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Message-id: <199306010518.AA19682@dante.cs.uiuc.edu> Date: Tue, 1 Jun 1993 00:22:04 +0000 To: cg@cs.umn.edu, interlingua@ISI.EDU, sowa <sowa@turing.pacss.binghamton.edu> From: phayes@cs.uiuc.edu X-Sender: phayes@dante.cs.uiuc.edu Subject: Re: Getting back to the notes of May 10th Cc: eileen@turing.pacss.binghamton.edu, jerry@turing.pacss.binghamton.edu, sowa@turing.pacss.binghamton.edu

Hi John This reply is a little late as I have been offline for two weeks. >I deferred responding to your two notes of last Monday, since Len's >notes addressed similar issues without raising the question of who was >being muddled, confused, or eccentric. As a result of that discussion, >I believe that we reached a satisfactory conclusion that enables us >to agree on a notation for KIF that we both like, while allowing us to >maintain our own private metaphysical preferences. If that was the correspondence with Len that I was able to read, then I know that you misunderstood his position. There are several ideas which Len and I (and many others) want to distinguish but you seem to want to conflate. Unfortunately, the word 'model' has been used for several of them. One of these is an internal computational 'model' which is syntactically unlike a set of assertions in a logical formalism, but is thought of as a store of information which can be efficiently accessed. A color-wheel representation of chromographic relationships, or a commercial database, might be examples. Len endorsed the use of such things, as I do myself. Another idea is that of a 'possible world' in conventional model theory (don't react here to the 'any textbook' joke, I address this later). Len does not endorse conflating these ideas, which your 'compromise' involves doing. > .... >> But in any case, your alternative is even worse. In order to decide >> whether one of your set-theoretic models is a reasonable simlacrum of >> the world of cats and mats, I have to have a COMPLETE theory of cat/mat >> imagery, since everyone of these is an entire world-simulation. Thats >> Principia MatCatica. You may claim that this isnt necessary in order >> to do the model theory, but how do I know that one of your models isnt >> quite inappropriate as a way of interpreting cat-mat talk? > >It is questions like these that lead me to believe that situation >semantics with its preference for limited regions of space-time is >on the right track. It is quite possible to talk about 'limited regions of spacetime' in ordinary logic: in fact, its the normal case. Situation semantics is original not in this way but in that it allows possible worlds to be incomplete, ie relations need not have a truthvalue on every sequence of individuals. ( Actually, I should say that this refers to rather oldfashioned situation semantics. More recent work in this area is more radical.) > Our use of language and logic doesn't depend >on things that neither we nor anyone else has ever observed. I'm not sure what you mean here, but in referring to the future (not to mention such things as one's goals in life, the State of Israel or the number 26) we do seem to use language quite successfully to refer to unobserved things. And the correctness or otherwise of logic applied to such entities seems to be largely independent of observation. > And >I believe that a system of semantics that is based only on observable >regions of space time is preferable to one that claims to speak about >the entire world as a completed whole. Just to keep things clear, model theory doesn't do this either. In fact, it was exactly the apparent need to have such completion that I was objecting to in your semantic program, if you recall. > For propositional attitudes, >plans, and talk about the future, the semantics must also take into >account imaginable situations, but again, these are still finite >(both in our brains and in our computers). NO! Again you simply assume as given the point I am arguing about. I can imagine that the big bang idea is wrong and the universe is in steady-state expansion: Hoyle's old idea. It might be wrong, but its correctness turns on subtleties of microwave astronomy. Its certainly *conceivable*. But I couldnt get it into my head. Imaginable situations need not be finite, but in any case are not (usually) in my head or in a computer. > >> ... Now, can we PLEASE stop referring to Tarski?? > >Fine. If you stop trying to push me out of the mainstream with claims >that I'm eccentric or not conforming to established usage, I will >stop citing and quoting "Big Names" who I believe support my position. 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. If you insist on an exact definition (using the debating ploy 'I'm being more exact than you') then by all means let us use the one you yourself cited from one of the standard textbooks. Choose your definition, I don't care, use any respectable undergraduate textbook on formal logic, there are probably dozens to choose from. There are minor differences but they are completely irrelevant to this discussion, as you know well. This entire business of refusing to use 'tarskian model' and citing Tarski at every turn and ridiculing my usage as 'mythical' is all a debating strategy designed to divert attention from the issues I am trying to discuss. I will not refer to it again except to indicate when you are using it, if you do. > >> Most of what I talk and think about couldnt be accessed or recognised >> in these ways, anyway. "Julius Caesar" denotes some guy who is long >> since passed from my sight, and 'tomorrows lunch' doesnt even exist >> yet, but I can refer to it. Commercial databases don't usually encode >> propositions that require pattern recognition to confirm. > >There are very complex issues in psychology of language about how >children learn the denotations of words, in philosophy of science >about how theoretical terms and symbols are related to observation, >and in philosophy about the denotations of names of historical >figures and about references to as yet nonexistent futures. True, but completely irrelevant to the point I was making, which was a response to your assertion thast only observable things can be referred to. You have not addressed the point, but simply taken an opportunity to repeat yourself. 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. Again, you seem to be inconsistent. On the one hand you insist that I must not ignore these matters, then you tell me I should. >My point is that model theory solves only one problem: the relation >between formal symbols and mathematical constructions. Although this >is a very important part of semantics, it does not and cannot by itself >address the question of how those mathematical constructions are >related to the real world. Claiming that the real world things are >somehow included in those constructions is just begging all the most >difficult questions. By assuming that those models contain symbolic >surrogates instead of actual physical objects, I have openly admitted >that model theory doesn't solve those problems. Then I can begin to >address the separate question of how those surrogates map to the world. >(Or I can, like you, just ignore that question if it is not of interest >to me at the moment.) We are tantalisingly close to agreement, except that I don't believe that the things that model theory describes the relation of formal symbols to, must be 'mathematical constructions'. This seems to be one focus of our disagreement and I will return to it later: for now, let me just remark that earlier correspondence had I thought clarified the distinction bewteen mathematical entitites ( eg, integers, finite groups or categories) and mathematical *constructions*, which are the kind of set-theoretic conglomerations posited by the foundational set-theorists in the spirit of Principia. >As for commercial databases, you would be amazed at the kinds of garbage >they contain. And much of that garbage is caused by a failure to make >clear philosophical distinctions about the reference of various terms. >One of my favorite examples is from a system that came up with the >following response: > > Q: What is the largest state in the U.S.? > A: Wyoming. > >For numbers and character strings, this system used the greater-than >relation to answer questions about size. Therefore, it found the >last state in alphabetical order. Nice example of the need to not confuse symbol and denotation and keeping one's quotes clean. >> NO!! The denotation functions do not recognise (and are not computed, >> and do not access or DO anything else). They are simply a mathematical >> way of talking about correspondences between names and things. > >Exactly!!! Model theory is a system of pure mathematics. The only >thing it can do is relate mathematical symbols to mathematical things. Why 'pure'?? It uses mathematical ideas and terminology in what is now a reasonably respectable intellectual tradition, but that only means that it is, if you like, a mathematical theory of reference. To say that the things it describes must therefore somehow be made of mathematical stuff seems to me to be a category error, even if it can be made coherent. (What are 'mathematical things'?? And even if one were inclined, on philosphical grounds, to accept this idea, THESE mathematical abstractions should not be mixed up with anything mental or computational.) 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. But this distitnction (between pure and applied) is not one of subjectmatter. In one sense (Russel's) all of mathematics is not about anything, since the theorems are 'true' independently of their subjectmatter. But it would be equally correct (and make the same philosophical point) to say that all of mathematics is about anything that it can be fitted to, ie mathematics is about anything. Einstein went looking for some way of talking about his relativity ideas and found a piece of mathematics that was considered 'pure' because it had been investigated for its formal properties with no application in mind. Tensor calculus is now considered 'applied'. But the change was not one of taking abstract mathematical 'things' called tensors and providing a tensor-recogniser, or a theory of tensor epistemology, or whatever, but simply using tensor calculus to describe the world appropriately. Thats what we DO with mathematics: we use it to talk about things. >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. >> 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. >But as I pointed out many >times, philosophy of science is a separate subject that is not included >in and cannot be presupposed by anyone's claim to be using "conventional >model theory". I agree: as I have repeatedly insisted, I am not presupposing it and do not include it (or epistemology or psychology) in semantics. > >> I reject this distinction between 'formal, mathematical' and 'messy, >> real-world'. Mathematics can refer to the real world. When I use >> arithmetic in carpentery I am using mathematics to reason about >> the lengths of real pieces of wood I hold ion my hands. Similarly >> for much of engineering. Being a formalist, you probably disagree: >> but I am not a formalist. > >Pure mathematics doesn't refer to anything in the world (as in your >quote from Bertrand Russell). In order to refer to physical objects, >you must APPLY mathematics. And that application always involves >methodological assumptions. A skilled carpenter might be able to >make those applications without much conscious thought, but that is >only because he or she has spent long years of apprenticeship. >When I try to do carpentry, it's much harder for me to make accurate >measurements -- not because I don't know mathematics, but because >I don't have the experience of applying mathematics to that task. I profoundly disagree with all of this paragraph, as you will guess. 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 (unless we are discussing the semantics of a theory of measuring, of course, but then for other reasons.) Let me emphasise that I am not claiming that these matters are trivial or unintersting or incapable of analysis, only that semantics does not assume them. As you correctly observe, model theory is not concerned with such matters, just as it should not be. I was careful (and it took care) in the above to talk of numerals rather than numbers. It is easy to slip into the usage in which one talks of doing arithmetic on numbers, as though these 'things' were the reality. And this does, I admit, sound awfully like your four-part diagram, in which there is a layer of abstract ontology standing between the formalism and the world. This is harmless for arithmetic, although I think a philosphical error, but it becomes daunting when we think of the mathematical work that would need to be done to build such an abstraction for cats and mats and forests. And these things aren't mental or computational (one can't get numbers into a computer, only numerals) > >> ... This makes no claim to magic; only plain, ordinary >> interpretations of the usual definitions to be found in any logic >> textbook :-) > >As Ronald Reagan said, "There you go again." I promise to stop >mentioning Tarski if you promise to stop mentioning your mythical >"logic textbook". I have repeatedly asked you to find one of those >textbooks and quote exactly what it says about the "ordinary >interpretations of the usual definitions". (I thought you would recognise that this was meant to be a joke. But as humor is sadly lacking, let me simply refer to my earlier comments on Tarski.) Use the textbook you quoted earlier yourself (Mendelson, I believe). Most textbooks, as you pointed out, simply refer to a universe as consisting of a set, without specifying what that is supposed to be a set of. Each of us seemed think that that made his point, so we must be misunderstanding one another. But on the subject of whether sets (or 'classes', using an older terminology) could contain physical objects, try these quotes: Carnap, writing in the 1950s: "The extension of a predicate is the class of individuals having the property designated by the predicate. (E.g. the extension of "Book' is the class of books, of "Blue" is the class of blue things.)" (This is from 'Introduction to Symbolic Logic and its Applications', half of which is devoted to describing physics in logic.) Quine, in the introduction to "Set theory and its Logic" (discussing extensionality): "..If the universe is taken as that of the real numbers, then....But if the universe is taken as that of persons, and the predicates are interpreted in ways depending on nothing but people's incomes, then the proposed way of defining 'x=y' will equate persons having equal incomes...it might be better to be so rectified as to construe the members of the universe as whole income groups..." or from "Mathematical Logic", introducing the idea of function on page 120: "The class of marine mammals living in 1940 is the same as the class of whales and porposes living in 1940.." 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.) > >> Your new talk of 'depictions' is just an example of the kind of muddle >> that is going to emerge if we let you get away imposing your constructivist >> religion on our formalisms. > >This is the kind of verbal escalation that drives me to "pseudo- >scholarly" quotations and citations. As I have pointed out many times, >computer science in general and AI in particular not only deal with >constructive techniques, but finite and even small finite (i.e. polynomial) >techniques. And I can cite any number of examples you please to show >that my religion is just as popular, established, or respectable as yours. John, I have already apologised to you for "pseudo", so let me do so again in public. As I said in our correspondence, "amateur" would be more appropriate. On religions, computer science's concern with constructive techniques is irrelevant to the discussion we are having about semantics, since semantics is not concerned with computational issues but with matters of meaning. These are related to one another, of course,in all kinds of fascinating ways, but they should not be identified. >> These things are not properly defined >> anywhere, are supposed to be made of symbols but to be something like >> images; like databases but able to play the role of possible worlds, >> to be built of datastructures but in 1:1 correspondence with reality, >> and to act as a kind of unifying information blackboard in a robot. > >To avoid accusing you of being confused about what I have said, I will >charitably assume that you are using a debating ploy. No, every one of these was culled from one of your messages in this extended correspondence. I am completely confused about what these 'depictions' could possibly be meant to be. Let me spend a little time on this. 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). But my first point is only that 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'm sorry, I am completely confused as to what the hell you can possibly be talking about. This is not debating rhetoric, but a genuine cry of despair. ......... > >> First, a TMT model (a possible world) is not a >> representation, it is an account of how a representation might be >> understood to mean something. > >I will stop mentioning Tarski if you stop talking about this mythical >TMT that isn't defined anywhere. As I have been trying to get across >many times in these notes, there are two very distinct issues: the >formal, mathematical operations defined by Tarski and the mathematical >logicians and the question about how that formalism is to be applied >to the real world. Tarski never addressed that second question. If >your TMT adresses it, please quote the mythical "any textbook" that >defines TMT. No, it doesn't address it. It shouldn't address it, as I have insisted over and over again, so to criticise it for not doing is ridiculous. All it needs to do is take it that it can be so applied, and I have still not seen you produce a single argument why it cannot. TMT is just conventional model theory, which defines a 'model' ('possible interpretation' is much better) as consisting of a domain D (which is a set) and a mapping from each individual name to an element of D and from each n-ary predicate symbol to a subset of D^n, etc. etc.. And let me emphasise (again) that these 'mappings' are not (in general) 'operations' to be evaluated or computed or have any operational interpretation whatever. >> ... This distinction, between assertion and counterexample, >> has been understood and used in reasoners for 25 years. > >Nobody is questioning that distinction. A model, like most mathematical >constructions, can be used for multiple purposes. Evaluating the >denotation of a formula was Tarski's original reason for inventing >model theory. You can hardly claim that using a DB for the purpose >of evaluating denotations is "not appropriate for a model". Im not sure what 'evaluating' means here. I thought that on your account, the DB *was* the denotation.(??) 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 (?TMT? What terminology will not cause some rhetorical reaction from you?) model. That something is true in a such a model is not sufficient grounds for asserting it, and does not constitute having it represented. >> We can argue about philosphical matters for ever, but the central issue >> is that you refuse to let me talk about how representations represent, >> and I insist on keeping this clear. > >On the contrary, I have been begging you to explain that. And please >be clear about what kind of representations you are discussing. Are >you talking about logic-like or language-like representations such as >KIF, or are you talking about model-like representations consisting of >nothing but sets of symbols and relations over them (but no quantifiers >or Boolean operators). 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. So your usage here seems to me to be simply incoherent, if taken at face value. However, I think I see what you are talking about: you mean here 'model' in the first sense I used earlier. I think this exhibits another difference betwen us, which has not been uncovered very clearly until recently. I classify representations chiefly on the basis of what they represent, while you seem to classify them chiefly on the basis of their syntactic details. For example, a database seems to me to be a compact implementation of a set of ground assertions, what has been called a 'vivid' representation. This is because it carries the same semantic weight as such a set of assertions: in a model-theoretic account, they have the same models. So the differences between them are not in their meaning, but in matters concerned with their computational characteristics. Semantically they are indistinguishable. Thus, I attach little semantic importance to the lack of negation or of quantification in such representations. You however call these 'model-like', and seem to really mean this in the sense of model theory. 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. >> ... and exhibit a zealot's fierceness in defending him [Tarski]. > >I am not trying to defend Tarski. I am trying to defend myself against >your charge that I am "eccentric" or outside the mainstream or contrary >to years of AI practice. All I was trying to do by quoting Tarski >was to prevent you from using him as a cover for your position. See earlier comments on this. I will however say that I have never tried to use Tarski as a 'cover' for any position, and have never cited him in this correspondence (except to note half-humorously that he used 'snow' in a famous example), so I reject your implicit charge of intellectual dishonesty. ...... >I did discuss this point with Ray at a conference on AI and DB a couple >of years ago. He prefers to think of a DB as a theory, but I prefer to >think of it as a model. 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. 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). >> and you cite robotics, virtual reality, etc etc without any authority. > >What are you trying to do? Push me out of the mainstream again? >Do you want to provoke another barrage of quotes and citations? No, merely observing an aspect of your rhetorical style which I deplore. I bow (with respect) to your reasons for mentioning virtual reality, now you have explained them. 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'. Pat ---------------------------------------------------------------------------- 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 hayes@cs.stanford.edu or Phayes@cs.uiuc.edu