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Sequent calculus for classical logic probabilized

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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.

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Authors and Affiliations

  1. Faculty of Organizational Sciences, University of Belgrade, Belgrade, Serbia

    Marija Boričić

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  1. Marija Boričić
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Correspondence to Marija Boričić.

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The results contained in this paper were partially presented at the European Summer Meetings of Association for Symbolic Logic, Logic Colloquium 2012, held in Manchester on July 12–18, 2012, and Logic Colloquium 2014, held in Vienna on July 14–19, 2014. This research was supported in part by the Ministry of Education, Science and Technology of Serbia, Grant number 174026.

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Boričić, M. Sequent calculus for classical logic probabilized. Arch. Math. Logic 58, 119–136 (2019). https://doi.org/10.1007/s00153-018-0626-3

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  • DOI: https://doi.org/10.1007/s00153-018-0626-3

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