[SCL] abstract vocabulary and interpretation

[Robert E. Kent]

If we define object position, for either names or terms, as anything
potentially denoting, then by following the somewhat elaborate definitions
in my previous message, we can define simpler abstract notions of vocabulary
and interpretation.

The (concrete) vocabulary of any SCL text T is the triple (O, R, F)
where O is the set of terms appearing in T,
where R is the set of terms of T in relation position, and
where F is the set of terms of T in function position,

For any set of names N,
an (abstract) vocabulary is a triple (O, R, F)
where O is a set of N-terms, both R and F are subsets of O
and for any composite N-term (t, t1, ... tn) in O,
t is in F and ti is in O for 1 <= i <= n.

For any set of names N,
an interpretation for a vocabulary (O, R, F) has
a nonempty set U called the universe,
three maps
int : O --> U, rel : R --> rel(U), and fun : F --> fun(U),
and two partial maps (folds)
relation : U --> rel(U) and function : U --> fun(U),
where
the value of int on any composite term (t, seq) in O
is the value of fun[t] applied to the sequential extension int[seq],
the image of R under int is a subset of the domain of relation
and relation[int[t]] = rel[t] for any t in R,
the image of F under int is a subset of the domain of function
and function[int[t]] = fun[t] for any t in F.

Robert E. Kent
rekent at ontologos.org