Legal & DebateNEUTRAL — Rhetoric

Reductio ad Absurdum

What it is

Refuting a claim by assuming it true, deriving a consequence that is false, contradictory, or unacceptable, and concluding that the claim must be rejected — the oldest and most reliable form of indirect proof.

How it works

The form is ancient and exact. Assume the proposition; show that it leads, by steps the opponent accepts, to a contradiction or to something plainly false; conclude that the proposition is false. Euclid used it to prove there is no largest prime — suppose there is, multiply all the primes together and add one, and the result is divisible by none of them. Galileo used it to dispose of Aristotle's claim that heavier bodies fall faster: tie a light body to a heavy one, and the light body should both slow the heavy one down (as a drag) and speed it up (as added weight), which cannot both be true. In argument the move is powerful because it works entirely from the opponent's commitments; nothing is asserted that they have not granted. It has two counterfeits. The first is the slippery slope, which replaces derivation with speculation — the consequence does not follow, it is merely feared. The second is the appeal to ridicule, which presents a consequence as absurd because it sounds strange rather than because it contradicts anything; a genuine reductio ends in a contradiction or in a proposition the opponent already rejects, not in a laugh. The test of an honest reductio is that every step from premise to absurdity can be written down and checked.

Real-world examples

  • •Euclid, Elements IX.20: assume finitely many primes, form their product plus one, and derive a contradiction — a reductio that has stood for twenty-three centuries.
  • •Galileo's Two New Sciences (1638): the tied-bodies argument that turns Aristotle's rule about falling weights against itself without dropping anything from a tower.
  • •In a policy round: “Their interpretation of the resolution would make a case about parking fines topical; since they agree that would be absurd, the interpretation fails.”
  • •Workplace: “If we accept that every customer request must be built, then the request to remove the login page must be built too — so the rule cannot be what we actually mean.”
  • •The counterfeit in a family argument: “If I let you stay out until eleven, next you will want to stay out all night” — a slippery slope wearing the reductio's clothes, since nothing about the second follows from the first.

Ethical guidelines

  • ●Derive, do not speculate: each step from the premise to the absurdity must be one the other side accepts.
  • ●The absurdity must be a genuine contradiction or a proposition the other side already rejects — not something that merely sounds odd.
  • ●Apply the reductio to the position as held, with its qualifications; reducing an unqualified version the opponent never stated is a straw man.
  • ●Accept the reductio when it is run on you; if your premise really does entail something you reject, revise the premise.

How to defend against it

  • ►Ask for the chain: “Show me the steps. Which of my premises, combined with what, yields that?” A real reductio can produce them; a slippery slope cannot.
  • ►Check whether the absurdity is yours: “I do not accept that the consequence is absurd — here is why it is acceptable.” Biting the bullet is a legitimate reply if you mean it.
  • ►Check the premise attributed to you: most failed reductios work on a stronger claim than the one you made. Restate yours with its qualifier and ask whether the derivation still goes through.
  • ►Distinguish contradiction from discomfort: a consequence that contradicts something you believe forces revision; one you merely dislike does not.

References

  1. Euclid (trans. T. L. Heath) (-300). The Thirteen Books of Euclid's Elements (Book IX, Proposition 20). Cambridge University Press, 1908; Dover reprint, 1956
    The proof that there are infinitely many primes by reduction to contradiction.
  2. Galilei, G. (trans. H. Crew & A. de Salvio) (1638). Dialogues Concerning Two New Sciences. Macmillan, 1914
    The tied-bodies thought experiment refuting the claim that heavier bodies fall faster.
  3. Copi, I. M., & Cohen, C. (2009). Introduction to Logic (13th ed.). Pearson Prentice Hall
    Indirect proof and reductio ad absurdum as a formal method of refutation.
  4. Walton, D. N. (1992). Slippery Slope Arguments. Clarendon Press
    The distinction between a derived consequence and a merely feared one in slippery-slope reasoning.
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