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This reports introduces a novel sound and complete semantics for first orderintuitionistic logic, in the framework of category theory and by thecomputational interpretation of the logic based on the so-called Curry-Howardisomorphism. Aside, a sound and complete semantics for the correspondinglambda-calculus is derived, too. This semantics extends, in a way, the moretraditional meanings given by Heyting categories, the topos-theoreticinterpretation, and Kripke models. The feature which justifies the introductionof this novel semantics is the fact that it is 'point-free', i.e., there is nouniverse whose elements are used to interpret the logical terms. In otherwords, terms do not denote individuals of some collection but, instead, theydenote the 'glue' which keeps together the interpretations of statements,similarly to what happens in formal topology. Since the proposed semantics canbe trivially extended to all the first-order logical theories based on theintuitionistic system (and, with some care, to minimal systems as well), thesemantics covers also all the predicative theories, even if some peculiaraspects of these theories should be remarked.

Intuitionistic First-Order Logic: Categorical Semantics via the Curry-Howard Isomorphism