1 3 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 c3 . ~ (p ∨ q) 7. (p & q) ∨ (p & r)
p ∴ p & (q ∨ r)
∴ q4. ~ (p ∨ q)
p
∴r
5. ~ (p & q)
q
∴~ p
6. ~ (p & q)
~q
∴p
8. (p ∨ q) & (p ∨ r)
∴ p & (q ∨ r)
9. p & q
∴ (p ∨ r) & (q ∨ r)
10. p ∨ q
∴ (p & r) ∨ (q & r)
Some Further Connectives
We have developed the logic of propositions using only three basic notions
corresponding (perhaps roughly) to the English words “and,” “or,” and
“not.” Now let us go back to the question of the two possible senses of the
word “or”: one exclusive and the other inclusive. Sometimes “or” seems to
rule out the possibility that both alternatives are true; at other times “or”
seems to allow this possibility. This is the difference between exclusive and
inclusive disjunction.
Suppose we use the symbol “∨” to stand for exclusive disjunction. This is
the same as the symbol for inclusive disjunction except that it is underlined.
(After this discussion, we will not use it again.) We could then give two truth
table definitions, one for each of these symbols:
Inclusive Exclusive
p q p ∨ q p ∨ q
T T T F
T F T T
F T T T
F F F FWe could also define this new connective in the following way:
(p ∨ q) = (by definition) ((p ∨ q) & ~(p & q))
It is not hard to see that the expression on the right side of this definition
captures the force of exclusive disjunction. Because we can always define
exclusive disjunction when we want it, there is no need to introduce it into
our system of basic notions.97364_ch06_ptg01_111-150.indd 132 15/11/13 10:15 AM
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