1.1.1 Vocabularies

Introduction of vocabularies telling us which first-order languages and models belong together.

Our ultimate goal in this lecture is to define how first-order formulas are evaluated in first-order models. In general terms, the purpose of the evaluation process is to tell us whether a description is true or false in a situation.

We shall look at this in a moment - but first, it's important to point out a relevant issue. 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. For example, if we examine a formula (that is, a description - see above) from a first-order language intended for talking about various relations and properties like loving, being a moron, and being a therapist that hold between the characters Mary, Anna, John, and Peter, while being provided with a model (that is, a situation - see above) recording information about something entirely different (say, about which household detergents are the best choice to get rid of nasty stains) then it makes no sense at all to evaluate this particular formula in that particular model. But a vocabulary (or a signature , as a vocabulary is also called) allows us to avoid such problems: It tells us which first-order language belongs to a given model.

Here is our first vocabulary:

Vocabularies connect Formulas and Models.

Intuitively, the vocabulary tells us the language the conversation is going to be conducted in. It tells us in what terms we will be able to talk about things. To be a bit more precise:

1. The first set in a vocabulary tells us what symbols we can use to name certain entities of special interest. In the case of the vocabulary we have just established, we are informed that we will be using four symbols for this purpose (we call them constant symbol s or simply name s), namely , , , and .

2. The second set tells us with what symbols we can speak about certain properties and relations (we call these symbols relation symbol s or predicate symbol s). With our example vocabulary, we have one predicate symbol of arity 2 (that is, a 2-place predicate symbol) for talking about one two-place relation, and two predicate symbols of arity 1 ( and ) for talking about (at most) two properties.

As such, the vocabulary we've just seen doesn't yet tell us a lot about the kinds of situations we can describe. We only know that some entitities, at most two properties, and one two-place relation will play a special role in them. But since we're interested in natural language, we will use our symbols ``suggestively''. For instance, we will only use the symbol for talking about a (one-sided) relation called loving, and the two symbols and will serve us exclusively for talking about therapists and morons. With this additional convention, the vocabulary gives us all the information needed to define the class of models of interest (that means the kinds of situations we want to describe) and the relevant first-order language (that means the kinds of descriptions we can use).

Obviously, to really understand all of this we need to know something about first order models and how they're used to interpret formulae. So let's next have a look at what first-order models and languages actually are.