In this article we present a p-adic valued probabilistic logic equation image which is a complete and decidable extension of classical propositional logic. The key feature of equation image lies in ability to formally express boundaries of probability values of classical formulas in the field equation image of p-adic numbers via classical connectives and modal-like operators of the form Kr, ρ. Namely, equation image is designed in such a way that the elementary probability sentences Kr, ρα actually do have their (...) intended meaning—the probability of propositional formula α is in the equation image-ball with the center r and the radius ρ. Due to modal nature of the operators Kr, ρ, it was natural to use the probability Kripke like models as equation image-structures, provided that probability functions range over equation image instead of equation image or equation image. (shrink)

In this article we present a p-adic valued probabilistic logic equation image which is a complete and decidable extension of classical propositional logic. The key feature of equation image lies in ability to formally express boundaries of probability values of classical formulas in the field equation image of p-adic numbers via classical connectives and modal-like operators of the form Kr, ρ. Namely, equation image is designed in such a way that the elementary probability sentences Kr, ρα actually do have their (...) intended meaning—the probability of propositional formula α is in the equation image-ball with the center r and the radius ρ. Due to modal nature of the operators Kr, ρ, it was natural to use the probability Kripke like models as equation image-structures, provided that probability functions range over equation image instead of equation image or equation image. (shrink)

We offer an alternative approach to the existing methods for intuitionistic formalization of reasoning about probability. In terms of Kripke models, each possible world is equipped with a structure of the form $$\langle H, \mu \rangle $$ that needs not be a probability space. More precisely, though H needs not be a Boolean algebra, the corresponding monotone function (we call it measure) $$\mu : H \longrightarrow [0,1]_{\mathbb {Q}}$$ satisfies the following condition: if $$\alpha $$, $$\beta $$, $$\alpha \wedge \beta $$, (...) $$\alpha \vee \beta \in H$$, then $$\mu (\alpha \vee \beta ) = \mu (\alpha ) + \mu (\beta ) - \mu (\alpha \wedge \beta )$$. Since the range of $$\mu $$ is the set $$[0,1]_{\mathbb {Q}}$$ of rational numbers from the real unit interval, our logic is not compact. In order to obtain a strong complete axiomatization, we introduce an infinitary inference rule with a countable set of premises. The main technical results are the proofs of strong completeness and decidability. (shrink)