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Modus Tollens[edit]

\mathbf{T7.} \quad (\mathrm{P} \rightarrow \mathrm{Q}) \land \lnot \mathrm{Q} \rightarrow \lnot \mathrm{P}\,\!
1.     (\mathrm{P} \rightarrow \mathrm{Q}) \land \lnot \mathrm{Q}\,\!   Assumption    [(\mathrm{P} \rightarrow \mathrm{Q}) \land \lnot \mathrm{Q} \rightarrow \lnot \mathrm{P}]\,\!
2.       \mathrm{P}\,\!   Assumption    [\lnot \mathrm{P}]\,\!
3.       \mathrm{P} \rightarrow \mathrm{Q}\,\!   1 KE
4.       \mathrm{Q}\,\!   2, 3 CE
5.       \lnot \mathrm{Q}\,\!   1 KE
6.     \lnot \mathrm{P}\,\!   2–5 NI
7.   (\mathrm{P} \rightarrow \mathrm{Q}) \land \lnot \mathrm{Q} \rightarrow \lnot \mathrm{P}\,\!   1–6 CI

Now we use T7 to justify the following rule.

Modus Tollens (MT)
(\varphi \rightarrow \psi)\,\!
\underline{\lnot \psi \quad \quad \ }\,\!
\lnot \varphi\,\!

Modus Tollens is also sometimes known as 'Denying the Consequent'. Note that the following is not an instance of Modus Tollens, at least as defined above.

\lnot \mathrm{P} \rightarrow \lnot \mathrm{Q}\,\!
\underline{\mathrm{Q} \quad \quad \quad \quad}\,\!

The premise lines of Modus Tollens are a conditional and the negation of its consequent. The premise lines of this inference are a conditional and the opposite of its consequent, but not the negation of its consequent. The desired inference here needs to be derived as below.

1.   \lnot \mathrm{P} \rightarrow \lnot \mathrm{Q} \,\!   Premise
2.   \mathrm{Q}\,\!   Premise
3.   \lnot \lnot \mathrm{Q}\,\!   2 DNI
4.   \lnot \lnot \mathrm{P}\,\!   1, 3 CE
5.   \mathrm{P}\,\!   4 DNE

Of course, it is possible to prove as a theorem:

(\lnot \mathrm{P} \rightarrow \lnot \mathrm{Q}) \land \mathrm{Q} \rightarrow \mathrm{P}\ .\,\!

Then you can add a new inference rule—or, more likeley, a new form of Modus Tollens—on the basis of this theorem. However, we won't do that here.