Rashed Ahmad

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.

There have been recent arguments against the idea that substructural solutions are uniform. The claim is that even if the substructuralist solves the common semantic paradoxes uniformly by targeting Cut or Contraction, with additional machinery, we can construct higher-level paradoxes (e.g., a higher-level Liar, a higher-level Curry, and a meta-validity Curry). These higher-level paradoxes do not use metainferential Cut or Contraction, but rather, higher-level Cuts and higher-level Contractions. These kinds of paradoxes suggest that targeting Cut or Contraction is not enough for solving semantic paradoxes; the substructuralist must target Cut of every level or Contraction of every level to solve the paradoxes. Hence, the substructuralists do not provide as uniform of a solution as they hoped they did. In response, we argue that the substructuralists need not admit these additional machineries. In fact, they are redundant in light of the validity predicate (i.e., there is no gain in terms of expressive power). The validity predicate is powerful enough to creep these paradoxes in the object level. The substructuralist does not need to ascend to metainferences to construct higher-level paradoxes. Moreover, there is a reading available to the substructuralist such that all the higher-level structural rules would collapse to instances of the object-level structural rules (e.g., meta \(_n\) Cut and meta \(_n\) Contraction would become instances of Cut and Contraction). We then address Barrio et al.’s worry that the validity predicate has its shortcomings; the substructuralist cannot internalize some of its metarules. We claim that the validity of metarules can be internalized without the need to strengthen the validity predicate. However, a problem raised by Barrio et al. is still present—the problem of internalizing unwanted instances of Cut in Cut-free approaches. We argue that this internalization problem is not unique to the validity predicate; the same problem is present with other problematic predicates, such as the truth predicate and the provability predicate.

In this paper, we provide a recipe that not only captures the common structure of semantic paradoxes but also captures our intuitions regarding the relations between these paradoxes. Before we unveil our recipe, we first talk about a well-known schema introduced by Graham Priest, namely, the Inclosure Schema. Without rehashing previous arguments against the Inclosure Schema, we contribute different arguments for the same concern that the Inclosure Schema bundles together the wrong paradoxes. That is, we will provide further arguments on why the Inclosure Schema is both too narrow and too broad. We then spell out our recipe. The recipe shows that all of the following paradoxes share the same structure: The Liar, Curry's paradox, Validity Curry, Provability Liar, Provability Curry, Knower's paradox, Knower's Curry, Grelling-Nelson's paradox, Russell's paradox in terms of extensions, alternative Liar and alternative Curry, and hitherto unexplored paradoxes. We conclude the paper by stating the lessons that we can learn from the recipe, and what kind of solutions the recipe suggests if we want to adhere to the Principle of Uniform Solution.