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C00002 00002	know[f83,jmc]		Notes on non-knowledge
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know[f83,jmc]		Notes on non-knowledge

	The difficulty in proving non-knowledge is essentially the
phenomenon demonstrated by Goedel's theorem.  Proving that a particular
sentence is not a logical consequence of one's beliefs immediately leads
to the conclusion that the beliefs are consistent, since if the
beliefs were consistent, then any sentence would be a logical consequence.
However, Goedel's theorem shows that systems powerful enough to do
elementary syntax or arithmetic cannot prove their own consistency.
Yet people often have firm convictions that they do not know some
fact and could not determine it by any amount of reasoning.

	This conviction usually will be justified by an assertion
that it could be either way as far as the person knows.  For example,
if I ask you whether President Reagan is standing or sitting at this
moment, and you answer that you don't know, you will resist my
suggestion that you might know if you thought harder.

	My current idea for getting around the dilemma is to use
non-monotonic reasoning, especially circumscription.  The idea is
this.  Suppose we jump to the conclusion that only a certain set A of facts
together with the reasoning methods of first order logic is relevant
to whether Reagan is standing or sitting.  Our total knowledge goes
beyond these facts, and, in particular, contains metamathematical
knowledge and reasoning methods.  Therefore, we can hope to construct
two models of A, one in which Reagan is standing and another in
which he is sitting.  If these are indeed all that is relevant, then
we don't know which he is doing.

	In order to reason formally about non-knowledge, it looks
like we need a language which can reason about the models of sets
of sentences in that language.  It does not have to reason about
the models of the set of all its axioms.