19 Mar 2009

Vergauwen, A Metalogical Theory of Reference, 2.3.1 Model Theory

by Corry Shores
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Roger Vergauwen

A Metalogical Theory of Reference: Realism and Essentialism in Semantics

Chapter 2: Reference and Theory of Reference
2.3 Model Theory and Model Theoretic Semantics

2.3.1 Model Theory

Formals systems such as logic are constructed with syntactical rules which determine what formulas are considered well-formed (WFFs). The models provide the semantic function of determining what the WFFs refer-to. In order to determine whether or not a proposition is true, we will need to know for example what predicates such as P and Q stand-for.

So the model might say that J refers to the individual John, and that P refers to the set of sleeping individuals. Then, when we construct the formula P (J), we know this means "John is sleeping." If John really is sleeping, then P (J) is true. Hence P (J) expresses the truth conditions for the proposition 'John is sleeping.' Model theory studies this reference relation between formal languages and and their possible models or interpretations. We may represent a model as a structure,
It is made-up of two components.

D is the universe of discourse. It is made-up of a set of elements. These elements will function as the values or the denotations for the WFFs.

F is function. It determines the elements that expressions in D refer-to. Functions take an element from the set of expressions and assigns to the elements from the domain of the function. The element obtained by means of the function is called the value of the function. The particular formulation that we are interpreting is called the argument. So we take an example:

F (x) = 2x + 1

We know that this takes a number from the set of numbers and maps it onto another number in the set of numbers. In this case, we double the first number and add 1 to it.

We might take the D and the F of the model together. They form its ontology.

We will now consider an example from predicate logic.

(Pa V Qb)

At this point, we do not know whether or not the above formulation is true. We know that it is well-formed, because it follows our rules of syntax. The model will tell us what these symbols are referring-to. We will consider the universe to be the natural numbers. Predicate P will be 'is an even number. Predicate Q will be 'is an odd number'. We will say that a refers to the number 2. And b refers to 3. We see then that this formula says "either 2 is an even number, or 3 is an odd number." Of course, this is a true statement.

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

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