Complete the truth table for p ∧ q. Note that we have to use four lines in this table to include all possible combinations of truth values of p and q.

 

 

The connective or is used to form a compound statement known as a disjunction. In common English the word or can have two meanings. In the sentence We are going to paint our house yellow or green. the intended meaning is yellow or green, but not both. This is known as the exclusive meaning of or. On the other hand, in the sentence Do you want cake or ice cream for dessert? the intended meaning may include the possibility of having both. This inclusive meaning is the only way the word or is used in logic. Thus, if we denote the disjunction p or q by p ∨ q, we have the following truth table:

is called an implication or a conditional statement. The if-statement p in the implication is called the antecedent and the then-statement q is called the consequent. To decide on an appropriate truth table for implication, let us consider the following sentence: If it stops raining by Saturday, then I will go to the football game. If a friend made a statement like this, under what circumstances could you call him a liar? Certainly, if the rain stops and he doesn’t go, then he did not tell the truth. But what if the rain doesn’t stop? He hasn't said what he will do then, so whether he goes or not, either is all right.

Although it might be argued that other interpretations make equally good sense, mathematicians have agreed that an implication will be called false only when the antecedent is true and the consequent is false. If we denote the implication “if p, then q” by p ⇒ q, we obtain the following table:

It is important to recognize that in mathematical writing the conditional statement can be disguised in several equivalent forms. Thus the following expressions all mean exactly the same thing: