A p‐adic probability logic

Abstract

Abstract In this article we present a p ‐adic valued probabilistic logic \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {L}_{\mathbb {Q}_p}$\end{document} which is a complete and decidable extension of classical propositional logic. The key feature of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {L}_{\mathbb {Q}_p}$\end{document} lies in ability to formally express boundaries of probability values of classical formulas in the field \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {Q}_p$\end{document} of p ‐adic numbers via classical connectives and modal‐like operators of the form K r , ρ . Namely, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {L}_{\mathbb {Q}_p}$\end{document} is designed in such a way that the elementary probability sentences K r , ρ α actually do have their intended meaning—the probability of propositional formula α is in the \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {Q}_p$\end{document} ‐ball with the center r and the radius ρ. Due to modal nature of the operators K r , ρ , it was natural to use the probability Kripke like models as \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal {L}_{\mathbb {Q}_p}$\end{document} ‐structures, provided that probability functions range over \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {Q}_p$\end{document} instead of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {R}$\end{document} or \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$^*\mathbb {R}$\end{document} .