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Within classical propositional logic, assigning probabilities to formulas is shown to be equivalent to assigning probabilities to valuations by means of stochastic valuations. A stochastic valuation is a stochastic process, that is a family of random variables one for each propositional symbol. With stochastic valuations we are able to cope with a countably infinite set of propositional symbols. A notion of probabilistic entailment enjoying desirable properties of logical consequence is defined and shown to collapse into the classical entailment when the propositional language is left unchanged. Motivated by this result, a decidable conservative enrichment of propositional logic is proposed by giving the appropriate semantics to a new language construct that allows the constraining of the probability of a formula. A sound and weakly complete axiomatization is provided using the decidability of the theory of real closed ordered fields.
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Amílcar Sernadas, João Rasga, Cristina Sernadas, On probability and logic. Port. Math. 74 (2017), no. 4, pp. 267–313DOI 10.4171/PM/2007