Reference patterns of semantic paradoxes and the problem of their graph-theoretic characterization

Beringer, Timo (2025): Reference patterns of semantic paradoxes and the problem of their graph-theoretic characterization. Dissertation, LMU München: Fakultät für Philosophie, Wissenschaftstheorie und Religionswissenschaft

Abstract

This thesis is a contribution to the field of semantic paradoxes. It joins a tradition of works that investigate the questions of why certain referential structures of sentences or sentence systems lead to semantic paradoxes. The approach presented in Chapter 2 is language-independent. The fundamental concept it is based on is that of a Boolean network. Other frameworks of the tradition that could be described as graph-theoretic approach to the semantic paradoxes can be embedded in ours straightforwardly. As in various other accounts, reference patterns are formally conceived as directed graphs. A question every graph-theoretic account has to answer is what graphs should be conceived as being potentially paradoxical. The answer is not straightforward, since graphs that occur as reference graphs of paradoxical sentences usually do so not exclusively, but occur also as reference graphs of sentences that are not paradoxical. I will suggest three candidates for the class of all potentially paradoxical directed graphs: dangerous digraphs, digraphs of infinite character and not strongly kernel-perfect digraphs. Each of them captures a different aspect of the intuition one might have about potentially paradoxical graphs and each gives rise to a characterization problem, i.e., the problem of finding a graph-theoretic property that is a necessary and sufficient condition for a graph to be a member to this class. To each of the three characterization problems I conjecture a solution. A directed graph is conjectured to be dangerous if and only if it contains a directed cycle or a finitary inflation of the Yablo-graph (i.e., the reference pattern of Yablo's paradox); it is conjectured to be of infinite character if and only if it contains a finitary inflation of the Yablo-graph; and it is conjectured to be not strongly kernel-perfect if and only if it contains an odd directed cycle or an odd finitary inflation of the Yablo-graph. It will be investigated how these conjectures are interrelated. The goal of Chapter 3 is to show that any Boolean network (and the question of whether it has a fixed point in particular) can be analyzed in terms of an associated directed graph. Such a graph is called a characteristic digraph of the Boolean network and contains more information about it than a reference graph. This reduces the question of whether a sentence system is paradoxical to a purely graph theoretic one. In Chapter 4 it is shown that all three criteria conjectured as sufficient and necessary are sufficient indeed. In Chapter 5 it is shown that the criterion for dangerous digraphs is necessary under certain additional assumptions.

• Dokumententyp: Dissertationen (Dissertation, LMU München)
• Keywords: semantic paradoxes, Yablo’s paradox, Boolean networks, kernels of directed graphs, graph minors
• Themengebiete: 100 Philosophie und Psychologie ; 100 Philosophie und Psychologie > 160 Logik
• Fakultäten: Fakultät für Philosophie, Wissenschaftstheorie und Religionswissenschaft
• Sprache der Hochschulschrift: Englisch
• Datum der mündlichen Prüfung: 10. Februar 2025
• 1. Bericht­erstatter:in: Leitgeb, Hannes