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Category-theoretic approaches to semantic technologies
Published in James Juniper, The Economic Philosophy of the Internet of Things, 2018
Using linear combinatory algebra, Abramsky, Haghverdi and Scott (2002) apply traced monoidal categories to the representation of the logical connectives that feature in linear (i.e. resource-using) logic. Linear logic has found application to proof nets, functional computing, including various lambda calculi, and the modelling of resource-using processes (e.g. as seen in business process modelling). This is both a rich and promising focus of on-going research.
Fuzzy intensional semantics
Published in Journal of Applied Non-Classical Logics, 2018
As mentioned, all -core fuzzy logics are expansions (by the axiom of prelinearity and possibly further connectives and axioms) of intuitionistic logic without contraction, also known as the full Lambek calculus with exchange and weakening or intuitionistic affine linear logic IALL (without exponentials). This makes them part of the family of substructural logics, or logics of residuated lattices. Consequently, besides their standard -valued semantics, -core fuzzy logics also have a general algebraic semantics given by classes of (expanded) commutative bounded integral residuated lattices; the prelinearity axiom moreover ensures the completeness of -core fuzzy logics with respect to linear algebras of truth values. The results on weighted structures formalised in -core fuzzy logics thus automatically carry over to more general algebras of abstract, possibly non-numerical, degrees.
Monoidal logics: completeness and classical systems
Published in Journal of Applied Non-Classical Logics, 2019
It might be argued that there is a lack of discussion regarding the co-tensor within the substructural logic literature. This might be due to the fact that many substructural logics lack a multiplicative disjunction. Actually, multiplicative disjunctions generally arise within classical substructural logics, such as in multiplicative linear logic (Girard, 1987). Hence, when speaking of an intuitionistic system of substructural logic, one is usually only referring to the tensor fragment. Some notable exceptions are full intuitionistic linear logic (Hyland & De Paiva, 1993), bilinear logics (Lambek, 1993) and the general proof theory of bunched implications logics, which are defined on the grounds of display logics (cf. Brotherston, 2010).