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Suppes-style sequent calculus for probability logic. (English) Zbl 1444.03085
Summary: In order to treat the deduction relation \(\vdash\) in the context of probabilistic reasoning, we introduce a system \(\mathbf{LKprob}(\varepsilon)\) making it possible to work with expressions of the form \(\Gamma \vdash^n \Delta\), a generalization of Gentzen’s sequents \(\Gamma \vdash \Delta\) of classical propositional logic \(\mathbf{LK}\), with the intended meaning that ’the probability of the sequent \(\Gamma \vdash \Delta\) is greater than or equal to \(1 - n \varepsilon\)’, for a given small real \(\varepsilon > 0\) and any natural number \(n\). The system \(\mathbf{LKprob}(\varepsilon)\) can be considered a program inferring a conclusion of the form \(\Gamma \vdash^n A\) from a finite set of hypotheses of the same form \(\Gamma_i \vdash^{n_i} A_i\) (\(1 \leq i \leq n\)). We prove that our system is sound and complete with respect to the Carnap-Popper-type probability models.

MSC:
03B48 Probability and inductive logic
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