Modus Tollendo Tollens
Contents |
Theorem
The modus tollendo tollens (or modus tollens) is a valid deduction sequent in propositional logic:
- $p \implies q, \neg q \vdash \neg p$
That is:
- 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.
It can be written:
- $\displaystyle {p \implies q \quad \neg q \over \neg p} \text{MTT}$
Its abbreviation in a tableau proof is $\mathrm{MTT}$.
This is sometimes known as denying the consequent.
Proof
Proof by Natural Deduction
By the tableau method:
| Line | Pool | Formula | Rule | Depends upon | Notes | |
|---|---|---|---|---|---|---|
| 1 | 1 | $p \implies q$ | P | (None) | ||
| 2 | 2 | $\neg q$ | P | (None) | ||
| 3 | 3 | $p$ | A | (None) | Assume $p$ ... | |
| 4 | 1, 3 | $q$ | $\implies \mathcal E$ | 1, 3 | ... and derive $q$ ... | |
| 5 | 1, 2, 3 | $\bot$ | $\neg \mathcal E$ | 2, 4 | ... and thence derive a contradiction ... | |
| 6 | 1, 2 | $\neg p$ | $\neg \mathcal I$ | 3 - 5 | ... so our assumption of $p$ must have been false. |
$\blacksquare$
Proof by Truth Table
We apply the Method of Truth Tables to the proposition.
As can be seen for all models by inspection, where the truth value under the main connective on the LHS is $T$, that under the one on the RHS is also $T$:
$\begin{array}{|cccccc||cc|} \hline
(p & \implies & q) & \land & \neg & q & \neg & p \\
\hline
F & T & F & T & T & F & T & F \\
F & T & T & F & F & T & T & F \\
T & F & F & F & T & F & F & T \\
T & T & T & F & F & T & F & T \\
\hline
\end{array}$
Hence the result.
$\blacksquare$
Note that the two formulas are not equivalent, as the relevant columns do not match exactly.
Also see
The following are related argument forms:
The Rule of Transposition is conceptually similar, and can be derived from the MTT by a simple application of Conditional Proof.
These are classic fallacies:
Linguistic Note
Modus tollendo tollens is Latin for mode that by denying, denies.
Modus tollens means mode that denies.
Sources
- Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{I}: \S 3$
- E.J. Lemmon: Beginning Logic (1965): $\S 1.2$: Theorem $5$
- Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.2.1, \ \S 1.2.2$
