Sequent calculus for classical logic probabilized

Abstract

Gentzen’s approach to deductive systems, and Carnap’s and Popper’s treatment of probability in logic were two fruitful ideas that appeared in logic of the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized in the sequent calculus, we introduce the notion of ’probabilized sequent’ $$\Gamma \vdash _a^b\Delta $$ with the intended meaning that “the probability of truthfulness of $$\Gamma \vdash \Delta $$ belongs to the interval [a, b]”. This method makes it possible to define a system of derivations based on ’axioms’ of the form $$\Gamma _i\vdash _{a_i}^{b_i}\Delta _i$$ , obtained as a result of empirical research, and then infer conclusions of the form $$\Gamma \vdash _a^b\Delta $$ . We discuss the consistency, define the models, and prove the soundness and completeness for the defined probabilized sequent calculus.