Peirce's 1870 “Logic Of Relatives ” • Comment 9.3

Post   : Peirce's 1870 “Logic Of Relatives” • Comment 9.3
http://inquiryintoinquiry.com/2014/02/26/peirces-1870-logic-of-relatives-%e2%80%a2-comment-9-3/
Posted : February 26, 2014 at 10:30 pm
Author : Jon Awbrey

Peircers,

An “idempotent element” x in an algebraic system is one that obeys the idempotent law, that is, it 
satisfies the equation xx = x.  Under most circumstances it is usual to write this as x² = x.

If the algebraic system in question falls under the additional laws that are necessary to carry out 
the required transformations then x² = x is convertible into x - x² = 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 
regarding his dimension-augmenting comma operation.

Regards,

Jon

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