The truth value assigned to P AND Q also ref lects common understand-
ing of the word and, which requires both P and Q to be true in order for
the statement P AND Q to be true. The last two columns require a little
more explanation.
The word or is used in two different senses in the English language: the
exclusive sense and the inclusive sense. When assigning truth values to
the statement P OR Q, doing so for the “exclusive or” would result in the
statement being true precisely when exactly one of the two statements P
and Q were true, whereas doing so for the “inclusive or” would result in
the statement being true if at least one of the two statements P and Q
were true. The example that I give to Math for Liberal Arts students to
distinguish between the two occurs when your waiter or waitress asks
you if you would like coffee or dessert after the meal. Your server is going
with the inclusive or, because you will never hear a server say, “Sorry, you
can only have one or the other,” when you say, “I’d like a cup of coffee and
a dish of chocolate ice cream.” Propositional logic has adopted the “inclu-
sive or,” and the table above ref lects this.
Finally, the truth values assigned to the statement IF P THEN Q are
motivated by the desire to distinguish obviously false arguments: those
that start from a true hypothesis and end with a false conclusion. This
has a tendency to cause some confusion, because both the following com-
pound statements are defined to be true.
If London is the largest city in England, then the sun rises in the east.
If Yuba City is the largest city in California, then 2 2 4.
The objection students make to the first statement being true is that
there’s no connecting logical argument, and the objection to the second is
that it’s impossible to reach the arithmetic conclusion just because the
hypothesis is false. IF P THEN Q does not mean (in propositional logic)
that there is a logical argument starting from P and ending with Q. One
of the original goals of propositional logic was to distinguish obviously
fallacious arguments from all others; there’s something clearly wrong
with an argument that goes 2 2 4, therefore the sun rises in the west. It’s
tempting to think of IF P THEN Q as an implication (which means there
is some underlying connecting argument), but it’s not the way proposi-
tional logic regards it.
Propositional logic incorporates a method of computing the true/false
value of a compound statement, just as arithmetic can compute a value for
xyz when numerical values of x, y, and z are given. For instance, if P and
Q are TRUE and R is FALSE, the compound statement (P AND NOT Q)
OR R is evaluated according to the above table in the following fashion.
118 How Math Explains the World