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

Abstract According to Lewis’s Triviality Results (LTR), conditionals cannot satisfy the equation (E) P(C if A) = P(C | A), except in trivial cases. Since 1935 [9], de Finetti showed that by three-valued truth-tables, the equation might be satisfied in a general way (at least for indicative conditionals). This result is not at odds with LTR, because one of its premises, namely that conditionals are two-valued sentences, is dropped. Unfortunately, de Finetti did not equip his truth-tables by a proper notion of logical consequence. Ernst Adams [1], however, provided a probabilistic semantics for the so-called, simple conditionals (that is for those sentences of the form ‘if A then C’ where both A and C are two-valued ordinary sentences) that also satisfies equation (E) and provides a probabilistic counterpart of logical consequence (called pentailment). Adams’s logic is coextensive to Stalnaker’s and Lewis’s logic as far as simple conditionals are concerned. A theorem, proved by McGee [22], shows that no truth-functional many-valued logic allows a relation of logical consequence coextensive with Adams’s p-entailment. This result applies to de Finetti’s truth-functional semantics. This paper presents amodifiedmodal (Kripke-style) version of de Finetti’s semantics that escapesMcGee’s result and provides a general truth-conditional semantics for indicative conditionals, without being affected by LTR. The new framework encompasses and extends Adams’s probabilistic semantics (APS) to compounds of conditionals. Like APS, the present truth-conditional semantics does not deal with counterfactual conditionals. From the philosophical side, this theory challenges the view (endorsed by many authors) that indicative conditionals lack truth-conditions and show that a truth conditional semantics bypassing LTR is possible. Moreover, a comparison of the present theory with Stalnaker’s theories about compounds of conditionals shows that the present theory presents inherent advantages, especially regarding the import-export law.