1 4 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 c- (p ∨ q) & (p ∨ r)
~r
∴ ~q - (p ⊃ q) & (p ⊃ ~r)
q & r
∴ ~p - (p ∨ q) ⊃ p
∴ ~q
18. (p ∨ q) ⊃ (p & q)
∴ (p ⊃ q) & (q ⊃ p)
19. (p & q) ⊃ (p n q)
∴ (p ⊃ q) n (q ⊃ p)
20. r
∴ (p ⊃ q) ∨ (q ⊃ p)
Logical Language and Everyday Language
Early in this chapter we started out by talking about such common words as
“and” and “or,” and then we slipped over to talking about conjunction and
disjunction. The transition was a bit sneaky, but intentional. To understand
what is going on here, we can ask how closely these logical notions we have
defined match their everyday counterparts. We will start with conjunction,
and then come back to the more difficult question of conditionals.
At first sight, the match between conjunction as we have defined it and
the everyday use of the word “and” may seem fairly bad. To begin with, in
everyday discourse, we do not go about conjoining random bits of informa-
tion. We do not say, for example, “Two plus two equals four and Chile is in
South America.” We already know why we do not say such things, for unless
the context is quite extraordinary, this is bound to violate the conversational
rule of Relevance. But if we are interested in validity, the rule of Relevance—
like all other conversational (or pragmatic) rules—is simply beside the point.
When dealing with validity, we are interested in only one question: If the
premises of an argument are true, must the conclusion be true as well? Con-
versational rules, as we saw in Chapter 2, do not affect truth.
The truth-functional notion of conjunction is also insensitive to another
important feature of our everyday discourse: By reducing all conjunctions to
their bare truth-functional content, the truth-functional notion often misses
the argumentative point of a conjunction. As we saw in Chapter 3, each of
the following remarks has a different force in the context of an argument:
The ring is beautiful, but expensive.
The ring is expensive, but beautiful.
These two remarks point in opposite directions in the context of an actual
argument, but from a purely truth-functional point of view, we treat them
as equivalent. We translate the first sentence as “B & E” and the second as97364_ch06_ptg01_111-150.indd 142 15/11/13 10:15 AM
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