# Higher-Order KIF and Conceptual Graphs

fritz@rodin.wustl.edu (Fritz Lehmann)
```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
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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

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