Truth, Reflection, and Hierarchies

A common objection to hierarchical approaches to truth is that they fragment the concept of truth. This paper defends hierarchical approaches in general against the objection of fragmentation. It argues that the fragmentation required is familiar and unproblematic, via a comparison with mathematical proof. Furthermore, it offers an explanation of the source and nature of the fragmentation of truth. Fragmentation arises because the concept exhibits a kind of failure of closure under reflection. This paper offers a more precise characterization of the reflection involved, first in the setting of formal theories of truth, and then in a more general setting. truthprooffinal.tex: January 27, 2003 (12:03) It is often noted that Tarski’s [1935] hierarchy of languages and metalanguages fragments the concept of truth. Instead of one concept, we have infinitely many, arranged in a hierarchy. Subsequent work along Tarskian lines [e.g. Parsons, 1974b; Burge, 1979; Barwise and Etchemendy, 1987] has proposed, in various ways, to base hierarchies on what speakers can express in a given context, rather than on multiple languages or concepts of truth per se. In my own work [2001; MS], I argue that a hierarchy of domains of propositions is in fact generated by the workings of linguistic context. I thus argue that natural language considerations lend some plausibility to a hierarchical approach. But the objection of fragmentation still stands. One way or another, hierarchical theories all require that speakers cannot in any one instance express the entirety of a unified concept of truth. In this essay, I shall defend hierarchical approaches in general against the objection of fragmentation. I shall do so in two ways. First, I shall argue that the fragmentation required is familiar. We see just the same sort of fragmentation in the concept of mathematical proof. In the case of mathematical proof, we find the fragmentation banal, or at least unthreatening, and perhaps even a source of richness of the concept. It is not problematic there, and it is not problematic in the case of truth either. Second, in the course of this argument, I shall attempt to make clearer the source and nature of the fragmentation of these concepts. It has been observed both for proof [Kreisel, 1970] and for truth [Parsons, 1974a,b] that fragmentation arises because these concepts exhibit a kind of failure of closure under reflection. Very roughly, any sufficiently precise articulation of either concept allows for a kind of reflection upon the correctness of that articulation that leads to a distinct, stronger one. I shall here investigate this reflection. Building on the work of Kreisel and Parsons, I shall offer a more precise characterization of it in the setting of formal theories of truth, and I shall show that this helps us to understand the hierarchical nature of the concept of truth in a more general setting. My arguments below will focus on a particularly strong and rarely noticed form of the Liar paradox, which I have dubbed the Fortified Liar paradox. As the Strengthened Liar paradox shows that a natural answer to the Liar paradox leads back to paradox, the Fortified Liar shows that a natural answer to the Strengthened Liar leads back to paradox as well. I shall argue that the Fortified Liar helps to make clear why the concept of truth is hierarchical, and helps explain the sense in which the hierarchy is generated by a process of reflection.