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Re: [ontolog-forum] cyclic and acyclic definitions

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Fri, 24 Apr 2009 08:28:24 -0400
Message-id: <49F1B068.9060908@xxxxxxxxxxx>
Bart,    (01)

Definitions of the following kind are trivial substitutions:    (02)

> Momo = Man who has Only Male Offspring
> (forall x (Momo(x) iff exists y s.t. Man(x) ^ Momo(y) ^ hasChild(y,x)))    (03)

Following is the more common form of definition in mathematics:    (04)

A dyadic relation R is transitive iff the following axiom is true:    (05)

    For all x, y, z,
       R(x,y) and R(y,z) implies R(x,z).    (06)

More generally, the basic form of a definition in mathematics
names the type of entity (e.g. 'transitive relation') and
states a symbol such as R that represents an arbitrary instance.
Then that statement is followed by one or more axioms that
specify the constraints on any such entity R.    (07)

Mathematicians have been using such definitions since Euclid,
and they have never caused any trouble at all.    (08)

Perhaps you're thinking of the following paradoxical statement
in set theory:    (09)

    The set of all sets that are not elements of themselves.    (010)

But that is a result of a peculiarity caused by the 'element of'
relation.  There is no paradox with the following statement:    (011)

    The set of all sets that are not subsets of themselves.    (012)

Since every set is a subset of itself, the set defined by
that sentence happens to be empty.  No paradox.    (013)

Suggestion:  I recommend that we following the well established
practice of mathematicians for stating definitions.    (014)

John Sowa    (015)

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