Abstract
The present paper focuses on Graham Priest’s claim that even primitive recursive relations may be inconsistent. Although he carefully presented his claim using the expression “may be,” Priest (2006, p. 240) made a definite claim that even numerical equations can be inconsistent. His argument relies heavily on the fact that there is an inconsistent model for arithmetic. After summarizing Priest’s argument for the inconsistent primitive recursive relation, I first discuss the fact that his argument has a weak foundation to explain that the existence of a model for some relations does not guarantee that they are primitive recursive. Moreover, since his identity relation is a combination of a standard identity and a congruence relation, it does not simply represent the standard identity function. Then, I argue that his identity relation cannot be both inconsistent and primitive recursive. Furthermore, I extend the argument to the general case that there is no inconsistent primitive recursive relation. These arguments show that the standard notions of “function” and “primitive recursive function” do not fit Priest’s inconsistent model. New definitions are necessary to show the existence of a dialetheic primitive recursive relation.
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Notes
In answer to Priest, Shapiro (2019) recently stated that it was his mistake in Shapiro (2002) to think that the soundness of \(PA^{*}\) could be proved straightforwardly through the induction. He argued that \(PA^{*}\) fails to prove its own soundness due to Curry’s Paradox. However, it does not mean that the unsoundness of the system has been established. Also, for the consistency of primitive recursive relations, he did not directly address Priest’s claim that there exists an inconsistent primitive recursive relation. In addition, no argument against Priest’s inconsistent primitive recursive relation has been made before. Therefore, a discussion on this point is most necessary.
It should be noted that Priest used the identity symbol “\(=\)” ambiguously. His identity relation is not the same as the standard identity relation but combines that with the congruence relation. The standard identity, the congruence relations, and Priest’s identity relation are distinguished in Sects. 3 and 4.
There are two standard notions of consistency. Roughly put, syntactic consistency means that for any system S, S is consistent if there is no \(\varphi \) such that S derives \(\varphi \) and \(\lnot \varphi \); otherwise, it is inconsistent. Semantic consistency means that, for any set S of formulas, S is consistent iff there is a case that every formula in S is true; otherwise, it is inconsistent. Both notions apply a set of formulas (or a system), not a single relation. It is an interesting point that the standard notion of semantic consistency does not rule out some true contradictions. I am grateful to the anonymous referee for noting this point.
For some m and n, the remainder is as defined in the division theorem that if \(x=ym+n\) where \(0 \le n <y\), then n is the remainder. Then, rem is primitive recursive. (Cf. Davis (1958).)
I am grateful to the anonymous referee here. Even when engaging in discussions beyond the discourse of mathematics, there is no question that Priest’s notion of identity is not the same thing as what people typically mean by identity. Priest (2005, Ch. 5) interestingly argued that there are no problems even when the identity relation is applied to non-existent objects or vague objects. Moreover, Priest (2014) defended the idea that the identity does not obey all sorts of standard rules such as transitivity. However, discussions of these topics are beyond the scope of the present paper, and the reader is referred to the literature for further details in Priest (2005, 2014).
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Acknowledgements
This research has been supported by the AMOREPACIFIC Foundation. An early version of this paper was presented at the Summer Regular Conference of the Korean Association for Logic, the Fall Regular Conference of the Korean Society of Analytic Philosophy, and the 2nd Korea Logic Day. I would like to thank two anonymous referees, Byungdeok Lee, Chulmin Yoon, Dongwoo Kim, Eunseok Yang, Hanjong So, Inkyo Chung, Junguk Lee, and Junhee Kim for their helpful comments.
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Choi, S. Is there an inconsistent primitive recursive relation?. Synthese 200, 418 (2022). https://doi.org/10.1007/s11229-022-03900-x
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DOI: https://doi.org/10.1007/s11229-022-03900-x