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Date: Wed, 28 Apr 93 17:53:03 EDT From: sowa <sowa@turing.pacss.binghamton.edu> Message-id: <9304282153.AA25149@turing.pacss.binghamton.edu> To: interlingua@ISI.EDU, phayes@cs.uiuc.edu, cg@cs.umn.edu Subject: Re: Practical effects of all this discussion Cc: sowa@turing.pacss.binghamton.edu, dietrich@turing.pacss.binghamton.edu

Pat Hayes made some comments on my recent notes that I cannot leave unanswered: >> John Sowa writes: > >The recent flurry of notes about possible worlds, models, etc., > >seems to have reached the point where all the discussants (Pat Hayes, > >Len Schubert, Chris Menzel, Jim Fulton, and I) are too exhausted to > >say anything more about the subject. > > Sorry, Ive just been away. And I agree that these issues are important, > which is why I want to try to get some points straight. > > It is important to get clear that John Sowa has a very idiosyncratic and > singular philosophical position, and not let this be simply accepted as > established fact or dogma. I do not want to argue that his view is wrong - > these are issues on which one can consistently take any of several views, > and debates between them belong in the philosophical literature - but since > the context of this entire discussion is the design of a standard > representational language, it would be unfortunate if that design were to > be unduly influenced by these ideas, especially if their details are not > entirely worked out. Anyone who confidently tosses aside almost a century > of work in the foundations of mathematics should be treated with a certain > amount of caution. I haven't "confidently tossed it aside". As an impressionable undergraduate math major, I was amazed and enthralled by "the paradise that Cantor opened up to us" as Hilbert put it. But I now sympathize with Wittgenstein that the entire body of work on transfinite set theory is a "swamp of confusions." I have no intention of stopping anyone from working in that swamp if they want to. But I believe that an agnostic attitude towards uncountable sets is a much sounder, more realistic attitude. There is nothing in either conceptual graphs or KIF that commits us to assuming uncountable sets, nor is there any feature of either one that prevents us from talking about them if we want to. All that I am saying is that if we want a solid foundation for mathematics, knowledge representation, or anything else, we should avoid talk about uncountable structures unless we explicitly intend to venture into the swamp. > John does seem to misunderstand, and certainly mis-states, some of the > basic ideas of conventional semantics, notably that of a Tarskian model or > interpretation. I beg to differ. I distinguish Tarski's actual writings from so-called "Tarskian" or "Tarski-style" metaphysics that goes far beyond anything that Tarski himself dared to say. > Tarskian model theory - TMT - defines how the denotations of symbols > interact to give the denotations of more complex expressions. Thus for > example it says that if we interpret a relation symbol 'Bigger' to refer to > a certain relation between things, and interpret the name 'New York' to > refer to a thing, and 'Arthur' to refer to another, then 'Bigger(Arthur, > NewYork)' is true just if that relation holds between those things; and so > forth. That is all it says, simple things like that. A 'model' of some > collection of symbols is simply a complete specification of appropriate > denotations for them all. This statement sounds innocuous enough until you get to the phrase "just if that relation holds between those things." Tarski never addressed the issue of how the symbols in his "formal languages" ever related to real "things." His models were always set theoretic constructions. When you go beyond Tarski to his students like Montague, you find that they sweep all those issues under the rug by saying that the intension of "tree" is a "function" that yields the value "true" when applied to a tree, and yields the value "false" when applied to a non-tree. Never do they say how those functions are computed. People like Carnap in his _Logische Aufbau_ and Gooodman in his _Structure of Appearance_ wrestled with those ideas without solving them even to their own satisfaction. Wittgenstein believed that he had solved that problem in his _Tractatus_, but he spent the rest of his life explaining the difficulties and limitations of his earlier views. > This is usually described set-theoretically, > largely because thats the way mathematics is done these days, but that has > no deep significance, and certainly doesnt mean that the models must be > somehow unreal or abstract, or of course that they must be mental > structures or data structures (where on earth did THAT idea come from?) It has a very deep significance. Sets and data structures are neat, clean manageable entities where the basic elements can be clearly distinguished >From one another. But if you take a walk in Muir woods, you'll discover that it is very hard, or even impossible to say clearly when two redwood trees are distinct or when they happen to be the same individual. For aspen forests, the problem is even worse, because you might have an entire "forest" that is actually a single individual with one root system and hundreds of trunks. The point is that Tarski-style model theory can only be applied with any degree of rigor to set theoretic constructions, data structures in a digital computer, or similar abstractions. When you attempt to apply them to things in the real world, you encounter exception after exception, and fuzzy "judgment calls" at every turn. For further examples, read any of Wittgenstein's later philosophy. > To emphasise the point, imagine a tiny language with three names 'Arthur', > 'Joe' and 'C', and one binary relation name 'Bigger'. Then I might > interpret 'Arthur' to refer to the Queen Elizabeth 2, 'Joe' to refer to my > sister Mary, 'C' to refer to the dirty stain on the window of my office > (assuming of course that these can all be accepted as things) and 'Bigger' > to refer to the relation which holds between physical things when the first > is at least as old as the second. Then 'Bigger(C,Joe)' is false and > '((Forall ?x) (Exists ?y) Bigger(?x,?y))' is true, for example. Now, I just > defined a Tarskian model. The things in its universe were real, not data > structures. To say that it is a 'possible world' is just a way of saying > that it is a possible way to interpret that language in the world. To > insist that there is only one world, does not rule out all kinds of ways of > mapping symbols into it. If you are willing to admit that "possible world" is a short-hand way of saying "a possible way to interpret that language in the world", then I would be happy to drop my objections. But Chris Menzel was citing David Lewis, who keeps making statements that "possible worlds are real" -- something that I find totally unintelligible except as a metaphor. > TMT does not assume that its models are in any sense complete > specifications of the world. It does assume that quantifiers range over all > the things that are being taken (in this model) to constitute the domain, > and this is an assumption which Situation Theory chose to reject, allowing > essentially partial quantifiers. McCarthy's idea of 'contexts' is another > way of making quantifiers range over part of what is being taken to be the > domain of things. One can argue about the utility or power of these > modifications to TMT, but these arguments have nothing to do with whether > or not the things being quantified over are real. There are two uses for the word "model": (1) in logic, a structure for which some axioms have denotation "true"; (2) in engineering, a structure that has parts and relationships that correspond to some aspects of a real world system or subsystem. Although these two uses arose from different academic disciplines, they are not inconsistent. I prefer, in fact, to identify these two kinds of models: the denotation of any formula in any formal language can only be computed in terms of some abstract model of the world, never in terms of the world itself. A civil engineer who is building a road and a mechanical engineer who is building a car to drive on that road may have very different models of exactly the same road. Both may be good approximations for their purposes, but the formulas used by the two engineers may be inconsistent or incompatible with one another. Yet for each engineering model, the formulas for that discipline may have denotation "true". > TMT does assume that the concept of 'thing' is somehow specified, and the Aye, there's the rub! "Somehow specified"! My complaint is that the theorists like Montague never do any actual specification. And the engineers who really do specifications always do them in terms of an abstract model of the world -- never the world itself. > universe of things is specified, and what counts as a relation is > specified. John observes that these specifications might be given in all > sorts of ways: the world does not come with crisp edges to 'things' given > to us, we have to impose them. True, but this makes not a whit of > difference to TMT. It can even be stated as the observation that a given > piece of the world might be used as a TMT interpretation in many ways. It > is merely to observe that TMT assumes that ontological issues are somehow Yes. And I am simply insisting that those assumptions be made explicit. Whenever anyone attempts to make them explicit, it becomes very clear that they are only talking about a very limited, finite abstraction >From the world -- more like a Barwise & Perry "situation type" than anything that resembles the real world in its entirety. > settled, and makes very few committments in this direction. (It does assume > that names refer to individuals, which is itself not uncontroversial; > readers of this email will no doubt be familiar with the chief alternative, > mereology, which takes expressions to refer to disconnected pieces of a > continuum. An exactly similar debate can be had about whether mereological > models are made of reality or must be mental.) Yes. Mereology and set theory are both at the same level of abstraction. > John also seems to think that (perhaps because TMT is usually phrased in > set-theoretic terms) these things and the relations between them, and so > on, are data structures. This is a much more peculiar idea. On the face of > it is simply wrong, since as I observed in an earlier message, data > structures are finite, but universes need not be (and some, eg the > integers, are usually taken to not be.) John's reaction to this point is to > simply reject all of modern mathematics since Hilbert. This reveals > considerable selfconfidence, but I do not find it very convincing. Most > rebels in mathematical philosophy - even the intuitionists - have agreed > that there are infinitely many integers. If anyone wants to use KIF or CGs to represent transfinite set theory, they are welcome to do so. But I claim that no mathematical theory with application to anything in the real world need ever talk about an infinite set. I am willing, as the 19th century mathematicians did, to say that the integers are infinite. By that, I mean that for any integer you can name, I can find another one that is larger. But I also agree with the 19th century mathematicians that it is inappropriate to talk of an infinite set as a completed whole. Instead of saying, let x be a member of the set of integers, I would simply say "let x be an integer". > But even if we keep away from mathematics, it is a strange idea. Much of > what we say, and certainly what we want to encode in representation > languages, quite clearly has nothing to do with mental or data structures. > A Kbase might consist of descriptions of the stress patterns in bridge > girders, or the current balances in bank accounts, or the whereabouts of > military units, etc. etc.. None of this is mental structure or data > structure. (Footnote: of course the things in the Kbase ARE data > structures, and my beliefs ARE mental structures, but they REFER to other > kinds of thing.) I have no quarrel with this statement. All I am saying is that every model is an abstract construction. It may have parts that are in correspondence with things in the real world, but the model itself is not the world. > It makes sense in one case. If we want to express knowledge about other > knowledge - as in discussing beliefs or reasoning about possible > misunderstandings, for example - then the problem arises of how to describe > these things. If we want to describe them in the same way that we would if > the system itself believed them - so that to represent 'harry believes > ...', we can write something with the content 'harry believes this:'...'' - > and if in addition we want to be able to quantify into these quoted > expressions; if all this, then indeed some problems arise with > straightforward TMT, since it is not able to account for the apparent > meaning of the quantifiers in this case. This is now very well understood, > and modal logics handle the resulting compexities quite well. John is > centrally concerned with this case, has philosphical objections to the > modal logic semantics, and wishes to modify the semantics of KIF to make > this case easier. His proposed solution is idiosyncratic and has not been > fully worked out, but in any case is too eccentric for a proposed knowledge > representation standard. On the contrary, there is no "fully worked out theory" that relates symbols in a formal language to things in the real world. As I point out above, Montague just postulates "tree" functions that are magically able to distinguish trees. By emphasizing the model as the mediating structure between word and symbol, I am simply returning to Aristotle's three-way distinction between words, things, and "experiences in the psyche". That was the established wisdom of the "perennial philosophy", and if anything is eccentric, it is the attempt to postulate magic denotation functions that ignore the essential role of the intervening model. John