The intended interpretation of the intuitionistic first-order logical operators.

The present thesis is an investigation on an open problem in mathematical logic: the problem of devising an explanation of the meaning of the intuitionistic first-order logical operators, which is both mathematically rigorous and faithful to the interpretation intended by the intuitionistic mathematicians who invented and have been using them. This problem has been outstanding since the early thirties, when it was formulated and addressed for the first time. The thesis includes a historical, expository part, which focuses on the contributions of Kolmogorov, Heyting, Gentzen and Kreisel, and a long and detailed discussion of the various interpretations which have been proposed by these and other authors. Special attention is paid to the decidability of the proof relation and the introduction of Kreisel's extra-clauses, to the various notions of 'canonical proof' and to the attempt to reformulate the semantic definition in terms of proofs from premises. In this thesis I include a conclusive argument to the effect that if one wants to withdraw the extra-clauses then one cannot maintain the concept of 'proof as the basic concept of the definition; instead, I describe an alternative interpretation based on the concept of a construction 'performing' the operations indicated by a given sentence, and I show that it is not equivalent to the verificationist interpretation. I point out a redundancy in the internal-pseudo-inductive-structure of Kreisel's interpretation and I propose a way to resolve it. Finally, I develop the interpretation in terms of proofs from premises and show that a precise formulation of it must also make use of non-inductive clauses, not for the definition of the conditional but -surprisingly enough- for the definitions of disjunction and of the existential quantifier