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.