Re: Getting back to the notes of May 10th
sowa <sowa@turing.pacss.binghamton.edu>
Date: Tue, 1 Jun 93 11:13:23 EDT
From: sowa <sowa@turing.pacss.binghamton.edu>
Message-id: <9306011513.AA21657@turing.pacss.binghamton.edu>
To: cg@cs.umn.edu, interlingua@ISI.EDU, phayes@cs.uiuc.edu,
sowa@turing.pacss.binghamton.edu
Subject: Re: Getting back to the notes of May 10th
Cc: eileen@turing.pacss.binghamton.edu, jerry@turing.pacss.binghamton.edu
Pat,
I was too hasty in saying that I agreed with your last note, since I
hadn't realized that there were more to come.
> You refer to Tarski only because of my usage in 'tarskian model', for which
> I have now apologised and explained my meaning several times. This is a
> conventionalised term throughout the community in which I work, but you
> objected to it, so I used the silly 'TMT' euphemism. You are now ridiculing
> this usage (in other messages), attempting to claim that the concept doesnt
> exist. This is simply irresponsible, as I know that you know that it does.
This is the crux of our disagreement. I don't believe that we have any
fundamental quarrels about model theory as a mathematical theory. Our
quarrels are solely about the APPLICATION of model theory. I kept
emphasing what Tarski and the mathematical logicians actually said because
they never applied model theory in a way that I objected to.
My original statement was very simple: the real world is not a model
in Tarski's sense. Any attempt to identify a model with the real world
is an APPLICATION of model theory. I am willing to allow such an
application, but only when the people who do the applying make it clear
that they are stepping outside of model theory per se and into a branch
of engineering or applied mathematics.
> a response to your assertion that only observable things can be
> referred to.
I didn't say that. I was trying to accommodate all such references
within a common semantic framework that allows me to refer to
fictional entities like Santa Clause, historical entities like
Julius Caesar, or entities I've actually observed like Pat Hayes.
I want to be able to construct (or postulate, if you prefer to allow
infinite or very large) models of all kinds, some of which may have
varying degrees of approximation to aspects of the current world
and others may be purely fictional or hypothetical.
>> When you blithely put such things into your "models", you have ignored
>>all of the most difficult questions about how words (or symbols in
>>a system of logic) can refer to things.
> Yes, exactly! I do ignore them! They are indeed irrelevant to semantics.
> Semantics takes it that symbols refer, and asks how such referring might
> be related to inference. It tries to make as few assumptions as possible
> about the nature of the things referred to.
The word "semantics" may be causing a problem here. I can accept your
last statement if I replace the word "semantics" with "model theory",
using model theory in the limited sense of a mathematical theory.
But the applied side of things is equally important: how do the
"individuals" in a model get associated with things in the world?
I would not consider that model theory, but it is a very important
question in fields ranging from psychology to robotics.
> I guess that you call it 'pure' because it uses techniques and ideas from
> set theory, which has traditionally and pedagogically regarded as part of
> Pure Mathematics, as opposed, say, to partial differential equations, which
> is part of Applied Mathematics.
No. The study of partial differential equations is just as pure as
set theory. It only becomes applied when you actually apply it to
electromagnetism or hydrodynamics. And when you do make such an application,
you have to face all the issues about measurement, approximation, experimental
error, etc. Those are the sames kinds of issues that Carnap was trying
to address in his Logische Aufbau: how are the symbols of logic related
to the things in the world?
> ... Tensor calculus is now considered 'applied'...
By itself, it is pure mathematics. But it is usually taught in
courses in physics or applied math. because of its usefulness to
those fields. But that does not change the nature of the subject.
> ... That's what we DO with mathematics: we use it to talk about things.
Yes, of course. But when we DO something with it, we are applying it.
>>To relate those mathematical things ("surrogates" in DB terminology)
>>to physical objects presupposes philosophy of science, psychology of
>>perception & language learning, or pattern recognition in AI.
> NO, it can be done simply by asserting that there is a 1:1 correspondence
> between them, if one has accepted such 'things'. Thats all the semantics needs.
> You came remarkably close to stating this in earlier messages, as I pointed
> out in comments that you never responded to.
I think that we are coming close to the point of disagreement. That
assertion (I would prefer to call it a hypothesis) that there is a
1:1 correspondence is where I claim that we step outside of model theory
as a mathematical discipline and into an application of model theory to
some aspect of the world.
For things like tables, chairs, people, and cats, that 1:1 correspondence
is relatively unproblematical. But when you get outside of that realm
into applications to very large, very small, or very fast things, the
question of how that correspondence can be established becomes much more
significant. You can say that is no longer "semantics", but "philosophy
of science". But whatever you call it, it is still part of the question
about how the symbols in logic can be used to refer to things in the world.
>>> Use conventional model theory. There is no problem!
>
>>My "depictions" are conventional models.
>Unfortunately, they are also vivid representations, data bases, analogical
>representations, mathematical idealisations and God knows what else. Thats
>my problem with them.
I was trying to separate the question of what they are from the
question of how they are used. Structurally, they are isomorphic to
the models of Tarski and the mathematical logicians. But they can be
used in a variety of ways.
> Of course you must apply mathematics. Now, how do I 'apply' arithmetic?
> Having made measurements, I manipulate numerals to discover the correct
> setting for the fence on my table saw, say. Had someone asked me what
> these symbols were, I would (correctly) have told them they were dimensions
> of a workpiece. I am simply construing the numerals to refer to aspects
> of the wooden world I am temporarily inhabiting. That construal does not
> affect any of the arithmetic I use on the numerals or its correctness, which
> is the business of semantic theories. The complicated issues of how
> measurement are taken and how their accuracy is ensured and so forth are
> not part of semantics...
At this point, the word "semantics" seems to have crept into the discussion.
There are so many different definitions of that term in different fields,
that it is very dangerous to make claims about what is or is not the
"business of a semantic theory". I would agree that the issues of how
the measurements are taken are not part of model theory nor of arithmetic.
But as soon as you apply arithmetic to some subject such as carpentry,
that application presupposes either a theory of measurement or at least
a body of practice in techniques of making accurate measurements.
> ... But on the subject of whether sets or 'classes', using an
> older terminology) could contain physical objects, try these
> quotes.... [from Carnap and Quine]
Those quotes are from examples where Carnap and Quine talk about how
set theory could be applied to mathematical things (whatever they may be)
and to concrete things like whales, porpoises, and people. But in their
other writings, where they address the question of how such an application
is possible, Carnap went into great detail in his Logische Aufbau, and
Quine went into detail in his talk about the "web of belief", which
contains many terms that are only very indirectly related to observables.
> Let me put the boot on the other foot and challenge you to find an
> authoritative quote that insists that sets *cannot* contain physical
> entities (as opposed to someone who is concerned only with mathematical
> abstractions and simply ignores such interpretations.)
I cannot claim to have done a complete literature search, but I would
like to refer to a book I happened to have on my shelf: _Reference,
Truth and Reality_, edited by Mark Platts, Routldge & Kegan Paul,
London, 1980, with contributions by Mark Platts, Colin McGinn,
Christopher Peacocke, Hartry Field, John McDowell, Donald Davidson,
Tyler Burge, Barry Taylor, Gareth Evans, and David Wiggins. As you
might guess, any book with so many contributors must have a wide range
of philosophical positions, which you or I might agree with or disagree
with to a greater or lesser extent.
Hartry Field, in "Tarski's Theory of Truth", claimed that "Tarski
succeeded in reducing the notion of truth to certain other semantic
notions; but that he did not in any way explicate these other notions,
so that his results ought to make the word 'true' acceptable only to
someone who already regarded these other semantic notions as acceptable"
(page 83). Among these, he includes "unreduced notions of proper names
denoting things, predicates applying to things, and function symbols
being fulfilled by things" (p. 89). Field continues with a lot more
subtle discussion, and McDowell responds to and criticizes some of
Field's analysis. I won't attempt to summarize all that discussion,
but the Field-McDowell discussion as well as other papers in that book
and other similar articles in the literature reinforce the point that
I was trying to make: model theory only explains how the denotation of
a complex sentence is related to the denotations of simpler terms; but
it does not explain how the denotations of those simple terms are
determined.
> ... I am completely confused about what these
> 'depictions' could possibly be meant to be.
> First, on definition. You proposed a sketch of a definition in a message
> circulated to a smaller working group, and it met with considerable skepticism,
> as you know, containing as it did seemingly arbitrary restrictions (eg no
> negative information could be included in one).
The only reason why I introduced the term 'depiction' is to avoid the
word 'model', which I was originally using. Therefore, when I introduced
the term 'depiction', I defined it to be exactly isomorphic to a conventional
model: only individuals and relations among them and no negations. We could
generalize that term to include something like Reiter's "open worlds", where
the depiction would contain two collections, one of positive information
and another of negative information. But to get the definition clear, it
is best to start with the simple case before generalizing it. The two
people who responded to that note (Len Schubert and Chris Menzel) were not
skeptical about the definition as a definition, but about whether I should
allow a more general definition. My response is yes, perhaps, but let's
get the simple form clear before generalizing it.
> ... this is a new idea, invented by you, which you have not yet adequately
> defined but are already claiming will serve as a unifying idea and rise
> above metaphysical disagreements. Earlier in this message you say they can
> be identified with conventional model-theoretic interpretations (what I was
> earlier calling TMT models), yet they also apparently can be identified
> with databases. It seems then that they might be infinite, but must be
> finite...? It also seems that they can be identified with what Len Schubert
> was referring to by his 'color wheel' example, i.e. an efficiently encoded
> vivid knowledge base. All of these are different ideas, with different
> properties, ontological stauses and possible functional roles. And now I am
> even more confused, because you now say they are apparently none of these
> things, but 'mathematical constructions', ie built only from the stuff
> of pure mathematics (or is that the pure stuff of mathematics?)
I was originally reluctant to introduce "this new idea" and did so only
to avoid using the term "model" in two different senses. You can think of
them as models (in Tarski's sense, if you like) where the primitive
individuals are lexical object types (i.e. numerals, character strings,
and structures built up from combinations of them). If you want to
represent them in a computer, they must be finite; if you would like to
postulate infinite depictions, you can do so. They can be put in a
1:1 correspondence with relational databases or with Len's color wheel.
> Im not sure what 'evaluating' means here. I thought that on your account,
> the DB *was* the denotation.(??)
To repeat: I said that a database is isomorphic to a model, which is
not itself a denotation, but the structure in terms of which the
denotation of a formula is computed.
> If a DB is used a source of information, ie if its contents are taken
> as having the semantic status of assertions, then it is *not* being
> treated as a tarskian ... model....
That introduces a new term "a source of information" -- anything can
be used as a source of information, including a model. As I pointed out
before, a collection of ground assertions is isomorphic to a conventional
model. So you can "think of" a DB as either one, depending on whether
you prefer to apply model-theoretic or proof-theoretic methods to it.
>> ... And please note that calling a relational DB
>>a model or a collection of ground-level assertions is purely a matter
>>of taste or convenience, since formally, they are isomorphic.
> NO! They are functionally very distinct. The distinction is rather like
> that between the existential and the universal quantifier. As I mentioned
> in my last message, a model-theory model can't be a representation (at
> least in any ordinary sense), since it has no assertional force. It
> simply exhibits one way the world might be: it does not, and cannot,
> state anything about the structure of other models.
I don't know what you mean by the word "functionally" that distinguishes
it from my word "formally". Are you suggesting something about the
usual purpose or intention? And there are a lot of other words in your
passage that could easily lead to misunderstandings: "representation",
"assertional force", and "exhibit". This is the first time that I have
ever heard anyone say that a "representation" has more or less
"assertional force" than a model. And a DB by itself doesn't "state"
anything. It is simply a collection of data that is used by an
SQL processor to generate answers to queries. And the operations
that such an SQL processor performs are formally (and functionally)
isomorphic to the operations that Tarski defined for evaluating
denotations of models.
> ... Thus, I attach little
> semantic importance to the lack of negation or quantification in such
> representations.
Computationally, this distinction is of the utmost importance. There
is simply no way to add negations and quantification to relational DB
and remain compatible with current implementations. It seems more
natural, not only to me, but to a many other people who talk and
theorize about databases to think of them as models. In an earlier
note, you suggested that I read Ray Reiter's papers, and I said that
I had talked with him about this point at a conference on DB & AI.
He said "I admit that it is common to talk about a database as a model,
but I prefer to think of it as a collection of assertions." Since Ray
was applying proof-theoretic techniques, that was a reasonable preference.
But since I wanted to apply model-theoretic techniques, I preferred the
alternate interpretation. They are both legitimate ways of thinking
about a database.
> .... I confess to finding this so
> extraordinary that I only came to this interpretation of your prose recently.
> In any case, as I have explained, if we take the functional role to reflect
> semantic issues such as validity, then it is just plain wrong.
Ray R. acknowledged that it was not "extraordinary", but common, even
though he preferred the other interpretation. You might also look at
Hintikka's "model sets", which he constructs out of collections of
ground-level atomic propositions (sorry, I don't have the reference
handy, but I'll send it when and if I come across it again). To me,
the interpretation of a DB as a model seemed so obvious that I didn't
think that it required much elaboration.
As far as other semantic notations are concerned (validity, satisfaction,
etc.), those are precisely why I wanted to treat a DB as a model. In
database land, people talk about "integrity constraints", which can be
interpreted as axioms for a collection of possible databases (i.e. models).
All the model-theoretic discussion carries over to such systems very nicely.
>> ... Since a theory consisting of nothing but
>>ground-level assertions is isomorphic to a model, there is no formal
>>(or functional) difference between us.
> Yes there is, as I have explained earlier. If you take it that an entry
> in a DB can refute an assertion in a theory, for example, then you are
> using the DB in a functional role which would not be permitted if it
> were only one among many possible interpretations.
But when I am using "model-talk", I wouldn't say that. I would simply
say that this DB does not satisfy that theory. If I had reason to
believe that the DB accurately corresponded to some relevant aspect
of the world, then I would conclude that the theory was not true
about that aspect.
> Such an interpretation can only be used as a counterexample to a line
> of reasoning or an entailment, not to an assertion. Getting these mixed up,
> which you SEEM to be doing here, is an example of what I was referring to
> in the last message as 'muddle'. But if you are not muddled, please excuse
> me (and maybe explain why not).
I believe that my terminology was consistent with my interpretation of
a DB as a model, and your terminology was consistent with your interpretation
of a DB as a theory.
> ... However, let me observe that it seems that
> 'depictions' are being used here as a form of intermodal communication
> device, ie what I was calling (admittedly in haste) "a kind
> of unifying information blackboard", and reiterate that this seems to me
> to be yet another idea which is different from the four or five different
> ideas that are somehow incomprehensibly conflated into 'depiction'.
As I said above, the term can be defined formally very simply,
but it can have many possible uses. I actually like your suggestion
of "a kind of unifying information blackboard" because that is roughly
how I would use it in our project of developing a hybrid system with
both a symbolic and a computational side. In fact, in most of the systems
that do have things that they call "blackboards", the kind of information
that is put in the blackboard is usually of the form of ground-level
assertions without negations or quantifiers. Sometimes, people do put
simple negations in blackboards, but that could also be accommodated
by a generalization of "depictions" to open worlds with collections of
simple positive and simple negative information.
John