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

[Robert E. Kent]

OK, I will try to provide an answer for this fellow.

Dear Robert,

Your first example below is right on the mark. A composite term in the relation position is intended for constructing relations, either from other relations or other kinds of things. I understand the intent of your second example below, to represent an FOL interpretation, but your solution is flaw; however, it can be represented. Your first attempt
 
((less-than (square-root x) (times x y)) 4 1)

was not successful because it contains free variables and it is unclear how to bind the arguments in the proper order. This can be done with equations

(forall (x y) (and (less-than (square-root x) (times x y)) (= x 4) (= y 1)))

However, this does not follow the form for relational position. A successful approach is to abstract to relation/function operators:

((substitution less-than (pairing (composition projection1 square-root) times)) 4 1)

where (substitution R f) substitutes function f into relation R, (composition projection1 square-root) is the function composition of f1 and f2, and (pairing g1 g2) pairs the output of functions g1 and g2 (assuming common arity).

And finally, about your second question: in the first atom the term "(union equal-to less-than)" is in relation position within the atom "((union equal-to less-than) 3 4)", whereas the name "union" is in function position within the term "(union equal-to less-than)" and the names "equal-to" and "less-than" are in object position (denoting) within the term "(union equal-to less-than)".
 
Robert E. Kent
rekent at ontologos.org


----- Original Message ----- 
From: "Robert E. Kent" <rekent at ontologos.org>
To: "SCL" <scl at philebus.tamu.edu>
Sent: Wednesday, March 17, 2004 4:08 PM
Subject: [SCL] question about composite terms in relation/function position


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

If only the first kind 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, since they are relations which evaluate to the "less-than-equal-to" relation?

Robert E. Kent
rekent at ontologos.org



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