Andrew Schumann NON - ARCHIMEDEAN VALUED PREDICATE LOGIC

In this paper I propose the non-Archimedean multiple-validity. Further, I build an infinite-order predicate logical language in that predicates of various order are considered as fuzzy relations. Such a language can have non-Archimedean valued semantics. For instance, infinite-order predicates can have an interpretation in the set [0, 1] of hyperreal (hyperrational) numbers. Notice that there exists an effectively axiomatizable part of non-Archimedean valued predicate logic, namely the class of higher-order formulas such that all their predicate quantifiers are universal (or existential). There exist various many-valued logical systems (see [9]). However non-Archimedean valued predicate logic has not been constructed yet. The idea of non-Archimedean multiple-validity is that (1) the set of truth values is uncountable infinite and (2) this set isn’t well-ordered. This idea can have a lot of applications in probabilistic reasoning (see [7], [13], [8]). In this paper I show that infinite-order predicate logic can have non-Archimedean valued semantics. 1. Non-Archimedean valued matrix The ultrapower Θ/U , where I is infinite and U is filter that contains all complements for finite subsets of I, is said to be a proper nonstandard extension of Θ and it is denoted by ∗Θ. There exist two groups of members of ∗Θ: (1) functions that are constant, e.g. f(α) = m ∈ Θ for an infinite index subset {α ∈ I}; a constant function [f = m] ∈ Θ/U is denoted by