Two Prices for the Liar: Tarski's Hierarchy Against Kripke's Truth-Value Gaps — Epoche C1
Say the sentence out loud: this sentence is false . The usual reaction is that a careful speaker can decline to be troubled, since the sentence is a trick that never arises in serious talk. That reaction is comfortable and it is mistaken. The liar is not a curiosity but a derivation, and what it derives is that no consistent theory in classical logic which can describe its own syntax can also contain a truth predicate obeying the obvious rule about truth. The question is therefore not whether to pay but in which currency. Alfred Tarski and Saul Kripke pay in different coins, and setting the two payments beside each other shows what each buys. Why the sentence cannot be waved away The derivation needs exactly three ingredients, and seeing each of them work is what makes the later choices intelligible, since every escape consists in refusing one. The T-schema. Tarski's condition of adequacy on any theory of truth — his Convention T — requires that the theory prove, for every sentence $S$ of the language in question, the biconditional $T(\lceil S\rceil) \leftrightarrow S$, where $T$ is the predicate "is true" and $\lceil S\rceil$ is a name, available inside the language, for the sentence $S$. In words: "snow is white" is true if and only if snow is white. The schema is not a definition of truth but a constraint any definition must satisfy, and it is hard to reject because rejecting it means allowing that some sentence is true while what it says fails to be the case. Self-reference. A theory strong enough to represent its own grammar can construct a sentence that talks about itself. Gödel showed in 1931 how: assign a number to each symbol, code a finite string of symbols as a single number, and then syntactic relations such as "$x$ is the code of a sentence" or "$z$ is the code of the result of substituting the numeral for $y$ into the formula coded by $x$" become arithmetical relations that arithmetic itself can express. The consequence needed here is the diagonal lemma: for any formula $\varphi(x)$ with one free variable there is a sentence $\lambda$ such that the theory proves $\lambda \leftrightarrow \varphi(\lceil\lambda\rceil)$ — a sentence that asserts of its own code that it has the property $\varphi$. Classical logic. Every sentence is either true or false and not both, and from a contradiction everything follows. Apply the diagonal lemma with $\varphi(x) = \neg T(x)$ and obtain $\lambda$ with $\lambda \leftrightarrow \neg T(\lceil\lambda\rceil)$: a sentence saying of itself that it is not true. The relevant instance of the T-schema is $T(\lceil\lambda\rceil) \leftrightarrow \lambda$. Substituting one into the other gives $\lambda \leftrightarrow \neg\lambda$, from which classical logic derives $\lambda \wedge \neg\lambda$, and from a contradiction every sentence whatever. This is Tarski's theorem on the undefinability of truth, and what it establishes is not that the theory is odd but that it is trivial: it proves everything, and so says nothing. The step that does the work is the combination — taken singly, no ingredient is culpable. That is why a shrug is not a response. A shrug rejects nothing, and the derivation is still available. Tarski's coin: one truth predicate per level Tarski's response, in the monograph on truth in formalised languages he published in Polish in 1933 and in German in 1935, refuses the first ingredient — not the T-schema itself, but the assumption that a single language may host the T-schema for its own sentences (Tarski 1956). The distinction matters, because Tarski keeps the schema entirely; he merely denies that both sides of it can live in one place. What results is a ladder. The object language talks about snow and numbers, and contains no truth predicate. A metalanguage contains a translation of the object language plus the predicate "true-in-the-object-language", and in it every T-biconditional for object-language sentences is provable. A third language does the same for the second, and so upward. In this arrangement the liar cannot even be written down: forming it would require a sentence at level $n$ to contain the level-$n$ truth predicate, and the syntax of each language forbids this. Consistency is not restored by argument; it is secured by the grammar, which is both the merit of the proposal and the ground of the complaints against it. Tarski was candid about the price. Ordinary language is what he called semantically closed : it contains its own words "true" and "false" and its own means of naming its own sentences (Tarski 1944). His diagnosis is that a semantically closed language governed by classical logic is inconsistent, and that no rigorous semantics can be supplied for natural language as it stands. That is a considerable claim about the object most of us take ourselves to be using. It also has an awkward practical face: when a judge says "everything the witness said was true", the sentence must be assigned a level, and if the witness herself used the word "true" the levels have to be sorted out before anyone can tell what was claimed. Since the assignment is fixed by the form of the sentence and not by what the witness happened to say, one and the same utterance can be well formed or ill formed depending on facts about a conversation it does not mention. Kripke's coin: one language, some sentences without a truth-value Kripke's 1975 paper takes the opposite route, refusing the third ingredient rather than the first: keep one language containing its own truth predicate, and give up the assumption that every sentence must be true or false. The same fixed-point idea was reached independently by Martin and Woodruff (1975); Kripke's paper works it out in the detail that made it a research programme. The evaluation uses Kleene's strong three-valued scheme (Kleene 1952), in which a third value $u$ marks sentences with no truth-value assigned. Its defining feature is that a compound may be defined even when a part is not. $A$ $B$ $A \vee B$