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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 14:58, 7 April 2009
, 14:58, 7 April 2009→Commentary Note 9.3
===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'' <math>x\!</math> is one that obeys the ''idempotent law'', that is, it satisfies the equation <math>xx = x.\!</math> Under most circumstances, it is usual to write this as <math>x^2 = x.\!</math>
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.
If the algebraic system in question falls under the additional laws that are necessary to carry out the requisite transformations, then <math>x^2 = x\!</math> is convertible into <math>x - x^2 = 0,\!</math> and this into <math>x(1 - x) = 0.\!</math>
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.
If the algebraic system in question happens to be a boolean algebra, then the equation <math>x(1 - x) = 0\!</math> says that <math>x \land \lnot x</math> 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".
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.
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.
===Commentary Note 9.4===
===Commentary Note 9.4===