Actualism, Free Logic and First-Order Supervaluations

Abstract

In [Woodruff 1984] I showed that the supervaluation semantics introduced by [Van Fraassen 1966a, b] for Lambert’s system of free logic [Meyer and Lambert 1968] failed to have certain metatheoretic properties (compactness, upward and downward Löwenheim-Skolem, strong completeness and recursive axiomatizability) which attach to ordinary first-order logic. I also introduced the idea of secondary supervaluations, in which the class of conventions for asigning truth values to formulas with non-designating terms was restricted in some way or another. I was able to show that for a particular such restriction, the falsity condition (R(cx) false of all x when c doesn’t denote), the desirable properties were regained. I speculated that the indifference condition (R(cx) true either of all or of no xwhen cdoesn’t denote) would also suffice, and might be necessary.¹