A new semantics for first-order logic, multivalent and mostly intensional

Abstract

Cited by many as distinctive of first-order logic are the bivalence of its statements and the extensionality of its operators, among them the three operators '~', '&', and 'V'. It is to that bivalence and that extensionality, we are told, that the logical entailments and, hence, logical truths peculiar to the logic are due. But first-order logic is a far more diverse thing than its exponents usually allow. As proof I submit here a new semantics for it, one in which statements are susceptible of up to 2 t~~ values, the operators '&' and 'V' are intensional, and yet the logical entailments and logical truths that first-order logic acknowledges are all preserved. Credit should go to Karl Popper, from whom I largely borrowed Constraints D1-D6 on p. 57; to Kent Bendall, from whom I borrowed Constraint D7 on that page; and to Nicholas Rescher, who pointed out in 1963 that '&' - as interpreted here - is intensional in the presence of '~'.