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Re: [ontolog-forum] Axiomatic ontology

To: sowa@xxxxxxxxxxx, "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "Rob Freeman" <lists@xxxxxxxxxxxxxxxxxxx>
Date: Sun, 12 Oct 2008 19:49:59 +0800
Message-id: <7616afbc0810120449w35e53ccfycbeb7723bff3f05@xxxxxxxxxxxxxx>
John,    (01)

Thanks for your analysis. That puts it nicely. Anything you can say in
any logic, you should be able to say in English. In counting the
meanings of English we need to count them all.    (02)

I don't know if your observation that the number would be countably
infinite is true also. What matters to me is that, even for finite
sets, the number of possibly "meaningful" relations could be very
large (unless we impose an artificial limitation in some way.)    (03)

As Pat H. says:    (04)

"This is why for example classical higher-order logic is undecidable,
because its semantics requires that its interpretations contain all
relations in this mathematical sense (and when the base set is
infinite, this makes the set of relations much more infinite.)"    (05)

Now, Pat deals with this by limiting the relations he considers:    (06)

"...this doesn't matter because the logic isn't obliged to consider
all of these relations, only enough to provide denotations for all the
relation-naming terms in the language."    (07)

In short Pat assumes some kind of other semantic model, and that other
model limits the relations he needs to consider, so this explosion of
possibly meaningful sets "doesn't matter", QED.    (08)

Pat doesn't like the explosion of relations. He needs some other
system of meaning to tell him which ones are meaningful:    (09)

"Yes, these sets are all distinct. I don't know if they are distinct
in "meaningful ways" because I don't know what you count as a
'meaningful way'."    (010)

But Pat, nobody knows what to count as a "meaningful way". That is
what is at issue. What right do we have to be throwing out all the
combinatorial explosion of possible meaningful sets suggested by set
theory?    (011)

I don't blame you. It is normal to look at this combinatorial
explosion as a problem. Perhaps that is because of the
"intractability" of such a system from the point of view of logic. But
that same "intractability" has a flip side. It can also be seen as a
power. What it tells us is that even for a finite "base set", you
might elaborate a very large (countably infinite?) number of meanings.    (012)

So a finite number of examples can specify an infinite number of
categories. Not just infinite strings of bits. Finite strings
specifying infinite numbers of categories too.    (013)

Am I the only one to see that as quite powerful?    (014)

What I don't understand is why we are not using finite sets of
examples to specify (very large or countably) infinite sets of
categories in this way. Say, use one set of examples to represent
numerous (all?) possible logics as relations over it.    (015)

If you are looking for a point to what I have been saying, that is one
expression of it.    (016)

Pat Cassidy: By "Guo" do you mean Guo Jin formerly of Singapore
National University? I can't imagine what methodology he used to
decide if a word no. 1401 were "required", or if word no. 1400 was
really the last one...!!    (017)

-Rob    (018)

P.S. John, that "countable infinity" of yours is a nice result. How
did you get it? I've guessed that if you can find more than n sets
over n examples then in principle you could go on for ever generating
n+p sets over n, and then (n+p)+q sets over those n+p etc. So I
guessed an infinity, but I had no evidence the expansion would not top
out at some point (much larger than the set used to specify it.)    (019)

On Sat, Oct 11, 2008 at 9:01 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
> Rob and Chris,
> RF>> How many [relations] are needed for English?
> CM> Good heavens, the question doesn't even make sense.
> The main sense I would make of it is "How many different kinds
> of relations might be expressed in English?"
> My interpretation of that question would be:  What is the set
> of all possible relations that might be mentioned or defined in
> any version of any formal logic or any programming language or
> any similar formalism.
> Since any of those relations could also be mentioned or defined
> in English sentences, the number for English would be the
> cardinal number for the totality of all those for any formal
> language.
> The number of relations that are possible would be uncountably
> infinite, but the number that could actually be defined or
> specified in a finite statement (in any natural or artificial
> language) would be merely countably infinite.
> That is still very large.
> John    (020)

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