### Logic and logical words

Statements of the form, "IF P THEN Q" are called implications. P is frequently called the hypotheses or antecedent, and Q is called the conclusion or consequent. Other related expressions are:
IF Q THEN P. This is called the converse.
IF {not Q} THEN {not P}. This is called the contrapositive.
IF (not P) THEN (not Q). This is called the inverse.
Example

• Original implication: If pigs are green, then I can fly. So "P" is "pigs are green" and "Q" is "I can fly."
• Converse: If I can fly, then pigs are green.
• Contrapositive: If I can't fly, then pigs aren't green.
• Inverse: If pigs aren't green, then I can't fly.
The truth value of an implication is always the same as its contrapositive. The truth value of the converse and the inverse are always the same. The truth value of an implication and its converse may be different.
Examples
The logical universe here is the real numbers.
• Implication and converse both true
Implication: If x3 is a positive real number, then x is a positive real number. True!
Converse: If x is a positive real number, then x3 is a positive real number. True!
Contrapositive: If x is not a positive real number, then x3 is not a positive real number. This is True!, and has the same logical "content" as the original implication (although to me it is more difficult to understand!).
• Implication true and converse false
Implication: If x is an integer, then x is a real number. True!
Converse: If x is a real number, then x is an integer. False!
Contrapositive: If x is not a real number, then x is not an integer. Also True!