A conditional statement has the form, " if p, then q." In such a
statement, p is called the antecedent and q the consequent. The
connection between them, the if-then relationship, is represented
by the horseshoe symbol
A statement involving the conditional connective says that if the
antecedent is true, the consequent is true as well. In
propositional logic, such a statement is defined by the following
Consider the statement, "If I study hard, I'll pass the exam." The antecedent is "I study hard," and the consequent is "I'll
pass the exam."
Which combinations of truth values for the antecedent and
consequent are consistent with the truth of the conditional
statement as a whole?
Suppose first that I do study hard and I do pass the exam: p and
q are both true, as in the first line of the truth table. That's
certainly consistent with the truth of the conditional.
Likewise, if I don't study hard, I don't pass--the last line of
What about the second line? Suppose I don't study hard, but I
still pass. Would that prove the conditional statement false? No --
the statement doesn't say that studying hard is the only way to
pass; perhaps the exam was easy, or perhaps I already knew the
But if I do study hard and I don't pass--the situation
represented by the third line--then we'd have to conclude that
the conditional is false.
In propositional logic, therefore, pq is defined as false in
line 3, where p is true and q false, and defined as true in the other
A conditional statement in English need not take the form, "if p,
then q." Each of the following statements is also a conditional:
A. I will go camping this weekend if I finish my work.
B. I will go camping this weekend only if I finish my work.
C. We will go camping this weekend unless it rains.
Each of these statements can be translated systematically
into an if-then structure, and symbolized according to the
A. p if q = If q, then p = qp
B. p only if q = If p, then q = pq
C. p unless q = If ~q, then p = ~q p
Thus, A says that if I finish my work, then I will go camping. B
makes the different statement that if I go camping, I have
finished my work. And C says that if it does not rain, then I
will go camping.