Re: Higher-Order KIF + Conceptual graphs
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Date: Mon, 15 Nov 1993 10:47:22 +0000
To: (Fritz Lehmann)
Subject: Re: Higher-Order KIF + Conceptual graphs
Hi Fritz

Here is Pat's last word in this debate.

There are two different semantic stories that can be told about
higher-order predicate syntax. According to one - the classical one - the
quantifiers range over ALL functions, relations, etc., that is, all
uncountably many of them. According to the other - Henkin's model theory -
they may range over some subset of these uncountable sets (but not any old
subset; the ranges must at least contain a referent for each function,
predicate etc. which is definable in the language). It is the SAME language
in each case, with the SAME inference rules, but different semantic
accounts are being given of it. 

According to the classical story the logic is incomplete and according to
Henkin's story it is complete, which is just another way of saying that
Henkin's account is accurate and the classical story is not. And finally,
if we tell Henkin's semantic story, then the logic turns out to be
syntactic sugar of a first-order logic (well, a first-order theory with
rather a lot of axioms). I'm all for syntactic sugar, you understand, but I
think it a mistake to make semantic claims for it that *provably* don't

By the way, I havn't seen your interesting-sounding list of examples. Is it
available by email?


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