Completeness in Equational Hybrid Propositional Type Theory

Studia Logica 107 (6):1159-1198 (2019)
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Abstract

Equational hybrid propositional type theory ) is a combination of propositional type theory, equational logic and hybrid modal logic. The structures used to interpret the language contain a hierarchy of propositional types, an algebra and a Kripke frame. The main result in this paper is the proof of completeness of a calculus specifically defined for this logic. The completeness proof is based on the three proofs Henkin published last century: Completeness in type theory, The completeness of the first-order functional calculus and Completeness in propositional type theory. More precisely, from and we take the idea of building the model described by the maximal consistent set; in our case the maximal consistent set has to be named, \-saturated and extensionally algebraic-saturated due to the hybrid and equational nature of \. From, we use the result that any element in the hierarchy has a name. The challenge was to deal with all the heterogeneous components in an integrated system.

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References found in this work

Completeness in the theory of types.Leon Henkin - 1950 - Journal of Symbolic Logic 15 (2):81-91.
The completeness of the first-order functional calculus.Leon Henkin - 1949 - Journal of Symbolic Logic 14 (3):159-166.
A theory of propositional types.Leon Henkin - 1963 - Fundamenta Mathematicae 52:323-334.

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