Page 407 - Encyclopedia of Philosophy of Language

Encodings, the T predicate and the subx function,
are the ingredients of the recipe for the formal recon-
struction of the liar. The key formula that we need
now is —Tsubx(x, x). Call this formula cp. What q> formulas, as follows: means may be a bit hard to grasp at first reading. The
diligent reader is asked to check that cp is true of all
numbers which encode formulas that, when their own
code number is substituted in them for the variable x,
yield sentences with code numbers that do not satisfy
T.
Let n be the code number of <p, that is, n = [q>~\ . Next, let ^ be the formula [n/x](p. The formalized liar is sentence (3):
\II=—T\\I/~\ (3)
It will be shown that now sentence (3) is true in M. Truth of \l/ is equivalent to truth of [n/x]q>, because $ =[n/x](p. Truth of [n/x](p is equivalent to truth of —Tsubx(n, n), since <p= —Tsubx(n, n) and n is sub- stituted for x in (p.Truth of —Tsubx(n, h) isequivalent totruthof—Tsubx(fcp~|,h),sincen=[<p~\.Truth of —Tsubx( [(p~\ ,ri) is equivalent to truth of —T |~[n/x](p "I , from the definition of subx. Finally, truth of —T [[nlx](f>~\ is equivalent to truth of —T |~^1 , since if/ = [n/x](p. This shows the equiv- alence of ^ and —T f \l/ ~] in M, so (3) is true in M.
Next, (3) and the /"-principle can be used to derive a contradiction. Assume $ is true (in M). Then, because of (3), —T \\j/~\ is true, so T f ^ ] is false, and, because of (2), \l/ is false. Contradiction. Assume \l/ is false. Consequently, because of (3), —T f \jt ~| is false, so T f \l/ ~\ is true and, because of (2), ty is true. Contradiction again. This paradox is closely related, in fact, to the half-paradox used by Kurt Godel to show the incompleteness of the first order theory of arithmetic and to derive the undecidability of first order predicate logic.
As the logical representation languages used for natural language semantics have at least the same expressive power as first order predicate logic, the above result applies to them as well. Tarski has drawn the conclusion that providing a precise semantic account of the concept of truth in natural languages is a more or less hopeless enterprise.
2. TheRevisionTheoryofTruth
Fortunately, things are not quite as gloomy as they look. Consider an arbitrary language L with a truth predicate T, and a device for naming sentences of the language by means of encoding or by some other means; f <p~\ will continue to be used as a name for the code of <p. It turns out that a truth predicate T for L can be constructed by switching to a dynamic perspective on how Tgets its proper extension.
Initially there is a model where the one place predi- cate T has some arbitrary initial extension, and then iterates through an infinite series of T revision steps. In the course of these revisions the interpretation of
= l(0), then at the next stage,
For nonparadoxical sentences, it is seen that through the revisions, gradually the T predicate approaches closer and closer to the status of a real truth predicate. In the initial stage, only the interpretations of the sentences that do not involve truth are guaranteed to be right. At the starting point, sentences like Snow is white will be true, but It is true that snow is white; It is true that it is true that snow is white, and so on, will have arbitrary truth values, since the predicate T occurs in them. The interpretation of T has to be revised. In the first step all (codes of) sentences which are true at the starting point are put in the extension of T, and T will be false of all (codes of) sentences which are false at the starting point (and all objects which are not codes of sentences at all). The result is that at this stage sentences such as // is true that snow is white become true. This process is then iterated. To see that this has to go on for quite a long time, consider an infinite set of sentences Snow is white; It is true that snow is white; It is true that it is true that snow is white, and so on. Call this set S. It is desirable to be able to say that all sentences in S are true; this can certainly be asserted in natural language (and the assertion would of course be true). To take such cases into account the process of revising the interpretation of T must be carried through into the transfinite.
It can be shown that there is some transfinite stage a at which the process stabilizes: at this stage, all sentences that eventually stop flipflopping are stable already. At this level Tis still unstable for paradoxical sentences. To see why this is so, consider the for- malizedliar(3),theformulai//=-T |~<A~|.Bythe reasoning that was given above, (3) is true in the initial stage of our model, and in all further stages (the reasoning does not depend on the extension of T). The extension of T at the initial stage was arbitrary. Letussuppose[7 f^l ]=-/atthisstage.Thenat
T in the startup state of the model will become less and less significant. Every revision step influences the interpretation of the terms which are codes of
If at a given stage, [r
the initial stage, [^] = 0, because of what (3) says. 7
Then after one revision, [T [ij/~\ ] = 0, because of what the revision instruction says, and because of (3), = I at this stage. And so on. Thus, the values of and [ r [\l/~\ ] will continue to flipflop through
the revision stages.
All sentences whose truth or falsity depends ulti-
mately on factors that do not involve the concept of truth will be stable at the stabilization stage a, and their truth value at this stage will be independent of the T interpretation we started out with. All paradoxical sentences will be unstable at a. A sentence such as This sentence is true will be stable as well, but which truth value it will have will depend on the original
Truth and Paradox
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