Modus tollens

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Modus tollens, "the method of denial", is the name of a proof rule in logic. Also called "denying the consequent" the rule states that where you have a conditional and the negation of the consequent you may then negate the antecedent. It should not be confused with the superficially similar denying the antecedent, which is fallacious.

Common English

Take the conditional: "if this bird is a raven then this bird is black". "This bird is a raven" is called the antecedent, "this bird is black" is the consequent. The truth of the antecedent is dependent on the truth of the consequent. If the consequent is false, if "this bird" is not black, then we can be assured that "this bird" is not a raven.

Logical Notation

In logical notation the rule can be formalized. "This bird is a raven" can be replaced with a letter, say "R", whilst this "this bird is black" can be replaced by "B". The conditional can be rewritten "RﬤB" and our observation as "¬B" ("not-B").

  1. RﬤB (Assumption)
  2. ¬B (Assumption)
  3. ¬R (from 1 and 3, Modus Tolens)

Modus Tolens will operate on any argument of this form, any values of R and B that can be expressed as "RﬤB" where we can establish "¬B". E.g.:

  • Every Ancient Egyptian is dead (if they are an Ancient Egyptian then they are dead), Harry is not dead, therefore Harry is not an Ancient Egyptian.
  • No fish walk, this animal walks, therefore this animal is not a fish
  • If I eat food then I will fly, I have not flown, therefore I did not drink this beer

The last example, of course, has a false conclusion. This, however, due to the falsity of the first premise, not any logical failing. The argument remains valid but is not sound.