More on Conditionals Section 1.4

Introduction Given a conditional of the form “if p then q” p  q, we can then form 3 other forms of the conditional. “if q then p”, q  p is called the converse. “if not p then not q”, ~p  ~q, is called the inverse. “ if not q then not p”, ~q  ~p, is called the contrapositive.

Converse Example let us have 2 statements p: You play poker. q: You win money. The conditional p  q is “if you play poker, then you win money.” The converse of the conditional q  p is “if you win money then you play poker.” The converse is formed by interchanging the premise with the conclusion.

Inverse The inverse of the conditional p  q is the negation of the premise and the negation of the conclusion and keeping the same order. The inverse is ~p  ~q. Using the previous p and q. The inverse of the conditional is “if you do not play poker then you do not win money.”

Contrapositive When you negate the premise and conclusion of the conditional, and interchange the premise an conclusion, then you form the contrapositive of the conditional. The contrapositive of p  q is ~q  ~p. Using the previous p and q the contrapositive is “if you do not win money, the you do not play poker”.

Equivalent Conditionals The reason we look at these different forms of the conditional is that there are 2 pairs of equivalent statements. The conditional is equivalent to the contrapositive.( p  q ≡ ~q  ~p) The inverse is equivalent to the converse. ( q  p ≡ ~p  ~q)

Truth Tables conditionalcontrapositiveconverseInverse pq~p~qpqpq~q  ~pqpqp~p  ~q TTFFTTTT TFFTFFTT FTTFTTFF FFTTTTTT

The “only if” connective “You have a picnic only if today is a holiday.” This is translated as “if you have a picnic, the today is holiday.” The statement that follows the “only if” is the conclusion of the conditional. p only if q is equivalent to p  q or if p then q.

The if connective Be careful when reading conditionals that have the connective word if in the middle of the sentence. Example: Let’s look at a conventional conditional. If the Eagles beat the Falcons, then Mr. D will go to Jacksonville. This is of the form If p, then q. Where p is the simple statement: The Eagles beat the Falcons. q is the simple statement: Mr. D will go to Jacksonville. In symbols this is p  q. Here’s the tricky part, I can rewrite the original conditional with the word if in the middle of the sentence. Mr. D will go to Jacksonville, if the Eagles beat the Falcons. Notice that I reverse the p and q statements. In English the sentence is read q if p. This has the same meaning as If p, then q.

Biconditional p  q The biconditional is a two way conditional or two way implication. The symbol p  q is the same as saying p  q and q  p. We use the key words “if and only if”. “A citizen is eligible to vote if and only if the citizen is at least 18 years of age.” Which is the same as saying; “if the citizen is eligible to vote then the citizen is at least 18 years of age” and “the citizen is at least 18 years of age then the citizen is eligible to vote”.