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A GRAPH-THEORETIC ANALYSIS OF THE SEMANTIC PARADOXES

Published online by Cambridge University Press:  15 February 2018

TIMO BERINGER  and
THOMAS SCHINDLER
TIMO BERINGER
Affiliation:
MUNICH CENTER FOR MATHEMATICAL PHILOSOPHY LMU MUNICH MUNICH, GERMANYE-mail: timob2001@yahoo.de
THOMAS SCHINDLER
Affiliation:
CLARE COLLEGE UNIVERSITY OF CAMBRIDGE CAMBRIDGE, UKE-mail: thomas.schindler1980@gmail.com
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Abstract

We introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook [5, 6] and Rabern, Rabern, and Macauley [16]. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s [14] notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth values. It is shown that this notion of paradoxicality coincides with that of Kripke [13]. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo [21]), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs.

Keywords

03A0503B9905C20liar paradoxgraph theorytheories of truthself-referenceYablo’s paradox
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 23 , Issue 4 , December 2017 , pp. 442 - 492
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

REFERENCES

Aczel, P., An introduction to inductive definitions, Handbook of Mathematical Logic (Barwise, J., editor), North Holland, Amsterdam, 1977, pp. 739–782.CrossRefGoogle Scholar
Beringer, T. and Schindler, T., Reference graphs and semantic paradox, The Logica Yearbook 2015 (Arazim, P. and Dancak, M., editors), College Publications, London, 2016, pp. 1–15.Google Scholar
Bolander, T., Logical theories for agent introspection, Ph.D. thesis, Informatics and Mathematical Modelling (IMM), Technical University of Denmark, 2003.Google Scholar
Cantini, A., A theory of formal truth arithmetically equivalent to ID1 . The Journal of Symbolic Logic, vol. 55 (1990), pp. 244–259.Google Scholar
Cook, R. T., Patterns of paradox. The Journal of Symbolic Logic, vol. 69 (2004), no. 3, pp. 767–774.Google Scholar
Cook, R. T., The Yablo Paradox: An Essay on Circularity, Oxford University Press, Oxford, 2014.Google Scholar
Diestel, R., Graph Theory, Springer, Heidelberg, 1997.Google Scholar
Dyrkolbotn, S. and Walicki, M., Propositional discourse logic. Synthese, vol. 191 (2014), pp. 863–899.CrossRefGoogle Scholar
Gaifman, H., Pointers to truth. Journal of Philosophy, vol. 89 (1992), pp. 223–261.Google Scholar
Herzberger, H., Paradoxes of grounding in semantics. Journal of Philosophy, vol. 67 (1970), pp. 145–167.Google Scholar
Herzberger, H. G., Notes on naive semantics. Journal of Philosphical Logic, vol. 11 (1982), pp. 61–102.Google Scholar
Jongeling, T. B., Koetsier, T., and Wattel, E., Self-reference in finite and infinite paradoxes. Logique et Analyse, vol. 45 (2002), pp. 15–30.Google Scholar
Kripke, S., Outline of a theory of truth. Journal of Philosphy, vol. 72 (1975), pp. 690–716.Google Scholar
Leitgeb, H., What truth depends on. Journal of Philosphical Logic, vol. 34 (2005), pp. 155–192.Google Scholar
Meadows, T., Truth, dependence, and supervaluation: Living with the ghost. Journal of Philosophical Logic, vol. 42 (2013), pp. 221–240.Google Scholar
Rabern, L., Rabern, B., and Macauley, M., Dangerous reference graphs and semantic paradoxes. Journal of Philosophical Logic, vol. 42 (2013), no. 5, pp. 727–765.Google Scholar
Schindler, T., Axioms for grounded truth. Review of Symbolic Logic, vol. 7 (2014), pp. 73–83.Google Scholar
Schindler, T., Type-free truth, Ph.D. thesis, Ludwig-Maximilians-Universität München, 2015.Google Scholar
Walicki, M., Reference, paradoxes and truth. Synthese, vol. 171 (2009), pp. 195–226.Google Scholar
Welch, P., Games for truth. The Journal of Symbolic Logic, vol. 15 (2009), no. 4, pp. 410–427.Google Scholar
Yablo, S., Grounding, dependence, and paradox. Journal of Philosophical Logic, vol. 11 (1982), no. 1, pp. 117–137.Google Scholar
Yablo, S., Paradox without self-reference. Analysis, vol. 53 (1993), pp. 251–252.Google Scholar