Dangerous Reference Graphs and Semantic Paradoxes

Abstract

Abstract. The semantic paradoxes are often associated with self-reference or referential circularity. Yablo [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 [2004] and Yablo [2006] with respect to more restricted sentences systems, which we call F-systems.Keywords: Paradox, Hypodox, Reference structure, Circularity, Ungroundedness, Yablo's paradox, Liar paradox, Graph theory, Dangerous, Precarious, F-system, Kernel.The semantic paradoxes are often associated with self-reference or referential circularity. Yablo [1993] 1 , however, has shown that there are infinitary versions of the paradoxes that do not involve this form of circularity. 2 The attempts to purge the semantic antimonies by banning self-reference or by constructing sophisticated hierarchies only eliminate the class of paradoxes that rely on the circular reference structures-if cyclical reference is not essential to the semantic paradoxes, then the acyclical paradoxes remain unscathed. It remains an open question what relations of reference between collections of sentences afford the structure necessary for paradoxicality. Since "circularity" has traditionally been assumed to be essential, this issue has been underrepresented in the literature on truth and semantic paradoxes 3 -but it is clear that no such theory can lay claim to comprehensiveness until this question is answered. The resolution of this general question, then, has great import for ...

A Graph-Theoretic Analysis of the Semantic Paradoxes

“…Now let us turn to the classification problem for dangerous graphs. First, we can formulate and prove in our framework the following result of Rabern et al [16]: Proof. The left-to-right direction follows from Corollary 3.3.…”

“…There have been several quite interesting approaches to characterize the notion of a paradoxical sentence for infinitary propositional languages using (unsigned) reference graphs (i.e., graphs that do not distinguish between 'positive' and 'negative' reference). We mention here in particular the work of Cook [5,6] and, more recently, Rabern, Rabern, and Macauley [16]. 1 Our goal is to extend their work in several ways: first, to develop reference graphs (rfgs) for first-order languages; second, to develop new tools for investigating paradoxical reference patterns; third, to develop the notion of a signed rfg.…”

“…Based on Leitgeb's notion of semantic dependence, we will introduce a valuation scheme (called the Leitgeb valuation scheme, V L ) in terms of which we will define the notion of an acceptable decoration of an rfg. Then, generalizing on Rabern et al [16], we will say that a sentence φ is paradoxical if and only if there is no rfg of φ that admits an acceptable decoration (where the quantifier in the definition takes care of the fact that φ might have several rfgs). As we will see later, our notion of paradoxicality coincides (extensionally) with that of Kripke [13] with respect to the Leitgeb valuation scheme.…”

“…V ) iff there is no fixed point (of J V ) in which it receives a definite truth value. Moreover, adapting a notion by Rabern et al [16], we say that a sentence φ is Kripke-hypodoxical (w.r.t. V ) iff there are two fixed points of J V such that ϕ has one definite truth value in the first and a different definite truth value in the second.…”

Theories of truth based on four-valued infectious logics

“…In common terms, both τ and λ are therefore taken to be pathological sentences, where the pathology referred is some kind of semantic oddness. In more appropriate terms, it is said that inconsistent sentences like λ are paradoxical, whereas indeterminate sentences like τ are hypodoxical (see [30], [31], [32] and [66] for this terminology). These semantic conundrums led many scholars-who believe the truth predicate should behave transparently, and who also think that self-referential sentences like the above rightfully arise in natural languages-to go beyond the limits of classical logic and to embrace some other non-classical systems to reason with the truth predicate.…”