Argument
Modus tollens
Overview
In propositional logic, modus tollens () (MT), also known as modus tollendo tollens (Latin for "mode that by denying denies") and denying the consequent, 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. The first to explicitly describe the argument form modus tollens was Theophrastus.
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.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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The particular habit of mind which governs this or that inference may be formulated in a proposition whose truth depends on the validity of the inferences which the habit determines; and such a formula is called a _guiding principle_ of inference. Suppose, for example, that we observe that a rotating disk of copper quickly comes to rest when placed between the poles of a magnet, and we infer that this will happen…
Charles S. Peirce, Chance, Love, and Logic
States what Modus tollens is.
The copula is the sign denoting that there is an affirmation or denial, and thereby enabling the hearer or reader to distinguish a proposition from any other kind of discourse. Thus, in the proposition, The earth is round, the Predicate is the word _round_, which denotes the quality affirmed, or (as the phrase is) predicated: _the earth_, words denoting the object which that quality is affirmed of, compose the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Modus tollens is.
XXV. The arguments proper to these excuses, being derived from the topics which we have already set forth, have been explained in our oratorical rules. But the refutation of an accusation, in which there is a repelling of a charge, which is called in Greek [Greek: stasis], is in Latin called _status_. On which there is founded, in the first place, such a defence as may effectually resist the attack. And also, in the…
Marcus Tullius Cicero, The Orations of Marcus Tullius Cicero, Volume 4
States what Modus tollens is.
Philosophers who discussed Modus tollens
Related ideas
- First-order logic
- Latin
- Propositional logic
- Modus ponens
- Validity (logic)
- Reductio ad absurdum
- Conditional probability
- Rule of inference
- Truth table
- Set theory
- Theorem
- Inference
- Susanne Bobzien
- Logical form
- Stanford Encyclopedia of Philosophy
- Logical consequence
- Metalogic
- Formal system
- Tautology (logic)
- Bayes' theorem
- Ancient history
- Fallacy
- Deductive reasoning
- Material conditional
Place in the tradition
Part of Outline of logic.