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

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 x^2 = x.

If the algebraic system in question falls under the additional laws that are necessary to carry out the required 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 \land \lnot 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.

This entry was posted in Boole, Boolean Algebra, Boolean Functions, Logic, Logic of Relatives, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , , . Bookmark the permalink.

2 Responses to Peirce’s 1870 “Logic Of Relatives” • Comment 9.3

  1. Guttering Surrey says:

    I visited several websites however the audio quality for audio songs current at this web page is in fact wonderful.

  2. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.