Logics for Reasoning About Processes of Thinking with Information Coded by p-adic Numbers

Abstract

In this paper we present two types of logics (denoted $${L^{D}_{Q_{p}}}$$ L Q p D and $${L^{\rm thinking}_{Z_{p}}}$$ L Z p thinking ) 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,? ? and D ? ?,β which enable us to make sentences of the form "the probability of ? belongs to the p-adic ball with the center r and the radius ?", and "the p-adic distance between the probabilities of ? and β is less than or equal to ?", 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,? ? means "the p-adic code of the information ? belongs to the p-adic ball with the center r and the radius ?", while D ? ?,β means "the p-adic distance between codes of ? and β are less than or equal to ?". The corresponding strongly complete axiom systems are presented and decidability of the satisfiability problem for each logic is proved.