Bypassing Lewis’ Triviality Results. A Kripke-Style Partial Semantics for Compounds of Adams’ Conditionals

According to Lewis’ Triviality Results (LTR), conditionals cannot satisfy the equation
(E) P(C if A) = P(C j A), except in trivial cases. Ernst Adams (1975), however,
provided a probabilistic semantics for the so-called simple conditionals that also satisfies equation (E) and provides a probabilistic counterpart of logical consequence (called p-entailment). Adams’ probabilistic semantics is coextensive to Stalnaker-Thomason’s (1970) and Lewis’ (1973) semantics as far as simple conditionals are concerned. A theorem proved in McGee 1981, shows that no truth-functional many-valued logic allows a relation of logical consequence coextensive with Adams’ p-entailment. This paper presents a modified modal (Kripke-style) version of de Finetti’s semantics that escapes McGee’s result and provides a general truth-conditional semantics for indicative conditionals. It agrees with Adams’ logic and is not affected by LTR. The new framework encompasses and extends Adams’ probabilistic semantics (APS) to compounds of conditionals. A generalised set of axioms for probability over the set of tri-events is provided, which coincide with the standard axioms over the set of the two-valued ordinary sentences.