Probabilized Sequent Calculus and Natural Deduction System for Classical Logic

Abstract

By combining, on one side Carnap–Popper–Leblance and Suppes concepts of sentence probability, and Gentzen’s sequent calculus LK and natural deduction system NK for classical propositional logic, on the other, we obtain their probabilistic versions. Through an introductory review, we briefly present Gentzen’s calculi, Carnap–Popper–Leblance and Suppes probability semantics. Afterwards we introduce systems LKprob, LKprob(e) and NKprob, through axioms and inference rules and define the corresponding probabilistic models followed by examples of derivations in these systems. We define the notion of “probabilized sequent” \( \Gamma \vdash ^{b}_a\mathit\Delta \) with the intended meaning that “the probability of truthfulness of \( \Gamma \vdash \mathit\Delta \) is into the interval [a;b]”, and in a similar way the notion of “probabilized formula” A[a;b]. The soundness and completeness theorems are proved for all of the presented systems with respect to defined models.