A logical theory will specify all the logical properties of a certain (usually small) number of basic concepts. These are known as the primitives of the theory. The vocabulary of the theory can then be extended by means of definitions. For example, the `part' relation holding between spatial regions can straightforwardly be defined in terms of the `connectedness' relation. We stipulate that:
x is part of y if and only if everything
that is connected to x is connected to y.
In the notation of
1st-order logic this is written as follows:
This means that if we have a theory of `connectedness' we can also reason about the `part' relation without the need for any further logical axioms. Whenever `part' occurs in a query then it can be replaced by its definition in terms of connectedness.