# User:JMRyan/Test2

### Modus Tollens

${\displaystyle \mathbf {T7.} \quad (\mathrm {P} \rightarrow \mathrm {Q} )\land \lnot \mathrm {Q} \rightarrow \lnot \mathrm {P} \,\!}$

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

Now we use T7 to justify the following rule.

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

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.

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

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

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

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

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.