Models and Depictions

sowa <>
Date: Tue, 11 May 93 10:01:03 EDT
From: sowa <>
Message-id: <>
To:, interlingua@ISI.EDU,
Subject: Models and Depictions
I'm hoping to wind down this discussion, important as it is, because
we all have other work to do.  I still maintain that the principle
differences between us are based on misunderstandings, and I'm hoping
that if I comment on some of the comments and add more examples, the
misunderstandings will go away (probably to be replaced by fresh ones).
In this note, I'll start with some comments on Len's comments:

> ... When Tarski spoke of functions from individual constants to individuals,
> for instance, he surely was not restricting himself to functions
> whose values were either computable or empirically verifiable.

True.  But he was restricting himself to formalized languages (he made
that point very clear in his classic paper) where the individuals were
mathematical things like numbers and points.  The question of empirical
verifiability was brought in by people like Carnap who went to great
lengths to address the question of how such a system could be applied
to real world objects -- his Logische Aufbau der Welt is a classic
attempt to address those issues, and Pat's mythical "any textbook"
either avoids them altogether by using only mathematical systems
(e.g. Schoenfield) or by giving a few examples and hoping that the
reader will "understand" how to apply them  --  thereby reducing
something which for Carnap was extremely problematical to something
that is somehow "intuitively obvious".

> ... All it takes to make such an identification [of symbols to individuals]
> is a minimal realist assumption that there ARE objects in the world.

No.  It also presupposes the existence of some way to recognize objects.
I'm claiming that my depictions address several problems at once:  they
avoid the issues of what is an object, how does one recognize an object,
and how can one talk about "possible objects" or fictional objects that
don't exist in the real world.

> Certainly you may need to go out in the world and look at cats on mats 
> to verify am empirical claim (or you may be able to verify it 
> inferentially from other claims you have already accepted). But that's
> a separate issue from that of how (in principle) a symbolism may be
> interpreted.

Yes, indeed, they are separate issues.  That is why I want a nice, clean
intermediate level of depictions to separate them.  On the one side,
your KR languages map very neatly into discrete, well-defined depictions.
On the other side, the surrogates in the depiction are mapped by all
the complicated pattern recognition and robot manipulation systems to
the real world (or to a virtual world, if we are doing a simulation).

I also become extremely suspicious about parenthetical phrases like
"(in principle)".  It reminds me of Bismark's rule:  "Whenever you agree
to do anything in principle, that means that you have no intention to
do anything about it in reality."  All those people who claim that the
symbols of a formal language could, in principle, map to real objects
never say anything about how they might actually be mapped.  Carnap is
one of the few who tried.  The others ignored the problem altogether.

>> If you want to put cats and mats in your models, then you must formalize
>> the process of recogizing cats and mats.

> Absolutely not. This is true neither in logic nor in physics. I think 
> when Newton wrote F = ma, he was proposing this as applying to actual 
> objects subject to actual forces (he seemed to think, for instance, that
> this law applied to the motions of the planets). Do you fault him for 
> not having supplied formalized apple- or planet-recognition procedures?

Exellent example!  Newton didn't have to say that explicitly because
people like Bacon and Gallileo and Ptolemy had worked out very well
defined procedures for relating the variables in physics to real world
measurements and observations.  If you want to develop a formal model 
theory that relates the symbols of a logic to real world entities, you
are obliged to formalize all of the details of scientific method to 
the same level of detail as the mappings that Tarski formalized
in his writings.  (Jerry Aronson, from whom I borrowed the word
"depiction", happens to be a philosopher of science who is concerned
about exactly those issues.  He came to the problem from a different
direction than I did, but we both feel quite happy about that intermediate

My only claim about depictions is that they separate the problem into
two more manageable ones:  one of them is simply Tarski's model theory;
the other is what the pattern recognition and robotics people are
doing in AI, what the simulation people are doing with virtual reality,
and what philosophers of science are trying to do in mapping formal
symbols to the real world.

>  Though I heartily endorse the use of special representations
>  and/or "vivid" and/or "db-like" ones) as an aid to efficient
>  inference, I regard them as dispensable in principle.

There's that old phrase "in principle" again.  I regard what people
actually do in practice as far more significant and worthy of
formalization than what ivory-tower logicians prefer "in principle".

>   More importantly for the ongoing discussion, I see no SEMANTIC role
>   for them (in the denotational semantics sense) in the interpretation
>   of a KREP, language-like or otherwise. On the contrary, they are
>   themselves in need of semantic interpretation -- something that can
>   be done either directly (considering correspondences between the
>   depiction symbolism and the world), or indirectly (by specifying 
>   the class of logical formulas determined by the depiction symbolism,
>   and then conventionally interpreting the logical formulas.

I keep insisting on going back to what model theory actually is:  a
method of computing the denotations of a formal language in terms of
a mathematical structure.  Tarski never claimed that it was a way of
relating symbols to the world, and nobody but Carnap and Goodman ever
tried to extend it to that level.  The SEMANTIC role of my depictions
is exactly the same role that Tarski's mathematical structures played
for his system of model theory.  The separate task of mapping the
surrogates in those depictions to the world is not model theory;
it is pattern recognition.

>   In other words, I endorse "depictions" as useful computational aids
>   but would not want to clutter my theory of the meaning of (language-
>   like) KREPs with them.

Great!  You can think of them as "useful computational aids", and I
can think of them as an intermediate level in my theory of meaning.
Then as long as we don't air our private metaphysical beliefs in
public, we can work together in harmony.

>   I agree with Chris Menzel that we wouldn't want to restrict
>   such auxiliary representations to be non-quantificational and 
>   non-disjunctive. For instance, in the temporal reasoning area one
>   may want to use a graphical representation of relations between
>   time intervals as an aid to inference....

I wanted to make the depictions as model-like as possible, and models
are isomorphic to collections of ground-level atomic formulas with no
negations.  That subset of logic with no negations and conjunction as
its only Boolean operator is capable of describing everything that
exists in the world.  I also said no quantifiers, but I would be
willing to allow the existential quantifier in order to avoid having
to invent unique names for each individual.  If you allow types in
your depiction, then you can also get a low-level kind of disjunction;
i.e. when you say "There exists an animal" that is effectively saying
"There exists a cat or a dog or a frog...."

I agree that temporal reasoning is very important.  That is why I
suggested that depictions should contain three kinds of things:
literals, surrogates, and enclosures.  An enclosure could be used
to model several different kinds of phenomena.  For temporal reasoning,
it could contain a snapshot of a situation at one instant of time;
a sequence of snapshots linked by the successor relation gives you
a movie reel.  I also wanted to use enclosures to model the objects
of O-O languages.  The reason for using the word "enclosure" is that
I wanted to reserve the word "context" for use in KR languages; then
the denotation of a context in a KR language would be an enclosure in
a depiction.

In any case, if you grant me depictions in some form, I'm willing to
negotiate exactly what collection of features they should have.

In the most recent note, Len summarized my position fairly accurately.
There are only a couple of points that might need a bit of clarification:

> ... and he [John] contends
> that that's what the founders of model theory had in mind.

No.  I don't want to do what I've complained about other people doing:
putting words in Tarski's mouth.  I don't know whether he would endorse
my proposed depictions.  All I am claiming is that he never related his
symbols to any real world objects and that he explicitly disavowed
other people's use of the term "Tarski's theory of truth" in a wider
sense than he did.

> However, my whole point had been to distinguish these "models" in AI
> from the logicians' models.  From a logical perspective, they are
> themselves REPRESENTATIONS, and as such in need of a semantics....

Yes, but that is the main reason why I wanted to restrict my depictions
to such a very simple form:  no Boolean operators other than conjunction
and no quantifiers (although I'll allow the existential if you like).
This language, although tiny, is capable of describing everything that
exists in the world.  And furthermore, every depiction is guaranteed
to be consistent, since the language cannot even state any contradictions.
If you want to construct a separate model theory to give depictions some
"semantics", the task is trivial.