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Logics for reasoning about processes of thinking with information coded by \(p\)-adic numbers. (English) Zbl 1382.03045
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.

MSC:
03B48 Probability and inductive logic
03B25 Decidability of theories and sets of sentences
03B42 Logics of knowledge and belief (including belief change)
68T27 Logic in artificial intelligence
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