CONICET | Buscador de Institutos y Recursos Humanos

In a famous paper from 1967, Jean van Heijenoort distinguished between logic as calculus and logic as language in order to describetwo opposite trends in the earlier development of mathematical logic. The distinction was generalized by Jaakko Hintikka, whoapplied it to the interpretation of 20th century philosophy. According to the universalist conception of language, semantics cannotbe defined in our only languagewithout falling into a vicious circle. So, semantics cannot be expressible inthe language. This fact motivated Hintikka to speak of the ?ineffability ofsemantics?. This paper is an attempt to discuss these two notions in relationto the proof-theoretic semantics, as it was characterized and carried out byMichael Dummett, Dag Prawitz and Peter Schroeder-Heister, among many others.The case of proof-theoretic semantics is quite interesting not only because itis an alternative to model-theoretic semantics, but also because of its rootsin mathematical intuitionism. This school had its own conception about the roleof language, as ordinary as formalized, in foundational issues. For theintuitionists language was secondary in the construction and justification ofmathematics. Arendt Heyting introduced formalization stricto sensu in intuitionism, and therefore paved the way forproof-theoretical semantics for intuitionistic logical constants. In the paperHeyting?s conception of formalization will be connected with the tradition of symbolic knowledge in formal sciences.

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