Thanks for the suggestion, John. My background is in computer science,
so I'm not as well versed in mathematics and logic, as the members of
the forum. I'm always learning, and the feedback has been very
helpful. (01)
Thanks again.
--
Bart Gajderowicz
MSc Candidate, '10
Dept. of Computer Science
Ryerson University
http://www.scs.ryerson.ca/~bgajdero (02)
2009/4/24 John F. Sowa <sowa@xxxxxxxxxxx>:
> Bart,
>
> Definitions of the following kind are trivial substitutions:
>
>> Momo = Man who has Only Male Offspring
>> (forall x (Momo(x) iff exists y s.t. Man(x) ^ Momo(y) ^ hasChild(y,x)))
>
> Following is the more common form of definition in mathematics:
>
> A dyadic relation R is transitive iff the following axiom is true:
>
> For all x, y, z,
> R(x,y) and R(y,z) implies R(x,z).
>
> 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.
>
> Mathematicians have been using such definitions since Euclid,
> and they have never caused any trouble at all.
>
> Perhaps you're thinking of the following paradoxical statement
> in set theory:
>
> The set of all sets that are not elements of themselves.
>
> But that is a result of a peculiarity caused by the 'element of'
> relation. There is no paradox with the following statement:
>
> The set of all sets that are not subsets of themselves.
>
> Since every set is a subset of itself, the set defined by
> that sentence happens to be empty. No paradox.
>
> Suggestion: I recommend that we following the well established
> practice of mathematicians for stating definitions.
>
> John Sowa
>
>
>
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