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Anders Kock et Gonzalo Reyes

Cahiers de topologie et géométrie différentielle catégoriques, tome 40, no 2 (1999), p. 127-140.

Dans le contexte de la théorie constructive des locales ou des cadres (c’est-à-dire de la théorie de locales sur un locale de base), nous étudions quelques aspects des “distributions sur les cadres” , i.e., des applications sur un cadre à valeurs dans un cadre de base préservant les suprema arbitraires. Nous obtenons une relation entre certains résultats dus à Jibladze- Johnstone et d’autres dus à Bunge-Funk. De plus, nous donnons des descriptions de l’opérateur “intérieur de fermeture” défini sur les parties ouvertes d’un locale en termes des distributions sur les cadres ainsi qu’en termes des opérations de double négation généralisée.

A note on frame distributions

La Palme Reyes M., Macnamara J., Reyes G. E. and H. Zolfaghari (1999). Models for Non-Boolean Negations in Natural Languages based on Aspect Analysis. In Gabbay D. and H. Wansing (eds.). What is negation? Kluwer Academic Publishers. Dordrecht, Boston, London. pp 241-260.

Since antiquity two different negations in natural languages have been noted: predicate negation (`not honest’) and predicate term negation (`dishonest’). Aristotle tried to formalize them in his system of oppositions, distinguishing between affirmation and negation (`honest’ and `not honest’) and contraries (`honest’ and `dishonest’). The Stoics replaced Aristotle’s logic of terms by their logic of propositions. Although they considered three types of negation, none of them corresponded to Aristotle’s predicate term negation. Frege and modern logic have followed the Stoics in either identifying predicate term negation with predicate negation or in casting predicate term negation out of logic into the realm of pragmatics. Although an extensive literature has arisen on these issues, we have not found mathematical models. We propose category-theoretic models with two distinct negation operations, neither of them in general Boolean. We study combinations of the two (`not dishonest’) and sentential counterparts of each. We touch briefly on quantifiers and modalities. The models are based on an analysis of aspects. For instance, to give an overall, global judgement of John’s honesty we must agree on what aspects of John are relevant for that judgement: John qua person (global aspect), John qua social being (social aspect), John qua family man, John qua professional man, etc. We conceptualize this `Aristotelian’ analysis by means of a category of `aspects’. A model (for the negations) is obtained from the category of presheaves on this category. Although neither of the negations is Boolean, predicate negation turns out to be Boolean at the `global’ aspect (the aspect of the overall judgement) which may help to explain the persistent belief that logic is naturally Boolean.

Models for Non-Boolean Negations in Natural Languages based on Aspect Analysis

La Palme Reyes M., Macnamara J., Reyes G. E. and H. Zolfaghari (1999). Count nouns, mass nouns and their transformations: a category-theoretic unified semantics. Language, logic and concepts. Bradford Book, MIT Press, Cambridge, Ma, 1999, pp 427-452.

All natural languages seem to distinguish at the semantic level between count nouns (CNs) such as “dog” and mass nouns (MNs) such as “matter” (in the sense of physical stuff, not in the sense of concern or affair). Some mark the distinction at the syntactic level (e.g. one can say ” a dog”, “a portion of matter”, but not “a matter”). One syntactic difference is that usually CNs take the plural (‘dogs’) whereas MNs do not. We organize the nouns in two categories with adjoint functors between them: the plural, from CNs into MNs, and “portion of” in the opposite direction. We interpret these nominal categories into the category of kinds and the category of sup-lattices, respectively, and build adjoint functors between them which interpret the adjoint functors at the nominal level. This semantics is applied, among others, to study the 8 syllogisms, already considered in the literature, which result from “Claret is wine, wine is liquid, so claret is liquid”, by adding the particle “a” to each noun or keeping it as it is.

Count nouns, mass nouns and their transformations: a category-theoretic unified semantics

Kock A. and G. E. Reyes (1999). Aspects of fractional exponent functor. Theory Appl. Categ. 5, No. 10, 251-265. (Electronic)

We prove that certain categories arising from atoms in a Grothendieck topos are themselves Grothendieck toposes. We also investigate enrichments of these categories over the base topos; there are in fact often two distinct enrichments.