De Morgan's Laws
Strictly speaking, DeMorgan's Law is the rule about the transfer of negation over "and" and "or" (inclusive) that says change "and" to "or" and conversely and change the signs of all the parts and wholes. That is
- ~(p & q) <=> ~p v ~q
- ~(p v q) <=> ~p & ~q
- and all the other combinations where one or the other parts has a negation, for example ~(p & ~q) <=> ~p v q.
(Students of digital logic will recall the above from their introductory course in the subject.)
In practice, however, the name is often used for the almost entirely analogous rule for moving negations across quantifiers, especially universal and particular ones. The rule then is analogous: change initial sign and sign before the component and change the quantifier from the one to the other:
- AxFx <=> ~Ex~Fx
- ExFx <=> ~Ax~Fx
- and all the variations with different initial placements of negations, e.g., Ax~Fx <=> ~ExFx.
The reason it works with the universal and particular quantifiers is that
- ExFx (to use the same notation) is equal to Fx1 v Fx2 v Fx3 v ... v Fxn for all possible values of n
- AxFx is equal to Fx1 & Fx2 & Fx3 & ... & Fxn for all possible values of n
(Provided that every item in the universe of discourse is assigned at least one xi. The intuition is correct, however, just generalized to the (possibly) infinite case.)
The extended use is somewhat complicated by the differences between quantifiers with and those without existential import. Moving negation changes one of these into the other as well as changing sign and quantity. However, Lojban sumti are importing unless specially marked (and what that mark is to be is open to some discussion), so this problem can be ignored in most practical cases. The peculiar standard quantifier O ("Some S are not P") has some suggested expressions, which relieve a gap in Lojban: or are leading possibilities.
Fractional quantifiers have also been dealt with., since it names a single complete individual (unusual in Lojban), is unaffected by the passage of . and n generally is treated as though n were n and the changes occur on the outer quantifier.
The question of the effect of "negations" other thanhas arisen and is being worked over - for at least.