There are only two paradoxes

TL;DR

This paper demonstrates that semantic paradoxes like the liar and Yablo patterns are inherent in all classical logic representations, and introduces a graph-theoretic approach to analyze their solvability.

Contribution

It generalizes Richardson's theorem to arbitrary graphs, showing solvability conditions based on the absence of odd cycles and Yablo-like patterns, using a novel compactness-like theorem.

Findings

Semantic paradoxes are present in all classical logic graph representations.

Finite graphs without odd cycles are solvable, extended to certain infinite graphs.

A new compactness-like theorem aids in analyzing infinitary logic and paradox patterns.

Abstract

Using a graph representation of classical logic, the paper shows that the liar or Yablo pattern occurs in every semantic paradox. The core graph theoretic result generalizes theorem of Richardson, showing solvability of finite graphs without odd cycles, to arbitrary graphs which are proven solvable when no odd cycles nor patterns generalizing Yablo's occur. This follows from an earlier result by a new compactness-like theorem, holding for infinitary logic and utilizing the graph representation.