Philosophy of Mathematics Seminar


Philosophy of Mathematics seminar


Tim Williamson , Oxford

Impossible worlds and semantic compositionality


Conditionals with impossible antecedents are widespread in logic and mathematics. They are often used to articulate proofs by reductio ad absurdum, and to report consequences of logics other than one’s own—for example, by a classical logician in characterizing a dialetheist account of Russell’s paradox. Pre-theoretically, it is natural to judge some of these conditionals as true and some as false. But on standard accounts within the framework of extensional (truth-functional) or intensional (possible world) semantics, all conditionals with impossible antecedents are vacuously true. In order to avoid this result, some logicians have used a framework with impossible as well as possible worlds, where any arbitrary set of sentences can be the set of sentences true at a given world. Thus even an impossible antecedent can be true at some impossible worlds, allowing the conditional itself to be false at some possible worlds. By treating logical consequence as truth-preservation at possible worlds, such an account can preserve classical logic (a recent example is the article by Berto, French, Priest, and Ripley in Journal of Philosophical Logic 2018). However, such accounts can be shown to violate the fundamental principle of semantic compositionality. More specifically, let the intension of a sentence in a model be the set of worlds at which the sentence is true in the model, and a sentential operator be intensional if and only if in every model the intension of the output sentence is a function of the input sentences. Then the impossible world semantics makes negation, conjunction, and disjunction non-intensional operators. Redefining intensionality by restricting ‘intensions’ to possible worlds (while still using impossible worlds in the semantics) makes the conditional itself a non-intensional operator. The same problem arises for the use of metaphysically impossible but epistemically/doxastically possible worlds to avoid the problem of logical omniscience in the semantics of epistemic and doxastic logic. A more critical treatment is needed of the pre-theoretic judgments used to motivate impossible world semantics. Arguably, they are artefacts of heuristics in the psychological sense, fast and frugal methods of problem-solving which are typically reliable over a wide range of normal cases but predictably fallible in special cases. For belief ascriptions, the relevant heuristic reads off from agents’ reactions to sentences their attitudes to the propositions those sentences express (even though they may react differently to two sentences expressing the same proposition). For assessing conditionals, the relevant heuristic is the Ramsey test: assess ‘If A, C’ unconditionally as you assess C conditionally on A. The Ramsey test applies even relative to a background set of suppositions. As special cases, it delivers both the Adams-Stalnaker equation of credence in ‘If A, C’ with credence in C conditional on A, and the standard natural deduction rules for the conditional. Our reliance on it explains our pre-theoretic assessments of some conditionals with impossible antecedents. But the Ramsey test, though cognitively useful, is unreliable, for it can be shown to be inconsistent.