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Sequent calculus for classical logic probabilized. (English) Zbl 1446.03047
This paper presents a proof-theoretic study on a logic of probability. It introduces a Gentzen-type system LKprob, which is a probabilistic modification of Gentzen’s sequent calculus LK. LKprob manipulates sequents of the form $$\Gamma \vdash^b_a \Delta$$ to be read as “the probability of truthfulness of a sequent $$\Gamma \vdash \Delta$$ is in the interval $$[a, b]$$”. The author characterizes this system as “a kind of the interval-valued probability sequent calculus”, which allows to construct “logical inferences dealing with probabilities”. The axioms of the systems are $$\Gamma \vdash^1_0 \Delta$$, $$\vdash^0_0$$, $$A \vdash^1_1 A$$ for any $$\Gamma, \Delta, A$$. By calculating probabilities, some additional empirical axioms of the form $$\Gamma \vdash^{b_{i}}_{a_{i}} \Delta$$ can be accepted, based, say, on some statistical research. Extending LKprob by such empirical axioms results in a theory suitable for probabilistic reasoning over certain empirical data. LKprob-theory can be equipped with a probabilistic model, which essentially is a map from the set of sequents into a finite subset of reals $$[0, 1]$$ closed under addition and containing 0 and 1. Soundness and completeness for LKprob with respect to this semantics are proved.
##### MSC:
 03B48 Probability and inductive logic 03B50 Many-valued logic 03F03 Proof theory, general (including proof-theoretic semantics)
##### Keywords:
consistency; sequent calculus; probability; soundness; completeness
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##### References:
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