Negation Law

- DeMorgan's law: the intersection of any class of sets can be expressed as the complement of the union of the complements of those sets.
http://www.britannica.com/eb/topic-153819/De-Morgan-laws

- Not (A or not B) sounds equivalent to not A or B. But of course the two aren't equivalent; DeMorgan's Law says not (A or not B) is equivalent to not A and B.
http://volokh.com/posts/1205277114.shtml

- Proposition 12 Let a, b ∈ R. If a · b = 0 then either a = 0 or b = 0 (or both).
Proof. If a 6= 0 and b 6= 0 then
0 = (1/a · 1/b) · 0 by Proposition 11
= 0· (1/a · 1/b) by M1
= (a · b) · (1/a · 1/b) by hypothesis
= (b · a) · (1/a · 1/b) by M1
= ((b · a) · 1/a) · 1/b by M2
= (b · (a · 1/a)) · 1/b by M2
= (b · 1) · 1/b by M4
= b · 1/b by M3
= 1 by M4
This contradicts Z and hence a · b = 0 contradicts a and b both being non-zero. By De Morgan’s laws it follows that at least one of a and b is zero.
www.maths.ox.ac.uk/filemanager/active?fid=1075


- DeMorgan Law is quite intuitive once one is able to make the distinction between "or" and "exclusive or" (a distinction that doesn't always exist in common parlance).
- when A is a chain - Many forms of fuzzy logic have a truth-value algebra obtained by equipping the closed unit interval [0, 1] with basic operations of various types; often a De Morgan negation is present. In addition, there will typically be some binary operations meant to model some form of conjunction and disjunction. If these are related via the negation by De Morgan’s laws, then the algebra is referred to as a De Morgan system.
http://www2.maths.ox.ac.uk/~hap/GP3dqalg.pdf



I think not (A or not B) sounds equivalent to not A or not not B, but that's a pointless quibble. My real point is, I think there's a bit of a problem here in the difference between "not" as a logical operator and "not" as an element of how people think. One is strictly "not" means "everything other than", whereas the other allows for "not" to mean "everything other than" or to mean "the opposite of". Likewise, "or" in most people's minds can have an exclusive element that it lacks in formal logic: not so much "A union B" as "A, alternatively B".