In this paper we present two types of logics and \ ) where certain p-adic functions are associated to propositional formulas. Logics of the former type are p-adic valued probability logics. In each of these logics we use probability formulas K r,ρ α and D ρ α,β which enable us to make sentences of the form “the probability of α belongs to the p-adic ball with the center r and the radius ρ”, and “the p-adic distance between the probabilities of (...) α and β is less than or equal to ρ”, respectively. Logics of the later type formalize processes of thinking where information are coded by p-adic numbers. We use the same operators as above, but in this formalism K r,ρ α means “the p-adic code of the information α belongs to the p-adic ball with the center r and the radius ρ”, while D ρ α,β means “the p-adic distance between codes of α and β are less than or equal to ρ”. The corresponding strongly complete axiom systems are presented and decidability of the satisfiability problem for each logic is proved. (shrink)

Probability logics have been an active topic of investigation of beliefs in type spaces in game theoretical economics. Beliefs are expressed as subjective probability measures. Savage’s postulates in decision theory imply that subjective probability measures are not necessarily countably additive but finitely additive. In this paper, we formulate a probability logic Σ + that is strongly complete with respect to this class of type spaces with finitely additive probability measures, i.e. a set of formulas is consistent in Σ + iff (...) it is satisfied in a finitely additive type space. Although we can characterize Σ + -theories satisfiable in the class as maximally consistent sets of formulas, we prove that any canonical model of maximally consistent sets is not universal in the class of type spaces with finitely additive measures, and, moreover, it is not a type space. At the end of this paper, we show that even a minimal use of probability indices causes the failure of compactness in probability logics. (shrink)

In this article, we present the probability logic LFOCP which is suitable to formalize statements about conditional probabilities of first order formulas. The logical language contains formulas such as CP≥s and CP≤s with the intended meaning ‘the conditional probability of ϕ given θ is at least s’ and ‘at most s’, respectively, where ϕ and θ are first-order formulas. We introduce a class of first order Kripke-like models that combine properties of the usual Kripke models and finitely additive probabilities. We (...) propose an infinitary axiom system and prove that it is sound and strongly complete with respect to the considered class of models. In this article, the terms finitary and infinitary concern meta language only, i.e. the logical language is countable, formulas are finite, while only proofs are allowed to be infinite. We analyse decidability of LFOCP and provide a procedure which decides satisfiability of a given conditional probability formula, in the case when the underlying first-order theory is decidable. Relationships to other systems and possible extensions of the presented approach are discussed. (shrink)