Modus Ponendo Tollens

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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: