Dangerous Reference Graphs and Semantic Paradoxes

Journal of Philosophical Logic 42 (5):727-765 (2013)
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Abstract

The semantic paradoxes are often associated with self-reference or referential circularity. Yablo (Analysis 53(4):251–252, 1993), however, has shown that there are infinitary versions of the paradoxes 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. In this essay, we lay the groundwork for a general investigation into the nature of reference structures that support the semantic paradoxes and the semantic hypodoxes. We develop a functionally complete infinitary propositional language endowed with a denotation assignment and extract the reference structural information in terms of graph-theoretic properties. We introduce the new concepts of dangerous and precarious reference graphs, which allows us to rigorously define the task: classify the dangerous and precarious directed graphs purely in terms of their graph-theoretic properties. Ungroundedness will be shown to fully characterize the precarious reference graphs and fully characterize the dangerous finite graphs. We prove that an undirected graph has a dangerous orientation if and only if it contains a cycle, providing some support for the traditional idea that cyclic structure is required for paradoxicality. This leaves the task of classifying danger for infinite acyclic reference graphs. We provide some compactness results, which give further necessary conditions on danger in infinite graphs, which in conjunction with a notion of self-containment allows us to prove that dangerous acyclic graphs must have infinitely many vertices with infinite out-degree. But a full characterization of danger remains an open question. In the appendices we relate our results to the results given in Cook (J Symb Log 69(3):767–774, 2004) and Yablo (2006) with respect to more restricted sentences systems, which we call $\mathcal{F}$ -systems

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Brian Rabern
University of Edinburgh

Citations of this work

A graph-theoretic analysis of the semantic paradoxes.Timo Beringer & Thomas Schindler - 2017 - Bulletin of Symbolic Logic 23 (4):442-492.
Dangerous Reference Graphs and Semantic Paradoxes.Landon Rabern, Brian Rabern & Matthew Macauley - 2013 - Journal of Philosophical Logic 42 (5):727-765.
What Paradoxes Depend on.Ming Hsiung - 2018 - Synthese:1-27.
A Unified Theory of Truth and Paradox.Lorenzo Rossi - 2019 - Review of Symbolic Logic 12 (2):209-254.

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References found in this work

Outline of a theory of truth.Saul Kripke - 1975 - Journal of Philosophy 72 (19):690-716.
Paradox without Self-Reference.Stephen Yablo - 1993 - Analysis 53 (4):251-252.
Truth and reflection.Stephen Yablo - 1985 - Journal of Philosophical Logic 14 (3):297 - 349.
Yablo’s paradox.Graham Priest - 1997 - Analysis 57 (4):236–242.
Yablo's paradox.Graham Priest - 1997 - Analysis 57 (4):236-242.

View all 15 references / Add more references