Some intuitions behind realizability semantics for constructive logic: Tableaux and Läuchli countermodels Original Research Article

We use formal semantic analysis based on new constructions to study abstract realizability, introduced by Läuchli in 1970, and expose its algebraic content. We claim realizability so conceived generates semantics-based intuitive confidence that the Heyting Calculus is an appropriate system of deduction for constructive reasoning. Well-known semantic formalisms have been defined by Kripke and Beth, but these have no formal concepts corresponding to constructions, and shed little intuitive light on the meanings of formulae. In particular, the completeness proofs for these semantics do not generate confidence in the sufficiency of the Heyting Calculus, since we have no reason to believe that every intuitively constructive truth is valid in the formal semantics. Läuchli has proved completeness for a realizability semantics with formal concepts analogous to constructions. We argue in some detail that, in spite of a certain inherent inexactness of the analogy, every intuitively constructive truth is valid in Läuchli semantics, and therefore the Heyting Calculus is powerful enough to prove all constructive truths. Our argument is based on the postulate that a uniformly constructible object must be communicable in spite of imprecision in our language, and that the permutations in Läuchliʹs semantics represent conceivable imprecision in a language, while allowing a certain amount of freedom in choosing the particular structure of the language. We give a detailed generalization of Läuchliʹs proof of completeness for the propositional part of the Heyting Calculus, in order to make explicit constructive and algebraic content. In our treatment, we establish several new results about Läuchli models. We show how to extend the sconing and gluing constructions familiar from Kripke and Frame semantics and Topos theory, to Läuchli models, and use them to give an algebraic approach to countermodel construction. In particular, the Läuchli arguments are given without the restriction to the integers, Z, as a group of permutations, which makes much of the coding scheme used in Läuchliʹs original paper transparent. We also make use of a new propositions-as-types syntax for the Heyting calculus, with limited nondeterminism, in which validity of formulae can be decided without loop-detection.