Modus Tollendo Tollens/Sequent Form/Proof by Truth Table

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:
 * $\displaystyle {p \implies q \quad \neg q \over \neg p} \text{MTT}$

Proof
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.

Note that the two formulas are not equivalent, as the relevant columns do not match exactly.