Structural fixed-point theorems

TL;DR

This paper explores the mathematical foundations of semantic paradoxes, especially infinitary ones like Yablo's paradox, by analyzing fixed points of functions related to 'dangerous' directed graphs that enable paradoxical structures.

Contribution

It reformulates the problem of semantic paradoxes into a fixed-point mathematical framework, extending previous work to analyze infinitary paradoxes and their underlying graph structures.

Findings

Reformulation of semantic paradoxes as fixed-point problems

Identification of conditions for 'dangerous' graphs to produce paradoxes

Extension of prior work to infinitary versions of paradoxes

Abstract

The semantic paradoxes are associated with self-reference or referential circularity. However, there are infinitary versions of the paradoxes, such as Yablo's paradox, that do not involve this form of circularity. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality -- these are the so-called "dangerous" directed graphs. Building on Rabern, et. al (2013) we reformulate this problem in terms of fixed points of certain functions, thereby boiling it down to get a purely mathematical problem.