Modus Tollendo Tollens

Proof Rule

Modus tollendo tollens is a valid argument in types of logic dealing with conditionals $\implies$ and negation $\neg$.

This includes propositional logic and predicate logic, and in particular natural deduction.

Proof Rule

If we can conclude $\phi \implies \psi$, and we can also conclude $\neg \psi$, then we may infer $\neg \phi$.

Sequent Form

The Modus Tollendo Tollens can be symbolised by the sequent:

\(\ds p\)

\(\implies\)

\(\ds q\)

\(\ds \neg q\)

\(\)

\(\ds \)

\(\ds \vdash \ \ \)

\(\ds \neg p\)

\(\)

\(\ds \)

Explanation

The Modus Tollendo Tollens can be expressed in natural language as:

If the truth of one statement implies the truth of a second, and the second is shown not to be true, then neither can the first be true.

Also known as

Modus Tollendo Tollens is also known as:

• Modus tollens, abbreviated M.T.
• Denying the consequent.

Also see

• Definition:Conditional

The following are related argument forms:

• Modus Ponendo Ponens
• Modus Ponendo Tollens
• Modus Tollendo Ponens

The Rule of Transposition is conceptually similar, and can be derived from the MTT by a simple application of the Rule of Implication.

These are classic fallacies:

• Affirming the Consequent
• Denying the Antecedent

Linguistic Note

Modus Tollendo Tollens is Latin for mode that by denying, denies.

The shorter form Modus Tollens means mode that denies, or method of denying.

Sources

• 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.3$: Argument Forms and Truth Tables
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): modus tollens
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): modus tollens