Modus Tollendo Ponens/Proof Rule
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Proof Rule
The Modus Tollendo Ponens is a valid deduction sequent in propositional logic.
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:
- $\displaystyle {\left({\phi \lor \psi}\right) \quad \neg \phi \over \psi} \textrm{MTP}_1 \qquad \text{or} \qquad {\left({\phi \lor \psi}\right) \quad \neg \psi \over \phi} \textrm{MTP}_2$
Tableau Form
Let $\phi \lor \psi$ be a propositional 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.
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 invokedpool
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$ ... $
delimitersfirst
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 forModusTollendoPonens_1
, and 2 forModusTollendoPonens_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': $\S 3$
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $3.1$: Formal Proof of Validity