Truth and Logical Truth
105
In Example 2, if I is the interpretation with I(p) = T and I(q) = I(r) = F, then
(-p V q) -). (r -- p) is true in I.
Example 3. Let
0 = ((-p V q) -- (r -+ p))
Find I (0b) for all interpretations 1.
Solution. Three proposition letters-p, q, and r-are in the formula. Hence, the truth of
the formula depends only on I(p), I(q), and I(r). Each of I(p), I(q), and 1(r) can be
one of T or F, so there are 23 = 8 possible interpretations.
The calculation of the truth value for each of the eight interpretations can be shown
concisely in a truth table. Start out with a truth table that has eight rows, one for each
interpretation:
p
q
r
1o T
T T
I1
T
T F
12
T
F
T
13
T
F
F
14
F
T T
15
F
T F
16
F
F T
17
F
F
F
Next, assign truth values to larger and larger subformulas until the formula itself is evaluated.
We now repeat the evaluation of the formula
0 = (--p V q) -+ (r -+ p)
using this method. Evaluating -p and r -+ p, we get
p
q
r
-p
-- pVqr- p
(-ppvq)- (rp)
Io
T
T T
F
T
Il
T
T F
F
T
12
T
F
T
F
T
13
T F
F
F
T
14
F
T
T
T
F
15
F
T
F
T
T
16
F
F
T
T
F
17
F
F
F
T
T
and in two more steps, we complete the evaluation of the formula: