Yablo paradox in partial semantics and potentially infinite domains Michal Tomasz Godziszewski April 22, 2015 Abstract The paper describes properties of Yablo sequences over growing domains of finite arithmetical models and over partial models of Kripke truth theory. We show that for any partial fixed-point model and for the Strong Kleene, Weak Kleene and Supervaluation valuation schema, all Yablo sentences Y (n) are neither true nor false under these schema or equivalently: the truth-value of all Yablo sentences Y (n) in fixed-point partial models under any of the above valuation scheme, is indeterminate. Furthermore, we show that under the logic of sufficiently large finite models (logic of potential infinity) all the Yablo sentences are false in the limit. The main philosophical conclusion is that a finitist, not accepting the notion of actual infinity may adopt the sl-theory of arithmetics as an adequate explication of the concept of potential infinity and simply give an answer to the Yablo paradox that in the limit (i.e. under the sl-semantics on the FM-domain of a given arithmetical model) all the Yablo sentences are false. 1 Preliminaries In 1993, Stephen Yablo gave in [27] an example of a semantic paradox that, according to the author, does not involve self-reference. He began his seminal paper with a question: Why are some sentences paradoxical while others are not? Since Russell the universal answer has been: circularity, and more especially self-reference. It is clear that the appearance of self-reference in a sentence is not a sufficient condition for the sentence to be paradoxical. However, it has been widely believed among philosophers that self-reference or circularity is a necessary feature of a sentence or a set of sentences that we claim to be paradoxical or antinomial. We will now present the paradox and define formally the sequence of sentences that generates the paradox within the framework of arithmetic. Let us consider the following infinite sequence of sentences: Y 0 = ∀k> 0 ¬ Tr(Y k ) Y 1 = ∀k> 1 ¬ Tr(Y k ) 1