# Difference between a theorem and a law

There are plenty of theorems out there as well as laws within mathematics. For example, in Boolean algebra:

Theorems

• Idempotent
• Involution
• Theorem of Complementarity

Laws

• Commutative
• Associative
• Distributive

There are countless other examples out there, but my real question is this: What makes a theorem a theorem and a law a law?

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In these examples, "laws" are really identities that are satisfied/imposed on the algebra a priori, while the theorems are propositions that are deduced from those identities and other axioms. –  Arturo Magidin Mar 3 '11 at 5:39
@ArturoMagidin Perhaps these are bad examples... this question arose as I was studying for my embedded circuits class... but I'm looking for a "in general" answer... if one exists. –  KronoS Mar 3 '11 at 5:40
@KronoS: It's nomenclature; what is the difference between "Theorem", "Proposition", "Lemma", and "Corollary"? Answer: the importance the author places on them. But generally speaking, "laws" are a priori restrictions/rules/identities, while "Theorems" are invariably "derived truths": conclusions that follow logically from whatever you are starting from. That is, "laws" are a priori, "theorems" are a posteriori. "Generally" speaking, anyway. –  Arturo Magidin Mar 3 '11 at 5:43
@Arturo: I think "law" is used much more generally, e.g. law of exponents, law of sines, parallelogram law, quadratic Reciprocity Law, Sylvester's law of inertia. "Law" is a bit old-fashioned, and tends to be used more frequently in applied math, e.g. physical laws. –  Bill Dubuque Mar 3 '11 at 5:55
@KronoS: As you see in comments, the distinction between “law” and other notions is not precise, unlike other things in mathematics. I even collected a list of 7 synonyms of the word “theorem”. It's going mad. :) The real distinction exists between a logical formula and a theorem. A theorem is a proven logical formula. –  beroal Mar 3 '11 at 17:16