Lambek calculus and its relational semantics: Completeness and incompleteness [Book Review]

The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while that version of the Lambek Calculus where we admit the empty sequence as the antecedent of a sequent is strongly complete w.r.t. those relational models whereW=U×U for some setU. We will also look into extendability of this completeness result to various fragments of Girard's Linear Logic as suggested in van Benthem (1991), p. 235, and investigate the connection between the Lambek Calculus and language models
Keywords Lambek Calculus  relational semantics  language models  algebraic logic  residuated semigroups  representation theorems
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DOI 10.1007/BF01066355
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References found in this work BETA
The Mathematics of Sentence Structure.Joachim Lambek - 1968 - Journal of Symbolic Logic 33 (4):627-628.
Language in Action.Johan Van Benthem - 1991 - Journal of Philosophical Logic 20 (3):225-263.
Cylindric Algebras. Part II.Leon Henkin, J. Donald Monk & Alfred Tarski - 1988 - Journal of Symbolic Logic 53 (2):651-653.
A Brief Survey of Frames for the Lambek Calculus.Kosta Došen - 1992 - Mathematical Logic Quarterly 38 (1):179-187.

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Citations of this work BETA
Models for the Lambek Calculus.Mati Pentus - 1995 - Annals of Pure and Applied Logic 75 (1-2):179-213.

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