A Weakly-Intuitionistic Logic I1

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J.  Ciuciura

Abstract

In 1995, Sette and Carnielli presented a calculus, $I1$, which is intended to be dual to the paraconsistent calculus $P1$. The duality between $I1$ and $P1$ is reflected in the fact that both calculi are maximal with respect to classical propositional logic and they behave in a special, non-classical way, but only at the level of variables. Although some references are given in the text, the authors do not explicitly define what they mean by ‘duality’ between the calculi. For instance, no definition of the translation function from the language of $I1$ into the language of $P1$ (or from $P1$ to $I1$) was provided (see [4], pp. 88–90) nor was it shown that the calculi were functionally equivalent (see [13], pp. 260–261).
The purpose of this paper is to present a new axiomatization of $I1$ and briefly discuss some results concerning the issue of duality between the calculi.

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References

Batens, D. “Paraconsistent extensional propositional logics”, Logique et Analyse, 1980, vol. 23, no 90-91, pp. 195–234.
N.C.A. da Costa, “On the theory of inconsistent formal systems”, Notre Dame Journal of Formal Logic, 1974, vol. 15, no 4, pp. 497–510.
Ciuciura, J. “Paraconsistency and Sette’s calculus P1”, Logic and Logical Philosophy, 2015, vol. 24, pp.265–273.
D’Ottaviano, I.M.L., Feitosa, H.A. “Paraconsistent logics and translations”, Synthese, 2000, vol. 125, no 1-2, pp. 77–95.
Imai, Y., Iseki, K. “On Axiom Systems of Propositional Calculi. I”, Proc. Japan Acad., 1965, vol. 41, no 6, pp. 436–439.
Jaskowski, S. “A Propositional Calculus for Inconsistent Deductive Systems”, Logic and Logical Philosophy, 1999, vol. 7, no 1, pp. 35–56.
Jaskowski, S. “On the Discussive Conjunction in the Propositional Calculus for Inconsistent Deductive Systems”, Logic and Logical Philosophy, 1999, vol. 7, no 1, pp. 57–59.
Marcos, J. “On a Problem of da Costa”, Essays on the Foundations of Mathematics and Logic [http://sqig.math.ist.utl.pt/pub/MarcosJ/05-MP12.pdf, accessed on 01.05.2015]
Sette, A.M. “On the propositional calculus P1”, Mathemetica Japonicae, 1973, vol. 18, no 3, pp. 173–180.
Sette, A.M., Alves, E.H. “On the equivalence between two systems of paraconsistent logic”, Bulletin of the Section of Logic, 1995, vol. 24, no 3, pp. 155–157.
Sette, A.M., Carnielli, W.A. “Maximal weakly-intutuinistic logics”, Studia Logica, 1995, vol. 55, no 1, pp. 181–203.
Slater, B.H. “Paraconsistent logics?”, Journal of Philosophical Logic, 1995, vol. 24, pp. 451–454.
Tomova, N.E. “Natural p-logics”, Logical Investigations, 2011, vol. 17, pp. 256–268. (In Russian)