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Title: Completeness in Equational Hybrid Propositional Type Theory
Author: Manzano, Maria
Martins, Manuel
Huertas, Antonia
Keywords: Propositional type theory
Hybrid logic
Equational logic
Issue Date: Dec-2019
Publisher: Springer Verlag
Abstract: Equational Hybrid Propositional Type Theory (EHPTT) 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 (a nonempty set with functions) 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: (i) Completeness in type theory (ii) The completeness of the first-order functional calculus and (iii) Completeness in propositional type theory. More precisely, from (i) and (ii) 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 EHPTT. From (iii), 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.
Peer review: yes
DOI: 10.1007/s11225-018-9833-5
ISSN: 0039-3215
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