Labelled Natural Deduction

Abstract

The functional interpretation of logical connectives is concerned with a certain harmony between, on the one hand, a indexfunctional! calculus functional calculus on the expressions built up from the recording of the deduction steps (the labels), and, on the other hand, a logical calculus on the formulae. It has been associated with Curry’s early discovery of the correspondence between the axioms of intuitionistic implicational logic and the type schemes of the so-called ‘combinators’ of Combinatory Logic [12], and has been referred to as the formulae-as-types interpretation. Howard’s [80] extension of the formulae-as-types paradigm to full intuitionistic first-order predicate logic meant that the interpretation has since been referred to as the ‘Curry-Howard’ functional interpretation. Although Heyting’s [75, 76] intuitionistic logic did fit well into the formulae-as-types paradigm, it seems fair to say that, since Tait’s [117, 118] intensional interpretations of Godel’s [69] Dialectica system of functionals of finite type, there has been enough indication that the framework would also be applicable to logics beyond the realm of intuitionism. Ultimately, the foundations of a functional approach to formal logic are to be found in Prege’s [47, 50, 51] system of ‘concept writing’, not in Curry, or Howard or, indeed, Heyting.