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FIBRED ALGEBRAIC SEMANTICS FOR A VARIETY OF NON-CLASSICAL FIRST-ORDER LOGICS AND TOPOLOGICAL LOGICAL TRANSLATION

Part of: General logic Categories and theories Categories with structure Algebraic logic

Published online by Cambridge University Press:  10 June 2021

YOSHIHIRO MARUYAMA
YOSHIHIRO MARUYAMA*
Affiliation:
RESEARCH SCHOOL OF COMPUTER SCIENCE THE AUSTRALIAN NATIONAL UNIVERSITYCANBERRA, ACT, AUSTRALIAE-mail:[email protected]
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Abstract

Lawvere hyperdoctrines give categorical algebraic semantics for intuitionistic predicate logic. Here we extend the hyperdoctrinal semantics to a broad variety of substructural predicate logics over the Typed Full Lambek Calculus, verifying their completeness with respect to the extended hyperdoctrinal semantics. This yields uniform hyperdoctrinal completeness results for numerous logics such as different types of relevant predicate logics and beyond, which are new results on their own; i.e., we give uniform categorical semantics for a broad variety of non-classical predicate logics. And we introduce an analogue of Lawvere–Tierney topology and cotopology in the hyperdoctrinal setting, which gives a unifying perspective on different logical translations, in particular allowing for a uniform treatment of Girard’s exponential translation between linear and intuitionistic logics and of Kolmogorov’s double negation translation between intuitionistic and classical logics. In the hyerdoctrinal conception, type theories are categories, logics over type theories are functors, and logical translations between them, then, are natural transformations, in particular Lawvere–Tierney topologies and cotopologies on hyperdoctrines. The view of logical translations as hyperdoctrinal Lawvere–Tierney topologies and cotopologies has not been elucidated before, and may be seen as a novel contribution of the present work. From a broader perspective, this work may be regarded as taking first steps towards interplay between algebraic and categorical logics; it is, technically, a combination of substructural (or Lambekian) algebraic logic and hyperdoctrinal (or Lawverian) categorical logic, as the hyperdoctrinal completeness theorem is shown via the integration of the Lindenbaum–Tarski algebra construction with the syntactic category construction. As such this work lays a foundation for further interactions between algebraic and categorical logics.

Keywords

tripos theorytopos theoryhyperdoctrineLawvere-Tierney topologycategorical logicalgebraic logicsubstructural logic

MSC classification

Primary: 03G30: Categorical logic, topoi
Secondary: 18C10: Theories (e.g. algebraic theories), structure, and semantics 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 18D30: Fibered categories
Type
Article
Information
The Journal of Symbolic Logic , Volume 86 , Issue 3 , September 2021 , pp. 1189 - 1213
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Copyright
© Association for Symbolic Logic 2021

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