Grishin Algebras and Cover Systems for Classical Bilinear Logic

Abstract

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.

Cover Systems for the Modalities of Linear Logic

“…In particular, this implies that Up(S) is closed under ⇒ l and ⇒ r , and so by (4.3), these operations are left and right residuals to • on Up(S). A residuated cover system was defined in (Goldblatt, 2011b) to be a structure of the form…”

“…Detailed discussion of this kind of cover system semantics, and associated completeness theorems axiomatising their valid sentences, are presented in (Goldblatt, 2006) for the logic of nonmodal FL-algebras; in (Goldblatt, 2011a) for intuitionistic modal first-order logics; in (Goldblatt, 2011c, Chapter 6) for propositional and quantified relevant logics; and in (Goldblatt, 2011b) for a 'classical' version of bilinear logic that we also discuss below in Section 9.…”

“…This is equivalent to requiring that Prop(S) be a Grishin algebra/classical FL-algebra, and is also equivalent to the requirement that (9.1) holds just for all up-sets X (Goldblatt, 2011b, Theorem 4.2). We showed in (Goldblatt, 2011b) that every Grishin algebra has an isomorphic embedding into the algebra of all propositions of some strong classical residuated cover system, by a map that preserves all existing joins and meets. The method, involving MacNeille completion, can be combined with the constructions of this paper to give a representation of any classical modal FL-algebra as an algebra of propositions of some strong classical modal FL-cover system.…”

The Distributive Full Lambek Calculus with Modal Operators

“…Algebraically, exponential modalities were studied by Ono as additional exponential operators on FL algebras [36]. The Lambek calculus and their modal extensions also have the cover semantics proposed by Goldblatt [28] [29]. The abstract polymodal case of such an extension of the full Lambek calculus was recently studied by Kanovich, Kuznetsov, Scedrov, and Nigam [32].…”