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Reply-To: cg@cs.umn.edu Date: Fri, 8 Oct 93 08:10:59 CDT From: fritz@rodin.wustl.edu (Fritz Lehmann) Message-id: <9310081310.AA15617@rodin.wustl.edu> To: phayes@cs.uiuc.edu Subject: Equality; higher order KIF/CGs Cc: boley@dfki.uni-kl.de, cg@cs.umn.edu, chrikey1.ucthpx@f4.25.fidonet.org, interlingua@isi.edu

Dear Pat Hayes, phayes@cs.uiuc.edu Thanks for describing to me the reduction of equality-definition to First-Order via a finite schema made of "Relation-symbol X Argument-place" instances, depending on a finite vocabulary of available relation-symbols. This seems like an unappealing kludge which nonetheless works for some systems. But there's no solace for me in relying on a finite vocabulary of relation-symbols in the syntax to make semantics (for equality or anything else) First Order. As soon as you allow what I require for KRep, and what ordinary discourse already uses (relations among relations, and among different-ordered relations, and individuals), then an infinite zoo of these relation-symbols springs up. This even in a universe having just two atomic 'individuals' and without allowing singletons or empties in the models. Forms are: a; b; R(a,b); R'(a,R); R''(b,R,R'); etc. Of course this begs the original logic-order question. And it looks similar to your description of nested formulae in "beta-conversion" -- have you a reference for this? (I say infinite rather than transfinite since I still balk at diagonalization and all its progeny.) When an email message gets no answer I never know whether it was too boring, too lame to warrant a response, or too convincing to question. It may just be that frank discussion of KIF occurs in a more exclusive email forum than Interlingua. I recommend higher- order KIF and Conceptual Graphs for practical convenience. Suppose there were some decent reduction to First Order via types, schemas, Henkin or what-not (though I doubt it); then perhaps the handy higher-order and mixed-order constructs which people need could be "syntactic sugar", as long as the semantics is done _right_. (An uninterpreted "holds(p,x)" relation/link won't do.) But give us these constructs, _with_ semantics, dumping decidability if necessary. [A similar point was reached by the KL-ONErs on tractability (and decidability too); even purists have now dumped worst-case tractability of subsumption, to let people actually say what they want to say. Tractability is good for getting a PhD, though.] At present, KIF and CGs are unable even to express -- with semantics -- Peirce's (or Leibniz's) basic sentence: "Given any two things, there is some character which one posseses and the other does not." Poignant, since all the First-Order model theory of KIF and CGs depends on there being such distinct things. I don't know whether the CYC Epistemological Language can say it. It could simply be that the major KIF players decided on limiting KIF's logic to First-Order a long time ago and "the deal is done" (although, if so, one should wonder at their openness to the nonmomonotonic stuff). An interlingua should be grounded on classical logic, but classical logic is n-adic, n-order. Yours truly, Fritz Lehmann fritz@rodin.wustl.edu 4282 Sandburg, Irvine, CA 92715 714-733-0566