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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)
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