# What's the difference between Principle of bivalence and law of excluded middle?

My understanding is excluded middle stay in the syntactic level $P\,\vee \,\neg P$ while Principle of bivalence interpret $P$ and $\neg P$ to some certain meaning.
But I'm not sure about it.

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Probably the easiest way to understand this is via boolean valued models. Let $\mathbb{B}$ be a boolean algebra. Then a boolean valued model is a function $f$ from proposition variables to $\mathbb{B}$. (That is, if $P_1,P_2,\ldots$ are proposition variables, then $f(P_1), f(P_2),\ldots$ are elements of $\mathbb{B}$.) We can extend $f$ to every formula by setting $f(\phi \wedge \psi) = f(\phi) \wedge f(\psi)$ and similarly for the other logical symbols.
We can call the elements of $\mathbb{B}$ "truth values" and say that $1 \in \mathbb{B}$ is "true". Boolean valued models then have the nice property that whenever $\phi$ is a theorem of propositional calculus, $f(\phi) = 1$. That is, every theorem of the propositional calculus is "true" in every boolean valued model. We call this soundness.
Excluded middle is the formula $P \vee \neg P$ of the propositional calculus. It is an axiom of the propositional calculus and as such has to hold in every boolean valued model by soundness. (Although there are more general models where excluded middle does not always hold).
The most basic (non trivial) boolean algebra has only two elements, $0$ and $1$. This is sometimes referred to as $2$. Boolean valued models built using this two element boolean algebra have only two truth values, and hence can be referred to as bivalent. However, there are many other boolean algebras with more than two elements. These give non bivalent models where excluded middle holds.