A family of deductive systems intended to provide an analysis of subjunctive conditionals and other types of conditional for which the truth value of a formula is not wholly determined by the truth values of and . For many natural assertions of conditional statements , the intended evaluation of involves the consideration of states (e.g., possible worlds or situations) that do not obtain. For example, it is natural to judge the counterfactual:
• Had the Pulitzer Board not intervened, Saul Bellow’s Henderson the Rain King would have been awarded the Pulitzer Prize for Fiction in 1960.
to be true due to considerations of a non-actual situation, namely, the situation or alternative in which the Pulitzer Board did not overrule the Pulitzer Committee in 1960.
A consequence of this intuition is that the models described for conditional logics frequently appeal to possible worlds in the evaluation of formulae . However, it is arguable that the interpretation of the conditional as preservation of truth at all possible worlds is insufficient for the analysis of subjunctive conditionals. In the case of the subjunctive conditional:
• Should Sally pluck the string of this violin, then a sound will be made.
the conditional may be considered true despite the fact that one can concoct possible worlds in which the antecedent obtains and the consequent fails, e.g., it is within the bounds of metaphysical possibility that there exists no air and Sally plucks her violin in vain. That this state fails to falsify the conditional is sometimes explained within the framework of conditional logic by appealing to a notion of similarity between worlds. The possible world in which there is no air in which sound is conveyed is not the type of world against which one naturally analyses the conditional. The worlds in which the conditional is evaluated are those that share nearly all their features with the actual world—for example, those in which the laws of nature hold—but the situation in which Sally plays the violin in a vacuum would be very unusual, that is, less similar to the actual world. This notion of similarity is modelled by various semantic devices, e.g., by special cases of accessibility relations, selection functions, or systems of spheres.