Boolean-like and frequentistic nonstandard semantics for first-order predicate calculus without functions

Abstract

 Well-formed formulas of the classical first-order predicate language without functions are evaluated in such a way that truthvalues are subsets of the set of all positive integers. Such an evaluation is projected in two different ways into the unit interval of real numbers so that two real-valued evaluations are obtained. The set of tautologies are proved to be identical, in all the three cases, with the set of classical first-order predicate tautologies, but the induced evaluations meet the properties of probability and possibility measures with respect to nonstandard supremum and infimum operations induced in the unit interval of real numbers.