Abstract
In this paper, we generalize the ordered-pair semantics advanced by Matthew Clemens for the Logic of Paradox (LP) to n-tuple semantics, for each fixed n. Moreover, we show that the resulting semantics can accommodate not only LP, but also classical logic as well as strong Kleene logic depending on the set of designated (or distinguished) values that one chooses. Building on the technical observations, we offer intuitively plausible readings for the semantics, and we also discuss some weaknesses of the original intuitive reading advanced by Clemens.
The research of HO was supported by a Sofja Kovalevskaja Award of the Alexander von Humboldt-Foundation, funded by the German Ministry for Education and Research. The research of JRBA was also supported by the Alexander von Humboldt-Foundation through the Experienced Researcher Fellowship program, funded by CAPES-Humboldt. We would like to thank the referees for their careful comments.
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Notes
- 1.
We are using the subscript 3 just to indicate that there are three designated values. We will consider other possibilities later.
- 2.
The idea of n-tuple semantics is not new. Diderik Batens [1], for one, presents one such semantics, but it goes along different direction than the one we introduce later.
- 3.
The two-part semantics that suits better for dialetheists is described by use of relational semantics with two truth values rather than functional semantics with three truth values (see [15, pp. 148–149] for the details). The most faithful semantic presentation of LP by dialetheists advanced so far, which relies on a dialetheic metatheory, is presented by Zach Weber, Guillermo Badia and Patrick Girard in [18].
- 4.
As observed by Newton C. A. da Costa, Otávio Bueno and Steven French in [6], the logic of quasi-truth is closely related to discussive logic, one of the first paraconsistent systems, introduced by Stanisław Jaśkowski in [9,10,11]. Interestingly, the conditional of the three-valued paraconsistent logic LPT introduced in [5] is that of Sette’s system \(\mathbf{P^1}\) (cf. [17]), which enjoys a discussive semantics, as observed in [13]. Therefore, by making some simple changes for the negation, we obtain a discussive semantics for the conjunction-disjunction free fragment of LPT. Moreover, seen from this perspective, the semantics by Clemens, as well as our generalization, can be understood as a discussive semantics for LP.
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Omori, H., Arenhart, J.R.B. (2021). A Generalization of Ordered-Pair Semantics. In: Ghosh, S., Icard, T. (eds) Logic, Rationality, and Interaction. LORI 2021. Lecture Notes in Computer Science(), vol 13039. Springer, Cham. https://doi.org/10.1007/978-3-030-88708-7_12
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