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Revision as of 00:45, 21 October 2007
, 00:45, 21 October 2007→Commentary Note 9.3: markup
===Commentary Note 9.3===
===Commentary Note 9.3===
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''<sup>2</sup> = ''x''.
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
If the algebraic system in question falls under the additional laws that are necessary to carry out the requisite transformations, then ''x''<sup>2</sup> = ''x'' is convertible into ''x'' – ''x''<sup>2</sup> = 0, and this into ''x''(1 – ''x'') = 0.
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,
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.
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
We have already seen how Boole found rationales for the commutative law and the idempotent law by contemplating the properties of "selective operations".
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
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.
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.
===Commentary Note 9.4===
===Commentary Note 9.4===
