Anti-intuitionism And Paraconsistency

This paper aims to help to elucidate some questions on the duality between the intuitionistic and the paraconsistent paradigms of thought, proposing some new classes of anti-intuitionistic propositional logics and investigating their relationships with the original intuitionistic logics. It is shown here that anti-intuitionistic logics are paraconsistent, and in particular we develop a first anti-intuitionistic hierarchy starting with Johansson's dual calculus and ending up with Gödel's three-valued dual calculus, showing that no calculus of this hierarchy allows the introduction of an internal implication symbol. Comparing these anti-intuitionistic logics with well-known paraconsistent calculi, we prove that they do not coincide with any of these. On the other hand, by dualizing the hierarchy of the paracomplete (or maximal weakly intuitionistic) many-valued logics (In) n∈ω we show that the anti-intuitionistic hierarchy (In*)n∈ω obtained from (In)n∈ω does coincide with the hierarchy of the many-valued paraconsistent logics (Pn)n∈ω. Fundamental properties of our method are investigated, and we also discuss some questions on the duality between the intuitionistic and the paraconsistent paradigms, including the problem of self-duality. We argue that questions of duality quite naturally require refutative systems (which we call elenctic systems) as well as the usual demonstrative systems (which we call deictic systems), and multiple-conclusion logics are used as an appropriate environment to deal with them. © 2004 Elsevier B.V. All rights reserved.31161184Avron, A., Natural 3-valued logics-characterization and proof theory (1991) J. Symbolic Logic, 56 (1), pp. 276-294Avron, A., Simple consequence relations (1991) Inform. and Comput., 92, pp. 105-139Batens, D., Paraconsistent extensional propositional logics (1980) Logique et Anal., 90-91, pp. 195-234Benson, H.H., The priority of definition and the Socratic elenchos (1990) Oxford Stud. Ancient Philos., 8, pp. 19-65Brunner, A.B.M., Carnielli, W.A., Kripke semantics for anti-intuitionistic logics, abstract (2002) Bull. Symbolic Logic, 8 (1), p. 168Caicedo, X., A formal system for the non-theorems of the propositional calculus (1996) Notre Dame J. Formal Logic, 19, pp. 147-151Carnielli, W.A., Possible translations semantics for paraconsistent logics (2000) Frontiers of Paraconsistent Logic: Proceedings of the I World Congress on Paraconsistency, pp. 159-172. , D. Batens, C. Mortensen, G. Priest, J.-P. van Bendegem (Eds.), Logic and Computation Series, Baldock Research Studies Press, King's College PublicationsCarnielli, W.A., Lima-Marques, M., Society semantics for multiple-valued logics (1999) Advances in Contemporary Logic and Computer Science, 235, pp. 33-52. , W.A. Carnielli, I.M.L. D'Ottaviano (Eds.), Contemporary Mathematics Series, American Mathematical SocietyCarnielli, W.A., Marcos, J., Limits for paraconsistent calculi (1999) Notre Dame J. Formal Logic, 40 (3), pp. 375-390Carnielli, W.A., Marcos, J., A taxonomy of C-systems (2002) Paraconsistency: The Logical Way to the Inconsistent, pp. 1-94. , http://www.cle.unicamp.br/e-prints/abstract_5.htm, W.A. Carnielli, M.E. Coniglio, I.M.L. D'Ottaviano (Eds.), Proceedings of the 2nd World Congress on Paraconsistency, held in Juquehy, Brazil, May 8-12, 2000Carnielli, W.A., Coniglio, M.E., Marcos, J., Logics of Formal Inconsistency Handbook of Philosophical Logic, 12. , D. Gabbay, F. Guenthner (Eds.) Kluwer Academic Publishers, The Netherlands, in presshttp://www.cs.math.ist.utl.pt/ftp/pub/MarcosJ/03-CCM-lfi.pdf, Preprint available at the The Center for Logic and Computation, IST, Lisbonda Costa, N.C.A., On the theory of inconsistent formal systems (1974) Notre Dame J. Formal Logic, 15 (4), pp. 497-510Doyle, A.C., The Sign of Four, , http://etext.lib.virginia.edu/toc/modeng/public/DoySign.html, Electronic Text Center, University of Virginia LibraryDummett, M., A propositional calculus with denumerable matrix (1959) J. Symbolic Logic, 24, pp. 97-106Epstein, R., (1995) The Semantic Foundations of Logic, 1. , (with the assistance collaboration of W.A. Carnielli, I.M.L. D'Ottaviano, S. Krajewski and R.D. Maddux), second ed. Oxford University PressFernández, V.L., Coniglio, M.E., Combining valuations with society semantics (2003) J. Appl. Non-Classical Logics, 3 (1), pp. 21-46. , http://www.cle.unicamp.br/e-prints/abstract_11.html, Preprint available at CLE e-Prints 2(2) (2002) (Section Logic)Fitting, M.C., (1969) Intuitionistic Logic, Model Theory and Forcing, , Amsterdam: North-HollandGödel, K., Zum intuitionistischem Aussagenkalkül (1932) Akad. Wissensch. Wien, Math.-natur. Klasse, 69, pp. 65-66Goodman, N., The logic of contradiction (1981) Z. Math. Logik Grundlag. Math., 27, pp. 119-126Johansson, I., Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus (1937) Compositio Math., 4, pp. 119-136Marcos, J., Ineffable inconsistencies, in preparation http://www.cs.math.ist.utl.pt/ftp/pub/MarcosJ/04-M-ii.pdf, Preprint available at the Center for Logic and Computation, IST, LisbonMiller, D., The logic of unfalsified hypotheses Book of Abstracts, , http://logica.cle.unicamp.br/wcp/BookOfAbstracts/BookOfAbstracts.htm, II World Congress on Paraconsistency, Juquehy, BrazilMiller, D., Paraconsistent logic for falsificationists (2000) Proc. 1st Workshop on Logic and Language (Universidad de Sevilla), pp. 197-204. , Editorial Kronos s.a., SevillaMortensen, C., James, W., Categories, Sheaves and Paraconsistent Logic, , PreprintPopper, K.R., Conjectures and Refutations (1989) The Growth of Scientific Knowledge, , fifth ed. London: Routledge & Kegan PaulPopper, K.R., On the theory of deduction (1948) Indag. Mat., 10 (PART I-II), pp. 173-183Popper, K.R., (1959) The Logic of Scientific Discovery, , London: Hutchinson & Co., Ltd. English transl. Logik der Forschung 1936 Julius Springer Verlag ViennaQueiroz, G.S., Sobre a Dualidade entre Intuicionismo e Paraconsistncia (1998) On the Duality Between Intuitionism and Paraconsistency, , Ph.D. thesis, State University of Campinas, IFCH (in Portuguese)Rasiowa, H., Sikorski, R., (1963) The Mathematics of Metamathematics, , Warszaw: Państwowe Wydawnictwo NaukoweRauszer, C., Applications of Kripke models to Heyting-Brouwer logic (1977) Studia Logica, 34, pp. 61-71Rauszer, C., An algebraic and Kripke-style approach to a certain extension of intuitionistic logic (1980) Diss. Math., 167Scott, D.S., Completeness and axiomatizability in many-valued logic (1974) Proceedings of the Tarski Symposium, , L. Henkin, et al. (Eds.), Amer. Math. Soc., Providence, RISette, A.M., Carnielli, W.A., Maximal weakly-intuitionistic logics (1995) Studia Logica, 55, pp. 181-203Shoesmith, D.J., Smiley, T.J., Multiple-Conclusion Logic (1978), Cambridge University PressSłupecki, J., Bryll, G., Wybraniec-Skardowska, U., Theory of rejected propositions, I and II (1971) Studia Logica, 29, pp. 75-115Tarski, A., Fundamentale Begriffe der Methodologie der deduktiven Wissenschaften I (1930) Monats. Math. Phys., 37, pp. 361-404. , English transl.: S.R. Givant, R.N. McKenzie (Eds.), Collected Papers, Birkhäser, 1986 341-390Urbas, I., Dual-intuitionistic logic (1996) Notre Dame J. Formal Logic, 37 (3), pp. 440-45

Paraconsistent Algebras

The prepositional calculi Cn, 1 ≤n ≤ ω introduced by N.C.A. da Costa constitute special kinds of paraconsistent logics. A question which remained open for some time concerned whether it was possible to obtain a Lindenbaum's algebra for Cn. C. Mortensen settled the problem, proving that no equivalence relation for Cn. determines a non-trivial quotient algebra. The concept of da Costa algebra, which reflects most of the logical properties of Cn, as well as the concept of paraconsistent closure system, are introduced in this paper. We show that every da Costa algebra is isomorphic with a paraconsistent algebra of sets, and that the closure system of all filters of a da Costa algebra is paraconsistent. © 1984 Polish Academy of Sciences.431-2798

New Dimensions On Translations Between Logics

After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: (conservative) translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another. © Birkhäuser Verlag Basel/Switzerland 2009.31118Béziau, J.Y., Recherches sur la Logique Universelle Excessivité, Négation, Séquents) (1994), 7. , Ph.D Thesis, ParisBueno-Soler, J., Possible-translations semantic algebraizability (2004), http://libdigi.unicamp.br/document/?code=vtls000337884, in Portuguese. Master Dissertation, IFCH, State University of Campinas Available atBrown, D.J., Suszko, R., Abstract Logics (1973), 102, pp. 9-41. , Dissertationes MathematicaeCarnielli, W.A., Many-valued logic and plausible reasoning (1990) Proceedings of the 20th International Congress on Many-Valued Logics, , In: University of Charlotte, North Carolina, pp. 328-335. IEEE Computer Society, New YorkCarnielli, W.A., Coniglio, M.E., Gabbay, D., Gouveia, P., Sernadas, C., (2008) Analysis and Synthesis of Logics. How to Cut and Paste Reasoning Systems, 1. , Springer, DordrechtConiglio, M.E., Recovering a logic from its fragments by meta-fibring (2007) Logica Universalis, 1 (2), pp. 377-416. , http://www.cle.unicamp.br/e-prints/vol_5,n_4,2005.html, Preprint available as: The Meta-Fibring environment: Preservation of meta-properties by fibring, CLE e-Prints, vol. 5, n. 4 (2005). Available atConiglio, M.E., Carnielli, W.A., Transfers between logics and their applications (2002) Studia Logica, 72 (3), pp. 367-400da Silva, J.J., D'Ottaviano, I.M.L., Sette, A.M., Translations between logics (1999) Models, Algebras and Proofs, Lectures Notes in Pure and Applied Mathematics, 203, pp. 435-448. , In: Caicedo, X., Montenegro, C.H.(eds) Marcel Dekker, New YorkD'Ottaviano, I.M.L., (1973) Fechos Caracterizados Por Interpretações (Closures Characterized By Interpretations), , in Portuguese. Master Dissertation, IMECC, State University of CampinasD'Ottaviano, I.M.L., Feitosa, H.A., Conservative translations and model- Theoretic translations (1999) Manuscrito - Revista Internacional De Filosofia, 2 (22), pp. 117-132D'Ottaviano, I.M.L., Feitosa, H.A., Many-valued logics and translations (1999) J. Appl. Non-Class. Log., 9 (1), pp. 121-140D'Ottaviano, I.M.L., Feitosa, H.A., Paraconsistent logics and translations (2000) Synthèse, 125, pp. 77-95. , DordrechtD'Ottaviano, I.M.L., Feitosa, H.A., Translations from Lukasiewicz logics into classical logic: Is it possible? (2006) Essays in Logic and Ontology, Poznan Studies in the Philosophy of the Sciences and the Humanities, 91, pp. 157-168. , In: Malinowski, J., Pietrusczak, A. (eds.)D'Ottaviano, I.M.L., Feitosa, H.A., Deductive systems and translations (2007) Perspectives on Universal Logic, pp. 125-157. , In: Béziau, J.-Y., Costa-Leite (Org.), A. (eds.) Polimetrica International Scientific PublisherEpstein, R.L., (1990) The Semantic Foundations of Logic, Vol. 1. Propositional Logics, , Kluwer, DordrechtFeitosa, H.A., Conservative translations (1997), in Portuguese. Ph.D Thesis, IFCH, State University of CampinasFeitosa, H.A., D'Ottaviano, I.M.L., Conservative translations (2001) Ann. Pure Appl. Logic, 108 (1-3), pp. 205-227Fernández, V.L., Fibring of Logis in Leibniz Hierarchy (2005), http://libdigi.unicamp.br/document/?code=vtls000365017, in Portuguese. Ph.D. Thesis, IFCH, State University of Campinas Available atGentzen, G., On the relation between intuitionist and classical arithmetic (1933) (1969) The Collected Papers of Gerhard Gentzen, pp. 53-67. , In: Szabo, M.E.(eds) North-Holland, AmsterdamGlivenko, V., Sur quelques points de la logique de M. Brouwer (1929) Bulletins De La Classe De Sciences, 5 (15), pp. 183-188. , Académie Royale de BelgiqueGödel, K., On intuitionistic arithmetic and number theory (1933e) (1986) K. Gödel's Collected Works, 1, pp. 287-295. , In: Feferman, S.(eds) Oxford University Press, OxfordGödel, K., An interpretation of the intuitionistic propositional calculus (1933f) (1986) K. Gödel's Collected Works, 1, pp. 301-302. , In:, (eds) Oxford University Press, OxfordGoguen, J.A., Burstall, R.M., Introducing institutions (1984) Logics of Programs (Carnegie-Mellon University, June 1983), Lecture Notes in Computer Science, 164, pp. 221-256. , In: Springer, HeidelbergGoguen, J.A., Burstall, R.M., Institutions: Abstract model theory for specification and programming (1992) J. ACM, 39 (1), pp. 95-146Hoppmann, A.G., Closure and Embedding (1973), in Portuguese. Ph.D Thesis, FFCL, São Paulo State University, Rio ClaroHumberstone, L., Béziau's translation paradox (2005) Theoria, 2, pp. 138-181Humberstone, L., Logical discrimination (2005) Logica Universalis: Towards a General Theory of Logic, pp. 207-228. , In: Béziau, J.-Y.(eds) Birkhäuser, BaselJanssen, T., (2007) Compiler Correctness and the Translation of Logics, , http://www.illc.uva.nl/Publications/ResearchReports/PP-2007-14.text.pdf, ILLC Research Reports and Technical Notes 2007 Report PP-2007-14 Available atKolmogorov, A.N., On the principle of excluded middle (1925) (1977) Mathematical Logic 1879-1931, , In: Heijenoort, J.(eds) Harvard University Press, CambridgeŁoś, J., Suszko, R., Remarks on sentential logics (1958) Indagationes Mathematicae, 20, pp. 177-183Marcos, J., Possible-translations semantics (1999), http://libdigi.unicamp.br/document/?code=vtls000224326, in Portuguese. Master Dissertation, IFCH, State University of Campinas Available atMossakowski, T., Diaconescu, R., Tarlecki, A., What is a logic? (2005) Logica Universalis: Towards a General Theory of Logic, pp. 111-134. , In: Béziau, J.-Y.(eds) Birkhäuser, BaselPrawitz, D., Malmnäs, P.E., A survey of some connections between classical, intuitionistic and minimal logic (1968) Contributions to Mathematical Logic, pp. 215-229. , In: Schmidt, H.(eds) North-Holland, AmsterdamScheer, M.C., Towards a theory of translations between cumulative logics (2002), http://libdigi.unicamp.br/document/?code=vtls000284889, in Portuguese. Master Dissertation, IFCH, State University of Campinas Available atSzczerba, L., Interpretability of elementary theories (1977) Logic, Foundations of Mathematics and Computability Theory, pp. 129-145. , In: Butts, H., Hintikka, J.(eds) D. Reidel, DordrechtWójcicki, R., (1988) Theory of Logical Calculi: Basic Theory of Consequence Operations, Vol. 199 of Synthese Library, , Kluwer, Dordrech

Logics of Formal Inconsistency

In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory entails all possible consequences) are assumed inseparable, granted that negation is available. This is an effect of an ordinary logical feature known as `explosiveness': According to it, from a contradiction `$\alpha$ and $eg\alpha$' everything is derivable. Indeed, classical logic (and many other logics) equate `consistency' with `freedom from contradictions'. Such logics forcibly fail to distinguish, thus, between contradictoriness and other forms of inconsistency. Paraconsistent logics are precisely the logics for which this assumption is challenged, by the rejection of the classical `consistency presupposition'. The \textit{Logics of Formal Inconsistency}, \textbf{LFI}s, object of this chapter, are the paraconsistent logics that neatly balance the equation: \textsc{contradictions consistency = triviality}. The \textbf{LFI}s have a remarkable way of reintroducing consistency into the non-classical picture: They internalize the very notions of consistency and inconsistency at the object-language level. The result of that strategy is the design of very expressive logical systems, whose fundamental feature is the ability of recovering all consistent reasoning right on demand, while still allowing for some inconsistency to linger, otherwise