Re: Trying again to respond

sowa <>
Date: Sat, 1 May 93 13:05:47 EDT
From: sowa <>
Message-id: <>
Subject: Re: Trying again to respond
Cc:, interlingua@ISI.EDU,

I don't believe that we disagree about any of the formal operations
in Tarski's model theory nor about the way that AI programmers implement
those operations in working systems.  I keep insisting on a terminology
that allows me to distinguish the mathematical models from the things
that they model.  All of that discussion is at the metalevel, and it
would not affect any knowledge engineer who might use KIF or CGs to
represent some subject matter.  

There is only one tiny area in which this metaphysics would affect the
use of KIF, and that is in the simplification of ,(name ?x) to just ,?x.
My point was that everything stored in the computer is a lexical item,
and that the real-world entities are represented by surrogates (i.e.,
GENSYMs) inside the machine.  Therefore, ,?x would be a lexical item,
and it could occur inside a KIF expression.

Some comments on your note:

>>You seem to claim, however, that logic should be related directly
>>to real-world things without any intervening level of "concepts"
>>or "models".  

>Yes. (Well, strictly, I want to say that logic CAN be so related,
>not SHOULD be; but let this pass for now.) This is quite 
>consistent with the reasonable statement above. The expressions 
>in a Begriffsschrift - in modern terminology, a Krep formalism - 
>are intended to express the content of a thought or proposition. 
>Krep expressions ARE the intermediate level that your philosphical
>tradition puts between language and the world. Model theory IS an 
>account of how the 'correspondence' between that and the world it 
>purports to describe should be structured.

In my metalevel discussions, I make a sharp distinction between
four separate, but related levels:

  natural language  <-->  a system of logic  <-->  models  <-->  the world

My claim is that Tarski only discussed the relationships between the
two middle levels.  I don't believe that we have any disareements about
the formalism at that level.  I am also quite willing to accept Kripke's
semantics for modal logic, as long as we agree that all of his models
are mathematical constructions at the same level.  Even though Kripke
called his models "worlds", I can interpret all his formalism in terms
of mathematical models, where the ground elements are not physical
objects, but some kind of abstract mathematical constructions.
Bertrand Russell liked to use the empty set to build everything else,
with constructions like {{},{{}}} and multiple iterations of them.
In LISP, you would get something like (NIL . (NIL . NIL)).  I am happy
to let Russell, Kripke, and LISP programmers build such things.
I am even willing to let Kripke call one of his models w0 and say
that it has some privileged relationship with the real world.

But where I draw the line is with attempts to "identify" w0 with
the real world.  I am willing to let people say that there is a
mapping between w0 and some aspects of the world, but I won't let
them say that any of the elements of w0 are identical to things 
in the world.

>I know you disagree with this, since you want to identify krep
>with natural language. I disagree, and think that this is not 
>the right way to fit knowledge representation to work in language
>comprehension. I recognise that others disagree (Yorick Wilks, for 
>example, takes your position), and am willing to argue my side 
>until the cows come home. But my only point here is that any position
>on these issues should not be allowed to affect the syntax of a 
>Krep standard.

No.  As I pointed out above, I make a very sharp distinction
between natural language and any system of logic (or kn. rep.).
I believe that natural languages set the standard for expressive
power, and logic sets the standard for precision.  What I am trying
to do with CGs is to design a system of logic that is capable of
expressing the propositional content of any sentence in NL.
I am designing the syntax of CGs to simplify the mapping to and
>From NLs.  Mike G. and others are welcome to design the syntax of KIF
in any way they please.  All I want is to make sure that KIF provides
enough expressive power for me to map the CG constructs into it.

>>Although you use the word "model", you identify
>>at least some of those models with aspects of the real world.
>>I prefer to make a very clean distinction and say that none of
>>those models are the real world. 

>I know you prefer to make this distinction, but I don't want 
>your preferences to be incorporated into a language I might 
>be forced to use.

So far, I have asked for the following changes to KIF:  the option
of marking types in the quantifier lists; the option of putting sets
in the quantifier list so that one can say (forall (?x in ?s) ...;
and the ability to say ,?x instead of ,(name ?x).  This last 
simplification is the only one that raises any questions about what
the variables in KIF really "refer" to.  Len Schubert made a comment
that it shouldn't even be necessary to have a comma in that position,
and one should be able to write just ?x instead of ,?x or ,(name ?x). 
I like Len's suggestion best of all, and the nice thing about it is
that it keeps the underlying referent out of KIF altogether:  there
would be no possible way of using KIF that could distinguish between
your preferences and my preferences.  The only drawback is that it
might be necessary to have a parser for KIF that was different from
the Common LISP reader.  It's a pity that metaphysics keeps running
into such grungy aspects of reality.

>I don't want to insist that formalisms MUST be interpreted in 
>the 'real world' (whatever that really means), only that they
>CAN be. In fact, I want the semantic theory of the formalism to be
>as agnostic as possible about the intrinsic nature of the possible 
>interpretations. TMT makes very few such ontological assumptions;
>fewer in fact than any other semantic theory I have ever seen. 
>Thus, a TMT interpretation (I carefully avoid 'model', since 
>you confess to having deliberately  confused two distinct 
>meanings of this overloaded word) might be 
>over a domain of integers or bacteria or ocean liners 
>or concepts or whatever else one might decide to try to describe.

Your phrase "'real world' (whatever that really means)" brings us
to the crux of the matter.  There are so many metaphysical positions
about the "real world" that I want to keep it out of the knowledge
representation language.  I don't deny that the world exists.  I don't
deny that the ultimate purpose of our languages (natural and artificial)
is to talk about things in the world.  But when I am constructing a
model theory, I want to be able to relate a syntactic construction in
a formal language to a mathematical construction in a model.  I want
to avoid ontological, epistemological, and pragmatic questions about
whether two apparently distinct redwood trees in Muir woods are
"the same" or "different".  Those issues are very important, but they
shouldn't intrude on the formal mapping between the syntax and the model.

When you keep insisting that the sets in a model are or can be made
up of trees or ocean liners, you are bringing in a host of problems
about the nature of those elements, how one can distinguish them, etc.
When I am working on model theory, I want to have pure, clean models
constructed out of integers, empty sets, GENSYMs, NIL, and abstract
things like that.  Then when I talk about how those abstract models
relate to the world, I bring in the questions of pattern recognition,
measurements, measuring instruments, error rates, etc.  Those questions
are too important to be "postulated away" by magic functions that
make distinctions between redwood trees that even a biologist couldn't

When I identify the models of model theory with the mathematical
models of the physicists and engineers, I am not "confusing" two
issues.  I am instead making a distinction that the people who try
to identify models with the real world fail to recognize.  My claim
is that every model that any logician has ever used in any application
of model theory is in fact an abstract mathematical construction that
is of the same nature as the mathematical models used by physicists
and engineers.  I admit that people like Montague would like to
pretend that they are relating language to the world.  But by "postulating"
those magic functions, they are ignoring the most difficult issues of
science and engineering:  how do you measure those values that those
variables are supposed to represent?  how to you transform that neat
drawing of a bridge into a construction of ten-ton girders spannning
a raging river?  Tarski never addressed those issues.  Montague
postulated them away.  But I want to make them very clear and distinct.

>The mistake in this 'clean distinction' of yours, it seems 
>to me, arises when one notes that TMT refers only to the 
>structure of a possible interpetation. Thus, if we have an 
>'abstract' interpretation and a 'correspondence' between that 
>and some part of a more concrete world, this presumably means 
>that that part of the real world is similarly structured to 
>the abstract model. But if that is true, then that part of 
>the world IS a valid TMT interpretation (since to be one, it 
>only has to have the appropriate structure...) So if your notion 
>of 'correspondence', the last link in your three-way chain, 
>makes any sense, then it is simply TRUE that the formalism 
>could have been interpreted as referring directly to the 
>world, regardless of what your philosophical sensibilities 
>lead you to prefer. 

I agree with everything you say up to that "But if...."  The
crucial phrase is "similarly structured."  Two things can be
similar without being identical.  Similarity also admits of
degrees of approximation.  You can have two different models
that are both "similar to" the same part of the world, but
they might not be isomorphic to one another.  I also object to
your capitalized word "TRUE".  A logical formula has denotation
true or false in terms of some mathematical model.  That model
may be an approximation to some aspect of the world.  But the
formula is true of the world only to the degree of approximation
between the model and the world.

Making a clear distinction between mathematical models and the
world allows you to talk about degrees of approximation and
about the methods of measuring that approximation.  Physicists
and engineers do that all the time, and philosophers and logicians
who ignore that distinction make it impossible to talk about some
of the most crucial issues in the philosophy of science.

>Check my earlier reply for the error in talking of 'set-theoretic 
>constructions' in this way.

By "set-theoretic constructions" I meant Bertrand Russell's habit of
building up everything from the empty set.  Perhaps the term
"mathematical construction" is preferable.

>Lets be clear: I am very much in favor of distinguishing language, 
>thought and reality, which is what many of the philosophers you cite 
>were concerned with. Your dropping all these Big Names is a 
>debating ploy: we both know about debating. But that is not 
>the essential difference between us. You want to identify 
>Krep formalisms with language and their interpretations with 
>thoughts, while I want (to oversimplify somewhat) to identify 
>Krep with thoughts and its interpretation with the world: or, 
>in this context, I want a Krep standard to allow such an interpretation.

Re Big Names:  I cited that passage to counter your debating
ploy of "eccentric".  If you try to push me out of the mainstream
by an epithet, I can bring some allies into the swim.

Re interpreting the variables of a kn. representation:  If we adopt
Len Schubert's suggested notation, the question of what a variable
refers to never arises.  You can pretend that it refers to something
in the world, and I can pretend that it only refers to a GENSYM inside
the computer (which in fact is a good approximation to practice in AI,
database systems, and other computer applications).

>So, if we try to represent engineers or physicists knowledge, 
>we are encoding beliefs about reality (not about an abstraction 
>which 'corresponds' to reality). Exactly my point, thanks. 

Yes, of course.  When physicists talk about electrons and photons,
they are talking about things that they believe to be in the world.
But the meaning of those terms is established only through a complex
"web of belief" (to use Quine's term) in which only a small fraction
of the terms have a direct connection with anything observable.
Physics is a science that studies the world and how it works,
but the variables in a physical theory are related to the world
by very complex measuring instruments whose construction presupposes
other physical theories.

In order to avoid worrying about every possible detail and exception
simultaneously, physicists and engineers adopt a two stage process:
first, relate the mathematical formulas to an abstract model that
captures those aspects of reality they are trying to address; then,
worry about how to use measuring instruments to relate some parts
of the model to some parts of the world.

I don't believe that you and I disagree about what engineers and
physicists do, believe, or talk about.  What I would like to do is
to get people like Montague to make room in their theories for the
kinds of distinctions that are needed for the common practice in
physics, engineering, and AI.  (Actually, it is too late for
Montague, since he's dead, but some of his sympathizers, like
Link and Cocchiarella are quite happy to make the kind of
distinctions that I would like to see.  We in fact talked about
that at the conference on ontology in Padova, and it's too bad
that you couldn't get there.)

>Russell said that mathematics was the discipline in which we did not
>know what we were talking about, nor whether what we were saying 
>about it was true. I think this captures the essence of mathematics. 
>You can't say what much mathematics is about because it can be 
>interpreted in all sorts of ways. One might say that it has been 
>abstracted from specific content. But there's a subtle mistake 
>in moving from that, to saying that it is about something abstract, 
>so being forced to hypothesise a new domain of 'abstract things' 
>standing between the mathematical language and the world. When I measured
>the lengths of kitchen cabinets, added these numbers together and 
>concluded that I would have to move a wall, my calculations
>referred to the actual world, not an abstraction.

I think that we agree on the word "about".  I am quite happy to
say that language is used to make statements "about" the world.
But if we want to analyze that word "about", there are many ways
to go about doing so.  One way is psychology:  study how people
perceive things in the world, how children learn language, etc.
Another way is AI:  write computer programs that simulate things
in the world and relate those simulations to language.  And
another way is philosophy:  analyze the way people use language
to talk about the world construct formal theories that explicate
the details that are hidden in that word "about".  These three
approaches are different, but compatible.  My claim is that
physics, engineering, psychology, and AI all presuppose some
kind of mediating structure -- an engineering drawing, a mental
model or neural process, or some AI data structures constructed
>From pointers and GENSYMs.  I don't believe that an adequate
philosophical theory can be constructed without assuming some
kind of mediating structures between language and reality.
Quine is about as antimetaphysical and behavioristic as you
can get, but even he talks about the "web of belief" as the
structure that stands between words and reality.

> ...  The data structures in a Krep 
>system ARE (a suitable encoding of) the expressions of the 
>formal language. That mapping is not one between a logic 
>and its models (in the sense of TMT). 

No.  We have to distinguish the AI language from the data
structures that are manipulated by the programs that process
the language.  I agree that a kn. rep. language such as KIF,
CGs, frames, production rules, etc., is a formal language at
the same level of abstraction as logic.  If you are just doing
theorem proving, then you never get out of the syntactic level
of manipulating formulas.  But if you are trying to connect your
language to a robot, you must construct some data structures in
your computer that represent the current state of the world.
You may also have multiple models -- one corresponding to the
currently observed state, and another corresponding to some
desired state you are trying to construct by robot manipulations.

>This is just the kind of confusion which carelessness 
>about words like 'model' gets us into. (While we are giving 
>opinions, I find your careless lumping together of formal and 
>natural language quite unacceptable. Natural languages and 
>Krep formalisms must be carefully distinguished, in 
>my view.)

On the contrary, I was trying to make some precise distinctions
about models.  Your talk that lumps sets of trees and sets of integers
at the same level of abstraction is an open invitation to careless
confusions.  And as I pointed out above, I also make a very clear
distinction between natural languages and formal languages.  In my
previous messages, I wasn't even mentioning natural languages -- I was
only talking about KIF, CGs, and other formal systems.

>In order to make this case, you have to change the meaning of 'model' 
>at least twice. But I'm not arguing against your honorable tradition, 
>but the idiosyncratic (and I believe confused) conclusions you 
>are wanting to draw from it for a Krep standard.

I claim that my interpretation, far from being idiosyncratic,
is in fact the de facto standard that has been in use by AI
programmers for the past 30 years -- at least by those AI programmers
who try to go all the way from a language via a model to a robot
manipulator and vision system.  Most programers never attempt to
span the whole range; they usually address only one part of the
problem at a time.  But when you try to put it all together in a
single system that has language input and output, visual and/or
auditory sensors, and mechanical manipulators, there must be in
the middle of that system a "model of the world", built out of
pointers and GENSYMs (or their equivalent in some programming
language), which is distinct from the real world objects that
the robot is manipulating, distinct from the kn. rep. language,
and distinct from the natural language that may or may not be
used to talk to the robot. 

> ... As I said, I was referring to
>standard 'model theory' of logic, found in any textbook;

We can't argue about "any", so I pulled a specific textbook off
my shelf:  _Mathematical Logic_ by J. R. Schoenfield, Addison-Wesley, 1967.
On page 22, Schoenfield says

   By a _model_ of a theory T, we mean a structure for L(T) in which
   all the nonlogical axioms of T are valid.

Then on page 18, he says

   Let L be a first-order language.  A _structure_ A for L consists
   of the following things:

   i) A nonempty set |A|, called the _universe_ of A.  The elements
      of |A| are called the _individuals_ of A.

   ii) For each n-ary function symbol f of L, an n-ary function F
       from |A| to |A|.

   iii) For each n-ary predicate symbol p of L other than =, an n-ary
        predicate P in |A|.

If you notice, nowhere does Schoenfield say what those individuals
happen to be.  The silence about the nature of those individuals is
deafening.  Tarski started his famous paper by talking about the
German sentence "Schnee ist weiss", but when he got down to formulating
model theory, he was also silent about the nature of his individuals.

My claim is that those philosophers who attempt to identify those
individuals with things in the world are ignoring very significant
issues about measurement, pattern recognition, etc.  Montague was
guilty of that sin, and Barwise and Perry tried to correct it.\
Quine was not guilty of that sin, because he was very explicit
about the complexities in his "web of belief."

You accused me of an amateur-scholarly approach by citing actual
examples of what philosophers have said.  I have cited actual quotes
because I refuse to argue with a strawman.  This TMT you keep citing
is not what Tarski said, and if Schoenfield's book qualifies as
"any textbook", then it is not what Schoenfield said.  If you insist
in claiming that "standard TMT" is formulated in terms of physical 
objects, then please give an exact quote from that mythical "any textbook"
that defines TMT.