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

Abstract

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.