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Date: Fri, 11 Feb 1994 16:59:00 +0000
To: firstname.lastname@example.org (Fritz Lehmann), email@example.com, interlingua@ISI.EDU
Subject: Re: Intension/Sinn
Cc: firstname.lastname@example.org, email@example.com
Conversation between Fritz Lehmann and Pat Hayes on extensionality. Pat is
in the even corner, Fritz in the odd. Round three, highlights.
>>>P.P.S. Must we totally neglect intension? Sinn? Two facts
>>>are: A. There are too many different discordant notions of
>>>what "intensions" are and many are hard to formalize neatly,
>>>whereas everybody pretty much understands extensions as a formalism.
>>>B. Notwithstanding fact A., intensions really matter more than
>>>extensions. Triangularity is the thing to know about, not the set of
>>>triangles. Same goes for Democracy, puffinhood and FAX machines.
>>[this is so difficult]... that one begins to be suspicious that maybe the
>>primal intuition is
> What's often faulty in formalizations is failure to "do justice to what
>we know," in Hao Wang's words. (Are _primal_ intuitions faulty? Lesser
>intuitions are faulty if in conflict with primal ones.) Fact B.
When intuitions which seem overwhelmingly convincing nevertheless have
resisted the best efforts of some of the cleverest people for around a
century, I incline to re-examine them. History is full of incredibly
'intuitively obvious' truths turning out to be profoundly wrong. If you
trust your primal intuitions, read a book on quantum electrodynamics.
>>I have now reached the stage where I really have no idea what
>>'triangularity' means, and suspect it means many different things in
>>different philosopher's mouths.
> I'm still at the stage in which I have a better idea of what
>"triangularity" means than what "the set of triangles" means.
Not me. I can begin to visulaise the member sof the set of triangles, but I
dont even know what *kind* of thing 'triangularity' is supposed to be. It
is presumably not a shape or anything geometrical or topological. If we
make it a description then we need an account of the semantics of this
language, which takes us back to the beginning. Is it something found in a
geometric way-of-being-in-the-world ?
>>>Who really cares about arbitrary extensional sets of objects?
>>I do! ...
> Pat, I don't believe it. I think what you mean is that, as a logician,
>you care about a logic's _ability_to_denote_ arbitrary extensional sets, but
>do you really care about any arbitrary sets themselves? Nah. No-one does.
>If I list arbitrary objects or entities with no pattern of common
>qualities, you will be bored, not interested.
I didn't even think of this interpretation of your words. No, of course, if
you cook up a random set I am not going to hug it to my bosom or become
infatuated with it. I don't get all misty-eyed at the thought of 276,927
either, but I do 'care about' (in the sense that I care that they are
there) arbitrarily large integers. (But notice, if I believe that
*everything* is either soft or heavy, then I will believe this of the
things in your set, no matter how unrelated they might be. Their
relatedness or otherwise will have nothing to do with this belief.)
> More essentially, such sets
>are inherently useless.
They are just a mathematical way of referring to arbitrary things.
> Except in the "What is a set?" section of a math book, or in
>discussions like this one, no-one uses arbitrary sets at any time in
>real life. That's one reason why the set in "shoes--- and ships --- and
>sealing wax --- Of cabbages --- and kings --- And why the sea is boiling
>hot --- And whether pigs have wings." is funny. Sets are interesting or
>useful only when the (intensional) qualifications of membership are
>interesting or useful.
Your position seems to be that most sets are uninteresting, so we shouldn't
be so careless with them but instead focus on the interesting ones. To see
what this argument amounts to, apply it to integers. Most integers are
uninteresting; therefore, we should abandon arithmetic as it is now
practiced and develop a theory of interesting integers.
Look how sets are used in defining semantics. In the metatheory, set
language is used to define possible models ( "A model is a domain D and,
for each n-ary relation symbol a subset of D!n and ..."). But notice that
this does not imply that the model is any any nontrivial sense 'made' of
sets; it is simply a way of saying (using conventional mathematical usage)
that the universe could be made of anything. Its that 'any' in 'anything'
that the set language gets for you. Sets are about as ontologically
noncommittal as you can get.
Suppose we had an account of 'interesting' sets, and we defined a semantic
theory for a representational formalism by saying that the universe D had
to be such a set. Then in this representation, a correct and adequate
axiomatisation of the concept of 'interesting' would be simply (forall x
(interesting x)). There could be no other axioms since this is now a
>Even math books use only intensional "set-builder"
>notation rather than arbitrary lists, after the opening section. The
>operations of logic and set theory treat interesting and dull sets the same,
>which is a both a merit and a limitation of logic and set theory.
>Extensional logic is a valid _constraint_ on the logic that matters, that
>of meanings or "Sinn". (That constraint is a Galois connection as I said
"Sinn" is just Frege's word for things he thought had to be there because
his semantics didnt give the results he wanted. Let me suggest that it
belongs in the same intellectual category as "Fitzgerald contraction" and
> From Aristotle through Leibniz and up to Frege, intensions
>were considered more important. Then, especially with the influence of young
>Quine and a 40% chunk of Tarski, extensions became all the rage partly due to
>ease of formalization. There has always been a countercurrent, though. More
>recently a third view has emerged, that the basic concept is the _connection_
>of intensions to extensions, in "trope theory", Wille's "formal concept
>analysis", Russian "meronomy", and "fact-based ontology".
These sound fascinating, can you give references or pointers (probably off
the big lists)?
> To call something "knowledge representation" when it deals only with
>sets is a bit misleading.
As explained earlier, the use of set-theoretic language in the semantic
metatheory of a Krep language does not imply that the Krep 'deals only with
sets'. An engineer might talk of 'the set of girders' in a bridge without
committing herself to Platonism. One must not confuse the semantic goal
with the quite different Russel/Whitehead goal of using set theory as a
definitional base for all of mathematics.
> Knowledge is certainly about qualities. An
>amusing limitation of set-based (extensional) logic is that it is incapable
>of distinguishing purely arbitrary sets from sets whose members do have some
>quality in common. A knowledge representation which is entirely extensional
>necessarily fail to capture meaning, including even purely structural (i.e.
>combinatorial) parts of meaning. I think Bill Woods has often urged this
>point. This doesn't negate the value of current KIF, conceptual graphs or
>extensional logic; it just means that there is more to the story. The fact
>that there is no consensus on formalizing the rest of the story doesn't mean
>it isn't there and isn't important.
I entirely agree with this conclusion, and that there is more to the story.
I just wanted to question this familiar line that we need to somehow come
to terms with Real meanings, senses, Sinns, qualities or whatever other
Thing Beyond Set Theory has been proposed. These are perfectly legitimate
targets for formalisation, but if this is rejected on the grounds that the
formal tools to be used have extensional semantics and therefore will be
forever unable to grock the essential nature of these things, then I give
up. If we have to abandon extensional tools of formalisation then we are
going back to the 1890's, and I would really rather not do that if we can
possibly avoid it. I don't have the chutzpah, for one thing. Let me suggest
that we follow the insights of Kripke , Montague et. al. and try to
describe INtensionality by hypothesising EXtensional ontological
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