Orientador: Marcelo Esteban Coniglio / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias Humanas / Made available in DSpace on 2018-08-17T03:29:03Z (GMT). No. of bitstreams: 1 Rodrigues_TarcisioGenaro_M.pdf: 1141020 bytes, checksum: 59bb8a3ae7377c05cf6a8d8e6f7e45a5 (MD5) Previous issue date: 2010 / Resumo: A Programação Lógica nasce da interação entre a Lógica e os fundamentos da Ciência da Computação: teorias de primeira ordem podem ser interpretadas como programas de computador. A Programação Lógica tem sido extensamente utilizada em ramos da Inteligência Artificial tais como Representação do Conhecimento e Raciocínio de Senso Comum. Esta aproximação deu origem a uma extensa pesquisa com a intenção de definir sistemas de Programação Lógica paraconsistentes, isto é, sistemas nos quais seja possível manipular informação contraditória. Porém, todas as abordagens existentes carecem de uma fundamentação lógica claramente definida, como a encontrada na programação lógica clássica. A questão básica é saber quais são as lógicas paraconsistentes subjacentes a estas abordagens. A presente dissertação tem como objetivo estabelecer uma fundamentação lógica e conceitual clara e sólida para o desenvolvimento de sistemas bem fundados de Programação Lógica Paraconsistente. Nesse sentido, este trabalho pode ser considerado como a primeira (e bem sucedida) etapa de um ambicioso programa de pesquisa. Uma das teses principais da presente dissertação é que as Lógicas da Inconsistência Formal (LFI's), que abrangem uma enorme família de lógicas paraconsistentes, proporcionam tal base lógica. Como primeiro passo rumo à definição de uma programação lógica genuinamente paraconsistente, demonstramos nesta dissertação uma versão simplificada do Teorema de Herbrand para uma LFI de primeira ordem. Tal teorema garante a existência, em princípio, de métodos de dedução automática para as lógicas (quantificadas) em que o teorema vale. Um pré-requisito fundamental para a definição da programação lógica é justamente a existência de métodos de dedução automática. Adicionalmente, para a demonstração do Teorema de Herbrand, são formuladas aqui duas LFI's quantificadas através de sequentes, e para uma delas demonstramos o teorema da eliminação do corte. Apresentamos também, como requisito indispensável para os resultados acima mencionados, uma nova prova de correção e completude para LFI's quantificadas na qual mostramos a necessidade de exigir o Lema da Substituição para a sua semântica / Abstract: Logic Programming arises from the interaction between Logic and the Foundations of Computer Science: first-order theories can be seen as computer programs. Logic Programming have been broadly used in some branches of Artificial Intelligence such as Knowledge Representation and Commonsense Reasoning. From this, a wide research activity has been developed in order to define paraconsistent Logic Programming systems, that is, systems in which it is possible to deal with contradictory information. However, no such existing approaches has a clear logical basis. The basic question is to know what are the paraconsistent logics underlying such approaches. The present dissertation aims to establish a clear and solid conceptual and logical basis for developing well-founded systems of Paraconsistent Logic Programming. In that sense, this text can be considered as the first (and successful) stage of an ambitious research programme. One of the main thesis of the present dissertation is that the Logics of Formal Inconsistency (LFI's), which encompasses a broad family of paraconsistent logics, provide such a logical basis. As a first step towards the definition of genuine paraconsistent logic programming we shown, in this dissertation, a simplified version of the Herbrand Theorem for a first-order LFI. Such theorem guarantees the existence, in principle, of automated deduction methods for the (quantified) logics in which the theorem holds, a fundamental prerequisite for the definition of logic programming over such logics. Additionally, in order to prove the Herbrand Theorem we introduce sequent calculi for two quantified LFI's, and cut-elimination is proved for one of the systems. We also present, as an indispensable requisite for the above mentioned results, a new proof of soundness and completeness for first-order LFI's in which we show the necessity of requiring the Substitution Lemma for the respective semantics / Mestrado / Filosofia / Mestre em Filosofia

Text in Afrikaans, abstract in Afrikaans and English / Hierdie studie is daarop gemik om te bepaal tot welke mate wiskundeleerlinge op skool en onderwysstudente in wiskunde, onderrig in logika ontvang as agtergrond vir strenge bewysvoering. Die formele aspek van wiskunde op hoerskool en tersiere vlak is besonder belangrik. Leerlinge en studente kom onvermydelik met hipotetiese argumente in aanraking. Hulle leer ook om die kontrapositief te gebruik in bewysvoering. Hulle maak onder andere gebruik van bewyse uit die ongerymde. Verder word nodige en voldoende voorwaardes met stellings en hulle omgekeerdes in verband gebring. Dit is dus duidelik dat 'n studie van logika reeds op hoerskool nodig is om aanvaarbare wiskunde te beoefen. Om seker te maak dat aanvaarbare wiskunde beoefen word, is dit nodig om te let op die gebrek aan beheer in die ontwikkeling van 'n taal, waar woorde meer as een betekenis het. 'n Kunsmatige taal moet gebruik word om interpretasies van uitdrukkings eenduidig te maak. In so 'n kunsmatige taal word die moontlikheid van foutiewe redenering uitgeskakel. Die eersteordepredikaatlogika, is so 'n taal, wat ryk genoeg is om die wiskunde te akkommodeer. Binne die konteks van hierdie kunsmatige taal, kan wiskundige toeriee geformaliseer word. Verskillende bewystegnieke uit die eersteordepredikaatlogika word geidentifiseer, gekategoriseer en op 'n redelik eenvoudige wyse verduidelik. Uit 'n ontleding van die wiskundesillabusse van die Departement van Onderwys, en 'n onderwysersopleidingsinstansie, volg dit dat leerlinge en studente hierdie bewystegnieke moet gebruik. Volgens hierdie sillabusse moet die leerlinge en studente vertroud wees met logiese argumente. Uit die gevolgtrekkings waartoe gekom word, blyk dit dat die leerlinge en studente se agtergrond in logika geheel en al gebrekkig en ontoereikend is. Dit het tot gevolg dat hulle nie 'n volledige begrip oor bewysvoering het nie, en 'n gebrekkige insig ontwikkel oor wat wiskunde presies behels. Die aanbevelings om hierdie ernstige leemtes in die onderrig van wiskunde aan te spreek, asook verdere navorsingsprojekte word in die laaste hoofstuk verwoord. / The aim of this study is to determine to which extent pupils taking Mathematics at school level and student teachers of Mathematics receive instruction in logic as a grounding for rigorous proof. The formal aspect of Mathematics at secondary school and tertiary levels is extremely important. It is inevitable that pupils and students become involved with hypothetical arguments. They also learn to use the contrapositive in proof. They use, among others, proofs by contradiction. Futhermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practice Mathematics satisfactorily. To ensure that acceptable Mathematics is practised, it is necessary to take cognizance of the lack of control over language development, where words can have more than one meaning. For this reason an artificial language must be used so that interpretations can have one meaning. Faulty interpretations are ruled out in such an artificial language. A language which is rich enough to accommodate Mathematics is the first-order predicate logic. Mathematical theories can be formalised within the context of this artificial language. Different techniques of proof from the first-order logic are identified, categorized and explained in fairly simple terms. An analysis of Mathematics syllabuses of the Department of Education and an institution for teacher training has indicated that pupils should use these techniques of proof. According to these syllabuses pupils should be familiar with logical arguments. The conclusion which is reached, gives evidence that pupils' and students' background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what Mathematics exactly entails. Recommendations to bridge these serious problems in the instruction of Mathematics, as well as further research projects are discussed in the final chapter. / Curriculum and Institutional Studies / D. Phil. (Wiskundeonderwys)

Text in Afrikaans, abstract in Afrikaans and English / Hierdie studie is daarop gemik om te bepaal tot welke mate wiskundeleerlinge op skool en onderwysstudente in wiskunde, onderrig in logika ontvang as agtergrond vir strenge bewysvoering. Die formele aspek van wiskunde op hoerskool en tersiere vlak is besonder belangrik. Leerlinge en studente kom onvermydelik met hipotetiese argumente in aanraking. Hulle leer ook om die kontrapositief te gebruik in bewysvoering. Hulle maak onder andere gebruik van bewyse uit die ongerymde. Verder word nodige en voldoende voorwaardes met stellings en hulle omgekeerdes in verband gebring. Dit is dus duidelik dat 'n studie van logika reeds op hoerskool nodig is om aanvaarbare wiskunde te beoefen. Om seker te maak dat aanvaarbare wiskunde beoefen word, is dit nodig om te let op die gebrek aan beheer in die ontwikkeling van 'n taal, waar woorde meer as een betekenis het. 'n Kunsmatige taal moet gebruik word om interpretasies van uitdrukkings eenduidig te maak. In so 'n kunsmatige taal word die moontlikheid van foutiewe redenering uitgeskakel. Die eersteordepredikaatlogika, is so 'n taal, wat ryk genoeg is om die wiskunde te akkommodeer. Binne die konteks van hierdie kunsmatige taal, kan wiskundige toeriee geformaliseer word. Verskillende bewystegnieke uit die eersteordepredikaatlogika word geidentifiseer, gekategoriseer en op 'n redelik eenvoudige wyse verduidelik. Uit 'n ontleding van die wiskundesillabusse van die Departement van Onderwys, en 'n onderwysersopleidingsinstansie, volg dit dat leerlinge en studente hierdie bewystegnieke moet gebruik. Volgens hierdie sillabusse moet die leerlinge en studente vertroud wees met logiese argumente. Uit die gevolgtrekkings waartoe gekom word, blyk dit dat die leerlinge en studente se agtergrond in logika geheel en al gebrekkig en ontoereikend is. Dit het tot gevolg dat hulle nie 'n volledige begrip oor bewysvoering het nie, en 'n gebrekkige insig ontwikkel oor wat wiskunde presies behels. Die aanbevelings om hierdie ernstige leemtes in die onderrig van wiskunde aan te spreek, asook verdere navorsingsprojekte word in die laaste hoofstuk verwoord. / The aim of this study is to determine to which extent pupils taking Mathematics at school level and student teachers of Mathematics receive instruction in logic as a grounding for rigorous proof. The formal aspect of Mathematics at secondary school and tertiary levels is extremely important. It is inevitable that pupils and students become involved with hypothetical arguments. They also learn to use the contrapositive in proof. They use, among others, proofs by contradiction. Futhermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practice Mathematics satisfactorily. To ensure that acceptable Mathematics is practised, it is necessary to take cognizance of the lack of control over language development, where words can have more than one meaning. For this reason an artificial language must be used so that interpretations can have one meaning. Faulty interpretations are ruled out in such an artificial language. A language which is rich enough to accommodate Mathematics is the first-order predicate logic. Mathematical theories can be formalised within the context of this artificial language. Different techniques of proof from the first-order logic are identified, categorized and explained in fairly simple terms. An analysis of Mathematics syllabuses of the Department of Education and an institution for teacher training has indicated that pupils should use these techniques of proof. According to these syllabuses pupils should be familiar with logical arguments. The conclusion which is reached, gives evidence that pupils' and students' background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what Mathematics exactly entails. Recommendations to bridge these serious problems in the instruction of Mathematics, as well as further research projects are discussed in the final chapter. / Curriculum and Institutional Studies / D. Phil. (Wiskundeonderwys)