# Modus Ponendo Tollens

## Modus Ponendo Tollens

The modus ponendo tollens is a valid deduction sequent in propositional logic:

### Proof Rule

$(1): \quad$ If we can conclude $\neg \left({\phi \land \psi}\right)$, and we can also conclude $\phi$, then we may infer $\neg \psi$.
$(2): \quad$ If we can conclude $\neg \left({\phi \land \psi}\right)$, and we can also conclude $\psi$, then we may infer $\neg \phi$.

## Variants

The following forms can be used as variants of this theorem:

### Variant

#### Formulation 1

$\neg \left({p \land q}\right) \dashv \vdash p \implies \neg q$

#### Formulation 2

$\vdash \left({\neg \left({p \land q}\right)}\right) \iff \left({p \implies \neg q}\right)$

## Explanation

The Modus Tollendo Ponens can be expressed in natural language as:

If two statements cannot both be true, and one of them is true, it follows that the other one is not true.

## Linguistic Note

Modus Ponendo Tollens is Latin for mode that by affirming, denies.

## Also see

The following are related argument forms: