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Modus Tolens

Modus Tolens

A logical argument of the form shown below. This is essentially the argument employed in proof by contradiction.

 

Modus Tollens logical form: "If p then q. q is false. Therefore p is false.

Example of Modus Tollens: "If it is raining then the grass is wet. The grass is not wet. Therefore it is not raining.

 

 

See also

Modus ponens

Key Formula

If P⇒Q is true, and ¬Q is true, then ¬P is true.\text{If } P \Rightarrow Q \text{ is true, and } \neg Q \text{ is true, then } \neg P \text{ is true.}
Where:
  • PP = The antecedent (hypothesis) of the conditional statement
  • QQ = The consequent (conclusion) of the conditional statement
  • ¬\neg = Logical negation ('not')

Example

Problem: Given: (1) If it is raining, then the ground is wet. (2) The ground is not wet. What can you conclude?
Step 1: Identify the conditional statement and its parts.
P="it is raining",Q="the ground is wet",P⇒QP = \text{"it is raining"}, \quad Q = \text{"the ground is wet"}, \quad P \Rightarrow Q
Step 2: Note that the consequent Q is false: the ground is not wet.
¬Q is true\neg Q \text{ is true}
Step 3: Apply modus tollens: since the conditional is true and Q is false, P must be false.
¬Q and (P⇒Q)  ⟹  ¬P\neg Q \text{ and } (P \Rightarrow Q) \implies \neg P
Answer: It is not raining.

Why It Matters

Modus tollens is the logical foundation of proof by contradiction (indirect proof), one of the most powerful techniques in mathematics. When you assume a statement is true and derive a false conclusion, modus tollens lets you conclude the original assumption was false. This reasoning pattern appears throughout algebra, geometry, and higher mathematics whenever direct proof is difficult.

Common Mistakes

Mistake: Confusing modus tollens with the fallacy of denying the antecedent. Students sometimes reason: 'If P then Q; not P; therefore not Q.'
Correction: Modus tollens denies the consequent (not Q), not the antecedent. Denying the antecedent is a logical fallacy. For example, 'If it is raining then the ground is wet; it is not raining; therefore the ground is not wet' is invalid — the ground could be wet for another reason.

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