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14 ReferencesA probabilistic extension of intuitionistic logic
Abstract
We introduce a probabilistic extension of propositional intuitionistic logic. The logic allows making statements such as P≥sα, with the intended meaning “the probability of truthfulness of α 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.
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- "This paper deals with the problem of developing an axiomatic system with the class of big-stepped probability distributions as semantics. Strong completeness of the system, named LBSP, is proved using a Henkin-like construction, along the line of research presented in [4, 12, 13, 14, 15, 17]. By [2], the consequence relation of System P can be modelled in the logic LBSP. "
[Show abstract] [Hide abstract] ABSTRACT: We develop a sound and strongly complete axiomatic system for probabilistic logic in which we can model nonmonotonic (or default) reasoning. We discuss the connection between previously developed logics and the two sublogics of the logic presented here.- "There is an extensive study about formal systems with uncertainty. There are two main approaches: extending classical logic with probabilistic operators (such as modal logic of knowledge in [27]); combining probabilistic approach with non-classical logics (probabilistic extension of intuitionistic logic [72]). Below we review well-known probabilistic temporal logics. "
[Show abstract] [Hide abstract] ABSTRACT: Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for the formal specification of real-time and complex systems, an up to date comprehensive analysis of these logics does not exist in the literature. In this paper we analyse real-time and probabilistic temporal logics which have been widely used in this field. We extrapolate the notions of decidability, axiomatizability, expressiveness, model checking, etc. for each logic analysed. We also provide a comparison of features of the temporal logics discussed.- "A consequence is that, if we want the extended completeness theorem, we cannot obtain a finitary axiomatization. Building on [8] [12] [13] [14] [17], we define a system which we show to be sound and complete, using infinitary rules of inference (i.e., rules where a conclusion has a countable set of premises). "
[Show abstract] [Hide abstract] ABSTRACT: We investigate probability logic with the conditional probability operators. This logic, denoted LCP, allows making statements such as: Psα, CPs(α | β), CP0(α | β) with the intended meaning "the probability of α is at least s", "the conditional probability of α given β is at least s", "the conditional probability of α given β at most 0". A possible-world approach is proposed to give semantics to such formulas. Every world of a given set of worlds is equipped with a probability space and conditional probability is derived in the usual way: P(α | β )= P(α∧β) P(β) , P(β) > 0, by the (uncondi- tional) probability measure that is defined on an algebra of subsets of possible worlds. Infinitary axiomatic system for our logic which is sound and complete with respect to the mentioned class of models is given. Decidability of the presented logic is proved.
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