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.
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Supported by Hungarian National Foundation for Scientific Research grant Nos. 1911, 2258, and by TEMPUS JEP No. 1941-92/2.
The first report on the first part of this work was in the Banach Centre, Warsaw, 17th October 1991.
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Andréka, H., Mikulás, S. Lambek Calculus and its relational semantics: Completeness and incompleteness. J Logic Lang Inf 3, 1–37 (1994). https://doi.org/10.1007/BF01066355
- Lambek Calculus
- relational semantics
- language models
- algebraic logic
- residuated semigroups
- representation theorems