The unwinding that Cook, 767–774 2004) proposed is a simple but powerful method of generating new paradoxes from known ones. This paper extends Cook’s unwinding to a larger class of paradoxes and studies further the basic properties of the unwinding. The unwinding we study is a procedure, by which when inputting a Boolean modal net together with a definable digraph, we get a set of sentences in which we have a ‘counterpart’ for each sentence of the Boolean modal net and (...) each point of the digraph. What is more, whenever a sentence of the Boolean modal net says another sentence is necessary, then the counterpart of the first sentence at a point correspondingly says the counterparts of the second one at all accessible points of that point are all true. The output of the procedure is called ‘the unwinding of a Boolean modal net on a definable digraph’. We prove that the unwinding procedure preserves paradoxicality: a Boolean modal net is paradoxical on a definable digraph, iff the unwinding of it on this digraph is also paradoxical. Besides, the dependence digraph for the unwinding of a Boolean modal net on a definable digraph is proved to be isomorphic to the unwinding of the dependence digraph for the Boolean modal net on the previous definable digraph. So the unwinding of a Boolean modal net on a digraph is self-referential, iff the Boolean modal net is self-referential and the digraph is cyclic. Thus, on the one hand, the unwinding of any Boolean modal net on an acyclic digraph is non-self-referential. In particular, the unwinding of any Boolean modal net on \ is non-self-referential. On the other hand, if a Boolean modal net is paradoxical on a locally finite digraph, the unwinding of it on that digraph must be self-referential. Hence, starting from a Boolean modal paradox, the unwinding can output a non-self-referential paradox only if the digraph is not locally finite. (shrink)

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. (shrink)

We can classify the (truth-theoretic) paradoxes according to their degrees of paradoxicality. Roughly speaking, two paradoxes have the same degrees of paradoxicality, if they lead to a contradiction under the same conditions, and one paradox has a (non-strictly) lower degree of paradoxicality than another, if whenever the former leads to a contradiction under a condition, the latter does so under the same condition. In this paper, we outline some results and questions around the degrees of paradoxicality and summarize recent progress.

This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (2005), and the dependence digraph by Beringer & Schindler (2015). 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 (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 nested variant are non-self-referential, and neither McGee's paradox nor the omega-cycle liar paradox has circularity dependence. (shrink)

It is proved that Yablo’s paradox and the Liar paradox are equiparadoxical, in the sense that their paradoxicality is based upon exactly the same circularity condition—for any frame ${\mathcal{K}}$ , the following are equivalent: (1) Yablo’s sequence leads to a paradox in ${\mathcal{K}}$ ; (2) the Liar sentence leads to a paradox in ${\mathcal{K}}$ ; (3) ${\mathcal{K}}$ contains odd cycles. This result does not conflict with Yablo’s claim that his sequence is non-self-referential. Rather, it gives Yablo’s paradox a new significance: (...) his construction contributes a method by which we can eliminate the self-reference of a paradox without changing its circularity condition. (shrink)

We argue that Anselm’s ontological argument (or at least one reconstruction of it) is based on an empirical version of Berry’s paradox. It is invalid, but it takes some understanding of trivalence to see why this is so. Under our analysis, Anselm’s use of the notion of existence is not the heart of the matter; rather, trivalence is.

Although it was traditionally thought that self-reference is a crucial ingredient of semantic paradoxes, Yablo (1993, 2004) showed that this was not so by displaying an infinite series of sentences none of which is self-referential but which, taken together, are paradoxical. Yablo's paradox consists of a countable series of linearly ordered sentences s(0), s(1), s(2),... , where each s(i) says: For each k > i, s(k) is false (or equivalently: For no k > i is s(k) true). We generalize Yablo's (...) results along two dimensions. First, we study the behavior of generalized Yablo-series in which each sentence s(i) has the form: For Q k > i, s(k) is true, where Q is a generalized quantifier (e.g., no, every, infinitely many, etc). We show that under broad conditions all the sentences in the series must have the same truth value, and we derive a characterization of those values of Q for which the series is paradoxical. Second, we show that in the Strong Kleene trivalent logic Yablo's results are a special case of a more general fact: under certain conditions, any semantic phenomenon that involves self-reference can be emulated without self-reference. Various translation procedures that eliminate self-reference from a non-quantificational language are defined and characterized. An Appendix sketches an extension to quantificational languages, as well as a new argument that Yablo's paradox and the translations we offer do not involve self-reference. (shrink)

The goal of this paper is to present Yabloesque versions of Grelling’s and Zwicker’s paradoxes concerning the notions of “heterological” and “hypergame” respectively. We will offer counterparts of these paradoxes that do not seem to involve self-reference or vicious circularity.El objetivo de este artículo es ofrecer versiones de las paradojas de Grelling y de Zwicker inspiradas en la paradoja de Yablo. Nuestras versiones de estas paradojas no parecen involucrar ni autorreferencia ni circularidad viciosa.

The goal of this paper is to present Yabloesque versions of Grelling’s and Zwicker’s paradoxes concerning the notions of “heterological” and “hypergame” respectively. We will offer counterparts of these paradoxes that do not seem to involve any kind of self-reference or vicious circularity.