Re: Intensions/Sinnfritz@rodin.wustl.edu (Fritz Lehmann)
Date: Sun, 13 Feb 94 09:38:44 CST
From: email@example.com (Fritz Lehmann)
To: firstname.lastname@example.org, interlingua@ISI.EDU, email@example.com
Subject: Re: Intensions/Sinn
Dear Pat Hayes, February 13, 1994
>>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.
For knowledge representation, yes -- in particular, focus on
what it is (intensional qualities) that makes them interesting or
useful or definable.
>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.
Mathematics as it is now practiced (when about integers at
all) is entirely about 'interesting' sets of integers, like
primes, etc. No-one studies features true of all sets of
integers. You are right to analogize extensional logic to
arithmetic; extensional logic's role in knowledge representation
should be approximately that of arithmetic in mathematics: a
reliable low level tool. (Incidentally, I gather that Chaitin's
theorem shows that "almost all" integers are indeed uninteresting
in the sense that their shortest Turing machine definitions are no
shorter than the numbers themselves.)
>("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.
A 'model' in your sense is a directed hypergraph, no more, no
less. D must BE a set (you mention "subset"). If we discuss what
D's MEMBERS are -- or your "nontrivial" -- we'll rehash your long
email debate with John Sowa on "models" in which I sided with him.
>>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
Failure to "give the results he wanted" was a serious problem
for Frege's extensional semantics, not to be airily dismissed.
(It's no insult to group a theory with the Lorentz-Fitzgerald
contraction, in case you think it is. It is Special Relativity in
different words; either both are true or both are false -- the
choice between them is aesthetic. A fairly distinguished
"intellectual category" to be in.) What you meant to suggest was
>>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-
>These sound fascinating, can you give references or pointers (probably off
>the big lists)?
Some references are: TROPE THEORY: J. Bacon "Four Modelings",
J. Philosophical Logic, 17(2) 91-114, 1988; D C. Williams "The
Elements of Being" in Principles of Empirical Realism, Thomas,
Springfield, 1966. FORMAL CONCEPT ANALYSIS: R. Wille "Concept
Lattices and Conceptual Knowledge Systems" in Semantic Networks in
Artificial Intelligence, F. Lehmann, Ed., Pergamon Press, 1992; R.
Wille, "Restructuring Lattice Theory" in Ordered Sets, I. Rival,
Ed., NATO ASI Series C83, Reidel, Dordecht, 1983. RUSSIAN
MERONOMY: O. M. Polyakov & V.V. Dunaev "Classification Schemes:
Synthesis through Relations", Nauchno-Tekhnicheskaya Informatsiya,
Seriya 2, 19(5) 15-21, 1985 (translated as Automated Documentation
and Mathematical Linguistics, Allerton Press, NY), and many
previous and subsequent articles in that journal. FACT-BASED
ONTOLOGY: K. R. Olson "An Essay on Facts" CSLI Lecture Notes No.
6, Stanford, 1987.
Philosopher Barry Smith told me that trope theory is "big in
Australia now". Wille's survey in my Semantic Networks collection
can also be found in Computers & Math. with Applications (journal)
v 23 no.2-9, 1992, p493-515. There are other "fact-based ontology"
sources which I forget.
>> 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.
For KRep, sets are a tool. She commits herself to intensions
if the set of girders is to be distinguished somehow from the set
of kumquats. Len Schubert sort of made that point to me earlier.
Tarskian model theory is about models only "up to structural
isomorphism" -- alas this again raises the old Hayes-Sowa email
debate on models.
>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.
This pertains to the earlier HAYESISM/LEHMANNISM difference
in which you seemed to want completeness in KR so as to "ground"
everything in logic, whereas I'm content to have some logically
uninterpretable (externally interpretable) primitives, at least
for interlingua purposes. Your statement above tells me that I
might not understand HAYESISM.
>> 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 will 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
You're probably right, but since I haven't seen the "grok" arguments you
disdain I'll reserve judgement (and I'm uncaptivated by what I know of Kripke
and Montague). Formal (Peircean) semiotics or Husserlian "noemae" may be the
solution. (But if we really did need to go back to the 1890's, or the
1190's, I'd be ashamed to fail for lack of chutzpah.)
Yours truly, Fritz Lehmann
4282 Sandburg, Irvine, CA 92715 714-733-0566 firstname.lastname@example.org