Example of a 4+ boolean algebra

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

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.

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