Reference and Truth

Abstract

I apply the notions of alethic reference introduced in previous work in the construction of several classical semantic truth theories. Furthermore, I provide prooftheoretic versions of those notions and use them to formulate axiomatic disquotational truth systems over classical logic. Some of these systems are shown to be sound, proof-theoretically strong, and compare well to the most renowned systems in the literature.

Alethic Reference

“…Nonetheless, one might think that by ignoring t's subterms we could be overlooking dangerous reference patterns. I show this is not the case in my companion paper [22]. Moreover, note that allowing that Tt m-refers to all denotations of subterms of t would result in a grossly overgenerating notion.…”

“…With the right notions of alethic reference and self-reference in place, we will be in a position to properly evaluate the orthodox view on semantic paradoxes. Moreover, the new notions also allow us to formulate sensible restrictive criteria for instances of disquotation resulting in philosophically and technically appealing truth theories, as shown in my companion paper [22].…”

“…But this doesn't mean there is no use for the new notions of reference in the quest for the root of paradox or in the formulation of interesting truth theories. Although restricting our truth principles to non-selfreferential expressions will not always lead to sound truth systems, restricting them to well-founded sentences will, as I show in the companion piece, [22]. This suggests that it is not self-reference but unfoundedness which is behind the semantic paradoxes.…”

“…Recall the aim of this paper is to offer an appropriate account of reference for the study of the reference patterns underlying paradoxical expressions. However, the ultimate hope (realized in the companion paper [22]) is to find shared patterns in terms of which we can formulate sensible restricting criteria on the instances of disquotation. In turn, the truth principles we embrace will then determine the extension of the truth predicate.…”

“…footnote 8). 22 The salient features of Kripke's models are that each sentence ϕ and its truth ascription T ϕ receive the same truth value, and that paradoxical sentences are neither true nor false. In addition, the biconditional in the Strong Kleene evaluation scheme behaves in such a way that ϕ ↔ ψ is true only if ϕ and ψ are both true or both false; if either ϕ or ψ are neither true nor false, so is ϕ ↔ ψ.…”