A Contextual-Hierarchical Approach to Truth and the Liar Paradox

This paper presents an approach to truth and the Liar paradox which combines elements of context dependence and hierarchy. This approach is developed formally, using the techniques of model theory in admissible sets. Special attention is paid to showing how starting with some ideas about context drawn from linguistics and philosophy of language, we can see the Liar sentence to be context dependent. Once this context dependence is properly understood, it is argued, a hierarchical structure emerges which is neither ad hoc nor unnatural. Your bait of falsehood takes this carp of truth: And thus do we of wisdom and of reach, With windlasses, and with assays of bias, By indirections find directions out: Hamlet II.i.68–71 It is a perennial idea in the study of the Liar paradox, from Tarski [56] onwards, that its solution requires some kind of hierarchy. More recent, but also quite common, is the idea that the paradox is to be solved by some sort of appeal to context dependence. Both ideas face a fundamental objection: they appear to be ad hoc solutions, unmotivated by our prior understanding of the concepts involved, and incompatible with the applications we make of them. Hierarchical solutions appear to require multiple truth predicates, which fragments what is intuitively one concept into many. Moreover, they impose restrictions on the application of the truth predicate that are incompatible with many of its ordinary and apparently unproblematic uses (as seminal work of Kripke [32] made abundantly clear). The idea of context dependence appears to fare no better. It simply does not appear that the sentences which generate the Liar paradox contain context-dependent elements, nor do we have any clear idea what aspect of context might be involved in the paradox. In this essay, I shall attempt to show that a combination of the ideas of context dependence and hierarchy can provide a non-ad hoc, well-motivated, and intuitively attractive approach to the Liar paradox. I shall argue that we really do have good reason, independent of the paradox itself, to see the Liar sentence as context dependent. I shall also argue that once we have a proper understanding of the nature of the context dependence involved, we find that a hierarchical structure of semantic relations, including the truth predicate, emerges. I shall thus argue that neither context dependence nor hierarchy are ad hoc responses to the paradox, but genuine features of the concepts involved. The hierarchy that so emerges is more liberal than the sort suggested by Tarski (and so avoids the objections from Kripke), but is able to resolve the paradoxes in an essentially hierarchical way. Much of the work in this paper is directed towards identifying and explaining the context dependence that is at work in the Liar. Once we understand it well enough, we can move on to explain how it leads to hierarchical structures. The paper is divided into three parts. Part (1) will make heavy use of concepts from linguistics, particularly the linguistics of context dependence. It will identify the context-dependence phenomenon at work in the Liar, and sketch out an account of how it can have the kinds of effects