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Date: Sat, 13 Nov 93 08:20:05 CST From: fritz@rodin.wustl.edu (Fritz Lehmann) Message-id: <9311131420.AA14000@rodin.wustl.edu> To: boley@dfki.uni-kl.de, cg@cs.umn.edu, interlingua@ISI.EDU Subject: Higher-Order KIF and Conceptual Graphs

My higher-order candidate example 25 was: >>...Charles Peirce's (or Leibniz's) sentence: "Given any >>two things, there is some character which one posseses >>and the other does not." He was unable to represent >>this in his Beta (First-Order) Existential Graphs, on >>which Sowa's Conceptual Graphs are based. . . .The >>point of this statement, which is a theorem in some >>concept-systems, a possible axiom in others, and false >>in still others, is to give a criterion for the >>distinctness of two individuals. >>. . . In true second-order logic I'd say: >>"Ax,y (~(x=y) -> EP ((P(x) & ~P(y)) v (P(y) & ~P(x))))". >>The "Exists P" makes it second-order; the disjunction >>may be droppable. ...Apparently one cannot in the >>definition assume that distinctness of individuals is >>already defined. John Sowa responded: >In response to Fritz's example from Peirce, I suggest the following >representation in conceptual graphs: > If: [*x]->(DFFR)->[*y] > Then: [Relation: *r] > [ (rho r)->[?x] ~[ (rho r)->[?y] ] ]- > (OR)-[ ~[ (rho r)->[?x] ] (rho r)->[?y] ]. >The Greek letter rho is the operator that allows the relation >variable ?r to be used in a relation position instead of the >referent position. See my paper in the ICCS'93 proceedings for some >discussion of the rho and tau operators and their use in >metalanguage statements. . . . For those who may not be familiar >with the CG notation, you can read this graph as the English >sentence "If x differs from y, then there exists >a relation r where either r applies to x and not to y or r does not >apply to x and it does apply to y." This doesn't seem to accomplish it in First-Order. It asserts the existence of a predicate r distinguishing x >From y. The predicate r is either unrestricted (thus it could be a predicate like "is-finite" or "is-a-number" or "is a complete partial order feature" definable only in strongly higher-order semantics), or else it is only (weakly) pseudo-higher-order definable and ranges only over a fixed arbitrary vocabulary of available predicates. The latter of course does not convey the desired meaning since it would incorrectly identify two different things that merely happen to be indistinguishable under the currently available vocabulary; the sentence says when they are the different under ANY vocabulary. (In both senses the sentence is simply false in omniscient inchoative systems with symmetries -- i.e. with inherently distinct indistinguishables -- but in the First- Order interpretation it's a logically-true theorem. This is a problem.) Also, Conceptual Graphs get their formal semantics via Sowa's "phi-operator". The phi-operator translates this conceptual graph into a First-Order Predicate Calculus statement, which is is only interpreted with First-Order Tarskian model-theoretic semantics over a domain of distinct individuals. This presumes already-defined distinctness of those individuals. (Sowa made this point emphatically in his earlier model-theory debate with Pat Hayes, contrasting the mereology of trees having common roots.) This seems to violate the stated anti-circularity requirement: "Apparently one cannot in the definition assume that distinctness of individuals is already defined." The whole purpose of the statement is to BE the definition of individual distinctness. Other details: "[Relation: *r]" asserts the existence of an individual of type "Relation"; how does this differ >From "[CHAIR: *r]"? Can one say "If: [THING: *x] [CHAIR: *r] then: [ (rho ?r)->[?x] ]", and if not, why not (semantically)? Keep in mind that rho doesn't make _x_ a relation or chair. Rho marks a variable as ranging over generic relations. It also apparently doesn't range over _instances_ of relations; Sowa doesn't think much of these, as Conceptual Graphers may recall. (Identity of relational instances may now be derived from rho variables combined with individual identities of relata variables.) Formerly Sowa didn't use the form (xxx)->[yyy] for a simple predication; he would use [yyy]->(ATTR)->[xxx]. By the way, Charles Peirce's solution was to extend his First-Order "Beta" Existential Graphs to higher-order "Gamma" Existential Graphs. I concur. This allowed him to represent the sentence in example 25 above as well as example 1, "Aristotle has all the virtues of a philosopher." Yours truly, Fritz Lehmann fritz@rodin.wustl.edu 4282 Sandburg, Irvine, CA 92715 USA 714-733-0566