§1. Introduction. The problem raised by the liar paradox has long been an intriguing challenge for all those interested in the concept of truth. Many “solutions” have been proposed to solve or avoid the paradox, either prescribing some linguistical restriction, or giving up the classical true-false bivalence or assuming some kind of contextual dependence of truth, among other possibilities. We shall not discuss these different approaches to the subject in this paper, but we shall concentrate on a kind of formal construction which was originated by Kripke's paper “Outline of a theory of truth” [11] and which, in different forms, reappears in later papers by various authors.

The main idea can be presented as follows: assume a first order language ℒ containing, among other unspecified symbols, a predicate symbol T intended to represent the truth predicate for ℒ. Assume, also, a fixed model M = 〈D, I〉 (the base model)where D contains all sentences of ℒ and I interprets all non-logical symbols of ℒ except T in the usual way. In general, D might contain many objects other than sentences of ℒ but as that would raise the problem of the meaning of sentences in which T is applied to one of these objects, we shall assume that this is not the case.