5.1.1 Models

The task of logical semantics is to define how formulas are evaluated in models. In general terms, the purpose of the evaluation process is to tell us whether a description is true or false in a situation.

So what is a model? Actually, what we've just said already pretty much contains the answer to this: A model is like a situation - and a situation is a semantic entity, providing us with a certain amount of things we can talk about. Thus, a model should give us two pieces of information. First, it should tell us what kind of collection of entities we can talk about. This is the task of the so called domain , or for short. Secondly, a model should give us appropriate semantic entities, built from the items in , for the symbols in our language. The function carrying out this task is called the interpretation function .

What is a Model?

In set theoretic terms, a model thus is an ordered pair composed of a domain and an interpretation function specifying semantic values in .

However this definition hides one problem: Intuitively it doesn't make much sense to ask whether or not an arbitrary description is true in an arbitrary situation. Some descriptions and situations simply don't belong together. The same is true for the relation between formulas and models. The model used to evaluate a formula has to be a model for that formula (or, more precisely, for the language that formula is taken from). If we examine a formula , while being provided with a model recording information only about the symbols , and , then it makes no sense at all to evaluate this particular formula in that particular model. The element connecting a formula with the right models for it is the vocabulary (or a signature ) defining the language of that formula (see Section 1.2.1). So if we want to evaluate a formula in a model, we have to make sure that the model is a model for the vocabulary of the language that our formula belongs to.

We say that a given model is a model for a vocabulary if the domain of the interpretation function consists of the symbols specified in . So, according to our previous considerations, should assign appropriate semantic entities (built from the items in ) to the symbols of .

But what are appropriate semantic values? Given the arity information in , there's no mystery here. Since constants are names, each constant should be interpreted as an element of , the domain of the model. That is, for each constant symbol in the vocabulary, . And since -place relation symbols denote -place relations, each -place relation symbol should be interpreted as an -place relation on . That is, should be a set of -tuples of elements of .