Modus Ponendo Tollens

Modus Ponendo Tollens

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.

Proof Rule

$(1): \quad$ 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 $\neg \phi$.

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

$\vdash \paren {\neg \paren {p \land q} } \iff \paren {p \implies \neg q}$

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

• Definition:Conjunction

The following are related argument forms:

• Modus Ponendo Ponens
• Modus Tollendo Ponens
• Modus Tollendo Tollens