Modus ponendo tollens

Transformation rules
Propositional calculus
Rules of inference
  • Implication introduction / elimination (modus ponens)
  • Biconditional introduction / elimination
  • Conjunction introduction / elimination
  • Disjunction introduction / elimination
  • Disjunctive / hypothetical syllogism
  • Constructive / destructive dilemma
  • Absorption / modus tollens / modus ponendo tollens
  • Negation introduction
Rules of replacement
  • Associativity
  • Commutativity
  • Distributivity
  • Double negation
  • De Morgan's laws
  • Transposition
  • Material implication
  • Exportation
  • Tautology
Predicate logic
Rules of inference
  • Universal generalization / instantiation
  • Existential generalization / instantiation

Modus ponendo tollens (MPT;[1] Latin: "mode that denies by affirming")[2] is a valid rule of inference for propositional logic. It is closely related to modus ponens and modus tollendo ponens.

Overview

MPT is usually described as having the form:

  1. Not both A and B
  2. A
  3. Therefore, not B

For example:

  1. Ann and Bill cannot both win the race.
  2. Ann won the race.
  3. Therefore, Bill cannot have won the race.

As E. J. Lemmon describes it: "Modus ponendo tollens is the principle that, if the negation of a conjunction holds and also one of its conjuncts, then the negation of its other conjunct holds."[3]

In logic notation this can be represented as:

  1. ¬ ( A ∧ B ) {\displaystyle \neg (A\land B)}
  2. A {\displaystyle A}
  3. ∴ ¬ B {\displaystyle \therefore \neg B}

Based on the Sheffer Stroke (alternative denial), "|", the inference can also be formalized in this way:

  1. A | B {\displaystyle A\,|\,B}
  2. A {\displaystyle A}
  3. ∴ ¬ B {\displaystyle \therefore \neg B}

Proof

Step Proposition Derivation
1 ¬ ( A ∧ B ) {\displaystyle \neg (A\land B)} Given
2 A {\displaystyle A} Given
3 ¬ A ∨ ¬ B {\displaystyle \neg A\lor \neg B} De Morgan's laws (1)
4 ¬ ¬ A {\displaystyle \neg \neg A} Double negation (2)
5 ¬ B {\displaystyle \neg B} Disjunctive syllogism (3,4)

Strong form

Modus ponendo tollens can be made stronger by using exclusive disjunction instead of non-conjunction as a premise:

  1. A ∨ _ B {\displaystyle A{\underline {\lor }}B}
  2. A {\displaystyle A}
  3. ∴ ¬ B {\displaystyle \therefore \neg B}

See also

References

  1. ^ Politzer, Guy & Carles, Laure. 2001. 'Belief Revision and Uncertain Reasoning'. Thinking and Reasoning. 7:217–234.
  2. ^ Stone, Jon R. (1996). Latin for the Illiterati: Exorcizing the Ghosts of a Dead Language. London: Routledge. p. 60. ISBN 0-415-91775-1.
  3. ^ Lemmon, Edward John. 2001. Beginning Logic. Taylor and Francis/CRC Press, p. 61.