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Re: [ontolog-forum] Axiomatic ontology

To: "Rob Freeman" <lists@xxxxxxxxxxxxxxxxxxx>
Cc: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Fri, 17 Oct 2008 11:33:59 -0500
Message-id: <0508BB32-4445-4922-A5C6-5EB854062A9A@xxxxxxx>

On Oct 17, 2008, at 5:34 AM, Rob Freeman wrote:    (01)

> Pat,
> On Mon, Oct 13, 2008 at 12:21 AM, Pat Hayes <phayes@xxxxxxx> wrote:
>> On Oct 12, 2008, at 6:49 AM, Rob Freeman wrote:
>>> ... 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.)
>> But do you have any reason to suppose that it WILL be large, in fact?
> I think we have reason to suppose this, yes.
> At the very least the possibility should be studied carefully. I don't
> think it has been.    (02)

I don't think it is well-enough posed as a question to permit actual  
study right now. You still havnt said what you mean by 'meaningful',  
for example.    (03)

> Articles like the one by Chris Anderson I referenced earlier in this
> thread are one indication the number is beyond expression in any one
> model.    (04)

Sorry, but I have to ask: what sense of "model" do you mean here? We  
have gone around the houses many times on this (and other) threads  
chasing multiple meanings of this weasely word.    (05)

> Other people are beginning to say similar things.
>> I'm talking here about FO semantics as opposed to higher-order  
>> semantics.
>> Were you then arguing for the use of classical higher-order logic?
> I was trying not to assume any ab initio semantic model at all.    (06)

Then I fail to see how you can phrase a coherent question. After all,  
the question itself refers to "meaningful", which is surely a semantic  
notion is there ever was one.    (07)

>> Set theory does not suggest anything about 'meaningful' sets. It  
>> simply
>> talks about sets. Yes, there are a lot of sets. Nothing at all  
>> follows about
>> how many 'meaningful' sets there are. The fact that human beings do  
>> not seem
>> to be involved in combinatorial explosions when they think or speak
>> rationally, suggests that the number of meaningful sets might be a  
>> lot less
>> than the number of combinatorially possible sets. And one can give  
>> many
>> other reasons to suggest that there are far fewer 'meaningful' sets  
>> than
>> sets altogether: for example, the fact that people can do induction  
>> quite
>> successfully, or that one-shot learning often works very well. You  
>> may hold
>> other opinions, but it falls to you to come up with some reasons  
>> why this
>> might be an issue. Simply re-iterating elementary mathematical  
>> facts doesn't
>> cut it.
> I agree there must be "far fewer 'meaningful' sets than sets
> altogether". I'm not arguing that all sets are meaningful.    (08)

Good. But did you read my example of the laptop screen, suggesting  
that meaningfulness is very thinly spread, as it were?
> On the other hand I don't see that we have any justification for
> simply throwing away all sets which do not fulfill some arbitrary
> condition, e.g. conformance with one or other logic or model.    (09)

You seem to be laboring under some kind of delusion here. Take common  
or garden first-order Tarskian model theory. This does not 'throw  
away' any sets, not does it impose any arbitrary conditions on them.  
On the contrary, in fact: it is a basic hallmark of the Tarski  
construction that NO limitations are placed upon the entities in the  
universe of an interpretation, and NO limits are placed on the number  
of them (other than it is at least one, ie something exists) and NO  
limitations are placed upon the size or extent of any relational  
extensions. Nothing is "thrown away" or ruled out by some a priori  
conditions.    (010)

> Rather than defining "meaningful" as conformance with a model, why not
> take it from the other direction? Sets can be limited not by a model,
> but by say, a distance measure (like John's analogies.) Let that limit
> us. Then let the sets which result from clustering on a distance
> measure determine the model relevant to a particular case.    (011)

That all sounds vaguely interesting, but its too vague to respond to.  
Can you sharpen it up? What kind of "model" is being determined here,  
for example? Are you talking about a model in the semantic sense (from  
model theory) ie a satisfying interpretation, or in the everyday  
sense, as in 'architect's model'?    (012)

> What happens? Has anyone looked into it?    (013)

Probably. If you can sharpen up the question, we'd have a better  
chance of being able to find out.    (014)

>> You have just summed up one the central foundational notions
>> of computer science.
> Great. Yes. Thanks for making that association.
> It is somewhat my point that we do not use the full expressive power
> of these "foundational notions of computer science", a power beyond
> any single logic, when we attempt to represent knowledge.    (015)

We don't? Seems to me we use them all the time. The largest part of  
"Web science" (to use the latest buzzphrase) is concerned with ways of  
encoding information into byte streams, one way or another.    (016)

Why do you say that the idea of using character streams is a "power  
beyond any single logic"? There are results (analogous to Turing  
computability theory) showing that weak logics are 'universal' in this  
basic sense. All one really needs is a productive syntax and a  
conditional construction, and you have all the relevant "power" anyone  
can have.    (017)

Pat    (018)

> Why?
> -Rob
>    (019)

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