Re: SUO: Re: ONT Intension & Extension
Robert,
The terms intension and extension correspond to notions that have
a long history, often with somewhat different names. The term
"intension" is actually an unfortunate choice, since it is pronounced
exactly the same as "intention". The term "comprehension" is the
more traditional word, which is less likely to cause confusion.
But whatever term you use, "intension" or "comprehension", it is
most definitely true that the intension or comprehension of the
definition implies the extension of the set of instances.
The converse definitely does not hold in any philosophical or
logical discussion from Aristotle to the present.
JS>> I did the comparison you suggested, and the definition in the IFF
>> document is wrong.
> No, it is not wrong! The definitions, terminology and axiomatics in all of
> the IFF follow the definitions, standard terminology and axiomatics of
> well-known mathematical ideas.
Lattice theory, as an uninterpreted formalism, is not wrong.
It is only when you map formal notions from lattice theory
to terms that have an independent traditional usage that you
may create a conflict. The way of resolving the conflict is
either to change the mapping or to use different terminology.
> The definition in the IFF is precisely the definition of a formal concept in
> FCA (definition 20, page 18, "Formal Concept Analysis: Mathematical
> Foundations" by Bernhard Ganter and Rudolf Wille). The extent consists of
> all objects (IFF instances) that satisfy the attributes of the formal
> concept and the intent consists of all attibutes (IFF types) satisfied by
> the objects of the formal concept. A formal concept is a closed element of
> the Galois connection of FCA derivation. And more specifically, the
> equivalence noted in the IFF document is precisely stated in definition 21,
> page 19, of the Ganter-Wille text.
I have a copy of the G-W textbook somewhere, but I can't find it
right this minute (it seems to have been misplaced as a result of
some painting and redecorating that was done during the summer).
So I can't check the G-W definition. But if you are correct
in saying that you took the definition out of their text, then
Ganter & Wille are responsible for the error. (If you don't want
to call it an error, then it is an unfortunate confusion of a
traditional term with a mathematical construct that should be
given a different name.)
Bottom line: You can define a mathematical construct any way you
please, but if you identify it with "extension" and "intension"
(or "comprehension"), you are violating a 2500 year old tradition.
I strongly urge you to do one of two things: either revise the
definition so that intension implies extension (but is not equivalent
to it), or change the names so that there is no suggestion that
the predicates you are defining represent the concepts that are
traditionally called "extension" and "intension" (or "comprehension").
John Sowa