# 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

Theorems are results proven from axioms, more specifically those of mathematical logic and the systems in question. Laws usually refer to axioms themselves, but can also refer to well-established and common formulas such as the law of sines and the law of cosines, which really are theorems.

In a particular context, propositions are the more trivial theorems, lemmas are intermediate results, while corollaries are results deduced easily from others. However, lemmas and corollaries may be major results on their own.

Note that a system may be given axioms in more ways than one. For example, we can use the least upper bound axiom to define the real numbers, or we can consider this axiom as a theorem if we were to construct the reals from the rationals using Dedekind cuts and prove it instead. The difference here lies in which axioms we choose to start with.

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I am not so sure that laws "refer to axioms themselves." Doesn't Bills comment give at east 5 examples otherwise? To add a couple more, how about the law of large numbers, or the law of cosines. –  Eric Naslund Mar 3 '11 at 6:13
Also, there is De Morgan's Laws. –  Eric Naslund Mar 3 '11 at 6:20
Have a very, merry Christmas, Jasper Joy...oops, Jasper Loy! –  amWhy Dec 25 '12 at 0:03
@Jasper Perhaps a Christmas gift? ;-) –  amWhy Dec 25 '12 at 0:38