On semantics, next round

Message-id: <199305030254.AA28301@dante.cs.uiuc.edu>
Date: Sun, 2 May 1993 21:56:55 +0000
To: sowa <sowa@turing.pacss.binghamton.edu>
From: phayes@cs.uiuc.edu
X-Sender: phayes@dante.cs.uiuc.edu
Subject: On semantics, next round
Cc: dietrich@turing.pacss.binghamton.edu, interlingua@ISI.EDU,

Here I will respond to your latest arguments about semantics. Dots 
indicate omissions, and I have reordered some of the material to 
bring related points together.

>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. 

Unfortunately we disagree already. TMT doesn't have 'operations' in it, 
and isn't concerned with working systems. TMT is a theory of how 
languages relate to the world(s) they denote; a *semantic* theory. 
(this can be of use in discussing computational systems, of course, 
but should not be identified with them.)

> I keep insisting on a terminology
>that allows me to distinguish the mathematical models from the things
>that they model. 

If you wish to make a distinction between two categories, you can 
introduce predicates into your language. Thus you might have 
IsModel(?x) and Represents(?x,?y); and TMT will accomodate your 
metaphysics, as it will accomodate most of them.  I know this will 
not satisfy you, but that is because of your refusal to accept the 
power of set-theoretic mathematics. 

 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.  

You explained in an earlier message why these issues had practical 

>>>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

(I can't help noting that you have not answered my point, only told 
me (again) what your claims are and what you believe.)

Until now it was three levels, for which you cited the authority of 
philosophers from A to W. I agreed with your three (but not your 
interpretation of them) and you tell me there are four. Your 
diagram could be written thus:

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

ie the models (in the TMT sense) ARE mappings from the logic to the 
(better, a part of a) world.

>........ 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)). 

Care needed here. Russell, Whitehead and others indeed did create 
webs of such constructions, because they were concerned with the 
foundations of mathematics. so they wanted to show that one could 
define all of mathematics in terms of sets and so be sure that 
mathematics was consistent. This goal, which now seems naive, motivated 
this kind of 'mathematical construction, but Tarski's (and Montague's) 
goals make no reference to this kind of reductionist definitional game. 
TMT doesn't define 2 to be {{},{{}},{{},{{}}}}.

>>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.

I entirely agree. Of course: indeed, that is precisely why I like TMT,
because it makes so few such assumptions. I am puzzled why you think 
that by saying that a TMT model might be real, one must therefore have 
settled all these deep questions. 

>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.

No, Im not! I am exactly NOT bringing them in. On the contrary, I am 
explicitly electing to ignore them. 

Maybe this is the core of the difference between us. Your view here 
seems quite off the wall. Look at the textbook example you helpfully provide:
> _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. 

Yes, of course they are. Thats the point: model theory does NOT get 
involved with such issues. All it assumes is that a model consists of 
individuals, which is the 'minimal' assumption I have referred to.

You seem to have a double standard. On the one hand you imply 
("deafening silence", etc.) that model theory must be concerned
with such ontological issues, but then you insist that it should 
not be. I agree, it should not be. 

>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. 

I find it strange that you are happier with sets of such 'abstract' 
entities as the empty set than with sets of, say, pencils or bridge girders.

 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

I repeat, there is no magic involved. Biologists succeed in talking 
quite successfully about redwood trees, and small children can 
understand the idea that all the trees in the forest have their roots 
connected together in a single web, presumably withoput constructing 
mathematical theories of measurement. One does not need to have the 
boundaries of these things specified exactly in order to be able to 
reason about them. Nothing is being postulated away by the semantic 
functions. These aren't LISP 'functions', remember: this is not a 
theory of how anything can be computed. (Is there another pun at 
work here, on 'function'?)

TMT is not the least concerned with pattern recognition, measurement, 
and so forth. That is yet another collection of issues, really part of 
epistemology. I agree these are important areas, but I see no reason 
why I have to wait until they are solved before being allowed to talk 
about the world. (See below for why this matters to Krep.)

>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. 

But consider: the mathematical 'models' are expressed in language 
(or maybe languages: let me just say Mathematics, as though it were 
a single formalism - probably an idealisation, but beside the present 
point.) So they belong at the second node of your four-node diagram, 
not the third. So you seem to have your wires crossed here somewhere. 

(Or is this what you mean? Since (post-Principia) mathematical 
concepts have all been defined set-theoretically, these 'mathematical 
constructions' are what Mathematics really means: so the mathematical 
language indeed belongs at node 2, but its referent is a huge pile of 
abstract stuff built up from the empty set (and, more recently, NIL), 
which belongs at node three, and thats what engineers and physicists 
(and logicians) mean by a 'model'. In fact, just by using mathematical 
terminology, this is what they have been talking about, whether they 
liked it or not. Is that it? Thats certainly consistent with much of 
what you have been saying in these discussions.

If that's what you mean, let me know and I will respond to it.)

> 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.

Tarski was correct in never addressing such issues. Notice you 
say that the drawing is 'of' the bridge. Your question (..how does 
one transform...?) is a question for a civil engineer, but one 
doesn't need to be a civil enginer in order to *talk* about a bridge. 
Again, I think you have switched a computational meaning of 'function' 
for a mathematical one.
>>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.

It seems then that a piece of Krep, in your account, is *never* 
'really' true, since the world is *never* sufficiently similar 
to the model. That sounds 
like a plausible, if depressing, position, until we observe that 
the expression whose meaning we are discussing might not be concerned 
with qualitative matters. Suppose someone believes that the President's 
name is Clinton. In what way is the intervening 'mathematical model' 
more or less like the actual world? Surely, the kind of similarity 
involved is usually of a rather all-or-nothing character. And then 
my point holds: if TMT says that the mathematical model is a model, 
then it also says that the relevant part of the actual world is one, 
since it only talks of structure, and they have the same structure.

If you want to refer here to the web of belief, I agree: see my old 
"N.P. Maifesto" for an extended discussion of this. None of this argues 
against TMT as an account of how the web means.

>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.

I begin to understand what is bothering you. You want the 
final stage in your diagram to be centrally concerned with 
how we interact with the physical world and come to relate 
our mental 'models' to the reality we inhabit: how we are 
'embedded', to use fashionable terminology. And you think 
that by claiming that semantics connects to the world we 
have somehow shortcircuited this, and illegally assumed 
that these questions have been answered. If that were true 
I would agree that this way of talking would be dangerous. 
But it isnt true. 

To see this, notice that an agent's Krep might well itself 
contain assertions about accuracy and methods of measurement: 
certainly, if the agent is to reason about such matters - as 
we engineers and scientists do - then its mental representation 
had better have such assertions somehow represented. But now if 
the symbols in the 'mental model' of such a hypothetical agent 
could only refer to abstractions, or perhaps pieces of syntax, 
then such reasoning could not be represented in the agent's mind. 

But it is simpler to just insist that TMT does not make such an 
assumption, and challenge you to show why it does. Let me point 
out first that we humans surely do manage to refer to real world 
we inhabit, and think about it, without apparently being forced 
to be exact about strategies of measurement.

>>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.

This might be another source of confusion. OK, if that is what 
you mean, then to say that TMT is defined using 'set-theoretic 
constructions` is just plain wrong. These mathematical constructions 
(I agree, a better term) have nothing to do with the use of set 
theory in explaining semantics, any more than they have to do 
with the use of set language in, say, group theory. See earlier 
comment on this.

>>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.

This does not mean that the terms of physics are reducible to, 
or must indirectly refer to (or via) a language of measurements.

Introducing physics is potentially confusing. Much of experimental 
physics (science in general) is indeed concerned vitally with taking
measurements under difficult or even extreme conditions, using 
exotic instruments. And often, the situations being measured are 
such as to stretch the boundaries of our conceptual language. This 
is not a typical circumstance for an agent reasoning about its 
environment, still less for a reasoner reasoning about some aspect 
of the world that it is not closely connected with at that time, 
both of which are more typical of the problem confronting Krep. 

Just how much of our everyday 'web of belief' can be related to 
something perceptual is an interesting question. But in any case, 
these connections are not made by 'complex measuring instruments'. 
Much of Krep is totally unconcerned with this issue, we should say, 
but as you and I share an interest in it, we might as well use it 
as an arguing ground.

>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 just find this implausible. I don't believe physicists, any more 
than the rest of us, do adopt any such two-stage process; or indeed 
what this process could be such that anyone could decide to adopt it. 
(Could they have decided to adopt any other? But I thought we were 
talking about the fundamentals of semantics of mental representations.)

Imagine Fermis lab in Chicago in 1941. An assistant slowly withdraws 
a rod, and Fermi watches the screen, seeing spikes happen more and 
more rapidly, and the clicks merge into a hiss. The assistant quickly 
pushes the rod back into the pile of graphite blocks. Presumably (maybe
 you disagree with this??), things are happening in these folk's heads: 
activity in their 'webs of belief' is forming new beliefs in the way 
that our Krep tries to emulate by inference. Suppose Fermi knows he 
has to check the exact temperature before he is sure his calculations 
were entirely right: nevertheless, it is clear that some kind of 
chain reaction began, so the basic theory is correct: he never 
doubted it, but he is an empiricist and never fully believes anything 
until it is tested. He is also happy and relieved that the entire 
thing didn't melt or blow up, and is now more certain than he was that 
they will be able to control these things. And so on. All of this is 
somehow represented in his head, involving ideas about temperature, 
events that just happened, the sound of the hissing counter, graphite, 
neutrons, accuracy, whether theories are right or not, degrees of 
confidence, concepts of process growth rate which cannot be easily 
expressed in English or Italian, the past and the future, safety, etc., 
etc.. The physicist's case isn't essentially different from any other 
case, though probably more complicated than most.

I want to be able to say that this Krep in his head refers to what 
I just said it was about. You insist that I can't do that, but that 
it must refer to some mathematical abstraction which in turn 'corresponds' 
to that world I just (sketchily) described.  I don't believe in these 
'mathematical models' which you say stand between physicists and their 

>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.)

As Ive said, I don't care about Montague's concern with NL. 
If we are talking about Krep, there IS room for these distinctions. 
>.......... 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. 

The hypothesised 'mental models' ARE the web of belief which IS 
(a mental encoding of) the Krep assertions. 

>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.

I agree about the web of belief, and agree that it is what lies 
between language and the world. Now, where does Krep come into 
the picture? The 'web of belief' is precisely what Krep encodes. 
You interpose two layers of structure, not one: a web of belief 
and a web of 'model' . 

>> ...  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. 

Oh, don't start on THAT line. This is the line of 
thinking that leads Searle and his disciples to conclude that 
computers don't really exist.

> 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.

The data structures ARE, to repeat, the expressions of the Krep 
formalism. Consider the Boyer-Moore TP system. It is a program so, 
of course, it manipulates datastructures: these are encodings, 
in a suitable internal syntax, of first-order expressions, and 
it applies inference rules to them. But these expressions don't 
denote those datastructures, any more than the sentences I write 
in a letter denote the marks on the paper which constitute them: 
they ARE the datastructures. And yes, they represent the current 
state of the world; and how they do that is explained by model theory. 
Not what confirms that they do, or why they come to be there, 
or how they are connected to processes which construct them: none 
of that is explained by model theory, but a full explanantion of 
any of thse things will involve model theory, since it is the 
account of how these structures carry meaning.

Do you see the picture now?

>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.

Indeed. You may have descriptions of all kinds of states of affairs,
not all of which actually exist. These 'models' are descriptions of 
ways the world might be.

>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. 

That is almost nobody, and much of Krep in AI need not be concerned 
with such a wide span. But I agree it is the most interesting case.

 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. 

Wrong. That thing built of gensymns should be regarded as the 
internal encoding of the syntax of the Krep language: or anyway, 
if it is to be related to the world, it had better have a semantics, 
whether we call it 'sytax' or not. And that's all the actual 
system has: it doesnt have a Krep language that isn't somehow 
encoded inside it. So in your account, we might as well just 
throw away the Krep.


>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 are mixing up two issues. I agree about the web of belief (notice 
'belief' here, by the way, which usually implies some kind of expression). 
But that does not mean that accounts of meaning (especially of 
the meaning of the beleifs in the web) must be involved with 

>...... 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.

I insisted only that TMT *could* be concerned with physical objects, not that 
it was defined in terms of them. Certainly models can be defined over 
abstract universes also. But the quote you gave will do. Let me repeat 
it here, with your first comment:
>   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.

Precisely. He doesn't, does he? So they could be anything. In particular, they 
could be, say, water molecules, or Texan sherrifs, or the pieces of paper on 
my desk. 

The defense rests. 

Pat Hayes

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