This new column shows the effects of not on P, when P is true or false. So, when P is true the result of negating P will make that proposition false, and when P is false, negation will make P true. This truth table defines the "meaning" of not.

Consider the more complicated case of the conditional, which unlike negation involves two propositions (P, Q): if P then Q. On their own P and Q can each be true or false; when they are combined there are four possible states of affairs (see Table 16.1): both P and Q can be true, P can be true when Q is false and vice versa, and both can be false.

Now consider what happens when if... then is applied to these propositions. First, when P and Q are true then clearly if P then Q is true. If we

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