1. The best known solution of the Liar to have emerged in recent times is Alfred Tarski's. Tarski located the root of the semantical paradoxes in the semantic closure of language. For a discussion of this, and of Tarski's theory of truth, see S. Haack, Philosophy of Logics (1978), pp.99-114 and Scott Soames, Understanding Truth (1999), pp.49-64

In brief, Tarski aims to provide a theory of truth which will be both materially adequate and formally correct. What the material adequacy condition amounts to is that the definition of truth should have as a consequence all instances of the T schema

(T) S is true iff p

where what replaces `S' is any name of the sentence replacing `p'.

2. Presumably this would rule out any theory which, while ascribing truth and falsity to sentences, accepted that some sentences are truth-valueless. For, in such a case, substitution of one such sentence and its name in the T-schema would produce a false LHS and a truth-valueless RHS.

3. As to formal correctness, what this means is that we have to avoid e.g. circular definitions and inconsistency. Inconsistency looms in the shape of the Liar paradox, and Tarski, in §7 of his 1944 paper, discusses the Liar `[i]n order to discover some of the more specific conditions which must be satisfied by languages in which (or for which) the definition of truth is to be given'.

We can readily adapt his example. Take the sentence

The sentence on the OHP in M167 at 9.45 a.m. on 26 February 2004 is not true.

For brevity, replace this sentence by the letter `s'.

Now, by substitution of this in (T) we get

`s' is true iff the sentence on the OHP in M167 at 9.45 a.m. on 26 February 2004 is not true.

But, as a matter of empirical fact, `s' is identical with the sentence on the OHP in M167 at 9.45 on 26 February 2004.

Hence the sentence on the OHP in M167 at 9.45 on 26 February 2004 is true iff the sentence on the OHP in M167 at 9.45 on 26 February 2004 is not true.

And from this, on the assumption that either `s' is true or false, a contradiction follows. For a similar demonstration, see Soames, p.50

4. So the natural question to raise is `What is responsible for this unsatisfactory outcome?' If we look into the reasoning that led to contradiction, we can see that very few assumptions were made. The first is that the language under consideration contains not just its regular vocabulary, but also the means for referring to such linguistic expressions, and also such semantical terms as `true' and `false'. A language with these resources is what Tarski calls `semantically closed'. Second, the reasoning presupposed that the ordinary rules of classical logic hold good. Since Tarski is unwilling to abandon classical logic, he identifies the source of the paradoxes as the semantic closure of language. So Tarski concludes that, in order to produce a definiton of truth that is formally correct, we must use a language that is not semantically closed. Now, we could achieve semantical non-closure by restricting ourselves to a language in which the possibility of forming the name of every expression was somehow banned. But this seems draconian: surely, in any language, we want to have the means for talking about any expression of that language. So the way that Tarski avoids semantic closure is by insisting that the semantical predicates `true' and `false' be defined only for semantically open languages.

5. In Tarski's view, natural languages are semantically closed, and so are prone to contradiction, so that there is no prospect of defining the concept of truth in natural languages. So instead he offered a definition of truth for formalized languages which would avoid paradoxicality. Subsequent writers have suggested that the hierarchical structure proposed by Tarski is present in natural languages. As applied to English, the suggestion would be that the truth predicate in English is not univocal; that English is a hierarchy of semantically open languages. At any level i+1 the truth predicate at that level applies only to sentences at levels below (similar remarks apply to the satisfaction relation).

6. What is the hierarchy of languages that Tarski proposes? At the bottom there is an Object Language O, which can be thought of as containing just ordinary expressions - no quotations of expressions and no semantical terms. At the next level up is a metalanguage M₁ which contains the resources (such as quotation) for referring to the expressions of O, and contains also the predicates `true-in-O', `false-in-O'. At the next level up, there is a metametalanguage M₂ which, in a similar way, contains the resources for talking about expressions in M₁.

7. How does such a hierarchical arrangement avoid the Liar? I quote Haack (pp.143-4): `Since, in this hierarchy of languages, truth for a given level is always expressed by a predicate of the next level, the Liar sentence can appear only in the harmless form `This sentence is false-in-O' which must itself be a sentence of M, and hence cannot be true-in-O, and is simply false instead of paradoxical.'

8. Now, does Tarski's solution simply amount to an evasion of the problem. There may be a suspicion that all he has done is to forget about natural languages in which the Liar arises, and to confine his attention to a formal language which, by careful stipulation of its syntax, is kept hygienically free of contradiction. Tarski says of the Liar: `In my judgment, it would be quite wrong and dangerous from the standpoint of scientific progress to depreciate the importance of this and other antinomies, and to treat them as jokes or sophistries. It is a fact that we are here in the presence of an absurdity, that we have been compelled to assert a false sentence ..... If we take our work seriously, we cannot be reconciled with this fact. We must discover its cause, that is to say, we must analyze the premises upon which the antinomy is based; we must then reject at least one of these premises, and we must investigate the consequences this has for the whole domain of our research' (§7). Earlier (§5) he claimed that the methods advocated in his paper help overcome the difficulties associated with various paradoxes (he mentions the Liar, Richard's antinomy of definability and Grelling-Nelson's antinomy of heterological terms), the idea being that `[t]he word `true', like other words from our everyday language, is certainly not unambiguous' (§3), and so we need a more precise definition in order to capture the fundamental Aristotelian intuition about truth - in modern dress, that the truth of a sentence consists in its agreement with (or correspondence to) reality. It appears, then that what Tarski is doing is to urge a modest reform of our language, or at least, a reform of the language we use when pursuing the science of semantics.

9. Objections. First, consider `(x)(x is a sentence and I utter x in this lecture -> x is true.)' It does not belong in the hierarchy, since it attributes truth to itself. But is there anything wrong with that sentence? Second (Kripke's objection, anticipated by Cohen). How can I assign a level to my assertion `All of Nixon's utterances about Watergate are false' in advance of knowing the levels of all of Nixon's statements, so that I can ensure that mine is one level higher? Also, the level of my statement cannot be a matter of form alone, since what level it has will depend on empirical facts about the levels of Nixon's statements. Statements cannot have `intrinsic' levels. Suppose that Dean made the above statement, but that Nixon said `Everything that Dean says about Watergate is false'. Now the system of assigning levels breaks down. Third (Gupta) how do we assign the level of `true' in a sentence like `Nothing is both true and untrue.' or `Every truth in X's works is already to be found in Y's works.'? Fifth, if you say `Nixon's last statement was true' and I reply `That's not true' then, on the Tarskian theory I'm not contradicting you, since I'm using a different `true' from you. This relates to a point made by Gupta (p.205 of R.L. Martin (ed.), Recent Essays on Truth and the Liar Paradox) `... the fact that a sentence is decided to be true at stage n does not mean that occurrences of `true' in the sentence are to be interpreted to mean `true at stage m' (m<n). A logical law such as `no sentence is both true and untrue' can be known to be true at the first level but this is no reason for reading the word `true' in it to mean `true at level zero'... the level or stage at which a sentence gets decided is one thing and the interpretation of the truth predicates in it is another.' This is one of the considerations that leads Kripke, Herzberger, Gupta and others to adopt a hierarchical approach (or, as Gupta calls it, `a stage-by-stage picture') in which, although the extension of `true' alters at different levels, its meaning remains the same.

10. Tarski’s is a hierarchical theory, and we have considered various objections to it. For further discussion, see Soames, op. cit. chaps. 3 and 4. We now turn to consider another approach which is of quite a different sort – the indexical theory (of which there are several versions).

11. The Strengthened Liar has been the pitfall of many attempted solutions to the Liar paradox. Take, for example, the `truth-value-gap' solution. Given a strengthened Liar of the form

L. L is not true

the gap theorist wants to say that L. is truth-valueless, i.e., that it's neither true nor false. But, if it's neither true nor false, that means, at least, that it's not true. So the gap theorist should be willing to accept as true that L is not true, in other words, to accept that L. is true. But this contradicts what he originally wanted to say about L., viz., that it's neither true nor false.

12. In recent years, one way around this problem has been suggested by Charles Parsons and Tyler Burge (both essays reprinted in Martin (ed.), New Essays). Their idea (quite different from that of the hierarchical solution of Tarski) is that the extension of the predicate true (i.e., the range of things to which it truly applies) changes from context to context. So, in particular, in the above reasoning about the Strengthened Liar, the word `true' does not apply to L. when it first appears in the formulation of L., but it does apply to L. on subsequent occurrences when we are reasoning about L. Martin, in his Introduction (pp.6-7), summarizes the account as follows:

Burge accepts claim S [viz., that there is a sentence that says of itself only that it is not true] and his own hierarchical version of [Tarski’s Convention] T [Any sentence is true iff what it says is the case], with the truth-predicate indexed according to context, and rejects the incompatibility argument [the standard Liar reasoning which purports to show that S and T are incompatible]. The Liar Sentence, p, understood as denying truthⁱ of itself is not trueⁱ (as shown in the usual way); but the application of (T) to p yields only that p is trueⁱ⁺¹, due to a shift in context when we come to evaluate p. By this last move, Burge is able to account also for the intuition [brought out in reflecting on the Strengthened Liar] namely that the Liar sentence seems to be true after all.

13. In saying that the truth predicate has a `shifting extension', we are assimilating it to an indexical expression. We are all familiar with the fact that tokens of the word `I', `here' etc. have no absolute or fixed extension; what they refer to on any occasion of use is determined by context. Similarly, tokens of the phrase `these books' will typically differ in extension (e.g. to the books I am indicating which are on my desk, to the books in a certain section of a library etc.). Note that the meaning of each of these expressions is the same in whatever context it is used.

14. Keith Simmons, in his paper `On a Mediaeval Solution to the Liar Paradox', History and Philosophy of Logic 8 (1987), pp.121-140, and also in his book, Universality and the Liar, Chap. 5 argues that this idea of treating `true' as an indexical term was anticipated by some mediaeval authors. But that the approach taken by these authors is interestingly different from that of Parsons or Burge.

15. Simmons investigates approaches taken by William of Ockham, Walter Burley and Pseudo-Sherwood which have a common core. Each considers a version of the paradox in which the only utterance Socrates makes is `Socrates says a falsehood'. To escape the looming contradiction, all three authors invoke (Ockham and Burley with qualifications) the rule of the restringentes: a part never supposits for the whole of which it is a part. (To supposit means to stand for, or refer to.) So, the predicate `falsehood' in Socrates' utterance does not supposit for Socrates' utterance, but for all other falsehoods.