In the past sixty years or so, a real forest of intuitionistic models for classical theories has grown. In this paper we will compare intuitionistic models of first order classical theories according to relevant issues, like completeness (w.r.t. first order classical provability), consistency, and relationship between a connective and its interpretation in a model. We briefly consider also intuitionistic models for classical ω-logic. All results included here, but a part of the proposition (a) below, are new. This work is, ideally, (...) a continuation of a paper by McCarty, who considered intuitionistic completeness mostly for first order intuitionistic logic. (shrink)

, Rudolf Carnap became a chief proponent of the doctrine that the statements of intuitionism carry nonstandard intuitionistic meanings. This doctrine is linked to Carnap's ‘Principle of Tolerance’ and claims he made on behalf of his notion of pure syntax. From premises independent of intuitionism, we argue that the doctrine, the Principle, and the attendant claims are mistaken, especially Carnap's repeated insistence that, in defining languages, logicians are free of commitment to mathematical statements intuitionists would reject. I am grateful to (...) Nathan Carter, Gary Ebbs, Janet Folina, Luise Prior McCarty, Stewart Shapiro, Neil Tennant, Christopher Tillman, Beth Tropman, Wen-fang Wang, and two anonymous referees for their comments and suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

We constructively prove completeness for intuitionistic first-order logic, iFOL, showing that a formula is provable in iFOL if and only if it is uniformly valid in intuitionistic evidence semantics as defined in intuitionistic type theory extended with an intersection operator.Our completeness proof provides an effective procedure that converts any uniform evidence into a formal iFOL proof. Uniform evidence can involve arbitrary concepts from type theory such as ordinals, topological structures, algebras and so forth. We have implemented that procedure in the (...) Nuprl proof assistant.Our result demonstrates the value of uniform validity as a semantic notion for studying logical theories, and it provides new techniques for showing that formulas are not intuitionistically provable. Here we demonstrate its value for minimal and intuitionistic first-order logic. (shrink)

In 1996, Krivine applied Friedman's A-translation in order to get an intuitionistic version of Gödel completeness result for first-order classical logic and countable languages and models. Such a result is known to be intuitionistically underivable 559), but Krivine was able to derive intuitionistically a weak form of it, namely, he proved that every consistent classical theory has a model. In this paper, we want to analyze the ideas Krivine's remarkable result relies on, ideas which where somehow hidden by the heavy (...) formal machinery used in the original proof. We show that the ideas in Krivine's proof can be used to intuitionistically derive some crucial mathematical results, which were supposed to be purely classical up to now: the Ultrafilter Theorem for countable Boolean algebras, and the maximal ideal theorem for countable rings. (shrink)