Re: SUO: Re: ONT Intension & Extension
Robert,
I did the comparison you suggested, and the definition in the IFF
document is wrong.
REK> Compare Alonzo Church's discussion (below) of the intension and
extension of
> a concept with the mathematical (Formal Concept Analysis) (Rudolf Wille and
> Garrett Birkhoff) definitions of the intent and extent of a formal concept
> on the middle of page 30 in the IFF Classification Ontology document
> [http://suo.ieee.org/IFF/versions/20020102/IFFClassificationOntology.pdf].
The error is clearly seen on page 31 of the IFF document where it
says that a concept c1 is a subconcept of c2 when the extent of c1
is a subset of the extent of c2 "or equivalently" when the
intent of c1 includes (I don't know how you intend to read
the superset symbol when comparing intensions, so I used the
word "includes") the intent of c2.
That statement is false according to the common usage from Aristotle
to the present. The correct statement is the following:
If the intent of c1 includes the intent of c2, then the
extent of c1 is a subset of the extent of c2.
This a one-way implication, not a two-way equivalence. The common
counterexample is that the extension of "featherless biped" is
equal to the extension of "rational animal" but the neither intension
includes the other. That counterexample, by the way, was dramatically
illustrated by Diogenes the Cynic, who plucked a chicken and threw
it into the Acadmeny while shouting "Here is Plato's man!"
And by the way, Alonzo Church had a more detailed discussion of
the difference between intension and extension in his book on
lambda calculus. I scanned the first three pages of that text
into the following web page:
http://www.jfsowa.com/logic/alonzo.htm
And by the way, I agree with Pat Hayes that the IFF document is
overloaded with heavy formalism. Good mathematicians do not overload
their texts with unnecessary formalism. That kind of heavy-handed
notation does not make the presentation more precise -- it merely
makes it unreadable.
I recommend that you take a look at my KR book for a better level
of formalization. Every statement is as precise as necessary for
a mathematician to understand. But at the same time, I give a full
explanation of everything in a clear English statement, which can be
read even by people whose knowledge of mathematics is rather shaky.
I recommend that level of formalization for the IFF document.
The heavy-handed notation of the current draft is useless for a
professional mathematician and very off-putting for everybody else.
I also recommend that you take a look at Alonzo Church's presentation.
He was a very respectable logician, but he also uses a mathematical
style at about the same level of formalization that I recommend.
Bottom line: Use more English and thin out the amount of heavy
handed notation. Make sure that the definitions are correct, but
don't bother to write formulas that any mathematician would consider
trivial. However, do make sure that the equivalent information is
spelled out in English, not in jargon.
Fundamental principle: Whenever you state any formula in KIF or in
mathematical notation, always follow it with a clear English sentence
that says exactly the same thing. I consistently follow that
principle in every book and paper I write, and it is very much
appreciated, even by professional mathematicians and logicians.
John Sowa