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 (I^n*)_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.

Keywords: Dualizing logics; Anti-intuitionism; Paraconsistency; Dual-intuitionistic logics; Intuitionism; Paracompleteness

Mathematical subject codes: 03B50; 03B53; 03B55

Article Outline

1. Introduction

2. An environment for dualization

3. Heyting's and Johansson's dual calculi

3.1. The dual of Heyting's calculus

3.2. The dual of Johansson's calculus

4. The hierarchy of Gödel's dual calculi

5. The duals of the paracomplete logics (Iⁿ)_nω

6. Additional properties and comparisons

6.1. The anti-constructive logics AC

6.2. The anti-paracomplete logics AP

7. Summary and conclusions

Acknowledgements

References