Re: Good and Bad IS-A hierarchies (Peter Clark)
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From: (Peter Clark)
Date: Tue, 1 Aug 1995 09:37:21 -0600
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Subject: Re: Good and Bad IS-A hierarchies
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>>>>> There is no way to organize the concepts, or even the axioms, into neat 
>>>>> little packets so that the various alternatives can be assembled by 
>>>>> choosing some and ignoring others. [Pat Hayes]

>>>> Well, the above paragraph ignores the idea of being able to 
>>>> compose representations from components. [Peter Clark]

>>> It doesnt ignore it, it reports a sober conclusion that that is impossible.

>> It might help to distinguish picking-and-choosing axioms which have
>> different underlying assumptions about the world, and picking-and-choosing 
>> axioms which have the same underlying assumptions. Your 
>> time-interval example was an example of the former.

> Thanks, that gives a nice way to sum up my worry: EVERY set of axioms
> embodies a different set of assumptions about the world. What is a set of 
> assumptions other than a set of axioms?

Hmmm....yes, I see what you mean. I was worrying about that too. Let my
try again:

An assumption is a belief about how a theory's symbols map onto the real
world ***. Consider your temporal example again:
	A1: I starts J iff....
	A2: intervals are reversible
You say that adding A2 means A1 now needs to be revised. But why? It's
not that A1 has suddenly become false (axioms are all true in their theory,
by definition). It's because that the symbol "start" no longer reflects
what you intended it to mean.

Consider two (syntactically isomorphic) sets of axioms to illustrate this:

	A1: flowers are pretty		A1: cats are good-domestic-pets
	A2: roses are flowers		A2: lions are cats

S2 is meant to mimic your temporal example. After seeing A2, I say "oh, you
mean cat as in cat *species* (rather than small fluffy thing). Okay, that's
fine..," (I don't really mind, just as I don't really mind if temporal 
intervals are reversible or not), "....but I better go and revise A1 then."
Why does A1 in S2 need revising, but A1 in S1 not? There's two notions of 
"truth" here:
  1. truth within a theory (in or derivable from that theory)
  2. truth about whether a theory reflects (our beliefs about) the real world,
	ie. whether statements in a theory are true in the real world,
	under some mapping of the theory's symbols to the real world.

A good set of axioms (eg. S1), under some mapping, is a good approximation to 
the real world (ie. most true statements in the theory are also true in the
real world under that mapping). If I want to drop axioms from that theory,
then that's okay: I've just said less, and the theory will be a poorer
reflection of the world. I can pick-and-choose happily within this set.
I can't do that with S2 though -- the mappings from axiom symbols to the
real world conflict.

Maybe you *can* pick-and-choose from your temporal theories too. Suppose I 
take temporal theory T1, and I drop an axiom. Now T1 doesn't say everything I 
intended it say about time -- but that's okay, it's just now a poorer 
description of time. You might be tempted to reject the reduced T1 because it 
no longer says everything you intended to say about time, but that's unfair: 
how do you know when you've said everything anyway? (Though I take your point
about these temporal theories being "minimal" -- it might be that taking
out any single axiom will make the theory next-to-useless as nothing 
can be derived any more, but that doesn't invalidate it).

Saying an assumption is some non-axiom-based notion is probably going to
land me in trouble -- eg. how can I convey to you what I meant the symbols
to mean, other than write axioms? I'm not really sure of the answer;
though people can communicate and at least have some agreement on what
linguistic symbols mean. And you yourself introduced a mapping when you
stated your "I starts J" axiom needed revising.

Can I pick-and-choose from my S1-like axiom set now?

			Best wishes,


*** or, more precisely: An assumption is a belief about how a theory's symbols
    map onto my perception of the real world. Really, we're talking about
    a mapping between two symbol systems (the KB of axioms, and the 
    axioms/symbol system in an agent's head). But I'll just use "mapping onto
    the real world" for conciseness in this message.

Peter Clark (   Department of Computer Science
tel: (512) 471-9565                  University of Texas at Austin
fax: (512) 471-8885                  Austin, Texas, 78712, USA.
Some WWW pointers to KBS labs: