Vergauwen, A Metalogical Theory of Reference, Introduction, §10

by Corry Shores
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Roger Vergauwen

A Metalogical Theory of Reference: Realism and Essentialism in Semantics

Introduction: the Temperature of a Hot Topic

§10 Non-Standard Models and Gödel's Incompleteness Theorems

Let's suppose our previous considerations. From an epistemological point of view, a model-theoretic realist faces two dilemmatic choices. Either

1) he gives-up his position that language is a calculus requiring an interpretation so that it might refer to the world. Or,

2) he retains his realist view. But now he cannot represent the world with a single model.

In chapter three Vergauwen will show the possibility for a theory of references incorporating language's reference function, without also needing a unique standard model. He will base his approach on Gödel's Incompleteness Theorems.

If one takes model-theoretic intensionality as reference, then such paradoxes as the Grellingparadox result. This paradox shows that there cannot be unambiguous intensionaldeterminations in model-theoretic semantics. If in fact instead there were such unambiguous determinations, they would be "fixed-points" that allow for a unique link between words and things. But such fixed points seem not to be possible. If they did exist, they would provide the guarantee that meaning regards something in reality. But this is something that the reality outside of language should provide.

Here Gödel's theorems help. They demonstrate the existence of an undecidable (but true) sentence. Such a sentence is true in a standard model, but false in a non-standard one. In model-theoretic semantics, there is no true standard model. Language corresponds to the world through reference relations expressible with reference predicates that determine partial standard models. These in turn determine the nature of the partial standard model's reference relations.

The reference predicates express the absence of meanings' (intensions') specific trait. Then noreferentiality in the model made-up by a reference predicate.

However, the reference predicate in intensional logic can be likened to an undecidable sentence negated within a non-standard model. Thereby the non-standard model specifies the referencerelation to partial standard model. This would be a metalogical perspective. There would be a fixed point that determines reference. It would the be sequence of (partial) standard and non-standard models. The nature of the non-standard models would explain the standard models reference relation. The semantic metalanguage's incompleteness and non-categoricalness result from this. It would simulate the rift between words and real things. This also allows us to give external reality a place in our theory of reference.

(p. xvii-xviii )

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Vergauwen, Roger. A Metalogical Theory of Reference: Realism and Essentialism in