## 26 Feb 2009

### Vergauwen, A Metalogical Theory of Reference, 1.2, §20

by Corry Shores
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[The following is summary. Paragraph headings are my own.]

Roger Vergauwen

A Metalogical Theory of Reference: Realism and Essentialism in Semantics

Chapter 1.2 Primitive Reference and Satisfaction

§20 Tarski's Truth Definition and Sentential Functions

In the previously given examples, we introduce the truth concept directly. But Tarski was more indirect. For, we cannot directly speak of truth in formal language using quantifiers.

So consider the phrase

∀x (B (x) → V (x) )

It contains two one-place predicates B(x) and V(x). It also contains implication (→) and universal quantifier (∀).

Now let's interpret
B(x) as 'x is the president of the U.S.A.' and
V(x) as 'x is an American.'

We see that the variable x in these cases in not bound. So we cannot directly assign a truth value to these sentences. So we call such sentences as those above "sentential functions." In the sentence

∀x (B (x) → V (x) )

we see that the variable is bound. But compositionally built sentences are constructed from their constituent parts. And the meaning of the whole is determined by the meaning of these parts. So we cannot apply a truth definition like

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

to

'x is the president of the U.S.A.' and
'x is an American.'

To solve this problem, Tarski introduces the concept of "being satisfied by a sequence of objects," or just, "satisfaction."

According to this concept, composite sentences are not compounds of simple sentences. Elementary functions such as inclusion bring about sentential functions. But sentences are a special case of sentential functions. Thus there cannot be a method that would define truth by recursive means. But, if we introduce as more general concepts, we may directly obtain a concept of truth. This notion is "the satisfaction of a given sentential function by given objects."

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