Anti-intuitionism and paraconsistency |
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Authors: | Andreas B.M. Brunner Walter A. Carnielli |
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Affiliation: | aDepartment of Mathematics – UFBA, Salvador, BA, Brazil;bCentre for Logic, Epistemology and the History of Science and IFCH – UNICAMP, Campinas, SP, Brazil |
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Abstract: | 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)nω we show that the anti-intuitionistic hierarchy (In*)nω obtained from (In)nω does coincide with the hierarchy of the many-valued paraconsistent logics (Pn)nω. 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. |
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Keywords: | Dualizing logics Anti-intuitionism Paraconsistency Dual-intuitionistic logics Intuitionism Paracompleteness |
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