Involutive Negation Research Articles

Residuated fuzzy logics with an involutive negation

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant $\overline{0}$ , namely $\neg \varphi$ is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that $\neg$ is an involutive negation. However, for t-norms without non-trivial zero divisors, $\neg$ is Godel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation.

Idempotent uninorms

Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. Two broad classes of idempotent uninorms are fully characterized: the class of left-continuous ones and the class of right-continuous ones. In particular, the important subclasses of conjunctive left-continuous idempotent uninorms and of disjunctive right-continuous idempotent uninorms are characterized by means of super-involutive and sub-involutive decreasing unary operators. As a consequence, it is shown that any involutive negator gives rise to a conjunctive left-continuous idempotent uninorm and to a disjunctive right-continuous idempotent uninorm.

Internal Languages for Autonomous and ∗-Autonomous Categories

We introduce a family of type theories as internal languages for autonomous (or symmetric monoidal closed) and ∗-autonomous categories, in the same sense that the simply-typed lambda-calculus with surjective pairing is the internal language for cartesian closed categories. The rules for the typing judgements are presented in the style of Gentzen's Sequent Calculus. A notable feature is the systematic treatment of naturality conditions by expressing the categorical composition, or cut in the type theory, by explicit substitution. We use let-constructs, one for each of the three type constructors tensor unit, tensor and linear function space, and a Parigot-style mu-abstraction to express the involutive negation. We show that the eight equality and three commutation congruence axioms of the ∗-autonomous type theory characterise ∗-autonomous categories exactly. More precisely we prove that there is a canonical interpretation of the (∗-autonomous) type theories in ∗-autonomous categories which is complete i.e. for any type theory, there is a model (i.e. ∗-autonomous category) whose theory is exactly that. The associated rewrite systems are all strongly normalising; modulo a simple notion of congruence, they are also confluent. As a corollary, we solve a Coherence Problem a la Lambek: the equality of maps in any ∗-autonomous category freely generated from a discrete graph is decidable.

A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures. (II). The identity case

It has been shown in the first part of this paper that the concept of a fuzzy preference structure is only meaningful provided that the de Morgan triplet involved contains a continuous Archimedean triangular norm having zero divisors, or hence a φ-transform of the Łukasiewicz triangular norm. In this second part, additional arguments for this statement are supplied in what we call the ‘identity case’ ( φ τ is the identity mapping, and the involutive negator is the standard negator). First, it is shown that the use of a continuous non-Archimedean triangular norm having zero divisors in the definition of a fuzzy preference structure indeed is possible. Secondly, using such a triangular norm implies that at least in the square [0, 4 5 ] 2 it actually behaves as the Łukasiewicz triangular norm. Furthermore, an important transformation theorem indicates that any fuzzy preference structure with respect to a de Morgan triplet containing such a continuous non-Archimedean triangular norm having zero divisors can be transformed into a fuzzy preference structure with respect to the standard Łukasiewicz triplet. These additional arguments conclude in a convincing way our plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures.

A plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures. (I). General argumentation

The generalization of the concept of a classical (or crisp) preference structure to that of a fuzzy preference structure, expressing degrees of strict preference, indifference and incomparability among a set of alternatives, requires the choice of a de Morgan triplet, i.e., of a triangular norm and an involutive negator. The resulting concept is only meaningful provided that this choice allows the representation of truly fuzzy preferences. More specifically, one of the degrees of strict preference, indifference or incomparability should always be unconstrained to the preference modeller. This intuitive requirement is violated when choosing a triangular norm without zero divisors, since in that case fuzzy preference structures reduce to classical preference structures, and hence none of the degrees can be freely assigned. Furthermore, it is shown that the choice of a continuous non-Archimedean triangular norm having zero divisors is not compatible with our basic requirement: the sets of degrees of strict preference, indifference and incomparability in [0,1[ are always bounded from above by a value strictly smaller than 1. These fundamental results imply that when working with a continuous triangular norm, only Archimedean ones having zero divisors are suitable candidates. These arguments sufficiently support our plea for the use of Łukasiewicz triplets in the definition of fuzzy preference structures.

Hopf algebras and linear logic

It has recently become evident that categories of representations ofHopf algebrasprovide fundamental examples of monoidal categories. In this expository paper, we examine such categories as models of (multiplicative) linear logic. By varying the Hopf algebra, it is possible to model several variants of linear logic. We present models of the original commutative logic, the noncommutative logic of Lambek and Abrusci, the braided variant due to the author, and the cyclic logic of Yetter. Hopf algebras provide a unifying framework for the analysis of these variants. While these categories are monoidal closed, they lack sufficient structure to model the involutive negation of classical linear logic. We recall work of Lefschetz and Barr in which vector spaces are endowed with an additional topological structure, calledlinear topology. The resulting category has a large class of reflexive objects, which form a *-autonomous category, and so model the involutive negation. We show that the monoidal closed structure of the category of representations of a Hopf algebra can be extended to this topological category in a natural and simple manner. The models we obtain have the advantage of being nondegenerate in the sense that the two multiplicative connectives, tensor and par, are not equated. It has been recently shown by Barr that this category of topological vector spaces can be viewed as a subcategory of a certain Chu category. In an Appendix, Barr uses this equivalence to analyze the structure of its tensor product.