

Workshop on Proof Theory, Modal Logic and Reflection Principles
October 18, 2017 12:50–13:25, Moscow, Steklov Mathematical Institute






Systems of propositions referring to each other: a modeltheoretic view
D. Saveliev^{} 
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Abstract:
We investigate arbitrary sets of propositions such that some of them state that some of them (possibly, themselves) are wrong, and criterions of paradoxicality or nonparadoxicality of such systems. For this, we propose a finitely axiomatized firstorder theory with one unary and one binary predicates, $\mathrm{T}$ and $\mathrm{U}$. An heuristic meaning of the theory is as follows: variables mean propositions, $\mathrm{Tx}$ means that $\mathrm{x}$ is true, $\mathrm{Uxy}$ means that $\mathrm{x}$ states that $\mathrm{y}$ is wrong, and the axioms express natural relationships of propositions and their truth values. A model $\mathrm{(X,U)}$ is called nonparadoxical iff it can be expanded to some model $\mathrm{(X,T,U)}$ of this theory, and paradoxical otherwise. E.g. a model corresponding to the Liar paradox consists of one reflexive point, a model for the Yablo paradox is isomorphic to natural numbers with their usual ordering, and both these models are paradoxical.
We show that the theory belongs to the class $\Pi_2^0$ but not $\Sigma_2^0$ and is undecidable. We propose a natural classification of models of the theory based on a concept of collapsing models. Further, we show that the theory of nonparadoxical models, and hence, the theory of paradoxical models, belongs to the class $\Delta_1^1$ but is not elementary. We consider also various special classes of models and establish their paradoxicality or nonparadoxicality. In particular, we show that models with reflexive relations, as well as models with transitive relations without maximal elements, are paradoxical; this general observation includes the instances of Liar and Yablo. On the other hand, models with wellfounded relations, and more generally, models with relations that are winning in sense of a certain membership game are nonparadoxical. Finally, we propose a natural classification of nonparadoxical models based on the abovementioned classification of models of our theory.
This work was supported by grant 161110252 of the Russian Science Foundation.
Language: English

