[SCL] question about composite terms in relation/function position

[Pat Hayes]

>I am trying to understand the nature of the composite terms that 
>occur in function or relation position. Let us concentrate on 
>relation position here. These terms must represent relations of some 
>arities. Must they be built using relational operators such as the 
>"union" operator below, or can they be expressed as "ordinary" 
>terms; that is, terms that express a relation but whose arguments 
>are not relations. Here are two examples of possible SCL atomic 
>sentences.
>
>((union equal-to less-than) 3 4)
>
>((less-than (square-root (first z)) (times (first z) (second z))) (pair 4 1))
>  Since the second atom has a free variable, perhaps this is not allowed.

It is allowed, but may not mean what you may have intended. The 'z' 
is simply a name, since it is not bound by any quantifier.

>This means that we cannot express terms by implicitly using embedded 
>first order logic interpretations.

?? Sorry, you lost me there.

>For example the second atomic sentence above elaborated as:
>
>(= complex-relation (less-than (square-root (first z)) (times (first 
>z) (second z))))
>(complex-relation (pair 4 1))

OK, that is indeed equivalent in SCL. I still do not follow what 
point you are making here.

>
>If only the first kind

What kinds are you referring to?

>of atom is allow, perhaps this answers my question about the nature 
>of the composite relational position terms. But then a second 
>question crops up: in the first atom do I need to relegate the names 
>"equal-to" and "less-than" to relational position

Well, in that atom they do not occur in relation position. They may 
occur in relation position somewhere else in a larger text.

>, since they are relations which evaluate to the 
>"less-than-equal-to" relation?

Are they? Nothing in the SCL you presented would establish that 
interpretation. You could have also said

(forall (x y) (iff ((union equal-to less-than) x y) (or (equal-to x 
y)(less-than x y)))

in which case this meaning would be fixed; and of course here they do 
occur in relation position.

Pat

>
>Robert E. Kent
><mailto:rekent at ontologos.org>rekent at ontologos.org
>
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