The statement, "This sentence is false" poses the following problem: it is true if, and only if, it is false; and it is false if, and only if, it is true. Tarski proposes a definition of truth which he claims solves this paradox (or, in any case, renders it meaningless). Let us see how he does this.

He first mentions one of the most frequently used definitions of truth: the so-called "correspondence theory". This theory states, roughly, that a sentence is true if it corresponds to an existing state of affairs. Tarski says that this definition is inadequate. He does not say why, but with regard to the aforementioned paradox, we can clearly see that this definition does not in any way help us in explaining the contradiction. In other words, it is impossible to say whether or not "This sentence is false" corresponds to reality. Tarski says that our definition of truth should not contradict this seemingly obvious intuition, but it must go farther.

The first step Tarski takes in formulating this definition is to explain that the truth of a sentence depends on the language it is in, for a group of words may represent a true proposition in one language and be meaningless in another. Next, we have to understand that when we talk about the truth of a specific sentence (or say anything else about it), we cannot use the sentence itself, but we must name that sentence. In other words, when we talk about an object, we use the name of that object, not the object itself. For example, one way of naming sentences is to put quotations around it. So we can say, " ‘snow is white’ is true if, and only if, snow is white." The first occurrence of the sentence is the name of the sentence, and the second is the sentence itself. In order to generalize this, Tarski chooses ‘p’ as a sentence, and ‘X’ as the name of that sentence. He then defines what he calls "an equivalence of the form (T)", which is "X is true if, and only if, p."

(T) becomes the criterion under which any definition of truth must be tested. A definition of truth is deemed "adequate" if, and only if, all sentences that are called true by the new definition can be fit into the above equivalence.

After having set up these premises, we can finally get to the root of the problem and then solve it. Tarski says that in order to think that the above sentence ("this sentence is false") is meaningful, we have to make two assumptions: first, that the language with which we state the paradox is "semantically closed"; that is, that it contains, "in addition to its expressions, also the names of these expressions, as well as semantic terms such as the term "true", referring to sentences of this language." A semantically closed language also assumes that the definition of such concepts as truth can be stated in the language itself. The second assumption we make is that the ordinary laws of logic hold in the language that is used to state the paradox. Any language that holds both of these assumptions allows for the possibility of making a sentence like the liar’s paradox seem meaningful. In other words, a language like this is inconsistent.

Since Tarski is unwilling to admit the possibility that the second assumption may be false, he says that we have to do away with the first one. Therefore, our definition of truth must be about a language that is not semantically closed. This means that we have to use two different languages to define truth: one to express the sentences being talked about, and another to talk about them (specifically, to talk about their truth). The former Tarski calls the "object language", and the latter the "metalanguage." The object language has to be a part of the metalanguage, since the metalanguage has to be able to refer to the object language. However, the object language cannot be translatable into the metalanguage, since that would make the metalanguage superfluous. In other words, the metalanguage has to be "essentially richer."

Again, the definition of truth has to comply with the proposition that "X is true if, and only if, p." However, in this system of two languages, "X" (the name of the sentence) has to be in the object language, and the rest of the sentence has to be in the metalanguage.

We can now see why asking if "this sentence is false" is true or false doesn’t make any sense. Let us place this sentence in the equivalence formula: "‘This sentence is false’ is true if, and only if, this sentence is false." This is meaningless, since the sentence in the object language is doing something that can only be done in the metalanguage (since, as we saw above, the object language cannot be semantically closed) : that is, it is saying something about its own truth (or, in this case, falsity). In other words, the truth of "X" cannot be stated in the same language as "X", it has to be stated in the language of "p".

April 17, 1996