Modus Tollendo Tollens/Proof Rule

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Proof Rule

The Modus Tollendo Tollens is a valid deduction sequent in propositional logic.


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:

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


Tableau Form

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

The Modus Tollendo Tollens 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:    Modus Tollendo Tollens             
Depends on:    The line containing the instance of $\phi \implies \psi$             
The line containing the instance of $\neg \psi$             
Abbreviation:    $\text{MTT}$             


Explanation

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.


Also known as

Modus Tollendo Tollens is also known as:

  • Modus tollens
  • Denying the consequent.


Linguistic Note

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

The shorter form Modus Tollens means mode that denies.


Technical Note

When invoking Modus Tollendo Tollens 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 Modus Tollendo Tollens 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.


Sources