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A GRAPH-THEORETIC ANALYSIS OF THE SEMANTIC PARADOXES

Published online by Cambridge University Press:  15 February 2018

TIMO BERINGER  and
THOMAS SCHINDLER
TIMO BERINGER
Affiliation:
MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY LMU MUNICH MUNICH, GERMANYE-mail: [email protected]
THOMAS SCHINDLER
Affiliation:
CLARE COLLEGE UNIVERSITY OF CAMBRIDGE CAMBRIDGE, UKE-mail: [email protected]
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Abstract

We introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook [5, 6] and Rabern, Rabern, and Macauley [16]. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s [14] notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth values. It is shown that this notion of paradoxicality coincides with that of Kripke [13]. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo [21]), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs.

Keywords

03A0503B9905C20liar paradoxgraph theorytheories of truthself-referenceYablo’s paradox
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 23 , Issue 4 , December 2017 , pp. 442 - 492
Copyright
Copyright © The Association for Symbolic Logic 2018 

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