Grishin Algebras and Cover Systems for Classical Bilinear Logic

Studia Logica 99 (1-3):203-227 (2011)
Grishin algebras are a generalisation of Boolean algebras that provide algebraic models for classical bilinear logic with two mutually cancelling negation connectives. We show how to build complete Grishin algebras as algebras of certain subsets (“propositions”) of cover systems that use an orthogonality relation to interpret the negations. The variety of Grishin algebras is shown to be closed under MacNeille completion, and this is applied to embed an arbitrary Grishin algebra into the algebra of all propositions of some cover system, by a map that preserves all existing joins and meets. This representation is then used to give a cover system semantics for a version of classical bilinear logic that has first-order quantifiers and infinitary conjunctions and disjunctions
Keywords Grishin algebra  bilinear logic  residuated lattice-ordered monoid  quantale  cover system  orthogonality relation  Kripke-Joyal semantics  MacNeille completion
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DOI 10.1007/s11225-011-9360-0
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R. I. Goldblatt (1974). Semantic Analysis of Orthologic. Journal of Philosophical Logic 3 (1/2):19 - 35.

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