Generalized Kripke semantics for the Lambek-Grishin calculus

Abstract

properties. It turns out that this extension is precisely the algebra of Galois closed sets of the canonical frame as defined in section 2 of [Geh06]. Thus we get a simple abstract manner of working with the canonical frame. This makes it easy to treat additional operations and their interaction axioms. In particular, from A = (A,⊗, /, \,⊕,;, ) we get Aδ = (Aδ,⊗σ, / π, π,⊕π,; σ, σ) and Aδ is the algebra of Galois closed sets for some frame which we denote by (X,Y,6, R⊗σ , R⊕π) =: F(A). The central role of the canonical extension in the process of finding relational semantics is illustrated by the following diagramme. Lambek-Grishin logic LG-algebras A = (A,⊗, /, \,⊕,;, ) Canonical extensions Aδ ∼= (G(X,Y,6),⊗σ, / π, π,⊕π,; σ, σ) Lambek-Grishin frames F(A) = (X,Y,6, R⊗σ , R⊕π) Relational semantics // oo This process works for the Lindenbaum algebra A of the LG∅-logic in the sense that the canonical frame F(A) with the interpretation given by the embedding map is a model of precisely those sequents that are deducible in LG∅, as the following equivalences demonstrate. A ` B holds in LG∅ ⇐⇒ A 6 B holds in A ∗ ⇐⇒ [A]Aδ 6 [B]Aδ holds in A ⇐⇒ F(A) A ` B, where the subscript Aδ means that the formula is interpreted in Aδ. We call A = (A,⊗, /, \,⊕,;, ) an LG-algebra provided A is a poset, and the operations of A satisfy the rules of LG∅, i.e. / and are upper residuals of ⊗, while ; and are lower residuals of ⊕. The process of getting a canonical extension and from a canonical extension a Lambek-Grishin frame works for any LG-algebra. When dealing with a class of algebras that satisfy additional interaction axioms (e.g. one of Grishin’s groups, see next section), our aim is to find out which first-order condition is imposed by these axioms on the class of Lambek-Grishin frames. In order to prove the completeness of an axiomatic extension of LG∅, two components are needed: 1. We have to show that if we start with A, the Lindenbaum algebra for some extension of LG∅, then for each additional axiom, the equivalence indicated by ∗ still works. This