DeepAI AI Chat
Log In Sign Up

Dialectica Categories for the Lambek Calculus

by   Valeria de Paiva, et al.

We revisit the old work of de Paiva on the models of the Lambek Calculus in dialectica models making sure that the syntactic details that were sketchy on the first version got completed and verified. We extend the Lambek Calculus with a κ modality, inspired by Yetter's work, which makes the calculus commutative. Then we add the of-course modality !, as Girard did, to re-introduce weakening and contraction for all formulas and get back the full power of intuitionistic and classical logic. We also present the categorical semantics, proved sound and complete. Finally we show the traditional properties of type systems, like subject reduction, the Church-Rosser theorem and normalization for the calculi of extended modalities, which we did not have before.


page 1

page 2

page 3

page 4


Quantale semantics of Lambek calculus with subexponential modalities

In this paper, we consider the polymodal version of Lambek calculus with...

Semantic Analysis of Subexponential Modalities in Distributive Non-commutative Linear Logic

In this paper, we consider the full Lambek calculus enriched with subexp...

Categorical Vector Space Semantics for Lambek Calculus with a Relevant Modality (Extended Abstract)

We develop a categorical compositional distributional semantics for Lamb...

Logic of computational semi-effects and categorical gluing for equivariant functors

In this paper, we revisit Moggi's celebrated calculus of computational e...

A Formalization of Unique Solutions of Equations in Process Algebra

In this thesis, a comprehensive formalization of Milner's Calculus of Co...

Vector Space Semantics for Lambek Calculus with Soft Subexponentials

We develop a vector space semantics for Lambek Calculus with Soft Subexp...

The em-convex rewrite system

We introduce and study em (or "emergent"), a lambda calculus style rewri...