Modus Tollendo Tollens/Proof Rule

Proof Rule

Modus tollendo tollens is a valid argument in types of logic dealing with conditionals $\implies$ and negation $\neg$.

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

As a proof rule it is expressed in the form:

If we can conclude $\phi \implies \psi$, and we can also conclude $\neg \psi$, then we may infer $\neg \phi$.

It can be written:

$\ds {\phi \implies \psi \quad \neg \psi \over \neg \phi} \text{MTT}$

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

The is invoked for $\phi \implies \psi$ and $\neg \psi$ as follows:

			Pool:					The pooled assumptions of $\phi \implies \psi$
								The pooled assumptions of $\neg \psi$
			Formula:					$\neg \phi$
			Description:
			Depends on:					The line containing the instance of $\phi \implies \psi$
								The line containing the instance of $\neg \psi$
			Abbreviation:					$\text{MTT}$

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

If the truth of one statement implies the truth of a second, and the second is shown not to be true, then neither can the first be true.

Modus Tollendo Tollens is also known as:

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

Modus Tollendo Tollens is Latin for mode that by denying, denies.

The shorter form Modus Tollens means mode that denies, or method of denying.

When invoking in a tableau proof, use the {{ModusTollens}} template:

{{ModusTollens|line|pool|statement|first|second}}

or:

{{ModusTollens|line|pool|statement|first|second|comment}}

where:

line is the number of the line on the tableau proof where 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 $\phi \implies \psi$

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

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

1964: Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning ... (previous) ... (next): $\text{I}$: 'NOT' and 'IF': $\S 3$
1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $1$: The Propositional Calculus $1$: $2$ Conditionals and Negation
1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs: $2$
2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.1$: Rules for natural deduction