1.2.1 Vocabularies

Intuitively, a vocabulary tells us the language the ``first-order conversation'' is going to be conducted in. It tells us in what terms we will be able to talk about things. Here is our first vocabulary:

Generally, a vocabulary is a pair of sets:

1. The first set 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 entities, 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 really shows us what kind of situations the conversation is going to be about (formally, it gives us all the information needed to define the class of models of interest - but we said that we won't go into this topic here). Syntactically, it helps us define the relevant first-order language (that means the kinds of formulas we can use). So let's next have a look at how a first order language is generated from a vocabulary.