p q p q p q
T T T F
T F T T
F T T T
F F F F
We 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.
5. ~ (p & q) q
~ p
6. ~ (p & q) ~q
p
7. (p & q) (p & r)
p & (q r)
Construct a truth table analysis of the expression on the right side of the pre- ceding definition, and compare it with the truth table definition of exclusive disjunction.
8. (p q) & (p r)
p & (q r)
9. p & q
(p r) & (q r)
10. p q
(p & r) (q & r)
Actually, in analyzing arguments we have been defining new logical con- nectives without thinking about it much. For example, “not both p and q” was symbolized as “~(p & q).” “Neither p nor q” was symbolized as “~(p q).” Let us look more closely at the example “~(p q).” Perhaps we should have sym- bolized it as “~p & ~q.” In fact, we could have used this symbolization, because the two expressions amount to the same thing. Again, this may be obvious, but we can prove it by using a truth table in yet another way. Compare the truth table analysis of these two expressions:
p q ˜p ˜q ˜p & ˜q (p q) ˜(p q)
T T F F F T F
T F F T F T F
F T T F F T F
F F T T T F T
Under “~p & ~q” we find the column (FFFT), and we find the same sequence under “~(p q).” This shows that, for every possible substitution we make, these two expressions will yield propositions with the same truth value. We will say that these propositional forms are truth-functionally equivalent. The above table also shows that the expressions “~q” and “~p & ~q” are not truth-functionally equivalent, because the columns underneath these two expressions differ in the second row, so some substitutions into these expres- sions will not yield propositions with the same truth value.