The Language of Argument

1 3 4

C 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

truth-functionally equivalent, then it does not matter which one we use in

testing for validity. Of course, some translations will seem more natural than

others. For example, “p ∨ q” is truth-functionally equivalent to

~((~p & ~p) & (~q ∨ ~q))

Despite this equivalence, the first form of expression is obviously more

natural than the second when translating sentences, such as “It is either

cloudy or sunny.”

Use truth tables to test which of the following propositional forms are truth-

functionally equivalent to each other:

1. ~(p ∨ q)


2. ~(~p ∨ ~q)


3. ~p & ~q


4. p & q

Exercise XVIII

Use truth tables to determine whether the expressions in each of the following

pairs are truth-functionally equivalent:

1. “p” and “p & p”


2. “p” and “p ∨ p”


3. “p ∨ ~p” and “~(p & ~p)”


4. “p” and “p & (q ∨ ~q)”


5. “p” and “p & (q & ~q)”


6. “p” and “p ∨ (q & ~q)”


7. “p & (q ∨ r)” and “p ∨ (q & r)”


8. “p & (q & r)” and “(p & q) & r”
   9. “~(p ∨ q)” and “~p ∨ q”
   10. “~(p ∨ q)” and “~p & ~q”
   11. “(p ∨ q)” and “p & ~~q”
   12. “~(p & q)” and “~p ∨ q”
   13. “(p & q)” and “p ∨ ~~q”
   14. “~~p n ~~q” and “~(~p & ~q)”
   15. “~~p & ~~q” and “~(~p ∨ ~q)”
   16. “p & q” and “p & q”

Exercise XIX

Conditionals

So far in this chapter we have seen that by using conjunction, disjunction,

and negation, it is possible to construct compound propositions out of simple

propositions. A distinctive feature of compound propositions constructed in

these three ways is that the truth of the compound proposition is always

a function of the truth of its component propositions. Thus, these three

notions allow us to construct truth-functionally compound propositions.

Some arguments depend for their validity simply on these truth-functional

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