1 B The Conditional What is a Conditional

§ 1. B The Conditional

What is a Conditional Proposition? Sometimes a proposition is composed of two or more propositions that are contingent on one another. For example, in the proposition: “If the Nationals win the pennant then Joan will be a millionaire. ” is such a proposition. One proposition “Joan will be a millionaire. ” is a contingency to the proposition “The Nationals win the pennant. ” We note that the proposition says that it is sufficient that the Nationals win the pennant to ensure that Joan will be a millionaire. It does not say that this is the only way Joan will become a millionaire. So it is not necessary for the Nationals to win the pennant for Joan to be a millionaire.

What is a Conditional Proposition? (Continued) A proposition of the form “If p then q”, where p and q are propositions, is called a conditional proposition. This is written in symbolic logic as p q. The proposition p is called the antecedent and the proposition q is called the consequent. We see that a conditional proposition says that p is sufficient for q. It does not say that p is necessary for q, but it does say that q is necessary for p. Since p q is a proposition, it has its own truth value taken as a whole. Let’s try to find out what the truth value is. p q T T T F F F F T

What is a Conditional Proposition? (Continued) Let’s look at the truth table line by line to see why the results are as shown. Ø Line 1: We have both p and q true. So having p did indeed result in q. Thus p q is true. Ø Line 2: We have p false and q true. The result p q is still true because q still occurred independent of p. The statement p was not necessary for q. Ø Line 3: We have p true and q false. Then p q must be false since p happened but q didn’t happen. Thus p was not sufficient for q as the conditional statement implied. Ø Line 4. We have both p and q false. Then p q is true since the conditional statement doesn’t say what the truth of q is when p is false. The conditional statement is not contradicted.

What is a Conditional Proposition? (Continued) Observe that p q is true whenever q is true or when p is false. We can look at the truth table for ~p ˅ q. p q ~p ˅ q T T T F F F F T Note that the truth table for p q and ~p ˅ q have the same truth values for the same values of p and q. So p q is equivalent to ~p ˅ q.

Negation of a Conditional A conditional is a proposition. So it has a negation. Examples: Ø Proposition: “All cats have green eyes. ” This is really a conditional that can be restated: “If it is a cat, then it has green eyes. ” Negation: “It is a cat and it does not have green eyes. ” Ø Proposition: “If there is a rainstorm then there will be a flood. ” Negation: “There is a rainstorm and there is no flood. ” Ø Proposition: “No professor grades unfairly. ” This can be restated: “If the person is a professor then he/she grades fairly. ” Negation: “The person is a professor and he/she grades unfairly. ”

Negation of a Conditional (Continued) We can use De Morgan’s Laws coupled with the equivalence for conditional propositions to find a logical equivalent to the negation of a conditional. We have the conditional p q equivalent to ~p ˅ q. The negation of the conditional ~(p q) is equivalent to ~(~p ˅ q). We simplify ~(~p ˅ q) using De Morgan’s Laws: ~(~p ˅ q) = ~(~p) ˄ ~q. But ~(~p) is just p. So ~(~p ˅ q) = p ˄ ~q. Putting everything together we have: ~(p q) = p ˄ ~q.