4
Formal Logic
expression. Negation is a unary connective. The negation of A— symbolized by
A′—is read “not A.”
pRaCtiCe 4 Write the truth table for A′. (It will require only two rows.)
Table 1.4 summarizes the truth values for all of the logical connectives. This
information is critical to an understanding of logical reasoning.
tAbLe 1.4
A
B
A ` B
A ~ B
A S B
A 4 B
A′
T
T
T
T
T
T
F
T
F
F
T
F
F
F
T
F
T
T
F
T
F
F
F
F
T
T
ReMIndeR
A only if B means
A S B
Because of the richness of the English language, words that have different shades of meaning are nonetheless represented by the same logical connective. Table 1.5 shows the common English words associated with various logical
connectives.
tAbLe 1.5
english Word
Logical connective
Logical expression
and; but; also; in addition;
moreover
Conjunction
A ` B
or
Disjunction
A ~ B
If A, then B.
A implies B.
A, therefore B.
A only if B.
B follows from A.
A is a sufficient condition
for B.
B is a necessary condition
for A.
Implication
A S B
A if and only if B.
A is necessary and
sufficient for B.
Equivalence
A 4 B
not A
It is false that A ...
It is not true that A ...
Negation
A′
Suppose that A S B is true. Then, according to the truth table for implication,
the consequent, B, can be true even though the antecedent, A, is false. So while
the truth of A leads to (implies) the truth of B, the truth of B does not imply the
truth of A. The phrase “B is a necessary condition for A” to describe A S B simply