Law of Excluded Middle:
In logic, the law of excluded middle (or the principle of excluded middle) is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is. The principle should not be confused with the principle of bivalence, which states that every proposition is either true or false, and has only a semantical formulation.
Principle of Bivalence:
In logic, the semantic principle (or law) of bivalence states that every declarative sentence expressing a proposition (of a theory under inspection) has exactly one truth value, either true or false. A logic satisfying this principle is called a two-valued logic or bivalent logic. In formal logic, the principle of bivalence becomes a property that a semantics may or may not possess. It is not the same as the law of excluded middle, however, and a semantics may satisfy that law without being bivalent.
The principle of bivalence is related to the law of excluded middle though the latter is a syntactic expression of the language of a logic of the form "P ∨ ¬P".The difference between the principle and the law is important because there are logics which validate the law but which do not validate the principle.
I'm not quite sure I get the difference. It seems that 'excluded middle' is a syntatic problem and 'bivalence' would be a semantic one. Is this correct? Also, it seem that in the realm of bivalence, stating that "P" is false, doesn't necessarily mean "non-P" is true, which would be the case with the principle of the excluded middle. Is this correct?
I don't understand precisely in which situations one or the other principle are at play, it seems that they may appear together, but not necessarily. Can someone give me examples and help me clarify the differences?