Multi-dimensional Type Theory: Rules, Categories, and Combinators for Syntax and Semantics

We investigate the possibility of modelling the syntax and semantics of natural language by constraints, or rules, imposed by the multi-dimensional type theory Nabla. The only multiplicity we explicitly consider is two, namely one dimension for the syntax and one dimension for the semantics, but the general perspective is important. For example, issues of pragmatics could be handled as additional dimensions. One of the main problems addressed is the rather complicated repertoire of operations that exists besides the notion of categories in traditional Montague grammar. For the syntax we use a categorial grammar along the lines of Lambek. For the semantics we use so-called lexical and logical combinators inspired by work in natural logic. Nabla provides a concise interpretation and a sequent calculus as the basis for implementations.Comment: 20 page

Paraconsistency in hybrid logic

As in standard knowledge bases, hybrid knowledge bases (i.e. sets of information specified by hybrid formulas) may contain inconsistencies arising from different sources, namely from the many mechanisms used to collect relevant information. Being a fact, rather than a queer anomaly, inconsistency also needs to be addressed in the context of hybrid logic applications. This article introduces a paraconsistent version of hybrid logic which is able to accommodate inconsistencies at local points without implying global failure. A main feature of the resulting logic, crucial to our approach, is the fact that every hybrid formula has an equivalent formula in negation normal form. The article also provides a measure to quantify the inconsistency of a hybrid knowledge base, useful as a possible basis for comparing knowledge bases. Finally, the concepts of extrinsic and intrinsic inconsistency of a theory are discussed

Paraconsistência em lógica híbrida

Mestrado em Matemática e AplicaçõesThe use of hybrid logics allows the description of relational structures, at the same time that allows establishing accessibility relations between states and, furthermore, nominating and making mention to what happens at speci c states. However, the information we collect is subject to inconsistencies, namely, the search for di erent information sources can lead us to pick up contradictions. Nowadays, by having so many means of dissemination available, that happens frequently. The aim of this work is to develop tools capable of dealing with contradictory information that can be described as hybrid logics' formulas. To build models, to compare inconsistency in di erent databases, and to see the applicability of this method in day-to-day life are the basis for the development of this dissertation.O uso de lógicas híbridas permite a descrição de estruturas relacionais, ao mesmo tempo que permite estabelecer relações de acessibilidade entre estados, e, para além disso, nomear e fazer referência ao que acontece em estados específicos. No entanto, a informação que recolhemos está sujeita a inconsistências, isto é, a procura de diferentes fontes de informação pode levar a recolha de contradições. O que nos dias de hoje, com tantos meios de divulgação disponíveis, acontece frequentemente. O objetivo deste trabalho e desenvolver ferramentas capazes de lidar com informação contraditória que possa ser descrita através de fórmulas de lógicas híbridas. Construir modelos e comparar a inconsistência de diferentes bases de dados e ver a aplicabilidade deste método no dia-a-dia são a base para o desenvolvimento desta dissertação

Paraconsistent Query Answering Systems

Abstract. Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in query answering systems. We compare the paraconsistent and the non-monotonic solutions to the problem of con-tradictions. We propose a many-valued paraconsistent logic based on a simple notion of indeterminacy. In particular we describe the semantics of the logic using key equalities for the logical operators. We relate our approach to works on bilattices. We also discuss and provide formal-izations of two case studies, notably the well-known example involving penguins and a more interesting example in the domain of medicine. Paraconsistent logic can be seen as an alternative, for example, to non-monotonic logic. Non-monotonists reject monotony because they think that there are experiences (most of the time involving birds) which show that monotony is wrong and in particular leads to some contradictions. But one who thinks the paraconsistent way would reject the principle of non contradiction and not monotony. The strategy of the paracon-sistentist is more imaginative, he accepts to see penguins flying in the sky of Hawai’s beaches and pink floyds surfing on Antarctica’s permafrost. It seems to us that the future shall give the preference to paraconsistent logic taking in account the progress of genitical biology which already produces chicken without feathers, and in the future we may have flying pigs. In such an absurd world, it will make no sense to reason by default, because everything could be true by default

FQAS 2002 c ○ Springer-Verlag LNCS Paraconsistent Query Answering Systems

Abstract. Classical logic predicts that everything (thus nothing useful at all) follows from inconsistency. A paraconsistent logic is a logic where inconsistency does not lead to such an explosion, and since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in query answering systems. We compare the paraconsistent and the non-monotonic solutions to the problem of contradictions. We propose a many-valued paraconsistent logic based on a simple notion of indeterminacy. In particular we describe the semantics of the logic using key equalities for the logical operators. We relate our approach to works on bilattices. We also discuss and provide formalizations of two case studies, notably the well-known example involving penguins and a more interesting example in the domain of medicine. Paraconsistent logic can be seen as an alternative, for example, to nonmonotonic logic. Non-monotonists reject monotony because they think that there are experiences (most of the time involving birds) which show that monotony is wrong and in particular leads to some contradictions. But one who thinks the paraconsistent way would reject the principle of non contradiction and not monotony. The strategy of the paraconsistentist is more imaginative, he accepts to see penguins flying in the sky of Hawai’s beaches and pink floyds surfing on Antarctica’s permafrost. It seems to us that the future shall give the preference to paraconsistent logic taking in account the progress of genitical biology which already produces chicken without feathers, and in the future we may have flying pigs. In such an absurd world, it will make no sense to reason by default, because everything could be true by default