communication and representation
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Date: Fri, 17 Dec 1993 17:37:43 +0000
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Subject: communication and representation
Cc: interlingua@ISI.EDU
Hi Matt. Someone deftly removed me from interlingua so I didnt see the
recent discussion about consistency. By and large I agree with you that its
time to do a careful check of the foundations. However, there is one point
you didnt seem to get:

> Into this argument was injected the point that, while nonstandard
>  models of things like transitive closure are a problem for knowledge
>  REPRESENTATION, they are irrelevant to knowledge INTERCHANGE.  The
>  same conclusions can be drawn whichever semantics one chooses!
>In other words, the semantics are independent of the semantics.  I
>cannot imagine a context in which such a statement is defensible.

No, thats not the point. This arose in a discussion between Fritz Lehmann
and me on HOL vs FOL, in which Fritz dismissed completeness as a technical
result of little interest or importance , while I insisted it was central.
It emerged (I think) that we were talking at cross purposes: he was talking
about a languge for communication while I was talking about a
representation language. 

Suppose we have a logic suitably implemented which looks awfully like HOL.
There are two different semantic stories that can be told about it. In one,
it has 'classical' higher-order quantifers which range over the full
product power sets of the underlying classes in all models. On this story,
however, the logic is incomplete. In the other (Henkin's) story, it's
quantifiers range over suitably closed subsets of those power sets, so
there are a lot more possible models: and now the logic can be complete, in
fact its essentially first-order. Now, which is THE semantics of this
language? It depends on your taste and what you want to claim about it,
there isnt a single correct answer. You will get the same theorems proven
by your machine in either case. If you believe the first story then you
will have to learn to live with constant disappointment; on the other hand
you might feel that you have not compromised the integrity of your
intuitions about 'forall'.

Now, Fritz' answer is that the quantifiers mean what he says they mean, and
that's that. So when he writes higher-order expressions he MEANS the first
interpretation, and nobody, still less a machine, can remove that
assertional authority from him. Thats fine if we are talking about
languages being used by intelligent agents to communicate with one another,
but not (I believe) if we are talking about Krep languages, since here
there has to be a rather deep sense in which they contain no more than the
machine which uses them can, as it were, extract from them, so any claimed
semantics which defines uncomputable sets of theorems can only be a

This deserves longer and more careful discussion (and Id be grateful for
any input, comments, etc.), but I hope you get the idea of what issues are
involved and why its not just a simple confusion of 'semantics'.

Best wishes


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