A p-adic probability logic

• Published in: Mathematical logic quarterly 2012-08, Vol.58 (4-5), p.263-280
• Main Authors: Ilić-Stepić, Angelina, Ognjanović, Zoran, Ikodinović, Nebojša, Perović, Aleksandar
• Format: Article
• Language: English
• Subjects: 68T27; p-adic numbers. [msc2010]03B48; probability logic
• ISSN: 0942-5616; 1521-3870
• DOI: 10.1002/malq.201110006

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 Kr, 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 Kr, ρα 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 Kr, 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}.