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LOGICAL OPERATORS

RULES OF LOGIC

VALIDITY PROOFS

Modus Ponens

Modus Ponens is a rule of inference pertaining to the IF/THEN operator.

Modus Ponens states that if the antecedent of a conditional is true, then the consequent must also be true.

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.

Now if we know for a fact that it is raining, then we have to conclude that there are clouds in the sky. If the antecedent ("It is raining") is true, then the consequent ("There are clouds in the sky") must also be true, by Modus Ponens. Let's write our steps formally:

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

The conditional p -> q and the given p are above the line of dashes, and the conclusion q obtained by applying Modus Ponens is below the line.

Other examples of Modus Ponens

W -> C: If Yankees win today's game, they will be champions.
W: Yankees win today's game.
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C: Yankees are champions.

W -> B: If the weather is good, we can go to the beach.
W: The weather is good.
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B: We can go to the beach.

Links to Relevant Problems

These are links to validity proof problems whose solutions contain Modus Ponens.

2-step problem (A)
2-step problem (B)
3-step problem