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%3Concepts as objects%1

	We shall begin by discussing how to express such facts as
%2"Pat knows Mike's telephone number"%1, although the idea of
treating a concept as an object has application beyond the
discussion of knowledge.

	We begin by introducing notation.  %2mike%1 is the symbol
we shall use for the person Mike, and and %2telephone(p)%1 is our
notation for the telephone number of person %2p%1.  We shall suppose
that each person has exactly one telephone number.  We aren't much
interested in the domain of telephone numbers, but we shall take it
to be a string of digits with a dash in the right place, and, since
a telephone number is a string, we will write it in quotes.
Thus we can write

!!c2:	%2telephone(mike) = "321-2222"%1

as a formalization of the English %2"Mike's telephone number is
321-2222"%1.  Let us suppose that Mary lives with Mike (not in sin),
and so has the same telephone number, so that we can also write

!!c3:	%2telephone(mary) = telephone(mike)%1

from which the transitivity of equality in first order logic permits
concluding

!!c4:	%2telephone(mary) = "321-2222"%1.

	Now we want to translate %2"Pat knows Mike's telephone
number"%1.  If we were to express it as

!!c5:	%2knows(pat,telephone(mike))%1,

first order logic would let us conclude

!!c6:	%2knows(pat,telephone(mary))%1,

corresponding to %2"Pat knows Mary's telephone number"%1 which
mightn't be true.

	The way out of this difficulty currently most popular
among philosophers it to treat %2Pat knows%1 as a %2modal operator%1.
This involves extending the logic so that replacement of an expression
by an equal expression is not allowed in all contexts.  Knowledge
is not the only operator that admits modal treatment.  There is
also belief, wanting, and logical or physical necessity.  For
AI purposes we would need all the above modal operators and many
more in the same system.  This would make the semantic discussion
of the resulting modal logic extremely complex.  For this reason,
and because we want functions from material objects to concepts of
them, we have followed a different path - introducing concepts as
individual objects.  This is has not been popular in philosophy,
although I suppose no-one would doubt that it could be done.

	In the present case we introduce the symbol %2Mike%1 as
a name for the concept of Mike and the function %2Telephone%1 which
takes a concept of a person into a concept of his telephone number.
The second operand of the function %2knows%1 is now required to be a concept.
and we can write

!!c7:	%2knows(pat,Telephone(Mike))%1

to assert that Pat knows Mike's telephone number.  The previous trouble
is avoided so long as we can assert

!!c8:	%2Telephone(Mike) ≠ Telephone(Mary)%1,

which is quite reasonable, since we do not consider the concept
of Mary's telephone number to be the same as the concept of Mike's
telephone number even if the numbers themselves are the same.