JOÃO MARCOS LOGICS OF FORMAL INCONSISTENCY PhD Thesis FEB/2005 FICHA CATALOGRÁFICA ELABORADA PELA BIBLIOTECA DO IFCH – UNICAMP Almeida, João Marcos de AL64L Logics of Formal Inconsistency / João Marcos de Almeida. - - Campinas, SP : [s. n.], 2004. Orientador: Walter Alexandre Carnielli. Co-orientador: Carlos Caleiro Tese (doutorado) – Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas. 1. Lógica – Filosofia. 2. Lógica Matemática Não-Clássica. 3. Semânticas (Filosofia). 4. Modalidade (Lógica). 5. Inferência (Lógica) I. Carnielli, Walter Alexandre. II. Caleiro, Carlos. III. Universidade Estadual de Campinas, Instituto de Filosofia e Ciências Humanas. IV. Título. Ad parentes meos, sine qui non But we never admitted the birth of logic among us. Oswald de Andrade, ‘Anthropophagite Manifesto’, May 1928. Contents Resumo xiii Abstract xv Introdução xvii PROLEGOMENA TO ANY FUTURE PARACONSISTENCY xxiii THE BIRTH OF PARACONSISTENT LOGIC xxiii SEMANTIC INTUITIONS xxvii THE FUNDAMENTAL FEATURE OF LFIS xxx Agnosticism in logic xxx Recipe for a certain programme in paraconsistency xxxiii The fetish formula, and the Principle of (Non-)Contradiction xxxiv The replacement property xxxvi Good behavior and formal consistency xxxviii OTHER ASPECTS OF PARACONSISTENCY xli On set theory xlii On an infinity of logics xliii On maximality xliv On predicate logic xliv On databases and mechanized deduction for LFIs xlv Advertising LFIs xlvi On the duality between inconsistency and undeterminedness xlvi Paraconsistent mistakes xlvii The future xlviii SOME CONTRIBUTIONS OF THE PRESENT THESIS xlix TRAVELLING SALESMAN lii SHORT ADDENDUM ON UNNECESSARY EXPLANATIONS lviii LATE, AND HOPEFULLY ALSO UNNECESSARY, WARNING lviii Acknowledgements lix Bibliography lx Appendix: Brief historical note lxvii vii 1 MAP OF THE TERRITORY 1 RESUMO 1 CONTENTS 2 Byzantinisms 2 The meat 5 Parts that were promised and are missing, things that will change 8 BRIEF HISTORY 9 On coauthorship 11 Bibliography 12 1.0 A taxonomy of C-systems 15 1 THOU SHALT NOT TRIVIALIZE! 16 1.1 Contradictory theories do exist 17 1.2 Paraconsistent, but not contradictory! 21 1.3 What do you mean? 23 1.4 Standing on the shoulders of each other 28 2 A PARACONSISTENT LOGIC IS A PARACONSISTENT LOGIC IS… 31 2.1 A question of principles 33 2.2 The paraconsistency predicament 34 2.3 The trivializing predicament 36 2.4 Huge tracts of the logical space 39 2.5 DEFCON 2: one step short of trivialization 43 2.6 C-systems 45 3 COOKING THE C-SYSTEMS ON A LOW FLAME 46 3.1 Paleontology of C-systems 47 3.2 The basic logic of (in)consistency 50 3.3 On what one cannot get 53 3.4 Letting bC talk about (dual) inconsistency 56 3.5 The logic Ci, where contradiction and inconsistency meet 58 3.6 On a simpler presentation for Ci 62 3.7 Using LFIs to talk about classical logic 63 viii 3.8 Beyond Ci: The dC-systems 69 3.9 The opposite of the opposite 75 3.10 Consistency may be contagious! 77 3.11 Taking it literally: the Brazilian plan completed 84 3.12 Algebraic stuff 89 4 FUTUROLOGY OF THE C-SYSTEMS 96 5 REFERENCES 102 Errata to the paper ‘A Taxonomy of C-systems’, and more 109 2 POSSIBLE-TRANSLATIONS SEMANTICS FOR LOGICS OF FORMAL INCONSISTENCY 119 RESUMO DE PTSURVEY 119 RESUMO DE WEAKPTS 119 CONTENTS 120 One size fits all 120 How much is that in ‘real money’? 122 Não tem tradução 126 BRIEF HISTORY 128 Bibliography 130 2.1 Possible-translations semantics (extended abstract) 133 1 LOGICS, TRANSLATIONS, POSSIBLE-TRANSLATIONS 133 1.1 What is a logic? 135 1.2 What is the canonical notion of entailment? 137 1.3 What can be done with translations between logics? 138 1.4 What are possible-translations semantics? 140 1.5 Which logics have adequate semantics? 142 2 FURTHER ILLUSTRATIONS 146 3 SOME OTHER RELATED SEMANTIC STRUCTURES 146 REFERENCES 147 ix 2.2 Possible-translations semantics for some weak classically-based paraconsistent logics 151 1 LANGUAGES, BIVALUATIONS, AND SEQUENTS 151 2 SOME FUNDAMENTAL PARACONSISTENT LOGICS 153 3 BIVALUED ENTAILMENT, MODALITIES AND MATRICES 155 4 INTERPRETATIONS THROUGH POSSIBLE TRANSLATIONS 160 4 ADEQUACY OF EACH OF THE NEWLY PROPOSED PTS 162 REFERENCES 166 3 MODAL SEMANTICS FOR LOGICS OF FORMAL INCONSISTENCY 169 RESUMO DE LEA 169 RESUMO DE MODPAR 169 RESUMO DE PARANORMAL 170 CONTENTS 171 Some metaphysics 171 1. Understanding the problem 172 2. Devising a plan 173 3. Carrying out the plan 175 4. Looking back 175 Some esoterism 178 1. Of algebraization 179 2. Some puzzles 181 3. Of replacement 183 4. Of duality and modality 185 Coda 188 BRIEF HISTORY 189 Bibliography 192 3.1 Logics of essence and accident 199 1 THE WHAT-IT-IS-TO-BE 199 2 THE MINIMAL LOGIC OF ESSENCE AND ACCIDENT 201 3 EXTENSIONS OF (K)EA, AND DEFINABILITY OF S AND ◊S 203 x 4 CHARACTERIZABILITY OF CLASSES OF FRAMES 205 5 ON ESSENCE, AND BEYOND 206 ACKNOWLEDGEMENTS 209 REFERENCES 209 3.2 Modality and paraconsistency 211 1 WHAT IS A PARACONSISTENT LOGIC? 211 2 WHAT IS A MODAL LOGIC? 214 3 D2 IS NOT A MODAL LOGIC 215 4 MODAL LOGICS ARE PARACONSISTENT 216 REFERENCES 219 3.3 Nearly every normal modal logic is paranormal 221 1 AFFIRMATIVE AND NEGATIVE MODALITIES 221 1.1 Basic modal semantics 223 1.2 Modal negations? 226 2 VARIETIES OF PARANORMALITY 228 2.1 Duality, at last 231 2.2 The route from modality to paranormality, and the easy way back 232 3 IMAGINE THERE ARE NO SEA BATTLES… 235 ACKNOWLEDGEMENTS 238 REFERENCES 238 4 AN ABSTRACT PERSPECTIVE ON NEGATION 241 RESUMO DE PURELOCAL 241 RESUMO DE INEFFABLE 241 CONTENTS 242 Here is how you should do it 242 Here is how you should not do it 244 BRIEF HISTORY 248 Bibliography 249 xi 4.1 On negation: Pure local rules 251 PROPOSAL 252 0 BACKGROUND 254 1 RULES FOR ABSTRACT CONSEQUENCE RELATIONS 257 2 PURE RULES FOR NEGATION 261 3 CAUSES AND CONSEQUENCES FOR PARANORMAL LOGICS 274 4 OH YES, WHY NOT?… (BUT THEN AGAIN, WHAT IS NEGATION, AFTER ALL?) 276 5 DIRECTIONS 284 REFERENCES 287 4.2 Ineffable inconsistencies 291 1 INCONSISTENT CLASSICAL LOGIC 291 2 THE GENERAL RECIPE 293 3 PARACONSISTENCY IS NOT ENOUGH 296 ACKNOWLEDGEMENTS 299 REFERENCES 300 Conclusão 301 xii Resumo Segundo a pressuposi¸c˜aode consistˆenciacl´assica, as contradi¸c˜oes tˆem um car´a[c]terexplosivo: uma vez que estejam presentes em uma teoria, tudo vale, e nenhum racioc´ınio sensato pode ent˜aoter lugar. Uma l´ogica ´e paraconsistente se ela rejeita uma tal pressuposi¸c˜ao,e aceita ao inv´esque algumas teorias inconsistentes conquanto n˜ao-triviais fa¸camperfeito sentido. As L´ogicasda InconsistˆenciaFormal, LIFs, formam uma classe de l´ogicasparaconsistentes particularmente expressivas nas quais a no¸c˜ao meta-te´oricade consistˆenciapode ser internalizada ao n´ıvel da linguagem obje[c]to. Como consequˆencia, as LIFs s˜aocapazes de recapturar o racioc´ınio consistente pelo acr´escimo de assun¸c˜oesde consistˆencia apropriadas. Assim, por exemplo, enquanto regras cl´assicas tais como o silogismo disjuntivo (de A e hn˜ao-Ai-ou-B, infira B) est˜aofadadas a falhar numa l´ogicaparaconsistente (pois A e hn˜ao-Ai poderiam ambas ser verdadeiras para algum A, independentemente de B), elas podem ser recuperadas por uma LIF se o conjunto das premissas for ampliado pela presun¸c˜aode que estamos raciocinando em um ambiente consistente (neste caso, pelo acr´escimo de hconsistente-Ai como uma hip´otese adicional da regra). A presente monografia introduz as LIFs e apresenta diversas ilustra¸c˜oes destas l´ogicase de suas propriedades, mostrando que tais l´ogicas constituem com efeito a maior parte dos sistemas paraconsistentes da literatura. Diversas formas de se efe[c]tuar a recaptura do racioc´ınio consistente dentro de tais sistemas inconsistentes s˜aotamb´em ilustradas. Em cada caso, interpreta¸c˜oes em termos de semˆanticas polivalentes, de tradu¸c˜oes poss´ıveis ou modais s˜aofornecidas, e os problemas relacionados `aprovis˜aode contrapartidas alg´ebricas para tais l´ogicass˜aoexaminados. Uma abordagem formal abstra[c]ta ´eproposta para todas as defini¸c˜oes relacionadas e uma extensa investiga¸c˜ao´efeita sobre os princ´ıpios l´ogicos e as propriedades positivas e negativas da nega¸c˜ao. Palavras-chave: L´ogicaUniversal, nega¸c˜ao,paraconsistˆencia,semˆanticas de tradu¸c˜oes poss´ıveis, modalidades, filosofia formal. xiii Abstract According to the classical consistency presupposition, contradictions have an explosive character: Whenever they are present in a theory, anything goes, and no sensible reasoning can thus take place. A logic is paraconsistent if it disallows such presupposition, and allows instead for some inconsistent yet non-trivial theories to make perfect sense. The Logics of Formal Inconsistency, LFIs, form a particularly expressive class of paraconsistent logics in which the metatheoretical notion of consistency can be internalized at the object-language level. As a consequence, the LFIs are able to recapture consistent reasoning by the addition of appropriate consistency assumptions. So, for instance, while classical rules such as disjunctive syllogism (from A and hnot-Ai-or-B, infer B) are bound to fail in a paraconsistent logic (because A and hnot-Ai could both be true for some A, independently of B), they can be recovered by an LFI if the set of premises is enlarged by the presumption that we are reasoning in a consistent environment (in this case, by the addition of hconsistent-Ai as an extra hypothesis of the rule). The present monograph introduces the LFIs and provides several illustrations of them and of their properties, showing that such logics constitute in fact the majority of interesting paraconsistent systems from the literature.