Recorded Talks

A Case for Weak Kleene ST

Abstract: The substructural Strict/Tolerant logic based on a strong Kleene valuation (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 theories of truth and vagueness based on sST can avoid the semantic and soritical paradoxes, 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 a weak Kleene variation of the Strict/Tolerant logic (wST). We argue that wST 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 wST is more appealing than sST, for it has the same virtues as sST while it has an unmatched expressive power.

A Recipe for Paradox

Abstract: In this talk, we provide a recipe that not only captures the common structure between semantic paradoxes, but it also captures our intuitions regarding the relations between these paradoxes. Before we unveil our recipe, we first talk about a popular 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 the wrong paradoxes together. That is, we will provide alternative arguments on why the inclosure schema is both too broad for including the Sorites paradox, and too narrow for excluding Curry’s paradox.

We then spell out our recipe. Our recipe consists of three ingredients: (1) a predicate that has two specific rules, (2) a simple method to find a partial negative modality, and (3) a diagonal lemma that would allow us to let sentences be their partial negative modalities. The recipe shows that all of the following paradoxes share the same structure: The Liar, Curry’s paradox, Validity Curry, Provability Liar, a paradox leading to Löb’s theorem, Knower’s paradox, Knower’s Curry, Grelling-Nelson’s paradox, Russell’s paradox in terms of extensions, alternative liar and alternative Curry, and other new paradoxes.

We conclude the paper by stating the lessons that we can learn from the recipe, and what kind of solutions does the recipe suggest if we want to adhere to the Principle of Uniform Solution.

Paradoxes and Restricting Cut

Abstract: In a recent talk (above), we discussed the underlying common structure of semantic paradoxes which we called the Recipe for Paradox. In this talk, we will briefly sketch the Recipe for Paradox and focus on the possible uniform solutions the Recipe suggests. We will then provide our own revenge-immune uniform solution to semantic paradoxes. The upshot of our solution is to restrict the Cut rule to grounded sentences only. That is, if the Cut formula is ungrounded, then the Cut move is blocked. Since our solution depends on the notion of groundedness, we will present how we define “grounded” and “ungrounded” formally in a syntactic fashion. We will conclude the talk by discussing why restricting Cut is more appealing as opposed to getting rid of Cut completely.