What paradoxes depend on

Synthese 197 (2):887-913 (2020)
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Abstract

This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (J Philos Logic 34(2):155–192, 2005), and the dependence digraph by Beringer and Schindler (Reference graphs and semantic paradox, 2015. https://www.academia.edu/19234872/reference_graphs_and_semantic_paradox). Unlike the usual discussion about self-reference of paradoxes centering around Yablo’s paradox and its variants, I focus on the paradoxes of finitary characteristic, which are given again by use of Leitgeb’s dependence relation. They are called ‘locally finite paradoxes’, satisfying that any sentence in these paradoxes can depend on finitely many sentences. I prove that all locally finite paradoxes are self-referential in the sense that there is a directed cycle in their dependence digraphs. This paper also studies the ‘circularity dependence’ of paradoxes, which was introduced by Hsiung (Logic J IGPL 22(1):24–38, 2014). I prove that the locally finite paradoxes have circularity dependence in the sense that they are paradoxical only in the digraph containing a proper cycle. The proofs of the two results are based directly on König’s infinity lemma. In contrast, this paper also shows that Yablo’s paradox and its ∀∃\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \exists $$\end{document}-unwinding variant are non-self-referential, and neither McGee’s paradox nor the ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document}-cycle liar has circularity dependence.

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original Hsiung, Ming (2018) "What Paradoxes Depend on". Synthese 0():1-27

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Author's Profile

Ming Hsiung
Zhongshan University

Citations of this work

Paradoxes and contemporary logic.Andrea Cantini - 2008 - Stanford Encyclopedia of Philosophy.
Unwinding Modal Paradoxes on Digraphs.Ming Hsiung - 2020 - Journal of Philosophical Logic 50 (2):319-362.
Notes on Leitgeb’s What Truth Depends on.Edoardo Rivello - 2020 - Studia Logica 108 (6):1235-1262.
From Paradoxicality to Paradox.Ming Hsiung - 2024 - Erkenntnis 89 (7):2545-2569.

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References found in this work

Paradox without Self-Reference.Stephen Yablo - 1993 - Analysis 53 (4):251-252.
Axiomatic Theories of Truth.Volker Halbach - 2010 - Cambridge, England: Cambridge University Press.
Truth and paradox.Anil Gupta - 1982 - Journal of Philosophical Logic 11 (1):1-60.
Notes on naive semantics.Hans Herzberger - 1982 - Journal of Philosophical Logic 11 (1):61 - 102.
Truth and reflection.Stephen Yablo - 1985 - Journal of Philosophical Logic 14 (3):297 - 349.

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