Vergauwen, A Metalogical Theory of Reference, Introduction, §21

by Corry Shores
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Roger Vergauwen

A Metalogical Theory of Reference: Realism and Essentialism in Semantics

Chapter 1.2 Primitive Reference and Satisfaction

§21 Tarski's Truth Definition and Satisfaction

In Tarski's sense of satisfaction, there is a relation between words and things. Put more precisely, there is a relation between open formulas and sequences of objects.

So we recall

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

We will now say

an object a satisfies the sentential function 'X' if and only if p.

In this case, X names some certain sentential function. 'p' results from X when we substitute-in the sentential function 'a' for the variables in the sentential function's translation. For example, consider that X represents the sentential function in

'x is the president of the U.S.A.'

Thus we might say

an object a satisfies the sentential function 'x is the president of the U.S.A.' if and only if a is the president of the U.S.A.

Now lets suppose that we have found a general definition of 'satisfaction' for sentence functions. Next we will determine which objects satisfy the language's simplest sentential functions. Then we give the conditions that would allow for objects to satisfy composite sentential functions that are built-up from simple ones.

Lastly, we consider the well-formed formulae or well-formed sentences (WFFs) that no longer have free variables. There are two possibilities for such sentences: either

1) these sentences are satisfied by all sequences of objects, or

2) these sentences are satisfied by no objects at all.

We may then define truth for a sentence. It would be its satisfaction by all objects. The sentence would then be false if it is not satisfied by any object whatsoever.

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