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C00002 00002	know[e81,jmc]		Revising Montague's knowledge axioms
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know[e81,jmc]		Revising Montague's knowledge axioms

	In (Montague and Kaplan 1974), they proved inconsistent
the following axioms for knowledge:

	K(p') ⊃ p

	K((K(p') ⊃ p)')

	I(p',q') ∧ K(p') ⊃ K(q')

Here  p  stands for a sentence and  p'  is its Goedel number.  K(p')  is
the assertion that the knower knows  p  considered as a predicate on
its Goedel number.  I(p',q') is the relation that  q  is derivable
from  p  in predicate logic.  It is assumed that the system contains
enough arithmetic to represent elementary syntax.

	Montague and Kaplan suggest various ways of weakening the
axioms to get consistency.  One of them is to have a hierarchy of
languages.  Our proposal is a variant of this - to keep one language
but have a hierarchy of knowledge operators.

	We now write K(n,p') for the assertion that  p  is known
at the nth level.  Our axioms are

	K(n,p') ⊃ p

	K(n+1,(K(n,p') ⊃ p)')

	I(n,p',q') ∧ K(n,p') ⊃ K(n,q')

where  I(n,p',q')  is the assertion that  q  is derivable from   p  assuming
n  levels of the assertion that the previous level system is consistent.
Level 0 is Peano arithmetic.  It isn't clear that the consistency assumption
should be included in  I, because it may be that the second axiom already
contains the assumptions that the previous level is consistent.