Modus Tollens

A Modus Tollens is a Rule of Inference under which the Antecedent of a Conditional is rejected if and only if the Consequent of the Conditional is rejected.

• AKA: Denying the Consequent.
• Context:*
  • It can be expressed as If P, then Q. ¬Q. Therefore, ¬P.
• Example(s):
  • “If an intruder is detected, the alarm goes off. The alarm does not go off. Therefore, no intruder is detected.
• Counter-Example(s):
  • All men are mortal. Socrates is a man. Therefore, Socrates is mortal. (Affirming the Antecedent)
  • All men are mortal. Socrates is mortal. Therefore, Socrates is a man. (Affirming the Consequent).
• See: Modus Ponens, Valid Deductive Argument, Deductive Logic Framework.

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References

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Modus_tollens
  • In logic, modus tollendo tollens (Latin for "the way that denies by denying") is the formal name for indirect proof or proof by contraposition ...
  • In classical logic, modus tollens (or modus tollendo tollens)[1] (Latin for "the way that denies by denying")[2] has the following argument form: If P, then Q. ¬Q. Therefore, ¬P.
  • 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). (See also modus ponens or "affirming the antecedent".)
  • Modus tollens is sometimes confused with indirect proof (assuming the negation of the proposition to be proved and showing that this leads to a contradiction) or proof by contrapositive (proving If P, then Q by a proof of the equivalent contrapositive If not-Q, then not-P).
  • Consider an example: If an intruder is detected, the alarm goes off. The alarm does not go off. Therefore, no intruder is detected.
  • 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.
• http://en.wiktionary.org/wiki/modus_tollens
  • An valid form of argument in which the consequent of a conditional proposition is denied, thus implying the denial of the antecedent. ...
• CYC Glossary http://www.cyc.com/cycdoc/ref/glossary.html
  • modus tollens: A rule of inference which can be derived from modus ponens under which, given a knowledge base which contains the formulas "Not B" and "A implies B", one may conclude "Not A".
• http://www.logic-classroom.info/glossary.htm
  • modus tollens means "a way of destroying;" symbolically: "If p, then q; not-q; therefore,not-p. (Study 4)
• http://www.philosophy.uncc.edu/mleldrid/logic/logiglos.html
  • Modus Tollens: Consists of a conditional statement and one other premise. The second premise denies the consequent of the conditional, yielding the denied antecedent as the conclusion: IF (IF p THEN q) AND (not-q) THEN (not-p).

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