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Date: Sat, 15 May 93 23:20:41 EDT From: sowa <sowa@turing.pacss.binghamton.edu> Message-id: <9305160320.AA05183@turing.pacss.binghamton.edu> To: cmenzel@kbssun1.tamu.edu, sowa@turing.pacss.binghamton.edu Subject: Re: Getting back to the notes of May 10th Cc: cg@cs.umn.edu, interlingua@ISI.EDU, phayes@cs.uiuc.edu

Chris, Some comments: > ... your approach to understanding reference to cats and mats > would require a complete theory of cat/mat imagery. That is not just my approach. Any theory of reference -- psychological, philosophical, or computational -- must eventually address the question of how things in the world are related to symbols. The point that I was trying to make in the notes to Pat and Len is very simple: 1. Model theory, as defined by Tarski and the mathematical logicians, relates formal languages to mathematical constructions. It is a branch of pure mathematics, not a theory of perception nor a theory of physics. 2. In order to use model theory to relate a language (formal or natural) to the world, some very complex issues must be addressed about the way that the mathematical constructions relate to the world. According to Pat, there is a mythical TMT, which is able to deal with models whose individuals ARE real world objects. I admit that there is a lot of loose talk in elementary textbooks about sets whose elements ARE people, trees, cats, and dogs. But those books mix mathematical constructions and physical objects in a very muddled way. In one of his earlier notes, Pat mentioned an anecdote where he was talking with Jon Barwise about a similar issue, and Barwise said "Look around this office. You see a lot of objects. But how many SETS do you see? Where are the sets?" Barwise was trying to make a similar distinction between sets as mathematical constructions and the things in the world. Last November, Barwise invited me to give a talk to his seminar in Indiana, and I showed my four-part diagram that I described in an earlier note to Pat: Natural language <--> Logic <--> Models <--> World Barwise very much approved of that approach, as did Godehard Link, a Montegovian, who was at the ontology workshop in Padova and who came up to me after my talk (in which I also showed the four-part diagram) and said that he completely agreed with the need to make such a distinction. > ... Taking a relation > in the usual model theoretic way as a set of ordered pairs, surely it > is false that "the only thing [model theory] can do is relate > mathematical symbols to mathematical things." If I've got the symbol > `John' in my language, then if the interpretation function (taking it > to be a kind of relation) for my language includes the pair <`John', > John Sowa>, then I have related a symbol to a nonmathematical thing. You're using mathematical language in the same way that I just objected to. You have to distinguish pure mathematics from applied mathematics. In pure mathematics, there is never such a thing as an "ordered pair" consisting of a symbol and a person, nor an ordered pair consisting of two physical objects of any kind. When you develop a theory of pure mathematics (as Whitehead, Russell, Tarski, and others did), every variable and every construction consists of purely mathematical entities: numbers, points, spheres, the empty set {}, and combinations of such things. After you have built such a theory, then you can talk about applying it to the real world. In that case, you associate with each formal symbol in the mathematical construction some physical object in the world. But that association is not part of the formal theory; it is part of the methodology for applying the theory. That is the crux of the disagreement between Pat and me. I maintain that model theory says nothing about how it can be applied to the world. That application is not part of model theory, but of some other subject, which could be psychology, AI, or philosophy of science. Pat mentioned Kripke, Montague, and others who wanted to use model theory to relate to the world. But Kripke's famous work on model theory for modal logic was just as much a theory of pure mathematics as Tarksi's. Montague did assume certain "functions" that could recognize trees and dogs. But his formal construction stopped short of specifying those functions; his actual accomplishment was not to relate language to the world, but to relate it to the names of certain functions that he never defined. He is entitled to do so, but it is not thereby correct to claim that he "solved" the problem of relating language to the world. His undefined function symbols were just as "ungrounded" as any GENSYM used in any AI program. I do not mean to belittle the work of Tarski, Kripke, or Montague. They accomplished some brilliant work that solved a very important part of the problem: relate a complex language with quantifiers, Boolean operators, and other things like modalities to much simpler model-like constructions. But I want to make it very clear that they did not even begin to address the question of how those models actually mapped to the physical world. John