Dangerous Reference Graphs and Semantic Paradoxes

Journal of Philosophical Logic 42 (5):727-765 (2013)
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
Keywords Paradox  Hypodox  Reference structure  Circularity  Ungroundedness  Yablo’s paradox  Liar paradox  Graph theory  Dangerous  Precarious   $\mathcal{F}$ -system  Kernel
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References found in this work BETA
Roy T. Cook (2004). Patterns of Paradox. Journal of Symbolic Logic 69 (3):767-774.
Saul A. Kripke (1975). Outline of a Theory of Truth. Journal of Philosophy 72 (19):690-716.
Graham Priest (1997). Yablo’s Paradox. Analysis 57 (4):236–242.

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Similar books and articles
O. Bueno & M. Colyvan (2003). Yablo's Paradox and Referring to Infinite Objects. Australasian Journal of Philosophy 81 (3):402 – 412.
Ming Hsiung (2013). Equiparadoxicality of Yablo's Paradox and the Liar. Journal of Logic, Language and Information 22 (1):23-31.
J. C. E. Dekker (1981). Twilight Graphs. Journal of Symbolic Logic 46 (3):539-571.
Stephen Yablo (2004). Circularity and Paradox. In Thomas Bolander, Vincent F. Hendricks & Stig Andur Pedersen (eds.), Self-Reference. Csli Publications. 139--157.
Timothy Williamson (2000). Semantic Paradox and Semantic Change. The Proceedings of the Twentieth World Congress of Philosophy 2000:113-124.
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