Modus tollens

Modus tollens ("mode of taking") is a logical argument, or rule of inference. (Compare with modus ponens, or "mode of putting.") It is also known as indirect proof or proof by contrapositive, and is a valid form of argument in formal logic.[1]

Cogito ergo sum
Logic and rhetoric
Key articles
General logic
Bad logic
v - t - e

As an argument

A modus tollens argument has the following form:

P1: If X, then Y. (i. e. Either not X or Y)
P2: Not Y.
C: Therefore, not X.

For example:

P1: If it is raining, the ground is wet. (i. e. It is not raining or the ground is wet.)
P2: The ground is not wet.
C: Therefore, it is not raining.

The contrapositive of "if X then Y" is "if not Y then not X"; if a proposition is true, then so is its contrapositive.[2]

As a rule of inference

In propositional logic:

In first-order logic:

Denying the antecedent
See the main article on this topic: Denying the antecedent

It can be contrasted with the fallacy of denying the antecedent, for instance (using the above example) "it is not raining, therefore the ground is not wet" (obviously untrue if you're standing in a lake). Denying the antecedent asserts not-X rather than not-Y.

gollark: You could carry around extra battery capacity in a backpack or something.
gollark: Maybe they punched someone they disagree with.
gollark: Nonaggression.... something?
gollark: I think the key to that with digital media is to try and make sure it's still accessible on modern stuff every few years, so you can convert it to newer formats and storage media and stuff.
gollark: A Raspberry Pi would *probably* work with an SSD or something hooked to it, but the server does other stuff.

See also

References
  1. Modus Tollens, Jordan Bell, Wolfram Mathworld
  2. Converse and Contrapositive, CS381 Discrete Structures/Discrete Mathematics Web Course Material, Old Dominion University
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