We provide here a computational interpretation of first-order logic based on a constructive interpretation of satisfiability w.r.t. a fixed but arbitrary interpretation. In this approach the formulas themselves are programs. This contrasts

Abstract. Since the pioneering work by Kripke and Montague, the term possible world has appeared in most theories of formal semantics for modal logics, natural languages, and knowledge-based systems. Yet that term obscures many questions about the relationships between the real world, various models of the world, and descriptions of those models in either formal languages or natural languages. Each step in that progression is an abstraction from the overwhelming complexity of the world. At the end, nothing is left but a colorful metaphor for an undefined element of a set W called worlds, which are related by an undefined and undefinable primitive relation R called accessibility. For some purposes, the resulting abstraction has proved to be useful, but as a general theory of meaning, the abstraction omits too many significant features. So much information has been lost at each step that many philosophers, linguists, and psychologists have dismissed model-theoretic semantics as irrelevant to the study of meaning. This article examines the steps in the process of extracting the pair (W,R) from the world and the way people talk about the world. It shows that the Kripke worlds can be reinterpreted as part of a Peircean semiotic theory, which can also include contributions from many other studies in cognitive science. Among them are Dunn's semantics based on laws and facts, the lexical semantics preferred by many linguists, psychological models of how the world is perceived, and philosophies of science that relate theories to the world. A full integration of all those sources is far beyond the scope of this article, but an outline of the approach suggests that Peirce's vision is capable of relating and reconciling the competing sources.

Formal countermodels may be used to justify the unprovability of formulae in the Heyting calculus (the best accepted formal system for constructive reasoning), on the grounds that unprovable formulae are constructively invalid. We argue that the intuitive impact of such countermodels becomes more transparent and convincing as we move from Kripke/Beth models based on possible worlds, to Lauchli realizability models. We introduce a new semantics for constructive reasoning, called relational realizability, which strengthens further the intuitive impact of Lauchli realizability. But, none of these model theories provides countermodels with the compelling impact of classical truth-table countermodels for classically unprovable formulae. We prove soundness of the Heyting calculus for relational realizability, and conjecture that there is a constructive choice-free proof of completeness. In this respect, relational realizability improves the metamathematical constructivity of Lauchli realizab...

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