The proposition p is called the hypothesis or antecedent. The proposition q is called the conclusion or consequence.

Transcription

1 The Conditional (IMPLIES) Operator The conditional operation is written p q. The proposition p is called the hypothesis or antecedent. The proposition q is called the conclusion or consequence. The Conditional (IMPLIES) Operator Understanding the conditional operation p q is essential for logical reasoning. The conditional operation p q can be read as p implies q or if p, then q. The semantics of the conditional statement p q is: If p is True, then q is True. (True True) = True In particular, when p is True and q is False the conditional statement is False. (True False) = False What does False Imply? The semantics of the conditional statement p q does not say anything about the truth or falsity of q when p is False. (False q) =? Some value must be assigned to False q. The correct assignment is (False q) = True. Let s see why.there are 4 cases to consider. 1

2 Should False IMPLIES Q be False? 1. What if, when p is False,the conditional operation is False. (False False) = False (False True) = False But then the input-output behavior of the conditional is identical with AND. p q p q? p q This is the not semantics we want for the conditional statement. Should False IMPLIES False be True? 2. What if, when p isfalse, the conditional operation is True when q is False, and False when q is True. (False False) = True (False True) = False But then the input-output behavior of the conditional is identical with EQUIV. p q p q? p q This is the not semantics we want for the conditional statement. 2

3 Should False IMPLIES True be True? 3. What if, when p is False, the conditional operation is False when q is False, and True when q is True. (False False) = False (False True) = True But then the input-output behavior of the conditional is identical with IDENTITY(Q). p q p q? q This is the not semantics we want for the conditional statement. Should False IMPLIES Anything be True? 4. What if, when p is False, the conditional operation is True (independent of the value of q). (False False) = True (False True) = True This input-output behavior does not conflict with any other operator. p q p q This does not contradict the semantics we want for the conditional statement. 3

4 The Conditional (IMPLIES) Operator The four potential ways in which truth values can be assigned to the conditional operation p q when p = False are summarized in the table below. Input Functions P IMPLIES Q Q EQUIV AND p q p q q p q p q If p is False, q can have any value, and p q will be True. A Natural Language Example To better understand the conditional operator, consider the conditional statement (1) If Ed scored 90+ on the final, then Ed earned an A. Let p name the proposition Ed scored 90+ on the final. Let q name the proposition Ed earned an A. Statement (1) above can be written as the conditional p q Consider a Natural Language Example 1. If Ed scored 90+ and Ed s grade is A, then statement (1) is True. 2. If Ed scored 90+ and Ed s grade is less than an A, then statement (1) is False. 3. If Ed scored less than 90 and Ed s grade is A, then statement (1) not patently False, so it must be True. 4. If Ed scored less than 90 and Ed s grade is below an A, then statement (1) is not patently False, so it must be True. 4

5 IMPLIES Written in Terms of NOT and OR The conditional statement can be written using the NOT and OR operators. (p q) ( p q) This can be seen by examining the truth tables for the two expressions. Input Expressions P IMPLIES Q NOT (P OR Q) p q (p q) ( p q) IMPLIES Written as a CONTRAPOSITIVE The conditional statement can be written as a contrapositive. (p q) ( q p) This can be seen by examining the truth tables for the two expressions. Input Operations P IMPLIES Q NOT Q IMPLIES NOT P p q (p q) ( q p)

6 Direct Proofs (Modus Ponens) The conditional statement is the basis for direct proofs. If p and p q are both True, then q is True. Input Modus Ponens p q (p (p q)) q st 3 rd 2 nd 5 th 4 th Indirect Proofs (Modus Tollens) The contrapositive of a conditional statement is the basis for indirect proofs. If p and q p are both True, then q is True. Input Modus Tollens p q (p ( q p)) q st 3 rd 2 nd 5 th 4 th 6