Modus Ponendo Tollens/Proof Rule

Proof Rule
Modus ponendo tollens is a valid argument in types of logic dealing with conjunctions $\land$ and negation $\neg$.

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

As a proof rule it is expressed in either of the two forms:
 * $(1): \quad$ If we can conclude $\map \neg {\phi \land \psi}$, and we can also conclude $\phi$, then we may infer $\neg \psi$.
 * $(2): \quad$ If we can conclude $\map \neg {\phi \land \psi}$, and we can also conclude $\psi$, then we may infer $\neg \phi$.

It can be written:
 * $\ds {\map \neg {\phi \land \psi} \quad \phi \over \neg \psi} \textrm{MPT}_1 \qquad \text{or} \qquad {\map \neg {\phi \land \psi} \quad \psi \over \neg \phi} \textrm{MPT}_2$

Also see

 * This is a rule of inference of the following proof systems:
 * Definition:Natural Deduction