Modus Tollendo Ponens

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Sequent

Modus tollendo ponens is a valid argument in types of logic dealing with disjunctions $\lor$ and negation $\neg$.

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


Proof Rule

$(1): \quad$ If we can conclude $\phi \lor \psi$, and we can also conclude $\neg \phi$, then we may infer $\psi$.
$(2): \quad$ If we can conclude $\phi \lor \psi$, and we can also conclude $\neg \psi$, then we may infer $\phi$.


Sequent Form

Case 1

\(\ds p \lor q\) \(\) \(\ds \)
\(\ds \neg p\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds q\) \(\) \(\ds \)

Case 2

\(\ds p \lor q\) \(\) \(\ds \)
\(\ds \neg q\) \(\) \(\ds \)
\(\ds \vdash \ \ \) \(\ds p\) \(\) \(\ds \)


Variants

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

Variant

Formulation 1

$p \lor q \dashv \vdash \neg p \implies q$

Formulation 2

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


Note that the form:

$\neg p \implies q \vdash p \lor q$

requires Law of Excluded Middle.

Therefore it is not valid in intuitionistic logic.


Explanation

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

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

Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.
-- Sherlock Holmes (Sir Arthur Conan Doyle, 1890: The Sign of the Four, ch. 6)


Also known as

The Modus Tollendo Ponens is also known as the Disjunctive Syllogism, abbreviated D.S.


Examples

Third World War

The following is an example of the use of a Modus Tollendo Ponens:

The United Nations will be strengthened or there will be a third world war.
The United Nations will not be strengthened.
Therefore there will be a third world war.


Also see

The following are related argument forms:


Linguistic Note

Modus Tollendo Ponens is Latin for mode that by denying, affirms.


Sources

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