Anti-intuitionism and paraconsistency

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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.

MSC

03B50
03B53
03B55

Keywords

Dualizing logics
Anti-intuitionism
Paraconsistency
Dual-intuitionistic logics
Intuitionism
Paracompleteness

Cited by (0)

1

Supported by a post-doc grant from FAPESP (São Paulo State, Brazil).

2

Supported by a research grant from CNPq (Brazil).