Vergauwen, A Metalogical Theory of Reference, 1.3, §23

by Corry Shores
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Roger Vergauwen

A Metalogical Theory of Reference: Realism and Essentialism in Semantics

Chapter 1.3 Semantic closedness of Languages and the Paradox of the Liar

§23 Tarski's Truth Definition, Metalanguage & Object Language, and the Liar Paradox

Tarski's truth definition functioned well for formal languages. But he did not believe that the concept of truth could be applied to natural languages. In fact, he claimed it would inevitably produce confusions and contradictions.

We saw that a truth definition could be:

"snow is white" is true in English if and only if snow is white.

So truth definitions take the general form:

The sentence X is true (in L) if and only if p.

Notice the quotation marks in "snow is white." We are no longer considering what snow is and what white is. We are treating the whole proposition as one symbol or object.

In such a truth definition, there are two sorts of language being used.

a) object language: the language whose meaning we are explicating by means of truth definitions. We put the sentences of the object language in quotation marks. This turns the whole proposition into a single term, name, or object, as it were. It is treated as no more than a series of symbols. That particular series of symbols will be true under certain conditions.

b) metalanguage: those truth conditions are given in a language that has more expressive 'power' than the object language. It can say more, because it can talk about everything that is in the object language, and it can say things about the object language.

Consider that grass is green. The topic or subject that we are talking about here is grass. And, we see that "being green" is the predicament that the subject 'grass' is in. So 'is green' is the predicate for the subject, 'grass.'

In our truth definition, we say "the sentence X is true..."

Here, the subject of the truth definition is 'the sentence X.' But it is also in a predicament. It is predicated by "is true." So "is true" we call the truth predicate.

So consider our example

"snow is white" is true if and only if snow is white.

We see that the predicate is true is not a part of the quoted name for the object-language sentence. So the truth predicate must belong to the metalanguage. And we apply this predicate to the object sentence only under the given conditions. So if snow is not white, we would not predicate it with being true. But since snow is white, we say that "snow is white" is true.

Now consider that English is a possible object language. It itself contains the truth predicate "is true." In this case we cannot establish a strict division between object language and metalanguage. So we call such a language semantically universal.

One consequence of this is the liar paradox. Consider if someone says, "I am lying." If they are telling the truth, then they are lying. If they are lying, then they are telling the truth. Let's render such a statement into a different form. We'll create a statement that we call "sentence (1)." But that is just its name. Now we will say what Sentence (1) says:

(1):

Sentence (1) is false.

So Sentence (1) says of itself that it is false. So if we wanted to use a truth definition to give its meaning, we would have:

The sentence "sentence (1) is false" is true if and only if sentence (1) is false.

For (1) to be true, it must be false. But, if (1) is false, then it is true on account of the truth definition.

It appears then that we cannot use the object language as the metalanguage as well. For, this allows the object language to refer to itself, and then to contradict itself.

Tarski was pessimistic about using the truth definitions to formalize the syntax of natural language.

But there are reasons to think that it is still possible.

1) We think that the language of mathematics to be univocal and safe from paradox. However Gödel showed otherwise.

2) We can begin by having a limited version of the object language, then have a hierarchy of languages where one serves as a metalanguage for another one below it.

Later Vergauwen will show that it is possible to have a semantic theory based on Tarski's approach.

[Other Entries in this Series]

Vergauwen, Roger. A Metalogical Theory of Reference: Realism and Essentialism in Semantics. London: University Press of America, 1993.