De Morgan’s laws
De Morgan’s laws
Are the two dually related theorems of the propositional calculus,
~[p ? q] = [~p ~q],
~[pq] = [~p v ~q],
or the two corresponding dually related theorems of the algebra of classes,
-(a ? b) = -a n -b,
-(a n b) = -a ? -b.
In the propositional calculus these laws (together with the law of double negation) make it possible to define conjunction in terms of negation and (inclusive) disjunction, or, alternatively, disjunction in terms of negation and conjunction. Similarly in the algebra of classes logical product may be defined in terms of logical sum and complementation, or logical sum in terms of logical product and complementation.
As pointed out by Lukasiewicz, these laws of the propositional calculus were known already (in verbal form) to Ockham. The attachment of De Morgan’s name to the corresponding laws of the algebra of classes appears to be historically more correct.
Sometimes referred to as generalizations or analogues of De Morgan’s laws are the two dually related theorems of the functional calculus of first order,
~(Ex)F(x) = (x)~F(x),
~(x)F(x) = (Ex)~F(x),
and similar theorems in higher functional calculi. These make possible the definition of the existential quantifier in terms of the universal quantifier (or inversely). — A.C.