0^0

[Jon Awbrey]

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more arcane stuff --

c.s. peirce, in his "description of a notation for the logic of relatives,
resulting from an amplification of the conceptions of boole's calculus of
logic", 'memoirs of the american academy', vol. 9, pp. 317-378 (1870),
reprinted in his 'collected papers', cp 3.45-149, defined "involution"
for relative terms so that, e.g., if f = friend and p = person then
f^p = friend of every person (cp 3.77).  if the relative terms are
reduced to absolute terms like 0 = nothing, 1 = everything, then
we get:  0^0 = 1, 0^1 = 0, 1^0 = 1, 1^1 = 1, which is just the
truth table for x^y being read as x <= y.  (cp 3.86).

for analogous reasons in category theory, some people notate
the space of functions of type X->Y as X=>Y, Y<=X, or Y^X.
e.g., lambek & scott, 'higher order categorical logic':

http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521356539

jon awbrey

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Jud McCranie wrote:
> 
> At 04:24 PM 5/4/2005, Jon Awbrey wrote:
> >
> > but logic says that 0^O = 1,
> > as x^y is analogous to x<=y,
> > and 0<=0 is true.
> 
> It is often most convenient in combinatorics to have 0^0 = 1.  See
> "Concrete Mathematics", (Graham, Knuth, and Patashnik) section 5.1, page
> 162 in my first edition: "We must define x^0=1 for all x if the binomial
> theorem is to be valid when x=0, y=0, and/or x=-y.  The theorem is too
> important to be arbitrarily restricted!"

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