What Are the Mysteries of Boolean Algebra Beyond Binary Values?

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http://img710.imageshack.us/img710/2314/booleanalgebra.jpg

AFAIK logic is all about "T"/"F" or 0/1, and boolean algebra is all about logical manipulation.
But there seems to be something wrong since there is a boolean algebra with more that 2 objects in it`s set. So, can I have some clarification?

 

[HallsofIvy]

I'm not sure what objects you are talking about. Boolean algebra can involve an infinite number of "objects" that all have a value of 0 or 1.

 

[TMM]

Interpreting the 0 or 1 as a value for each member is not the best way to visualize them, in my opinion. There is an elegant (and simple!) theorem called the Stone representation theorem that says any boolean algebra is isomorphic (as a ring) to an algebra of sets, specifically some subset of a power set containing the empty set (0) and the set itself (1). Joins and meets become unions and intersections.

 

[Tac-Tics]

It's not totally clear what your question is.

In boolean algebra, you have a system where the values of variables range over B instead of over R.

We could use the word "proposition" instead of "variable", too. Instead of "x" meaning "the length of a piece of string" or "the age of my dog Scrappy", like you have in standard algebra, in boolean algebra, x might represent "it's raining outside" or "my dog Scrappy ate my homework".

Just as in standard algebra, we have an unlimited set of variables to work with. But all variables, when evaluated, must be equal to either true or false.