Abstract

The article is a critical essay of Evgeny Borisov’s research, which examines the logical structure and meaning of infinite semantic paradoxes (in particular, Yablo’s paradox). According to his view, the strict formalization of the infinite sequence of sentences in Yablo’s paradox requires selfreferential circularity descriptions. This view is based on Priest’s argument that a uniform representation of the content for Yablo’s paradoxical sentences can only be given by means of the two-place predicate of satisfaction. But it guarantees the existence of a fixed point for each sentence of Yablo’s paradox. Thus, we need to agree that Yablo’s paradox does involve circularity of a self-referential kind. However, Borisov believes that Priest’s argument is not sufficient for such a conclusion. His disagreement with Priest’s conclusion is based on the consideration of Sorensen’s and Beall’s metalanguage arguments. According to Sorensen, the specific properties of our formal (technical) descriptions are not inherited by the objects of such descriptions. On the contrary, Beall states that finite beings such as ourselves can know nothing about the actual infinity by demonstration. We cannot know how Yablo’s paradox could be represented in an arithmetic language without the help of descriptions. Priest’s argument shows that any such description is circular. It means that any entity generated by self-referential circularity description is itself circular. Comparing Sorensen’s and Beall’s arguments, Borisov states his own skeptical argument. He claims that the need to use circular descriptions is not a reliable evidence of the circularity of Yablo’s paradox. An alternative to Borisov’s skeptical views is the dilemmic argument that Yablo’s paradox either does involve circularity of a selfreferential kind or is not an example of a genuine semantic paradox. This view is based on the arguments of Cook and Ketland. According to Cook, we can unwind any finite semantic paradox to an infinite structure. Unwinding is a paradoxicality-preserving operation for replacing a sentence which says of itself with an infinite sequence of sentences which say of their successors. It shows that Yablo’s paradox is just an original example of circulus vitiosus. Ketland claims that Yablo’s paradox can be strictly formalized as an infinite conjunction of sentence tokens without using the predicate of truth (or satisfaction). It allows showing that Yablo’s infinite sequence is not a genuine semantic paradox, but rather a ω-paradox.