Re: Intensions/Sinn
phayes@cs.uiuc.edu
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Date: Mon, 14 Feb 1994 14:43:29 +0000
To: fritz@rodin.wustl.edu (Fritz Lehmann), cg@cs.umn.edu, interlingua@ISI.EDU
From: phayes@cs.uiuc.edu
Subject: Re: Intensions/Sinn
Cc: phayes@cs.uiuc.edu
Dear Fritz-
>...
>>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.
>
> 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.
Anyone who uses arithmetical induction studies a feature which is true of
all the integers and therefore one which is true of all sets of the
integers.
Look, it is just OBVIOUS that we need all the integers, or even calculators
wouldn't work properly ("That answer is unobtainable because it I find the
numbers in it rather boring"). This discussion has gotten so silly that I
suspect we must not be understanding one another.
>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.
Let us agree to agree on this.
....
>
>>...
>>("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").
Yes, it is a set...so what? Your emphasis seems to indicate that you place
some significance on this observation, which for me is close to trivial.
Could you explain in simple terms what this significance is?
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.
>Let's not.
D's members can be ANYTHING. Anything at all. My talk of sets (or indeed
hypergraphs, abelian groups or any other mathematical term) is not intended
to refer to Platonic abstractions, but is simply a way of talking about
structure. Hence, your "no more, no less" is not, for me, an ontological
dismissal. The pieces of paper on my desktop, arranged by height,
constitute a directed hypergraph. Directed hypergraphs are everywhere.
I can see that if you find this unacceptable, then insistence on set talk
might well be unsettling. This is probably an old, old difference of
philosophical position. If you are a Platonist and I am an Aristotelian,
then by all means let us agree on this and not argue about it any more.
>>>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
>>>before.)
>
>>"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
>>"vital force".
>
> 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.
An aside, but the Fitzgerald contraction was not relativity in different
words, but a completely ad-hoc and quite implausible hack to make the
numbers come out right, based on no underlying theory or plausible
assumption, and assuming the old Newtonian idea of absolute space.
Fitzgerald said so himself, in fact. Relativity produces those numbers from
some new first principles by reasoning which is quite rigorous and very
surprising and has other surprising consequences, such as temporal
dilation.
..<thanks for the refs> ....
>>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.
Yes, of course.
>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.
No, hold on, thats a different point. Of course, I agree, model theory is
about models-up-to-isomorphism. I made the same point in my old naive
physics papers. Never mind kumquats: theres no way within pure first-order
model theory of preventing any consistent theory being interpreted in a
Herbrand model which is built entirely of symbols. The moral is, that if
one wishes a representation to be able to make such a distinction (say, to
ensure that its representations can only refer to things made of steel)
then some representational machinery outside ordinary logic is going to be
needed, perhaps some kind of 'groundedness'. But notice the difference
between this conclusion and the conclusion that since our representations
DO refer to steel things, that therefore we need a better semantic
(meta)theory to explain this connection. This seems to me to be wishful
thinking; and this is the old arguemnt between us, Fritz, not Sowa and me.
>>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.
Quickly: all I meant here was to distinguish the very particular and
special goal of defining mathematics *in terms of* sets ( eg the integer 3
is defined to be {{{{{}},{}}},{{}},{}}, etc) and *deriving* mathematics
>From set theory to provide a consistent foundation: the Russell/Whitehead
goal. Some people erroneously assume that the use of set theory committs
one to this very peculiar ambition, but it does not. One need not take 0 to
BE the empty set in order to talk of sets of integers.
>>>..... 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. ......
> 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.
Oh God, I hope not. I dont even know what the problem is that Husserlian
noemae might be a solution to.
On grok. This to-and-fro started with a PS of yours suggesting that we
should look at Sinn's. An on-going problem here, as I understand it, has
always been that formalisms proposed in this area tend to need some kind of
semantics; it is difficult to avoid using set theory in the specification
of such semantics; and (it is sometimes argued) such usage violates the
very intuitions that we are trying to capture, being extensional and all.
My only point was to suggest that rather than conclude that there is
something inherently wrong with the extensional tools we now have
available, such as set theory, it might be more useful to ask what kinds of
entity these intensional ways of thinking seem to involve, and try to
describe them extensionally. Kripke's possible-world approach to the
semantics of modal languages is a paradigm. It may not captivate you, but I
still find it a masterpiece of clarity and insight compared to the
confusion that reigned before it appeared (eg try reading some of the early
books by A.N. Prior). As you say, set theory is a tool: so let us use it
confidently.
But I also enjoy rummaging around in the intellectual fleamarkets, so by
all means lets all go on looking for bargains.
Best wishes
Pat Hayes
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