Re: Getting back to the notes of May 10th
Message-id: <>
Date: Tue, 1 Jun 1993 15:33:00 +0000
        sowa <>
Subject: Re: Getting back to the notes of May 10th

>I was too hasty in saying that I agreed with your last note, since I
>hadn't realized that there were more to come.

Oh, come come: that was about a different subject, and surely it 
is nice to find we agree about *something* :-)

I confess to letting a certain irritation show in my responses to 
your last message, which I will try now to studiously avoid. So, 
let us not quibble any longer about what Tarski said or why anyone 
is citing him.

I think indeed that we have reached a crux, a genuine point of 
disagreement that allows us (me, anyway) to understand how the other 
comes to hold the views he does, and it is about what it means to
'apply' mathematics. You insist that the act of applying mathematics 
to anything real must involve doing something NONmathematical: something 
involving epistemology or pattern-recognition or engineering or applied 
mathematics, but in any case something more than can be expressed or 
captured in mere set theory, which is pure maths. I disagree, as 
explained in other messages sent recently. But perhaps I should be a 
bit more careful, since in a sense you are of course correct: any account 
of how some symbols in some head actually come to refer to something real 
(or something abstract, for that matter) must involve such considerations. 
But that does not mean that discussions of possible interpretations of these 
symbols must be forbidden from considering interpretations of them in the 
real world. Let me draw an analogy with some engineers talking about the 
stresses in a bridge, using symbols to refer to angles between girders, 
thicknesses of beams, etc. etc.. Your position seems to be that in order 
for us to even contemplate the possibility that these symbols refer in the 
way indicated, we must first have a complete theory of bridge-referring, 
a detailed account of how the engineers came by those figures, and so forth. 
This might involve discussions of measuring processes, accounts of how 
information is transferred between databases, maybe even legal and
matters. All this in order not to establish the correctness of their 
thinking (for which it might indeed be necessary) but even to contemplate 
the possibility that they are simply referring to something made of steel: 
without all this, on your view, such talk is forbidden. I find this conclusion 

To apply mathematics requires 'fitting' it to the world: it has to be 
possible to find a way of interpreting it as talking about the world 
one is interested in applying it to. But there is nothing mysterious
here. The requirements this imposes on the world in question 
depend on the assumptions made by the mathematics in question. Differential 
calculus assumes certain kinds of 'smoothness' in the worlds it describes, 
so can be applied successfully only when the (relevant part of the) world 
is indeed sufficiently 'smooth'. Graph theory makes different assumptions 
and can be applied in areas where differential calculus would be useless. 
Model theory, whatever Tarski said about it, uses set theory, and so can 
be applied wherever the assumptions of set theory are satisfied, ie wherever 
the (relevant part of the) world consists of individuals, with appropriate 
criteria of identification. (These are discussed carefully by Quine and 
other authors and are familiar to anyone who has learned set theory.) Thats it. 
So of course model theory can be applied to discussions of how symbols 
might refer to bridge girders, treetrunks or people. Why not? Watch me do it. 
I will refer to the mapping between us and our names. Ready? 
Here goes: {<"PAT HAYES",Pat Hayes>,<"JOHN SOWA",John Sowa>}. Now, 
you will object that this is somehow illegal or inappropriate or impure or
unsanctioned by Carnap. Tough, I just DID it, John: I defined that mapping. 
If you don't like that, you are free to fume or ignore it, but not to impose 
your dislike on my theorisin'.

Now to turn to some quick replies (and a second crux):

>........... 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.

I agree, thats exactly what I want to do as well. Thats exactly what 
set theory's ontological agnosticism allows.

>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 agree, of course it is important. I don't mean to imply that this is trivial 
or unimportant, only that it is a different (set of) issues than semantics.

>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.

It is still part of the question of how our symbols can come to refer, yes: 
the 'grounding' problem. This is what Carnap's 'Aufbau' is concerned with.
But we do 
not need to have solved the grounding problem in order to contemplate the 
possibility that our symbols might refer to reality. Carnap spends pages 
157-225 (Dover edition) of his 'Introduction to Symbolic Logic' building 
logical theories for physics and  biology, without even nodding at his 
'aufbau' worries.

>>>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.

Another crux of disagreement between us. When we are talking about 
representations, what something 'is' seems to me to be inseperable from 
how it is used, ie its inferential role. I can encode the same information 
as a set of sentences, a network, an object-oriented datastructure, or a table. 
Semantically these are the same thing, re-encoded in different styles 
for computational (or maybe aesthetic) reasons. What matters for a semantic 
theory is what is being claimed about the world by a representation. To change 
how a datastructure is 'used' might completely change its content, so a DB
used as a vivid knowledge base , and the 'same' DB being used as a model, are 
different things which happen to have a structural similarity. 
.........  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.

Maybe this is part of the problem. Yes, of course, if you are going to give
a complete account of how arithmetic is used in carpentery, you will have to 
face all these issues. If *that* is what you mean by 'apply' then I agree 
with this conclusion, but then reject the claim that using set theory to talk 
about relations between numerals and lengths of pieces of wood is 'applying' 
anything. One can talk of sets of pieces of wood without doing this detailed 
a job of 'application'.

.....  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

Not *how* they are determined, no. But that does not mean that we cannot 
imagine that they might be determined, and discuss the consequences. That 
model theory does not explain *how* a name like 'Julius Caesar' comes to be 
attached to that old roman guy is not sufficient reason to forbid us to even 
think that it might so refer, and insist that it must refer to a mental 
surrogate of some mysterious kind. 

>> ... I am completely confused about what these
>> 'depictions' could possibly be meant to be.
OK, I have this clearer now (see my 'summary of the argument so far' 
message, written later). They are indeed supposed to be a whole lot of 
things that I want to distinguish but you want to conflate. Since you have 
them conflated, different ways of talking about them seem to you to co-refer, 
but to me to be incompatible, hence your talk tends to seem confusing. They 
seem to be simply Herbrand models, by the way, since they are constructed 
>From lexical items: is that correct? If so, we have a perfectly fine 
terminology for them, so lets stick with that. If not, I'd be interested in 
how they differ.

>.  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.

A very different way of thinking than mine: see earlier comment. I am puzzled 
about how, in your way of thinking, you can characterise what the content of 
a representation is.

By the way, what are 'model-theoretic methods'? Model theory doesnt provide 
us with methods. Deductive techniques might be considered model-theoretic 
methods, since the process of generating a proof can often be regarded 
equally well as the process of surveying a space of partial models for a 
possible counterexample, and showing that one does not exist. So I see 
no clear distinction between these kinds of method which you place in sharply 
different categories.

>>> ... 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. 

Clearly you havnt been talking to the right people. Look, its a simple point. 
A model is just one interpretation, and there are usually many possible 
interpretations, all having equal claim to be 'the' model of a representation. 
Model theory gives us a way to take a representation and a possible
of it, and say whether this interpretation could be permitted by the
So if I exhibit a single possible world, all that I can infer from that
is that this is one possible way the world might be. Now, this could be a
way to 
refute an argument or claim to validity, since it can show that something
is NOT 
true in all models, but it clearly cannot be a way (in any ordinary sense,
such matters as protypical reasoning or inductive reasoning) to establish or 
claim that something is true in all models, which is what an assertion claims. 
So if a representation is taken as affirming (what word can I use? claiming? 
asserting? stating?) some proposition, then it must be more than simply an 
exhibition of a model.  Of course, inference and models are 
closely related, and the process of inferring involves building and 
manipulating models. And indeed, a set of ground assertions 
can be regarded as defining a model (of itself). Any model of 
a suitably prepared set of assertions has a purely syntactic simalcrum 
which can stand in its place (a Herbrand model), and Hintikka's model sets 
are similar to these. One can get by, in a sense, by having model theory only 
allow such syntactic constructions. But my point has always been only that 
we are under no compulsion to make this restriction. And note (again) that 
these models are often infinite, so cannot be simply identified with anything 

 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.  

If it is taken to be a represenation, then it does state something. 
You seem to be talking at a level of computational detail which ignores the 
content of the representation, what McCarthy and I called the 'implementation' 
level. No quarrel with such talk, but semantics is irrelevant to it.

>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.

Yes, but notice that that is evaluation in one model, and there are lots 
of others. 
If an SQL query comes up with $100,000 to a query about someone's salary, 
are we to conclude that this guy's salary IS $100,000, or only that it 
is logically possible that it might be that?

>> ... 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. 

Sure, I agree it might well be. Notice I said semantic importance. We 
were discussing semantics and model theory, right?

>.......  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.

I can believe that, and it certainly seems to make sense when viewed in 
this way. Could I interpret the integrity constraints as being essentially 
consistency constraints?

>> ........ 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.

But surely (?) the anaylsis of what a representation means should not be 
a function of what kind of 'talk' we use in referring to it. If I say its 
implemented in LISP, would its meaning change yet again? I want to use 
semantics-talk to talk about what representations - all representations - mean.

>As I said above, the term can be defined formally very simply,
>but it can have many possible uses.  

You make it sound like 'a list', which can also be simply formally defined 
and has many uses. But thats why this idea of 'a list' is not much use in 
a semantic account of representation: you have to say how the list is 
being (semantically) understood, what its inferential role is supposed to be.

>....... our project of developing a hybrid system with
>both a symbolic and a computational side.  

This leaves me puzzled (again). Surely, the symbolic 'side' of any implemented 
project is also a computational 'side'. It isnt that there are two parts, the 
symbolic part (on a Mac, say) and the computational part (over here on the
and one is doing symbols while the other is doing computation. Rather there is 
a system, all of it 'computational', which uses various representations, and we 
could use a semantic theory to help us talk about what these representations
might mean. We might find symbols all over the place inside the system, 
computationally encoded in all kinds of ways, and be interested in how their 
possible meanings might be connected with aspects of the system's behavior: 
whether its conclusion-drawing is valid or not, for example. Not all the 
representations need be symbolic, of course, there may be maps and graphs and 
all kinds of things, but a good semantics  will provide a coherent account of 
what all these mean and how their meanings relate to one another. But it must 
provide for the fact that the same content can be recompiled into all kinds of 

Does this picture more or less correspond to your vision also? I suspect not, 
and this might help us understand our misunderstandings.


Beckman Institute                                    (217)244 1616 office
405 North Mathews Avenue        	   (217)328 3947 or (415)855 9043 home
Urbana, IL. 61801                                    (217)244 8371 fax  or