Modus Tollendo Ponens/Proof Rule

Proof Rule

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.

As a proof rule it is expressed in either of the two forms:

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

It can be written:

$\ds {\paren {\phi \lor \psi} \quad \neg \phi \over \psi} \textrm {MTP}_1 \qquad \text{or} \qquad {\paren {\phi \lor \psi} \quad \neg \psi \over \phi} \textrm {MTP}_2$

Tableau Form

Let $\phi \lor \psi$ be a well-formed formula in a tableau proof whose main connective is the disjunction operator.

The Modus Tollendo Ponens is invoked for $\phi \lor \psi$ in either of the two forms:

Form 1

			Pool:					The pooled assumptions of $\phi \lor \psi$
								The pooled assumptions of $\neg \phi$
			Formula:					$\psi$
			Description:					Modus Tollendo Ponens
			Depends on:					The line containing the instance of $\phi \lor \psi$
								The line containing the instance of $\neg \phi$
			Abbreviation:					$\text{MTP}_1$

Form 2

			Pool:					The pooled assumptions of $\phi \lor \psi$
								The pooled assumptions of $\neg \psi$
			Formula:					$\phi$
			Description:					Modus Tollendo Ponens
			Depends on:					The line containing the instance of $\phi \lor \psi$
								The line containing the instance of $\neg \psi$
			Abbreviation:					$\text{MTP}_2$

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.

Also see

• This is a rule of inference of the following proof systems:
  • Definition:Natural Deduction

Linguistic Note

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

Technical Note

When invoking Modus Tollendo Ponens in a tableau proof, use the {{ModusTollendoPonens}} template:

{{ModusTollendoPonens|line|pool|statement|first|second|1 or 2}}

or:

{{ModusTollendoPonens|line|pool|statement|first|second|1 or 2|comment}}

where:

line is the number of the line on the tableau proof where Modus Tollendo Ponens is to be invoked

pool is the pool of assumptions (comma-separated list)

statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters

first is the first of the two lines of the tableau proof upon which this line directly depends, the one in the form $p \lor q$

second is the second of the two lines of the tableau proof upon which this line directly depends, the one in the form $\neg p$

1 or 2 should hold 1 for ModusTollendoPonens_1, and 2 for ModusTollendoPonens_2

comment is the (optional) comment that is to be displayed in the Notes column.

Sources

• 1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{II}$: 'AND', 'OR', 'IF AND ONLY IF': $\S 3$