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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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