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Completeness in Hybrid Type Theory

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Abstract

We show that basic hybridization (adding nominals and @ operators) makes it possible to give straightforward Henkin-style completeness proofs even when the modal logic being hybridized is higher-order. The key ideas are to add nominals as expressions of type t, and to extend to arbitrary types the way we interpret \(@_i\) in propositional and first-order hybrid logic. This means: interpret \(@_i\alpha _a\), where \(\alpha _a\) is an expression of any type \(a\), as an expression of type \(a\) that rigidly returns the value that \(\alpha_a\) receives at the i-world. The axiomatization and completeness proofs are generalizations of those found in propositional and first-order hybrid logic, and (as is usual inhybrid logic) we automatically obtain a wide range of completeness results for stronger logics and languages. Our approach is deliberately low-tech. We don’t, for example, make use of Montague’s intensional type s, or Fitting-style intensional models; we build, as simply as we can, hybrid logicover Henkin’s logic.

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Correspondence to Patrick Blackburn.

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Research partially funded by the Spanish Ministry of Science and Innovation as part of the project Nociones de Completud (Conceptions of Completeness), Proyectos de Investigaci´on Fundamental, MICINN FFI2009-09345, Universidad de Salamanca, 2009–2011.

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Areces, C., Blackburn, P., Huertas, A. et al. Completeness in Hybrid Type Theory. J Philos Logic 43, 209–238 (2014). https://doi.org/10.1007/s10992-012-9260-4

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