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 \(\forall \exists \)-unwinding variant are non-self-referential, and neither McGee’s paradox nor the \(\omega \)-cycle liar has circularity dependence.
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Notes
This is not to reproach Leitgeb but to show a key concept in search of a more general notion of self-reference. After all, Leitgeb’s main purpose of using the dependence relation, is not to study self-reference of paradoxes but to give an adequate definition of truth in the first-order language of arithmetic with the unitary predicate T. See Leitgeb (2005, p. 171ff).
In a nutshell, Beringer and Schindler’s target is to characterize those ‘dangerous’ digraphs which can be a dependence digraph of some paradoxical sets of sentences. See also Beringer and Schindler (2017) for further elaboration of their program. I appreciate that one of anonymous reviewers reminds of me this reference after my manuscript has been accepted.
The notion of local finiteness, in the form of digraph, was offered by Rabern et al. (2013, p. 754) in the context of sentential language. See the discussion of the last section. By the way, by a paradox I always mean a paradoxical set of sentences.
Of course, this is merely a technical explanation, and for more philosophical motivations about paradoxicality in a digraph, please refer to Hsiung (2009a).
The digraph we define is actually the digraph without parallel directed edges. This restriction does not lose any generality for our purpose.
In the digraph \(\left\langle {\varSigma }, {\prec }_{f}\right\rangle \), \({\prec }_{f}\) is actually the restriction relation \({\prec }_{f}{\rceil }_{{\varSigma }\times {\varSigma }}\). This is always clear, and so the subscript is omitted.
The n-cycle liar is called the ‘Liar cycle’ in Leitgeb (2005, p. 164). It is also known as ‘n-liar’. Generally, we can define \(\alpha \)-liar for all ordinals \(\alpha \). Herzberger (1982, pp. 74–75) and Yablo (1985, p. 340). The case for \(\alpha = \omega \) will be given in Sect. 5. But we will use the term ‘\(\omega \)-cycle liar’ rather than ‘\(\omega \)-liar’, because the latter may cause confusing: for instance, the term ‘\(\omega \)-liar’ sometimes is also used for Yablo’s paradox. See Yablo (2004, p. 140).
For more details about Feferman’s dot notion, see for instance Halbach (2011, p. 32ff).
The dual of Yablo’s paradox, that is, the \(\exists \)-unwinding variant, was first given by Cook (2004, p. 771). And the dual of the \(\forall \exists \)-unwinding variant, i.e., the \(\exists \forall \)-unwinding variant, was put forward by Yablo (2004, p. 144). Yablo’s paradox and the above two variants were generalized in Schlenker (2007a). The notion of unwinding was first formulated by Cook (2004, p. 770) and the present nomenclature ‘\(Q_1\ldots Q_n\)-unwinding’ comes from Cook (2014, p. 155).
See Theorem 3.3.2 in Cook (2014).
There are other variants of Yablo’s paradox. For instance, Butler (2017) gave a recipe for constructing what he called ‘infinitely non*-variants’ of Yablo’s paradox. There are even continuum-many such variants, whose formalized counterparts in \({\mathscr {L}}^+\) (if any) can be proved to be non-self-referential by the similar method we use in Example 1. By the way, Butler also asserted that some of these paradoxes are non-self-referential. But his criterion, proposed by Priest (1997), is whether a circular predicate is involved in the construction of a paradox.
For any transfinite limit ordinal \(\gamma \), we can similarly set up the \(\gamma \)-cycle liar. See Herzberger (1982, pp. 74–75) and Yablo (1985, p. 340). Of course, only those \(\gamma \)-cycle liars consisting of countably sentences can be formalized in \({\mathscr {L}}^+\). Surprisingly, all of those transfinite \(\gamma \)-cycle liars have the same degree of paradoxicality. See Hsiung (2014, p. 36). Thus, all of these paradoxes are examples that can be paradoxical in a digraph without proper cycles.
The sentence \(\mu _0\) was first introduced by McGee (1985, p. 400).
The notion of sentence net was first put forward independently by Bolander (2002) and Cook (2002). My presentation is based upon Bolander (2003, p. 89), Cook (2004, p. 767) and Rabern et al. (2013, p. 734). As Bolander himself pointed out (Bolander 2003, pp. 108–109), the notion of sentence net has some precursors such as Visser’s ‘stipulation list’ in Visser (1989). It should be mentioned that Gupta and Belnap (1993, p. 72ff.) also developed some of Visser’s ideas about the stipulation lists.
Unlike Bolander, Rabern et al. (2013) did not give a definition of self-reference, but studied what a dependence digraph is like if it supports a paradox. For instance, they proved that if a locally finite dependence digraph is acyclic, then it can not supports any paradox. This is actually equivalent to the statement I just mentioned in the text. A sentence-net version of Theorem 1 was also proved independently by Hsiung (2009b).
It seems that the method of proving the equiparadoxicality of Yablos paradox and the Liar in Hsiung (2013) can be somewhat generalized to the paradoxes with digraph compactness. And so it might not hard to prove that the unwinding preserves the degree of paradoxicality for the locally finite paradoxes. But the situation is different and difficult for the \(\omega \)-cycle liar and McGee’s paradox.
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Acknowledgements
An earlier version of this paper was presented by the author in Delta 6 Logic Workshop at Sun Yat-sen University (March 2017). I am thankful to the organizers and the participants for their helpful suggestions and criticism. Thanks to two anonymous referees for their valuable comments and suggestions for improvement of this paper. I also appreciate Professor Roy T. Cook for his help.
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This research has been supported by Major Project of China National Social Science Foundation (No. 17ZDA025) and China Scholarship Council (No. 201706755008).
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Hsiung, M. What paradoxes depend on. Synthese 197, 887–913 (2020). https://doi.org/10.1007/s11229-018-1748-1
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DOI: https://doi.org/10.1007/s11229-018-1748-1