an:06447328 Zbl 1382.03045 Ili?? Stepi??, Angelina; Ognjanovi??, Zoran Logics for reasoning about processes of thinking with information coded by $$p$$-adic numbers EN Stud. Log. 103, No. 1, 145-174 (2015). 00342127 2015
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03B48 03B25 03B42 68T27 $$p$$-adic; probability logic; coding information Summary: In this paper we present two types of logics (denoted $$L^D_{Q_p}$$ and $$L^{\mathrm{thinking}}_{Z_p}$$) 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 $$K_{r,\rho}\alpha$$ and $$D_\rho\alpha,\beta$$ which enable us to make sentences of the form the probability of $$\alpha$$ belongs to the $$p$$-adic ball with the center $$r$$ and the radius $$\rho$$'', and the $$p$$-adic distance between the probabilities of $$\alpha$$ and $$\beta$$ is less than or equal to $$\rho$$'', 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 $$K_r$$,$$_\rho\alpha$$ means the p-adic code of the information $$\alpha$$ belongs to the $$p$$-adic ball with the center $$r$$ and the radius $$\rho$$'', while $$D_\rho\alpha$$, $$\beta$$ means the $$p$$-adic distance between codes of $$\alpha$$ and $$\beta$$ are less than or equal to $$\rho''$$. The corresponding strongly complete axiom systems are presented and decidability of the satisfiability problem for each logic is proved.