Abstract
There are various approaches to develop a system of paraconsistent logic, and those we focus on in this paper are approaches of da Costa, Belnap, and Nelson. Our main focus is da Costa, and we deal with a system that reflects the idea of da Costa. We understand that the main idea of da Costa is to make explicit, within the system, the area in which you can infer classically. The aim of the paper is threefold. First, we introduce and present some results on a classicality operator which generalizes the consistency operator of Logics of Formal Inconsistency. Second, we show that we can introduce the classicality operator to the systems of Belnap. Third, we demonstrate that we can generalize the classicality operator above to the system of Nelson. The paper presents both the proof theory and semantics for the systems to be introduced, and also establishes some completeness theorems.
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- 1.
Interestingly, a similar criteria is also considered by Jaśkowski (cf. [8, p. 38]) who is the other founder of modern paraconsistent logic.
- 2.
Note here that BS4 is not the first many-valued system that reflects da Costa’s idea. Indeed, there are systems such as LFI1 and LFI2, developed in [4], which are complete with respect to three-valued semantics. Therefore, we may say that the first objection was already settled then. But at the same time, we are widening our scope to deal with incomplete situations and therefore it must be fair to say that the argument against the first objection becomes more widely acceptable by the presence of the system BS4.
- 3.
Note that two questions are quite different and thus our approaches to these questions will be also quite different.
- 4.
Note here that there is an attempt [10] to develop systems which are both paraconsistent and paracomplete, following the line of research of da Costa. Recall that a system is called paracomplete when the law of excluded middle is not valid in the system.
- 5.
We may of course add the Peirce’s law, i.e. \(((A\!\supset \!B)\!\supset \!A)\!\supset \!A\), in place of \(A\vee (A\!\supset \!B)\) Indeed, these two formulas are equivalent in negation-less fragment of classical propositional calculus as \(A\vee B\) and \((A\!\supset \!B)\!\supset \!B\) are equivalent in general. The reason we employed \(A\vee (A\!\supset \!B)\) is because we are simply following the convention in the study of LFIs. For an example of an axiomatization of IPC \(^+\), see e.g. (1)–(9) of [6, pp. 498–499] which is the axiomatization given by Kleene.
- 6.
Note here that in the study of Nelson’s systems, we also use the name strong negation to refer to a different negation. But we will here follow the convention of LFIs, not of Nelson’s systems, since our main focus is on the systems that generalize the framework of LFIs. Note further that strong negation \(\lnot \) behaves as classical negation in BS.
- 7.
The truth tables there are known as the truth tables for the system P \(^1\) of Sette studied in [17].
- 8.
Namely, \(\mathop {\sim }(A\wedge \mathop {\sim }A)\!\supset \!(A\!\supset \!(\mathop {\sim }A\!\supset \!B))\) holds, just like (A2) of cBS.
- 9.
The outline of the proof for the equivalence is as follows. First, \(\lnot (A\wedge \mathop {\sim }A)\!\supset \!\mathop {\sim }(A\wedge \mathop {\sim }A)\) follows as a special case of \(\lnot A\!\supset \!\mathop {\sim }A\), and this is equivalent to \(A\vee \mathop {\sim }A\) which is assumed in concerned systems. For the other way around, we assume \(\mathop {\sim }(A\wedge \mathop {\sim }A)\) and \((A\wedge \mathop {\sim }A)\). Then, by the special role given to \(\mathop {\sim }(A\wedge \mathop {\sim }A)\), the conjunction of the formulas are explosive. So, we have \(\mathop {\sim }(A\wedge \mathop {\sim }A) \wedge (A\wedge \mathop {\sim }A)\!\supset \!\lnot (A\wedge \mathop {\sim }A)\) in particular, and finally by reductio with respect to \(\lnot \), we obtain the desired formula. For the stronger definition, see Theorem 1.
- 10.
More results on related systems of \(\mathcal {N}_{2}\mathbf {BS4}\) is proved in [16].
- 11.
The readers may wonder if we may define the semantic clause \(w \models ^{-} \circ A\) as the equivalence: \(w \models ^{+} A\) iff \(w \models ^{-} A\). However, this semantic clause will break the persistency requirement. This is one reason why we employ the current version of the semantic clause.
- 12.
We already made use of this kind of reformulation in the proof of Theorem 3.
- 13.
Note that even though many of the relevant (relevance) logics are paraconsistent, relevantists have a different view on classical logic in the sense that they will not necessarily agree to make use of classical logic even in consistent cases.
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Acknowledgments
The authors would like to thank the referees for their detailed and helpful comments which improved the paper in many ways. We would also like to thank Michael De who kindly proofread our final draft and made many helpful suggestions to improve our remarks related to relevantists’ perspective as well as our English. Finally, we would like to thank the audiences at the Trends in Logic XI conference who showed their interest, and encouraged us to pursue this research. The first author is a postdoctoral fellow of Japan Society for the Promotion of Science (JSPS), and the present work was partially supported by a Grant-in-Aid for JSPS Fellows. The work of the second author was partially supported by JSPS KAKENHI, Grant-in-Aid for Young Scientists (B) 24700146.
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Omori, H., Sano, K. (2014). da Costa Meets Belnap and Nelson. In: Ciuni, R., Wansing, H., Willkommen, C. (eds) Recent Trends in Philosophical Logic. Trends in Logic, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-319-06080-4_11
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