Archive for Mathematical Logic

, Volume 58, Issue 1–2, pp 119–136

# Sequent calculus for classical logic probabilized

• Marija Boričić
Article

## 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 [ab]”. 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.

## Keywords

Consistency Sequent calculus Probability Soundness Completeness

## Mathematics Subject Classification

03B48 03B50 03B05

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