Contextual-Hierarchical Reconstructions of the Strengthened Liar Problem

In this paper we shall introduce two types of contextual-hierarchical (from now on abbreviated by ‘ch’) approaches to the strengthened liar problem. These approaches, which we call the ‘standard’ and the ‘alternative’ ch-reconstructions of the strengthened liar problem, differ in their philosophical view regarding the nature of truth and the relation between the truth predicates T r ⁿ and T r ⁿ⁺¹ of different hierarchy-levels. The basic idea of the standard ch-reconstruction is that the T r ⁿ⁺¹-schema should hold for all sentences of (mathcal {L}^{n}). In contrast, the alternative ch-reconstruction, for which we shall argue in section four, is motivated by the idea that T r ⁿ and T r ⁿ⁺¹ are coherent in the sense that the same sentences of (mathcal {L}^{n}) should be true according to T r ⁿ and T r ⁿ⁺¹. We show that instances of the standard ch-reconstruction can be obtained by iterating Kripke’s strong Kleene jump operator. Furthermore, we will demonstrate how instances of the alternative ch-reconstruction can be obtained by a slight modification of the iterated axiom system KF and of the iterated strong Kleene jump operator.