[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 25

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In algebra, an "idempotent element" x is one that obeys the
"idempotent law", that is, it satisfies the equation xx = x.
Under most circumstances, it is usual to write this x^2 = x.

If the algebraic system in question falls under the additional laws
that are necessary to carry out the requisite transformations, then
x^2 = x is convertible into x - x^2 = 0, and this into x(1 - x) = 0.

If the algebraic system in question happens to be a boolean algebra,
then the equation x(1 - x) = 0 says that x & ~x is identically false,
in effect, a statement of the classical principle of non-contradiction.

We have already seen how Boole found rationales for the commutative law and
the idempotent law by contemplating the properties of "selective operations".

It is time to bring these threads together, which we can do by considering the
so-called "idempotent representation" of sets.  This will give us one of the
best ways to understand the significance that Boole attached to selective
operations.  It will also link up with the statements that Peirce makes
about his adicity-augmenting comma operation.

Jon Awbrey

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