Conditionals and Biconditionals

1. Conditionals and Biconditionals PHIL012 03/26/2001

2. Outline • Announcements • Exams will be handed back Wednesday • Next exam April 11 • 4.1 The Material Conditional → • 4.2 The Biconditional: ↔

3. 4.1 The Material Conditional → • In English, we often have occasion to express conditional statements such as, “If we’re out of rice, we’ll go to the store.” “If it rains, the game will be postponed.” “If it is Sunday, the stores are all closed.” • FOL expresses conditionals in this way: P → Q

4. The Material Conditional → • The Material Conditional (P→Q) consists of two terms, P and Q. • The first term, P, is called the antecedent. • The second term, Q, is called the consequent. • The definition of → says that for P→Q to be true, the consequent must be true whenever the antecedent is true.

5. Truth Table for → • The definition of → in FOL is: P Q P →Q TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE

6. P→Q and ¬P v Q • Note that the truth tables for P→Q and ¬P v Q are identical P Q P → Q ¬ P v Q T T T T T F T T T T F T F F F T F F F T F T T T F T T F F F T F T F T F • Thus, the expressions are logically equivalent.

7. Translation Issues • Note, however, that the P→Q is not quite the same as the English expression “If P then Q.” • Similar to English, P→Q is true whenever P is true and Q is also true and is false whenever P is true and Q is false. • So, “If it is Sunday, the stores are all closed.” will be false just in case it is Sunday and some stores are open.

8. Translation Issues • However, unlike English, the definition of → says that the expression P→Q is also true whenever the antecedent is false: P Q P →Q TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE TRUE FALSE FALSE TRUE • So, the statements “If we’re out of rice, we’ll go to the store.” “If it rains, the game will be postponed.” “If it is Sunday, the stores are all closed.” are all true in FOL when we have plenty of rice, it isn’t raining, and it isn’t Sunday.

9. Translation Issues • So, “If P then Q” is similar but not identical to P→Q. • Other similar English expressions are, • P only if Q • Q provided P • Q when P

10. If today is Monday, we will have class. If Philadelphia is the capital of PA, Al Gore is President. Translation Practice TRUE • If today is Tuesday, we will have class. TRUE • If Russell wins best actor, Gladiator wins best picture. TRUE • If Julia doesn’t win best actress, Erin Brockovich wins best picture. TRUE TRUE

11. The biconditional ↔ • The ordinary English definition of P↔Q is P if and only if Q (P IFF Q). • In other words, P↔Q is true whenever P and Q have the same truth values: P Q P↔Q TRUE TRUE TRUE TRUE FALSE FALSE FALSE TRUE FALSE FALSE FALSE TRUE

12. ↔ and  • Since P ↔ Q is true just in case P and Q are logically equivalent, it might be tempting to confuse ↔ and . • However, ↔ is a logical connective in FOL (like ^, v, ¬, and →) • Whereas  is a shorthand symbol for talking about two statements in FOL.

13. The biconditional ↔ • It is also worth noting that P↔Q is logically equivalent to (P→Q) ^ (Q→P) • In other words, P↔Q  (P→Q) ^ (Q→P) • This becomes important in 4.3.

14. Questions

15. Assignment • For Monday, do Ch 4: probs 1-22 (except 17 and 21). • For Wednesday, read 4.3.