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Reply-To: cg@cs.umn.edu Date: Fri, 5 Nov 93 03:35:37 CST From: fritz@rodin.wustl.edu (Fritz Lehmann) Message-id: <9311050935.AA03585@rodin.wustl.edu> To: phayes@cs.uiuc.edu Subject: Higher-order KIF & Conceptual Graphs Cc: boley@dfki.uni-kl.de, cg@cs.umn.edu, interlingua@isi.edu

Dear Pat Hayes, phayes@cs.uiuc.edu Yes, I agree that we have been talking past each other a bit, because I've said three or four times that A. we need higher-order and mixed-order constructs for practical reasons, B. if the mathematical logicians want to call these mere "syntactic sugar" via some kludgy reduction to FOL based on an equivalent countable number of first order sentences on some arbitrary finite vocabulary, that's fine with me, and C. if we need real higher-order model-theoretic semantics (with a domain closed under disjoint unions of direct products) as I suspect, that's fine with me too. So, there may be no dispute between you and me. Just make sure the language has the constructs and handles them right in all cases likely to be encountered in real life. KIF and Conceptual Graphs now make no provision for higher-order concepts with any semantics at all. Gruber's Ontolingua makes a stab at it, as do Boley's DRLHs. We need relations among relations, relations between relations and their own arguments, functions of functions, and so on, not just relations among individuals. (For some purposes one might conceivably want relations defined on connectives, as in the works of Zellweger, Menger or Stern.) I think I've cited examples of about five or six kinds of higher-order things that users might want to say for practical applications, which are simply impossible to say in current (First-Order) KIF or CGs. (If anybody out there is sympathetic, please email to these lists a few more examples of higher-order or mixed-order statements. If unsympathetic, send counterexamples showing how my prior examples would appear in FO KIF or CGs.) Somebody earlier (I forget who) said KIF is like a standard being developed without any user input. That seems right. I don't think the potential users will give a hoot about transfinite numbers, incompleteness, nonstandard arithmetic, etc., nor should they. We need to trade taxonomies, exchange industrial PDES/STEP descriptions, describe business enterprises, classify scientific documents, etc. -- as I said before, probably no-one will need numbers any higher than Graham's Number (a finite biggie from Ramsey Theory), let alone the transfinite. But users will indeed be bothered if they can't say what they need to say. [Digressive Note: Your messages keep raising too many interesting issues. Such as the distinction between the semantic reality of AND and uninterpreted VERY-BIG. "Semantic theft from the syntax" is a pet peeve of mine: definitons of AND tend to steal its meaning (directly or via intersection or truth tables) from a semantic feature of our syntax. Similarly, post-Princ.Math. purported definitions of 42 steal the essential 42-ness from the semantics of our _42_ syntactic meta-applications of successor (or pure-set parentheses or the equivalent). But we can't explore such topics in this KIF/CG discussion. (I'll mercifully skip my "Non-Existence of the Self" thesis.) Your expounding to me on the hugeness of uncountable numbers assumes that I'm recommending higher-order KIF/CGs due to an expressed concern of mine for consequences in uncountable domains. Absolutely not. I don't even believe in them. I said earlier that "I balk at diagonalization and its progeny"; uncountables are among the progeny. You said you balk too. But I mean it -- someday I'll either A. Accept Cantor's Theorem, B. Refute it, or C. Die without having done A or B. B is the outcome I prefer. Until then, like other skeptics (many of whom later became converts), I regard the diagonal argument as a specious tissue of arrant nonsense and I heartily endorse Sowa's phrase "uncountable garbage". If X is countably infinite then I'm currently happy with: X = 2+X = 2*X = 2^X = 2(^X)Xtimes, etc. so I'm Aleph-free and excluded from Hilbert's "paradise". As the support-group- aholics like to say, "joyous, happy and free". Mainstream Cantor-following math types would say "benighted ignoramus". I don't necessarily expect others to share my view -- at least not until event B. Meanwhile there's no point in you demonizing uncountability for me, since I (privately) won't even acknowledge that closure under classes and direct products makes a domain "uncountable" in the first place. (I might believe vaguely, like Peirce, in a "real infinity" beyond all alephs which is 1/0.) But back to KIF and CGs.] The potential success and dominance of an un- expressive knowledge sharing standard could have bad consequences. In AI, scruffies typically innovate and then later on the neats clean up the messes. This was true in semantic networks. I want KIF/CGs to be logically sound, but not to _stifle_ the future innovators. (If you want a field dominated by formalizing, take nonmonotonic logics -- please!) I fear that pure First-Order-ism will be stifling. Remember -- when I asked KIF author Mike Genesereth about the relation between a relation and one of its own arguments, he informed me that "there is no such relation". Is that the kind of (FO-inspired) answer which knowledge standards will offer would-be users? You say: >[R]eal systems will often be incomplete >for all sorts of practical reasons.... traceable >to, as it were, failures to search all of the space. But inmcompleteness of >the underlying representation language is a gap in the search space itself, >which is a more serious matter. Incompleteness in the space is practically serious only if the locus of incompleteness has some likelihood of actually being reached in practice. Think of a very long strip of linoleum; it's complete and intact for the first zillion miles from one end, but at that distance small cracks begin to appear. It is still a usable, "practically complete" piece if our activities take us at most a few yards along it from the good end. That is an adequate standard for practical knowledge interchange, if the alternative is crippling un-expressiveness. First- Order-ism gives us a truly crack-free infinite strip of linoleum, but only an inch wide and patterned with big holes. We're better off with the wide, locally solid piece, distant cracks and all. >I was interested that when challenged to go above third order you >cited category theory. Yes, I can believe that category theory >might find some of the more rarified heights useful. Good. I hope category theorists will not be denied the ability to communicate in a knowledge interchange language due to its being merely First-Order. One reason catgory theory is a promising addition to Krep is because it has an inherent "action-oriented" pragmatism: the theory doesn't care what something is, only what it does. >I would be very >interested to find out more about Ontek's use [of 3rd and >4th order logic]; is it in any way related to Montague? I don't know. I think there might be a few Montagovians around U.C.Irvine with whom Ontek has connections. Since Ontek people get these email lists, they can answer this. >You ask for 'REAL EXAMPLES' of the trouble caused >by incompleteness. But that misses the point. If you are creating a limiting standard for other people to use which requires them to make some sacrifice, it never misses the point to ask you for some practical examples which would justify their sacrifice. Yours truly, Fritz Lehmann fritz@rodin.wustl.edu 4282 Sandburg, Irvine, CA 92715 USA 714-733-0566