NEGATION APPLIED TO A CONJUNCTION

Let P and Q be sentences which are true or false, but neither of them is both. "Not(P and Q)" means the same thing as "(not(P)) or (not(Q))".

The Negation of a Conjunction is equivalent to an Alternation of the Negations of the conjuncts
P Q P and Q not(P and Q) not(P) not(Q) (not(P)) or (not(Q))
T T T F F F F
T F F T F T T
F T F T T F T
F F F T T T T


Some people understand this principle as follows. They know that "P and Q" is true only when both P and Q are true. For it to be false, P is false or Q is false (possibly both are false, which fits our understanding of "or"). P being false makes "not(P)" true; Q being false makes "not(Q)" true. So, "not(P)" is true or "not(Q)" is true (possibly both are true). This is exactly what is meant by "(not(P)) or (not(Q))".

The truth table to the right is a different approach to the same principle. Note that columns 4 and 7 have identical truth values.

This is one of the famous DeMorgan transformations developed by Augustus Demorgan.