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A probabilistic extension of intuitionistic logic. (English) Zbl 1022.03011
Summary: We introduce a probabilistic extension of propositional intuitionistic logic. The logic allows for making statements such as \(P_{\geq s} \alpha\), with the intended meaning “the probability of truthfulness of \(\alpha\) is at least \(s\)”. We describe the corresponding class of models, which are Kripke models with a naturally arising notion of probability, and give a sound and complete infinitary axiomatic system. We prove that the logic is decidable.

03B48 Probability and inductive logic
03B25 Decidability of theories and sets of sentences
68T37 Reasoning under uncertainty in the context of artificial intelligence
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