Jump Liars and Jourdain’s Card via the Relativized T-scheme

Abstract

A relativized version of Tarski's T-scheme is introduced as a new principle of the truth predicate. Under the relativized T-scheme, the paradoxical objects, such as the Liar sentence and Jourdain's card sequence, are found to have certain relative contradictoriness. That is, they are contradictory only in some frames in the sense that any valuation admissible for them in these frames will lead to a contradiction. It is proved that for any positive integer n, the n-jump liar sentence is contradictory in and only in those frames containing at least an n-jump odd cycle. In particular, the Liar sentence is contradictory in and only in those frames containing at least an odd cycle. The Liar sentence is also proved to be less contradictory than Jourdain's card sequence: the latter must be contradictory in those frames where the former is so, but not vice versa. Generally, the relative contradictoriness is the common characteristic of the paradoxical objects, but different paradoxical objects may have different relative contradictoriness.

Guest Editors’ Introduction

“…4], and to degrees of paradoxicality (pairing with graph theory) by Hsiung [31][32][33][34], on the logical and mathematical side; the application to epistemology by Gupta [23] and [25, ch. 7-8], to vagueness by Asmus [6], and to the theory of abstract objects by Wang [60], in a more philosophical direction.…”