Modus tollens

Modus tollens
Type
Field
Statement implies . is false. Therefore, must also be false.
Symbolic statement [1]

In propositional logic, modus tollens (/ˈmdəs ˈtɒlɛnz/) (MT), also known as modus tollendo tollens (Latin for "method of removing by taking away")[2] and denying the consequent,[3] is a deductive argument form and a rule of inference. Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.

The history of the inference rule modus tollens goes back to antiquity.[4] The first to explicitly describe the argument form modus tollens was Theophrastus.[5]

Modus tollens is closely related to modus ponens. There are two similar, but invalid, forms of argument: affirming the consequent and denying the antecedent. See also contraposition and proof by contrapositive.

  1. ^ Matthew C. Harris. "Denying the antecedent". Khan academy.
  2. ^ Stone, Jon R. (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London: Routledge. p. 60. ISBN 978-0-415-91775-9.
  3. ^ Sanford, David Hawley (2003). If P, Then Q: Conditionals and the Foundations of Reasoning (2nd ed.). London: Routledge. p. 39. ISBN 978-0-415-28368-7. [Modus] tollens is always an abbreviation for modus tollendo tollens, the mood that by denying denies.
  4. ^ Susanne Bobzien (2002). "The Development of Modus Ponens in Antiquity", Phronesis 47.
  5. ^ "Ancient Logic: Forerunners of Modus Ponens and Modus Tollens". Stanford Encyclopedia of Philosophy.