Kripke: Outline …(1975) • First improtant decision: the theory is about sentences, not about propositions. • Like Tarski, not like Barwise & Etchemendy. • *** • A variant of the Liar: • More than half of Nixon’s assertions about Watergate are false. (Said by Jones.) • Everything Jones says about Watergate is true. (Said by Nixon.) • Context: • (1) is Jones’ single assertion about Watergate. • Exactly half of Nixon’s assertions other than (2) are false, half of them true. • In this context, (1) is a liar sentence (and (2) as well).
Motivations for a theory of truth that extends to such sentences: No a priori warrant against the occurrence of such unfavourable contexts. Gödel’s proof shows that self-reference may occur even without context-dependency. A sentence is grounded iff its truth-value can be reduced to the truth-value of sentences not involving the notion of truth. Otherwise ungrounded. Intuition: There are some basic-level sentences,”Snow is white” and the like. Sentences referring to the truth-value of basic-level propositions are at level 1 etc. There is a truth-predicate for nth level sentences at level n+1. Sentences that don’t have a level in this hierarchy are ungrounded and have no truth-value. Informal definition. A formal one follows later.
(More than) analogous to Tarski’s hierarchy of metalanguages. Difficulties with the application to everyday language: The concept of truth splits into infinitely many predicates. The level of (1) can’t be determined without knowing the level of all sentences told by Nixon. Sentences may prove ungrounded even in innocent cases of circularity. Example:(4) All of Nixon’s utterances about Watergate are false. (Said by Dean.)(5) Everything Dean says about Watergate is false. (Said by Nixon.)Context: Dean said something true (different from (4)). How to extend it to transfinite levels? And why? Because it is useful, see later.
Solutions with truth-value gaps: never forget the Strenghtened Liar! Moral: truth-value gaps don’t help us too easily. **** Let us have a FOL L with the usual two-valued Tarskian semantics. It is supposed to be strong enough to express its own syntax. We shall extend it by possibly partially defined truth predicates. Hence we need some „gappy” semantics. We use strong Kleene evaluation schemes (see next slide). L is interpreted over some domain D. (D and the interpretation of L overDis fixed.) The sentence in the rectangle is not true.
Strong Kleene evaluation: • A unary predicate F has an extension and an antiextension (they are disjoint). It is undefined for members of D that are neither in the extension nor in the antiextension. Generalization for n-ary predicates is obvious but not needed. • is the symbol for „undefined”. xA(x) = • 1 if A(x) is true for some member of D; • 0 if A(x) is false for every member of D; • otherwise.
Extend L with a partially defined predicate T(x) (extension: S1, antiextension: S2) to a language L*(S1, S2). Let S’1 be the set of (codes of) true sentences of L*(S1,S2) in D and S’2 the set of members of D which are either no sentence(code)s or (codes of) false sentences. Suppose that S1=S’1 and S2=S’2. In such a case, T(x) is a real truth predicate (true for the codes of true sentences, false for falses and undefined for sentences without a truth value.) Such a pair (S1, S2) is called a fixed point. In other words, a language witha fixed point (as the interpretation of T(x)) contains its own truth predicate. If (S1, S2) is not a fixed point, then let us extend the extension resp. antiextension of T(x) with the codes of sentences made true by the interpretation of L*(S1,S2). On this way we get S’1 resp. S’2 and we can continue on the same way. The operation defined by (S1, S2)=(S’1,S’2) is an uniquely determined operation from the set of pairs of disjoint subsets of D into the same set. We have to prove that any such operation has fixed points.
(S1+, S2+) extends (S1, S2) (in symbols, (S1+, S2+) (S1, S2))iff (S1+S1 and S2+S2). Obvious: this is a (weak) partial ordering. Rather simple: is monotonic for this ordering. Let us define the following series of languages:L*0 =L*(, ) If L*n = L*(S1, S2), then L*(n+1) = L*(S1’, S2’) (for any natural number, i. e. finite ordinal n). But it makes sense for any successor ordinal n, too. To extend our definiton for transfinite ordinals, we need only to give a definition for limit ordinals. It may happen on the usual way: For a limit ordinal, let the extension resp. antiextension be the union of the extensions resp. antiextensions for all the smaller ordinals. From the monotonicity follows that both the extensions and antiextensions form (not strictly) monoton increasing set series. By simple set-theoretic considerations, such series should have a fixed point.
Closing remarks to the construction: This fixed point is minimal in the sense that other fixed points (coming from a different choice of initial S1 and S2) are all extensions of it. The language at the minimal fixed point is a language which contains its own truth predicate (in the weaker sense that if the interpretation makes a sentence true or false, then the truth predicate says about it that it is true resp. not true). The construction can be generalized for a satisfaction relation instead of the truth predicate. (In this case the language must certainly contain some device to encode n-tuples of objects into objects.) A sentence is ungroundedif it has no truth value at the smallest fixed point, grounded in the other case. This definition fits well the informal definition at the beginning. Another informal notion that can be now formalized : the level of a sentence is the smallest ordinal at which it has a truth value. But (4) and (5) can’t have a level because the level of (4) should be at once greater and smaller than the level of (5). Fruits from this tree: next time.