1 2 2C H A P T E R 6 ■ P r o p o s i t i o n a l L o g i cDisjunction
Just as we can form a conjunction of two propositions by using the connective
“and,” we can form a disjunction of two propositions by using the connective
“or,” as in the following compound sentence:
John will win or Harry will win.
Again, it is easy to see that the truth of this whole compound proposition de-
pends on the truth of the component propositions. If they are both false, then the
compound proposition is false. If just one of them is true, then the compound
proposition is true. But suppose they are both true. What shall we say then?
Sometimes when we say “either-or,” we seem to rule out the possibility of
both. When a waiter approaches your table and tells you, “Tonight’s dinner
will be chicken or steak,” this suggests that you cannot have both. In other
cases, however, it does not seem that the possibility of both is ruled out—for
example, when we say to someone, “If you want to see tall mountains, go to
California or Colorado.”
One way to deal with this problem is to say that the English word “or” has
two meanings: one exclusive, which rules out both, and one inclusive, which does
not rule out both. Another solution is to claim that the English word “or” always
has the inclusive sense, but utterances with “or” sometimes conversationally
imply the exclusion of both because of special features of certain contexts. It
is, for example, our familiarity with common restaurant practices that leads us
to infer that we cannot have both when the waiter says, “Tonight’s dinner will
be chicken or steak.” If we may have both, then the waiter’s utterance would
not be as informative as is required for the purpose of revealing our options,
so it would violate Grice’s conversational rule of Quantity (as discussed in
Chapter 2). That explains why the waiter’s utterance seems to exclude both.
Because such explanations are plausible, and because it is simpler as
well as traditional to develop propositional logic with the inclusive sense
of “or,” we will adopt that inclusive sense. Where necessary, we will define
the exclusive sense using the inclusive sense as a starting point. Logicians
symbolize disjunctions using the connective “∨” (called a wedge). The truth
table for this connective has the following form:
p q p ∨ q
T T T
T F T
F T T
F F FWe will look at some arguments involving this connective in a moment.Negation
With conjunction and disjunction, we begin with two propositions and con-
struct a new proposition from them. There is another way in which we can97364_ch06_ptg01_111-150.indd 122 15/11/13 10:15 AM
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