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Modus tollens
From Wikipedia, the free encyclopedia
In logic, Modus tollendo tollens[1] (Latin for "the way that denies by denying")[2] is the formal name for indirect proof or proof by contraposition (contrapositive inference), often abbreviated to MT or modus tollens.[3][4] It can also be referred to as denying the consequent, and is a valid form of argument (unlike similarly-named but invalid arguments such as affirming the consequent or denying the antecedent).
Modus tollens has the following argument form:
- If P, then Q.
- ¬Q
- Therefore, ¬P.[5]
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The modus tollens rule may be written in logical operator notation:
where 
represents the logical assertion.
Or in set-theoretic form:
("P is a subset of Q. x is not in Q. Therefore, x is not in P.")
It can also be written as:
The argument has two premises. The first premise is the conditional "if-then" statement, namely that P implies Q. The second premise is that Q is false. From these two premises, it can be logically concluded that P must be false.
Consider an example:
- If there is fire here, then there is oxygen here.
- There is no oxygen here.
- Therefore, there is no fire here.
Supposing that the premises are both true, if there is a fire here, then then there must be oxygen. It is a fact that there is no oxygen here. It follows, then, that there cannot be a fire here. An argument is valid if it is not possible for the premises to be true and the conclusion false.
Another example:
- If Lizzie was the murderer, then she owns an axe.
- Lizzie does not own an axe.
- Therefore, Lizzie was not the murderer.
Modus tollens became well known when it was used by Karl Popper in his proposed response to the problem of induction, falsificationism.
Every use of modus tollens can be converted to a use of modus ponens and one use of transposition to the premise which is a material implication. For example:
- If P, then Q. (premise -- material implication)
- If Q is false, then P is false. (derived by transposition)
- Q is false. (premise)
- Therefore, P is false. (derived by modus ponens)
Likewise, every use of modus ponens can be converted to a use of modus tollens and transposition.
- Modus ponens
- Modus tollendo ponens
- Modus ponendo tollens
- Affirming the consequent
- Denying the antecedent
- Falsificationism
- Non sequitur (logic)
- ^ Sanford, David Hawley. 2003. If P, Then Q: Conditionals and the Foundations of Reasoning. London, UK: Routledge: 39 "[Modus] tollens is always an abbreviation for modus tollendo tollens, the mood that by denying denies."
- ^ Stone, Jon R. 1996. Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London, UK: Routledge: 60.
- ^ Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning, 7:217-234.
- ^ Suppes, Patrick & Hill, Shirley A. 1964. First Course in Mathematical Logic. Dover Publications:54-55.
- ^ University of North Carolina, Philosophy Department, Logic Glossary. Accessdate on 31 October 2007.
- Modus Tollens at Wolfram MathWorld