Mathematical Logic Quarterly 43 (1):49-59 (1997)

Edward Haeusler
Pontifícia Universidade Católica do Rio de Janeiro
Marcelo Corrêa
Universidade Federal de Minas Gerais
We present a categorical/denotational semantics for the Lambek Syntactic Calculus , indeed for a λlD-typed version Curry-Howard isomorphic to it. The main novelty of our approach is an abstract noncommutative construction with right and left adjoints, called sequential product. It is defined through a hierarchical structure of categories reflecting the implicit permission to sequence expressions and the inductive construction of compound expressions. We claim that Lambek's noncommutative product corresponds to a noncommutative bi-endofunctor into a category, which encloses all categories of such hierarchical structure. A soundness theorem for LSC is shown with respect to this semantical framework
Keywords Categorical model  Lambek syntactic calculus  Typed λlD‐calculus
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DOI 10.1002/malq.19970430107
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References found in this work BETA

Natural Deduction: A Proof-Theoretical Study.Dag Prawitz - 1965 - Stockholm, Sweden: Dover Publications.
Natural Deduction: A Proof-Theoretical Study.Richmond Thomason - 1965 - Journal of Symbolic Logic 32 (2):255-256.
Proofs and Types.Jean-Yves Girard - 1989 - Cambridge University Press.

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