" is true in possible world i if and only if Q is true in possible world f(i,P). This is pretty much all there is to the semantics. All that remains to be done is to make explicit the formal constraints on the selection function. Here are three plausible con- straints: first, the function must always selects a possible world in which the antecedent is true (if there is one; if the antecedent is necessarily false, then the function is undefined, and a clause is added to the rule making the condi- tional vacuously true in that case)~ Second, the actual world (more generally, the world we are selecting from) must be selected if it is eligible (that is, if the antecedent is true there). The third condition is motivated by the following intuitive considerations:intuitively, it seems reasonable to take the counter- factual possibility one is talking about when one says "if P" to be a situation that differs from the actual situation only in ways that are required to make P true. If this is right, we can assume that the selection is based on an ordering of possible worlds with respect to the extent to which they differ fromthe act- ual world: the function selects the closest or most similar world from among those that are eligible. The abstract theory can say nothing substantive about the basis for the ordering, but the selection will be based in this way on some such ordering if and only if it conforms to the following condition: if Q is true in f(i,P) and P is true in f(i,Q), then f(i,P) = f(i,Q).