On Trivalent Logics, Compound Conditionals, and Probabilistic Deduction Theorems
Angelo Gilio, David E. Over, Niki Pfeifer, Giuseppe Sanfilippo
TL;DR
This paper advances a coherence-based, Level-2 probabilistic treatment of conditionals, extending de Finetti’s conditional-event analysis beyond the classic trivalent view. It establishes a Probabilistic Deduction Theorem, analyzes iterated conditionals, and introduces a General Import-Export principle that links p-consistency and p-entailment to iterated conditioning. By treating conjunctions and iterates as conditional random quantities with values in [0,1], the work preserves fundamental probabilistic properties (e.g., Fréchet-Hoeffding bounds) and derives probabilistic analogues of classical inference rules (System P). It also contrasts this approach with trivalent logics, illustrating how import-export and deduction behave differently under coherence-based semantics and highlighting implications for AI reasoning under uncertainty. The results offer a rigorous probabilistic framework for reasoning with conditionals, iterated conditionals, and nonmonotonic inference rules, with potential applications to knowledge representation and decision making under uncertainty.
Abstract
In this paper we recall some results for conditional events, compound conditionals, conditional random quantities, p-consistency, and p-entailment. Then, we show the equivalence between bets on conditionals and conditional bets, by reviewing de Finetti's trivalent analysis of conditionals. But our approach goes beyond de Finetti's early trivalent logical analysis and is based on his later ideas, aiming to take his proposals to a higher level. We examine two recent articles that explore trivalent logics for conditionals and their definitions of logical validity and compare them with our approach to compound conditionals. We prove a Probabilistic Deduction Theorem for conditional events. After that, we study some probabilistic deduction theorems, by presenting several examples. We focus on iterated conditionals and the invalidity of the Import-Export principle in the light of our Probabilistic Deduction Theorem. We use the inference from a disjunction, "$A$ or $B$", to the conditional,"if not-$A$ then $B$", as an example to show the invalidity of the Import-Export principle. We also introduce a General Import-Export principle and we illustrate it by examining some p-valid inference rules of System P. Finally, we briefly discuss some related work relevant to AI.