‘Sometime a paradox’, now proof: Yablo is not first order

Interesting as they are by themselves in philosophy and mathematics, paradoxes can be made even more fascinating when turned into proofs and theorems. For example, Russell’s paradox, which overthrew Frege’s logical edifice, is now a classical theorem in set theory, to the effect that no set contains all sets. Paradoxes can be used in proofs of some other theorems—thus Liar’s paradox has been used in the classical proof of Tarski’s theorem on the undefinability of truth in sufficiently rich languages. This paradox (as well as Richard’s paradox) appears implicitly in Gödel’s proof of his celebrated first incompleteness theorem. In this paper, we study Yablo’s paradox from the viewpoint of first- and second-order logics. We prove that a formalization of Yablo’s paradox (which is second order in nature) is non-first-orderizable in the sense of George Boolos (1984). This was sometime a paradox, but now the time gives it proof. —William Shakespeare (Hamlet, Act 3, Scene 1).

Paradox as a Guide to Ground

I will use paradox as a guide to metaphysical grounding, a kind of non-causal explanation that has recently shown itself to play a pivotal role in philosophical inquiry. Specifically, I will analyze the grounding structure of the Predestination paradox, the regresses of Carroll and Bradley, Russell's paradox and the Liar, Yablo's paradox, Zeno's paradoxes, and a novel omega plus one variant of Yablo's paradox, and thus find reason for the following: We should continue to characterize grounding as asymmetrical and irreflexive. We should change our understanding of the transitivity of grounding in a certain sense. We should require foundationality in a new, generalized sense, that has well-foundedness as its limit case. Meta-grounding is important. The polarity of grounding can be crucial. Thus we will learn a lot about structural properties of grounding from considering the various paradoxes. On the way, grounding will also turn out to be relevant to the diagnosis (if not the solution) of paradox. All the paradoxes under consideration will turn out to be breaches of some standard requirement on grounding, which makes uniform solutions of large groups of these paradoxes more desirable. In sum, bringing together paradox and grounding will be shown to be of considerable value to philosophy.1

Truth, Meaning, and Yablo’s Paradox – A Moderate Anti-Realist Approach

Yablo’s Paradox, an infinite-sentence version of the Liar Paradox, aims to show that semantic paradox can emerge even without circularity. I will argue that the lack of meaning/content of the sentences involved is the source of the paradoxical outcome.I will introduce and argue for a Moderate Antirealist (MAR) approach to truth and meaning, built around the twin principles that neither truth nor meaning can outstrip knowability. Accordingly, I will introduce a MAR truth operator that both forges an explicit connection between truth and knowability and distinguishes between truth and factuality. I will also argue that the meaning/content of propositions should be identified not with the set of possible worlds in which the propositions are true/factual, but rather in which they are known.I will show that our MAR framework dissolves Yablo’s Paradox and also confirms our intuition that these sentences are all devoid of content/meaning.