Towards a semantics of proofs for non-commutative logic: multiplicatives and additives

Non-commutative Logic for Compositional Distributional Semantics

Distributional models of natural language use vectors to provide a contextual foundation for meaning representation. These models rely on large quantities of real data, such as corpora of documents, and have found applications in natural language tasks, such as word similarity, disambiguation, indexing, and search. Compositional distributional models extend the distributional ones from words to...

Non-commutative logic III: focusing proofs

It is now well-established that the so-called focalization property plays a central role in the design of programming languages based on proof search, and more generally in the proof theory of linear logic. We present here a sequent calculus for non-commutative logic (NL) which enjoys the focalization property. In the multiplicative case, we give a focalized sequentialization theorem, and in th...

Categorical semantics of non-commutative logic

Previous work [9, 1, 10] was devoted to proof nets and sequent calculus for NL. Here we study the categorical semantics of NL. One can say that proof nets constitute already a semantics of proofs of NL, but on one hand the extension to non-multiplicative connectives is delicate (boxes) and on the other hand one would like to have a semantics that is more general than syntax (correctness) and cl...

Phase Semantics for Mixed Non-Commutative Classical Linear Logic

Kripke-Style Semantics for Non-Commutative Monoidal t-Norm Logic

A t-norm is a binary operation on [0, 1] that is associative, commutative, with identity 1 and non-decreasing in both argument. The notion of t-norm is very important in the context of fuzzy logic, for it is used in modeling a general form of the propositional conjunction. The basic fuzzy logic (BL logic) was introduced by Hájek in [9,10] and it was inspired by a continuous t-norm and its resid...