Modus Tollens

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The 'denying mode': If P then Q, Q is false, therefore P is false.

Modus Tollens (Latin for 'mode that denies') is a fundamental rule of inference that works by denying the consequent. Given a conditional 'If P then Q' and knowing that Q is false, we can validly conclude that P must also be false. This rule is logically equivalent to Modus Ponens applied to the contrapositive.

P → Q, ¬Q ⊢ ¬P

• If it is raining, the ground is wet. The ground is not wet. Therefore, it is not raining.
• If the battery is charged, the phone will turn on. The phone does not turn on. Therefore, the battery is not charged.
• If a number is prime and greater than 2, it is odd. 8 is not odd. Therefore, 8 is not a prime number greater than 2.

How to recognize it

• Look for an 'If...then...' statement (conditional)
• Check if the 'then' part (consequent) is denied/negated
• The conclusion negates the 'if' part (antecedent)

Common mistakes

• Denying the antecedent (invalid): 'If P then Q, P is false, therefore Q is false' - this does NOT follow
• Example of the fallacy: 'If it rains, the ground is wet. It is not raining. Therefore the ground is not wet.' (Wrong! Sprinklers could still make it wet)

Practice guidance

Logical Deduction