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Implication (or Conditional)

A compound statement of the form “if p then q” is called a conditional statement (pq 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  qq  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.




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