The Liar and Related Paradoxes: Fuzzy Truth Value Assignment for Collections of Self-Referential Sentences

TL;DR

This paper introduces a method to assign consistent fuzzy truth values to self-referential sentences like the Liar paradox by reducing the problem to nonlinear equations and providing algorithms for solutions.

Contribution

It formulates a novel approach using fuzzy logic and nonlinear systems to resolve self-referential paradoxes, including proof of solution existence and algorithmic methods.

Findings

Existence of solutions for fuzzy truth value assignments

Mid-point solution is always consistent under certain conditions

Algorithms can be viewed as generalized sequential reasoning

Abstract

We study self-referential sentences of the type related to the Liar paradox. In particular, we consider the problem of assigning consistent fuzzy truth values to collections of self-referential sentences. We show that the problem can be reduced to the solution of a system of nonlinear equations. Furthermore, we prove that, under mild conditions, such a system always has a solution (i.e. a consistent truth value assignment) and that, for a particular implementation of logical ``and'', ``or'' and ``negation'', the ``mid-point'' solution is always consistent. Next we turn to computational issues and present several truth-value assignment algorithms; we argue that these algorithms can be understood as generalized sequential reasoning. In an Appendix we present a large number of examples of self-referential collections (including the Liar and the Strengthened Liar), we formulate the corresponding truth value equations and solve them analytically and/ or numerically.