Yet another constructivization of classical logic

Abstract

The aim of this paper is to provide a way of extracting the constructive content of a certain family of classical proofs directly from the proofs themselves. The paper itself is written in a purely constructive style. Our work is inspired by the game interpretations of classical logic due to Novikov (1943) and Coquand (1995). These interpretations date back to Gentzen (1969) and Bernays (1970) and were recently studied by Coquand (1995) who made use of technical tools developed by Novikov (1943). We will introduce an interpretation which is a short and compact description of the meaning assigned to classical formulas by Coquand’s interpretation. Contrary to Coquand, we will completely avoid any game terminology, by making use of the intuitionistic notion of continuous computation. A posteriori, our interpretation turns out to be related to Kreisel’s no-counterexample interpretation (Kreisel 1957) but, compared with his, it provides simpler constructive proofs. The reader is referred to Baratella and Berardi (1997) for a number of examples of constructive proofs provided by our interpretation that can be used for a comparison. Indeed, our interpretation is a fragment of Coquand’s that can be easily expanded to a variant of his. However, we claim that our interpretation suffices as long as we are only interested in the constructive meaning of classical formulas (whilst we need Coquand’s if we are interested in computations lying behind the constructive meaning). We will support this claim by proving, as the main result, that our interpretation is intuitionistically complete, in the same way as Coquand’s (Herbelin 1995). That is, we will intuitionistically prove that a formula is derivable in infmitary classical logic if and only if its interpretation holds. Since infinitary classical logic is classically complete, loosely speaking we can restate our result as follows: the classical truth of a classical formula is intuitionistically equivalent to the intuitionistic truth of the constructive interpretation of the formula. We also recall that Godel’s Dialectica interpretation is not intuitionistically complete (see section 7). In this regard, see also Berardi (1997). In addition, we point out that, contrary to the game interpretations, our interpretation is not a sort of reformulation of what is going on in the sequent calculus.