A case for weak Kleene ST

Synthese 205 (5):1-13 (2025)
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Abstract

The substructural Strict/Tolerant logic based on strong Kleene valuations (_sST_) was motivated by its ability to express a fully transparent truth predicate and the tolerance principle without falling into the traps of semantic and soritical paradoxes. Even though _sST_ rejects the meta-inferential rule of Cut, it has been shown that many instances of Cut are recoverable. Thus, not only can theories of truth and vagueness based on _sST_ avoid the semantic and soritical paradoxes, but these theories stay very close to classical theories, which is counted as a virtue of _sST_. In a recent paper by Murzi and Rossi, the authors argue that the notion of (un)paradoxicality plays a major role in recapturing the “safe” instances of Cut. However, the theory of truth based on _sST_ cannot be extended to express the notion (un)paradoxicality on pain of revenge paradox. Similarly, in a recent paper by Bruni and Rossi, the authors argue that the theory of vagueness based on _sST_ cannot be extended to express the notion of determinateness on pain of revenge paradox, even though “determinateness” plays a major role in the theory. In this paper, we argue that given the analysis of these revenge paradoxes, the Strict/Tolerant logician should prefer the weak Kleene variation of the Strict/Tolerant logic ($$w$$_ST_). We argue that $$w$$_ST_ can express a fully transparent truth predicate and the tolerance principle as well as the notions of (un)paradoxicality and determinateness (though we prefer to use the notion of groundedness to encompass both of these notions) while still being immune to revenge. We conclude that the logic $$w$$_ST_ is more appealing than _sST_, for it has the same virtues as _sST_ while it has an unmatched expressive power.

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Rashed Ahmad
Kuwait University

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References found in this work

Introduction to Metamathematics.Stephen Cole Kleene - 1952 - Groningen: North-Holland.
Outline of a theory of truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
The Logic of Nonsense.Sören Halldén - 1949 - Uppsala, Sweden: Upsala Universitets Arsskrift.
ST, LP and Tolerant Metainferences.Bogdan Dicher & Francesco Paoli - 2019 - In Can Başkent & Thomas Macaulay Ferguson, Graham Priest on Dialetheism and Paraconsistency. Cham, Switzerland: Springer Verlag. pp. 383-407.

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