Page 406 - Encyclopedia of Philosophy of Language

Formal Semantics
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In natural languages, extensive use is made of a truth predicate. We continually say things like What John has just said is false; Most statements made by the government spokeswoman are true, etc. This article discusses the relevance of the paradox of the liar for the concept of truth in formal and natural languages. The first part of the article focuses on first order logic; it explains the problem posed by the liar for the incor- poration of a truth predicate in first order languages, or in any language with at least the same expressive power. The second part shows how a dynamic per- spective on arriving at truth through revision can yield a predicate which accurately reflects truth for all non- paradoxical sentences. This result applies to both for- mal and natural languages.
1. The Liar Paradox and the T-Principle
In its simplest guise, the liar paradox revolvesaround sentence (1):
This sentence is false. (1)
Sentence (1) is paradoxical, for both the assumption that it is true and the assumption that it is false lead to a contradiction. For instance, assume that (1) is true. Then because of what (1) says, it must be false. Contradiction. Now assume that (1) is false. Then what the sentence says is not true, that is, it is not true that (1) is false. In other words, (1) is true, and contradiction again.
Suppose first order language L has a predicate T, to be interpreted as 'true in model M,' where M is some intended model of L. To make this work, one has to assume that there is a function f ~| mapping the (closed or open) formula <pto a term [<p~| denot-
ing an individual in the domain of M. Given such a function, we can say that the term f <p~\ represents the formula (p. The representation trick can be pulled off by including the natural numbers in the domain of M and devising an encoding of formulas as numbers. The encoding can be made to work for all syntactic objects; for instance, f x] is the term representing the variable x. Note that fx~| is a closed term: the variable x does not occur freely in it, because fx~| is just a name for a number, namely the number rep- resenting x. More generally, f/~j is the closed term representing the open or closed term t.
Give the encoding function, one would expect the following principle to hold in model M for all sen- tences <JPof L:
(p=T [q>~\ (2)
We call this the T-principle; sometimes it is also referred to as the convention T. What the /"-principle says, in fact, is that the truth predicate holds precisely of the representations of the sentences which are true (in M). We now proceed to show that there is some- thing very wrong with the T-principle.
When (pis a formula with x among its free variables, and MS a term, then we use [t/x]<p for the result of substituting / for x in (p. Next, we mirror this sub- stitution operation at the level of encodings. If <pis a formula with x among its free variables and t is a term,thenweusesubx([<p~\,[t~\)forf['/*]<?1• Thus, sub., is a function mapping pairs of code num- bers to new code numbers. The proof that subx can be constructed from f 1 is outside our scope (see Enderton 1972). Note that x is not a free variable in sub*.
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