The Language of Argument

1 3 3

B a s i c P r o p o s i t i o n a l C o n n e c t i v e s

Construct a truth table analysis of the expression on the right side of the

preceding definition, and compare it with the truth table definition of exclusive

disjunction.

Exercise XVI

Use truth tables to test the following argument forms for validity:

1. p
   ∴ p ∨ q


2. p ∨ q
   p
   ∴ ~q


3. p & q
   ∴ ~(p ∨ q)
   4. ~(p & q)
   ∴ p ∨ q
   5. p ∨ q
   ∴ p ∨ q


4. p ∨ q
   ∴ p ∨ q

Exercise XVII

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

symbolized 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.

Given the notion of truth-functional equivalence, the problem of more

than one translation can often be solved. If two translations of a sentence are

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