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- W2108840809 abstract "We pursue the idea that predicate logic is a “fibred algebra” while propositional logic is a single algebra; in the context of intuitionism, this algebraic understanding of predicate logic goes back to Lawvere, in particular his concept of hyperdoctrine. Here, we aim at demonstrating that the notion of monad-relativised hyperdoctrines, which are what we call fibred algebras, yields algebraisations of a wide variety of predicate logics. More specifically, we discuss a typed, first-order version of the non-commutative Full Lambek calculus, which has extensively been studied in the past few decades, functioning as a unifying language for different sorts of logical systems (classical, intuitionistic, linear, fuzzy, relevant, etc.). Through the concept of Full Lambek hyperdoctrines, we establish both generic and set-theoretical completeness results for any extension of the base system; the latter arises from a dual adjunction, and is relevant to the tripos-to-topos construction and quantale-valued sets. Furthermore, we give a hyperdoctrinal account of Girard’s and Gödel’s translation." @default.
- W2108840809 created "2016-06-24" @default.
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- W2108840809 date "2013-01-01" @default.
- W2108840809 modified "2023-10-16" @default.
- W2108840809 title "Full Lambek Hyperdoctrine: Categorical Semantics for First-Order Substructural Logics" @default.
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- W2108840809 doi "https://doi.org/10.1007/978-3-642-39992-3_19" @default.
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