Patterns of paradox

Cook, Roy T.

doi:10.2178/jsl/1096901765

Cited by 52 publications

(63 citation statements)

References 3 publications

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“…Variants of this format were implicit in [9,22], and elaborated in [34]. It does not limit the expressive power, and provides a normal form for propositional theories, as shown in [5].…”

Section: Formalizationmentioning

confidence: 99%

“…A local kernel L gives thus a general concept of a "coherent subdiscourse", in the sense of a subset of statements, namely dom(α L ), which can be consistently assigned truth-values, obeying the rules (2.5), irrespectively of the assignment to all other statements. 9 For instance, the graph D from (2.4) has no kernel, but {d, b} is its local kernel, and so is {e} (the latter inducing the subdiscourse F from Example 2.13. (2).)…”

Section: Kernels Of Digraphsmentioning

confidence: 99%

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Propositional discourse logic

Dyrkolbotn

Walicki

2013

Synthese

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A novel normal form for propositional theories underlies the logic pdl, which captures some essential features of natural discourse, independent from any particular subject matter and related only to its referential structure. In particular, pdl allows to distinguish vicious circularity from the innocent one, and to reason in the presence of inconsistency using a minimal number of extraneous assumptions, beyond the classical ones. Several, formally equivalent decision problems are identified as potential applications: non-paradoxical character of discourses, admissibility of arguments in argumentation networks, propositional satisfiability, and the existence of kernels of directed graphs. Directed graphs provide the basis for the semantics of pdl and the paper concludes by an overview of relevant graph-theoretical results and their applications in diagnosing paradoxical character of natural discourses.

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“…Variants of this format were implicit in [9,22], and elaborated in [34]. It does not limit the expressive power, and provides a normal form for propositional theories, as shown in [5].…”

Section: Formalizationmentioning

confidence: 99%

Section: Kernels Of Digraphsmentioning

confidence: 99%

Propositional discourse logic

Dyrkolbotn

Walicki

2013

Synthese

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“…To give criteria for the identity of paradoxes is a substantial problem that would involve an analysis of the structure of paradoxical arguments that, as far as we know, has not been undertaken yet. A first step into what is, we think, a promising direction are recent works by Cook (2004) and Schlenker (2007aSchlenker ( , 2007b) that show that self-reference can be eliminated from certain non-quantificational languages in the following sense: we can systematically transform each self-referential sentence into an infinite set of sentences in a quantificational language satisfying these two conditions: (a) they are not self-referential; and (b) they preserve the truth-value of the…”

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confidence: 99%

Eliminating Self-Reference from Grelling's and Zwicker's Paradoxes

Fernández

Abad

2014

THEORIA

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“…the language of arithmetic)-in this way the only resource available is "potential infinities" in the form of recursive definitions that are circular by their very nature. By Yablo's paradox we mean the infinitary version that does not involve (strong or weak) "fixed point circularity".3 Although some preliminary investigations into paradox supporting structures have been conducted in Yablo [1982], Yablo [1993], Yablo [2006] and Cook [2004] on a special class of restricted languages (see Appendix D). …”

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confidence: 99%

“…3 Although some preliminary investigations into paradox supporting structures have been conducted in Yablo [1982], Yablo [1993], Yablo [2006] and Cook [2004] on a special class of restricted languages (see Appendix D).…”

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confidence: 99%

Dangerous Reference Graphs and Semantic Paradoxes

Rabern

Macauley

2012

J Philos Logic

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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 ...

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Patterns of paradox

Cited by 52 publications

References 3 publications

Propositional discourse logic

Propositional discourse logic

Eliminating Self-Reference from Grelling's and Zwicker's Paradoxes

Dangerous Reference Graphs and Semantic Paradoxes

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