Modus Tollendo Ponens

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Sequent

The modus tollendo ponens is a valid deduction sequent in propositional logic.


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

$p \lor q, \neg p \vdash q$

Case 2

$p \lor q, \neg q \vdash p$


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.


Also see

The following are related argument forms:


Linguistic Note

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


Sources