[SCL] the relational component of an SCL interpetation

[Robert E. Kent]

> >I have a question that may be just a niggle. If it has been gone over
> >thoroughly before, just point me to the related discussion thread. An SCL
> >interpretation I = (I,R,ext,V) uses a relation component R = (R_w, R_1,
R_2,
> >...) that is a little strange -- it seems to serve only as a middleman
> >between relations symbols of an associated lexicon L and subsets of
tuples
> >of the set of individuals I: relational denotation and extension is a
pair
> >of composable functions
> >     V o ext : PredCon --> R --> Pow(I^w).
> >Can R be eliminated and a single extension/denotation function be used
here.
> >This would seem to be a much simpler and more aesthetically pleasing
> >approach.
>
> It would be simpler but I do not accept that it would be more
> aesthetically pleasing. I think that it would be fair to say that it
> would be more *familiar* to many readers, however.
>
> >I believe that this is related to endnote 5: "It is possible to
> >model of the members of R extensionally as sets, though this will in
general
> >require non-well-founded set theory, since a relation, qua individual,
can
> >be in its own extension." Is this true?
>
> Yes
>
> >  Is the purpose of R to proscribe
> >self-membership? If so, how exactly does this work? Can we not just
simplify
> >as mentioned above and verbally proscribe self-membership?
>
> We do not wish to proscribe it: it has its uses.
>
> BUt in fact there is another reason for using this construction, in
> that it allows SCL to axiomatize an intensional theory of relations:
> for example, there can be several distinct universally-false
> relations (several empty classes, in class terminology.) This is a
> notable advantage in many applications.

Whenever this kind of suggestion arises it reminds me of the idea of a
*classification* A = [inst(A), typ(A), |=A] which consists of a set of
instances (or tokens) 'inst(A)', a set of types 'typ(A)' and a binary
incidence ("has") relation '|=A' on instances and types. Any two instances
can have the same set of types (i.e., have equal intents), and any two types
can have^op the same set of instances (i.e., have equal extents). So we
could also represent this part of an interpretation as a classification
[Pow(I^w), R, |=] between tuples and relations, where the incidence 't |= r'
holds between a tuple 't' and a relation 'r' when 't' is a member of
'ext(r)'. Effectively, the incidence relation '|=' represents the same
information as the extent function 'ext'.

Now looking at the composite function
    V o ext : PredCon --> R --> Pow(I^w)
the question is whether R is a syntactic/type thing (stuff on the left) or a
semantic/instance thing (stuff on the right). And the elimination I had in
mind was not to identify on the right R with Pow(I^w), but instead to
identify on the left PredCon with R.

Reasons for the identification are that predicate constants PredCon and
relations R seem to be used in the same fashion. To wit:

Predicate constants:
One component in a lexicon is a countable set PredCon of predicate
constants. This set will include a distinguished predicate Id. If arity(p) =
n, then p is said to be an n-place predicate; otherwise p is variably
polyadic. Variably polyadic predicates will be able to take any number of
arguments. We let PredCon_n be the set of n-place predicates, and PredCon_w
the set of variably polyadic predicates.

Relations:
One component in an interpretation for a lexicon is a set of relations R. R
is itself the union of countable sets R_w, R_1, R_2, R_3, .... All are
possibly empty with the exception of R_2, which contains a distinguished
element Id, intended to serve as the identity relation. Intuitively, R_w is
the set of variably polyadic relations, and each R_n is the set of n-place
relations.

Of course, PredCon is a component of the lexicon, whereas R is a component
of an interpretation for a lexicon. When we use the terminology
"interpretation for a lexicon" we seem to be regarding the lexicon as a
parameter for the interpretation. But another approach is to incorporate the
lexicon as the "type" component of an interpretation. In that way we could
identify PredCon with R.

Robert E. Kent
rekent at ontologos.org