Philosophy Index

Modus tollens

Modus tollendo tollens, usually simply called modus tollens or MT is a valid argument form in logic. It is also known as "denying the consequent".

The form of modus tollens is: "If P, then Q. Not Q. Therefore, not P." It may also be written as:

P → Q, ¬Q infers ¬P

P and Q may represent any proposition, or any other formula (using Greek letters to represent formulas rather than propositions, we may also express modus tollens as α → β, ¬β infers ¬α).

Examples of modus tollens

The following are examples of the modus tollens argument form:

If the cake is made with sugar, then the cake is sweet.
The cake is not sweet.
Therefore, the cake is not made with sugar.

If Sam was born in Canada, then he is Canadian.
Sam is not Canadian.
Therefore, Sam was not born in Canada.

Proof of modus tollens by contradiction

Modus tollens is not a basic rule of inference — that is, it may be described by more fundamental rules. To justify modus tollens using basic rules, we may construct a proof as follows:

1 P → Q Pr.
2 ¬Q Pr.
RTP ¬P
3 P AIP
RTP Contradiction
4 Q MP, 1, 3
5 ¬Q R, 2
6 ¬P ¬I, 3–5

This is an indirect proof by contradiction (reductio ad absurdum), showing, by lines 3–6, that the alternative, P→Q, ¬Q infers P, is absurd, since it would result in both Q and ¬Q as being true.