An implication consists of a pair of sentences separated by the ⇒ operator and enclosed in parentheses. For example, the implication of p and q is (p ⇒ q).
The sentence to the left of the ⇒ operator in an implication is called the antecedent, and the sentence to the right is called the consequent. For example, in the implication (p ⇒ q), p is the antecedent and q is the consequent.
The truth value of an implication is false if and only if its antecedent is true and its consequent is false; otherwise, the truth value is true.
p 
q 
(p ⇒ q) 
1 
1 
1 
1 
0 
0 
0 
1 
1 
0 
0 
1 
This semantics for the ⇒ operator is called material implication to distinguish it from other possibilities.
