Page 58 - HOW TO PROVE IT: A Structured Approach, Second Edition

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44 Sentential Logic
reasoning in the second argument? Recall that the argument went like this:
If today is Sunday, then I don’t have to go to work today.
Today is Sunday.
Therefore, I don’t have to go to work today.
What makes this reasoning valid?
It appears that the crucial words here are if and then, which occur in the
first premise. We therefore introduce a new logical connective, →, and write
P → Q to represent the statement “If P then Q” This statement is sometimes
called a conditional statement, with P as its antecedent and Q as its consequent.
If we let P stand for the statement “Today is Sunday” and Q for the statement “I
don’t have to go to work today,” then the logical form of the argument would be
P → Q
P
∴ Q

Our analysis of the new connective → should lead to the conclusion that this
argument is valid.

Example 1.5.1. Analyze the logical forms of the following statements:
1. If it’s raining and I don’t have my umbrella, then I’ll get wet.
2. If Mary did her homework, then the teacher won’t collect it, and if she didn’t,
then he’ll ask her to do it on the board.
Solutions
1. Let R stand for the statement “It’s raining,” U for “I have my umbrella,” and
W for “I’ll get wet.” Then statement 1 would be represented by the formula
(R ∧¬U) → W.
2. Let H stand for “Mary did her homework,” C for “The teacher will collect
it,” and B for “The teacher will ask Mary to do the homework on the board.”
Then the given statement means (H →¬C) ∧ (¬H → B).
To analyze arguments containing the connective → we must work out the
truth table for the formula P → Q. Because P → Q is supposed to mean that
if P is true then Q is also true, we certainly want to say that if P is true and
Q is false then P → Q is false. If P is true and Q is also true, then it seems
reasonable to say that P → Q is true. This gives us the last two lines of the
truth table in Figure 1. The remaining two lines of the truth table are harder
to fill in, although some people might say that if P and Q are both false then