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LOGICAL OPERATORS

RULES OF LOGIC

VALIDITY PROOFS

Contrapositive

The Contrapositive is a rule of inference pertaining to the IF/THEN operator.

The Contrapositive states that in a conditional, if the consequent is false, then the antecedent must be false also.

Imagine we have the following conditional sentence: "If it is raining, then there are clouds in the sky." Formally, we would write:

p -> q: "If it is raining, then there are clouds in the sky."

In this expression, "If it is raining" is the antecedent and "There are clouds in the sky" is the consequent.

If we apply the Contrapositive to this expression, we would obtain the following: "If there are no clouds in the sky, then it is not raining." This makes perfect sense. Let's write our steps formally:

p -> q: "If it is raining, then there are clouds in the sky."
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~q -> ~p: "If there are no clouds in the sky, then there is no rain."

The original conditional is above the line of dashes, and the new expression ~q -> ~p formed by applying the Contrapositive is below the line.

Other examples of the Contrapositive

R -> W: "If the light is off, then it is dark."
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~W -> ~R: "If it is not dark, then the light is not off."

S -> W: "If there is snow, then it is wintertime."
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~W -> ~S: "If it is not wintertime, then there is no snow."

Links to Relevant Problems

These are links to validity proof problems whose solutions contain the Contrapositive.

4-step problem