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I just read about how Boolean Algebras can have more than 2 elements (0, 1). And this shines some more light:

All finite boolean algebras have an even number of elements?

I am wondering though how this is possible, or what it looks like. I always was under the assumption that "boolean" was synonymous with true/false, but if there can be true/false/more/stuff/... then I wonder what that looks like.

Wikipedia shows this, but doesn't give any further explanation of how they came up with it.

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Hoping for an explanation of how to think about creating a "more than two element" boolean algebra, and what it would be used for.

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For every set $A$, the set $\mathcal{P}(A)$ of subsets of $A$ can be made a boolean algebra with the following operations:

Given $X, Y \in \mathcal{P}(A)$:

(1) $X \lor Y =_{def} X \cup Y$

(2) $X \land Y =_{def} X \cap Y$

(3) $\neg X =_{def} X^{c}$ - the complement of $X$ in $A$

(4) $0 =_{def} \emptyset$ and $1 =_{def} A$

Check for yourself that these definitions satisfy the axioms of a boolean algebra.

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  • $\begingroup$ In fact, you can even have infinite boolean algebas, such as the $\sigma$-algebras in Measure Theory. $\endgroup$ – Nuntractatuses Amável Mar 6 '18 at 19:50

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