Re: Availability of the ANSI standard proposal? (John F. Sowa)
Date: Sat, 23 Mar 1996 12:37:09 +0500
From: (John F. Sowa)
Message-id: <9603231737.AA04801@west>
Subject: Re: Availability of the ANSI standard proposal?
Precedence: bulk
Fritz Lehmann raised some questions about the relational hierarchy.
Following is the approach that I recommend:

 1. A relation may be defined by a lambda abstraction over a logical
    expression or graph by specifying one or more variables (or concept
    nodes) as formal parameters.

 2. Two different abstractions over the same expression or graph that
    specify different formal parameters or the same parameters in a different
    order are considered distinct.

 3. Implication induces a preordering (which can be converted to a partial
    ordering by factoring out logical equivalences) over expressions or
    graphs.  That partial ordering can be extended to lambda expressions
    in a straightforward way:

    If expressions x<y, then the lambda abstractions x'<y' ONLY IF the
    corresponding variables or concept nodes in both x and y have been
    designated as formal parameters.

 4. Point #3 implies that if you select different formal parameters (or
    the same formal parameters, but in a different order), you won't
    preserve the partial ordering x'<y'.

 5. Point #3 also implies that relations of different valence (arity,
    adicity, or whatever you want to call it) are not comparable in the
    partial ordering.

I agree with Fritz that converse relations are an unnecessary redundancy.
But if anyone throws them into the pot, they won't be placed in the same
point in the partial ordering.

John Sowa