In some related cases subscripting would not even be possible in principle, let alone feasible in practice. In Kripke's most famous example, Dean wants to say
(2a) Most of Nixon's Watergate-related statements are false.
while Nixon wants to say
(2b) Most of Dean's Watergate-related statements are true.
On the one hand, the subscript in (2a) would have to be higher than any subscript in any of Nixon's Watergate-related statements, including (2b), while on the other hand, the subscript in (2b) would have to be higher than any subscript in any of Dean's Watergate-related statements, including (2a); and this cannot be.
Yet intuitively (2a) and (2b), understood as in ordinary, unsubscripted English, might both well be true. If at least 75% of Nixon's Watergate-related statements did not use “true” at all, and at least 75% of those were false, then since 75% of 75% is more than 50%, intuitively (2a) would be true regardless of the status of (2b). And similarly with the roles of Nixon and Dean reversed.
The impossibility of subscripting does point to a risk of paradox, however, which might be realized in exceptionally unfortunate circumstances. If Nixon and Dean each made, apart from (2ab), an even number of Watergate-related statements, exactly half true and half false, then their saying (2ab) would produce a situation like that in the medieval example of Socrates and Plato mentioned in §1.3.
As Kripke notes, the case is similar to one of the oldest of all examples, where Epimenides the Cretan says
(3) Everything said by a Cretan is false.
Intuitively, this could easily be false. To make it so, it is enough for even one thing said by a Cretan to be true. There is a risk of paradox, but it is realized only in case everything else said by Epimenides or any other Cretan is false.
Kripke's point is that Tarskian subscripting or more generally any syntactic criterion sufficient to exclude all paradoxical cases will have to exclude many harmless and even useful cases as well. Syntax can only show whether there is a risk of paradox, so a syntactic criterion will have to ban all risky cases, many of which will not be paradoxical in actual circumstances.
Even if, like Tarski, one is only interested in producing a replacement for an intuitive notion of truth deemed inconsistent, one might hope to be able to produce a more flexible one than Tarski's. This is in the first instance what Kripke aims to do. He aims to develop rigorously a suggestion floated in some of the previous literature to the effect that one can have a truth predicate that is self-applicable, provided that one allows truth-value gaps.
7.2 THE MINIMUM FIXED POINT
Kripke's idea is that sentences—for simplicity Kripke, like Tarski, works with “true” as a predicate of orthographic sentence types—should not be assigned levels on syntactic grounds, but should be allow to “find their own level,” with those that fail to find one exhibiting a truth-value gap.
Sentences that make no mention of truth at all would be at level zero, and for these the equivalence principle is enough to determine which of them are to count as true. Thus
(4a) Snow is white.
is true at level zero. But since (4a) is true, so also is
(5a) Either snow is white or most of Nixon's Watergate-related statements are false.
and this regardless of the status of (2a), assuming as Kripke does that the truth of the first disjunct is sufficient for the truth of a disjunction, regardless of the status of the second disjunct. Thus Kripke's level zero contains more than Tarski's syntactically defined level zero. By contrast with (5a), the following
(6) Either snow is black or most of Nixon's Watergate-related statements are false.
would not be true at level zero, since its truth value (if any) would depend on the truth value (if any) of something itself mentioning truth, namely, the second disjunct (2a).
At level one come sentences not themselves at level zero whose truth value is determined by the truth values of sentences of level zero, for instance these two:
(4b) “Snow is white” is true.
(5b) Either “Snow is white” is true or most of Nixon's Watergate-related statements are false.
At level two come sentences not themselves at level one or lower whose truth value is determined by the truth values of sentences of level one or lower, such as
(4c) “‘Snow is white’ is true” is true.
Kripke's hierarchy, unlike Tarski's, extends beyond all finite levels. For instance, let us call the infinite sequence whose first three items are (4abc) the snow sequence. Then for Kripke the following would be true at the first level beyond all finite ones:
(7) Every sentence in the snow sequence is true.
But as one proceeds through higher and higher levels, one eventually comes to what Kripke calls the minimum fixed point, the first level where no new sentences get classified as true that were not already so classified at some earlier level. At the minimum fixed point, whenever “ ______ ” has been classified as true, so has “‘ ______ ’ is true,” and vice versa. As is said, one has closure under the rules of T-introduction and T-elimination.
To make these ideas rigorous, one thing that will be needed is a way of counting levels beyond all finite ones. That is provided by the notion of transfinite ordinal number from set theory, but we do not want to enter into such technicalities at this point. Another thing that will be needed is a rigorous account of when the truth values of certain components are sufficient to determine a truth value for a compound. For instance, in the examples (5ab) we assumed that truth of one disjunct is enough to make a disjunction true. What one needs is precisely a logic of truth-value gaps. We have already encountered the names of a couple of such logics (in §4.3) in connection with presupposition and vagueness. Kripke uses one of them, the Kleene strong trivalent approach (mentioning another, the Van Fraassen supervaluation approach, as a possible alternative).
7.3 UNGROUNDEDNESS
Various pathological sentences will never obtain truth values at any level. Such, for example, are the liar or falsehood-teller and the truth-teller:
(8) (8) is false. (9) (9) is true.
Why such examples receive no truth value is easy enough to see. In order for (8) to become true (respectively, false) at a given level, it already would have to have become false (respectively, true) at some earlier level. In order for (9) to become true (respectively, false) at a given level, it would already have to have become true (respectively, false) at some earlier level. Therefore there can be no first level where either gets a truth value, and hence there is no level at all at which either does. Similarly with
(10a) (10b) is true. (10b) (10a) is false.
In general, for a sentence mentioning truth to get a truth value, enough sentences it mentions will have to get a truth value earlier. And if any of these mentions truth, for it to get a truth value, enough sentences it mentions will have to get a truth value even earlier. The truth value of one sentence may depend on the truth value of another sentence, which may in turn depend on the truth value of yet another sentence, and so on. In pathological cases, a sentence may depend on itself, as with (8) or (9), or two sentences may depend on each other, as with (10ab), or there may be a vicious circle of three or more, or an infinite regress. (Examples of this last kind are implicit in Kripke's technical work, and an explicit nontechnical example was given by Steve Yablo: Think of an infinite sequence of sentences in which each says “All later sentences in this sequence are false.”) Kripke's construction makes rigorous sense of the notion of an “ungrounded” sentence, one for which the unpacking procedure never hits bottom because a vicious circle or infinite regress of dependence is encountered: The ungrounded sentences are ones that receive no truth value at the minimum fixed point.
Not all ungrounded sentences are equal, however. If we started by arbitrarily declaring (9) true (or false), we could still carry out Kripke's procedure, and would obtain a fixed point different from the minimum one, where we still had closure under T-introduction and T-elimination, but
(9) was true (false). By contrast, if we tried to declare (8) true (false) at the beginning, we would have to declare it false (true) at the next stage, and the resulting contradiction would obstruct the procedure. Falsehood-tellers have no truth value at any fixed point, while truth-tellers are true at some, false at others, and without truth value at yet others, including the minimum fixed point.
There are even subtler distinctions to be made, illustrated by
(11) Either (9) is true or (9) is false.
(12) Either (12) is true or (12) is false.
Each of these can be true at some fixed point, can be false at no fixed point, and is without truth value at the minimum fixed point. But (11) can only be made true by making true something that could have been made false (either (9) or its negation), while (12) can be made true without making true anything that could have been made false. Kripke calls examples like (12) intrinsically true, and shows that there is a maximum intrinsic fixed point, where all and only the intrinsically true sentences are made true.
But we have come about as far as one can go without assuming a little more technical background (including material from the later, more technical sections of chapter 2). Without assuming such background it is hardly possible to bring out Kripke's main achievement, which is that of making all the ideas we have been ever-more-sketchily sketching rigorous and precise, or to say much of anything about the main rival approach, revision theory, that was developed in the wake of Kripke's work. However, understanding of these matters is not really required to follow the more purely philosophical developments in the next chapter, and so the more technical discussion in the remaining sections of this chapter may be treated as optional reading.
7.4 * THE TRANSFINITE CONSTRUCTION
A test case is provided by the language L of arithmetic, as in §§2.4 and 2.5, and its expansion L' adding a predicate T. For an interpretation of L', Tarski would require an assignment of an extension for T, which since the variables range over natural numbers would just be some set of natural numbers. Such a set can conveniently be represented by the function T that takes the value one or zero for a given number as argument according as that number is or isn't in the set.
Tarski's definitions give a denotation |t| to each closed term t of the language, and a truth value to all sentences of L', in such a way that the following composition laws hold:
(13a) t0 = t1 is true (false) iff | t0 | and | t1 | are the same (different).
(13b) T (t) is true (false) iff T(| t |) is one (zero).
(13c) ~A is true (false) iff A is false (true).
(13d) A0A1 is true (false) iff each Ai is true (at least one Ai is false).
(13e) A0A1 is true (false) iff at least one Ai is true (each Ai is false).
(13f) xA(x) is true (false) iff A(t) is true (false) for each (some) term t.
(13g) xA(x) is true (false) iff A(t) is true (false) for some (each) term t.
Sentences A of L' can be assigned code numbers #A, each of which can in turn be assigned an appropriate closed term as a numeral denoting it, and the numeral for its code number can serve as a kind of quotation of the sentence. Accordingly we write “A” for it. With this notation, if T is assigned as its extension the set of all code numbers of Tarski-true sentences of L, then the following will hold in restricted form, applicable only to sentences A not involving T:
(14a) If T (“A”) is true (false), then A is true (false).
(14b) If A is true (false), then T (“A”) is true (false).
However, T (“A”) will not be true for any sentence A involving T. Kripke aims to prove that truth values can be assigned to some though not all sentences of L' in such as way that (13) holds and (14ab) hold unrestrictedly.
An interpretation, we have seen, can be represented by a valuation or function T assigning each natural number A value one or zero, indicating that the predicate T holds of some things and fails of the rest. A partial interpretation can be represented by A partial valuation, or function T assigning some natural numbers values one or zero, indicating that the predicate T holds of some and fails of some others, while leaving the rest unclassified. The composition laws (13), given an interpretation, determine an assignment of truth values to all of the sentences of L', and equally, given A partial interpretation, determine an assignment of truth values to some of the sentences of L'. For the partial case, the composition laws (13) encapsulate Kleene's strong trivalent logic.
Let us write T[A] for the truth value of A given the partial interpretation T. Then the truth laws (14ab) we seek can be restated as follows:
(15a) If T(#A) is one (zero), then T[A] is truth (falsehood).
(15b) If T[A] is truth (falsehood), then T(#A) is one (zero).
Kripke aims to prove the existence of a T for which (15ab) hold.
We call A partial interpretation T coherent if (15a) holds. Note that the empty valuation T0, with T0(n) undefined for all n, trivially fulfills this condition, though it fails badly to fulfill condition (15b). Towards getting a T for which (15b) as well as (15a) holds, Kripke defines the jump T* to be given by
(16) T*(#A) is one (zero) iff T[A] is truth (falsehood).
(We may suppose the coding has been so arranged that every number is the code for some sentence, so that (16) suffices to determine the value of T* for all numbers.) The goal of finding A T for which (15a) and (15b) both hold can be restated as that of finding a T for which T* = T. Such a T is a fixed point of the jump operation, and Kripke's aim is to obtain A fixed point in this rigorously defined sense. For valuations T and U, we say U extends T iff they are identical or differ only by U(n) being defined for some n for which T(n) is undefined. Coherence can be restated as the requirement that the jump T* extends T. Kripke actually aims to show, not merely that there exists A fixed point, but that every coherent partial valuation has an extension that is A fixed point.
To do so, he uses A general theorem from the mathematical theory of inductive definitions, together with the following observation. If U extends T, then the partial truth-valuation obtained by applying the composition rules (13) to U will extend that obtained from T. If U(#A) is defined while T(#A) is not, then we will get from U A truth value for the atomic sentence T (“A”) while we did not get one from T, and then other sentences will get truth values that did not have them; but for no sentence that already had A truth value will that value be altered or lost. It follows that if U extends T then U* extends T*, A property called the monotonicity of the jump operation. There is A general theorem to the effect that for any monotonic operation, every coherent T can be extended to A fixed point, and there is moreover a minimum fixed point, of which all others are extensions. Kripke does not just appeal to this general theorem, but indicates A proof for the case of his jump, making use of Cantor's countable ordinals, an extension of the sequence of natural numbers 0, 1, 2,…into the transfinite.
Cantor's two principles are:
(17a) For every ordinal α there are ordinals greater than α and a least among these, the successor of α.
(17b) For every increasing sequence α0 < α1 < α2 <…of ordinals there are ordinals greater than all of them and a least among these, their limit.
Thus the natural numbers 0 < 1 < 2 <…themselves have a limit called ω, which has A successor called ω + 1, which has a successor called ω + 2, and the sequence ω < ω + 1 < ω + 2 … has A limit called ω + ω or ω • 2, and similarly we get an ω • 3 and so on, and ω < ω •2 < ω •3 … has A limit called ω • ω or ω2, and similarly we get an ω3 and so on, and ω < ω2 < ω3 … has A limit ωω, and that's just the beginning.
Kripke assigns each countable ordinal α a coherent partial valuation T in such a way that if α < β then Tβ extends Tα. We start with the empty valuation T0, already noted to be coherent. Monotonicity implies that the jump of a coherent partial valuation is coherent. (If T is coherent, that means T* extends T; monotonicity then implies that T** extends T*, making T* coherent.) So at successors we can let Tα+1 be the jump of
Tα. Monotonicity also implies that if we have a sequence of coherent partial valuations Un with each extending the one before, then their union (which gives a number a value one or zero iff some Un and hence every Um with n ≤ m does so) is coherent. (Since the union U extends each Un, monotonicity implies that U* extends Un*, while the coherence of Un means that Un* extends Un; thus U* extends each Un and therefore their union U, making U coherent.) So at limits we can let Tβ be the union of the Tα for α < β.
Let A0 be the sentence 0 = 0, and for each n let An+1 be T (“An”). It is not hard to see that T0(#A0) is undefined but T1(#A0) = 1, T1(#A1) is undefined but T2(#A1) = 1, and so on. (These An are a formal analogue of the snow sequence (4abc) and so on.) So we keep getting new truths as we go through the natural numbers. It can be shown that we keep getting new truths beyond that—but not forever. For each A for which there exists any ordinal α for which Tα(#A) is defined, there is a least such α, which we may call αA. If we let βn be the largest of the αA for sentences A with code number #A < n, then the ordinals βn form a nondecreasing sequence, and have a limit γ. By the time we get to Tγ, everything that is ever going to get a value has got it already, so there can be nothing that will get a value for the first time with Tγ+1 . Thus Tγ = Tγ+1 and we have a fixed point, the minimum fixed point. If we start with A valuation assigning 1 only to the code number of a truth-teller (9), a little thought shows that its jump will also value that code number 1, so that we have coherence. Starting from such a valuation rather than the empty valuation, we get a fixed point in which the truth-teller is true, and similarly we can get one where it is false.
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