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)

This paper is a continuation of [Łukowski, 2019], where it is shown that just like sets, sentences can also be understood in two ways: distributively or collectively. A distributive understanding of sets leads to the Russell antinomy, and a distributive understanding of sentences to liar antinomy. A collective understanding of sets frees up the set theory from Russell’s antinomy. Taking a similar approach to sentences no liar like paradoxes appear. The aim of the paper is to examine Yablo’s problem from (...) this collective perspective. Given its nature, by using the content implication connective it becomes possible to assign logical values to all Yablo’s sentences. Moreover, it seems that Yablo’s problem is not a case of circularity. (shrink)