In KIF, we can describe specific functions and relations by naming them with function constants and relation constants and then writing sentences in which those names occur in functional or relational position. For most purposes, this is adequate; but in some cases it is also useful to describe functions and relations more generally - to name their properties (such as associativity and transitivity) and to write axioms relating these properties (possibly quantifying over the functions and relations possessing these properties). In order to do this, we need to treat functions and relations as objects in our universe of discourse.
By definition, functions and relations are sets of lists of objects from our universe of discourse. The immediately preceding chapters offer a vocabulary for describing lists and sets in general. However, functions and relations have enough special properties to warrant additional vocabulary.
In what follows, we begin by presenting the KIF vocabulary for abstraction and application of functions and relations. We then talk about the use of functions and relation constants in argument position of terms. Finally, we present some supporting vocabulary.
Note that the introduction of functions and relations into our universe of discourse comes with the threat of paradox, as with sets in general. In KIF, we sidestep such paradoxes by defining the sets comprising our functions and relations in terms of the set concepts introduced in the preceding chapter.