Implication (or Conditional)
A compound statement of the form “if p then q” is called a conditional statement (p, q being logical statements).
The statement “if p then q” is denoted by p → q (to be read as “p implies q”) or by p ⇒ q. Note that p → q also means (i) p is sufficient for q, (ii) q is necessary for p, (iii) p only if q, (iv) p^{ }leads to q, (v) q unless ~ p, (vi) q if p, (vii) q when p, and (viii) if p, then q.
For example, if p: It rains and q: The crops are good, then p → q means the statement “If it rains then the crops are good”. Note that this statement is contradicted only when it rains but crops are not good so that p → q is false only in one situation when p is true but q is false. The following truth table gives explicitly the truth value of p → q and q → p.
Truth table (p → q, q → p)


p

q

p → q

q → p

T

T

T

T

T

F

F

T

F

T

T

F

F

F

T

T

Rule: p → q is false only when p is true and q is false.

Observe that p → q π q → p.