When You Prove Something Is True by Showing the Opposite Is Not (2024)

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When You Prove Something Is True by Showing the Opposite Is Not (1)

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Richard Nordquist

Richard Nordquist

English and Rhetoric Professor

  • Ph.D., Rhetoric and English, University of Georgia
  • M.A., Modern English and American Literature, University of Leicester
  • B.A., English, State University of New York

Dr. Richard Nordquist is professor emeritus of rhetoric and English at Georgia Southern University and the author of several university-level grammar and composition textbooks.

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Updated on February 12, 2020

In argumentation and informal logic, reductio ad absurdum(RAA) is amethod of refuting a claim by extending the logic of the opponent's argument to a point of absurdity. Also known as the reductio argument and argumentum ad absurdum.

"Proofs By Contradictions"

Similarly, reductio ad absurdum may refer to a type of argument in which something is proved to be true by showing that the opposite is untrue. Also known as indirect proof,proof by contradiction, and classical reductio ad absurdum.

As Morrow and Weston point out in A Workbook for Arguments (2015), arguments developed by reductio ad absurdum are frequently used to prove mathematical theorems. Mathematicians "often call these arguments 'proofs by contradiction.' They use this name because mathematical reductio arguments lead to contradictions--such as the claim that N both is and is not the largest prime number. Since contradictions can't be true, they make for very strong reductio arguments."

Like any argumentative strategy, reductio ad absurdumcan be misused and abused, but in itself it is not a form of fallacious reasoning. A related form of argument, theslippery slopeargument, takesreductio ad absurdumto an extreme and is often (but not always) fallacious.

Etymology:From the Latin, "reduction to absurdity"

Pronunciation:ri-DUK-tee-o ad ab-SUR-dum

Reductio Ad Absurdum in Academics

Read MoreUnraveling Ad Hominem: More Than Personal AttacksBy Richard Nordquist

Academics and rhetoricians have offered various explanations as to what makes up reductio ad absurdum arguments, as the following quotes demonstrate.

William Harmon and Hugh Holman

  • - "Reductio ad absurdum. A 'reducing to absurdity' to show the falsity of an argument or position. One might say, for instance that the more sleep one gets the healthier one is, and then, by the logical reductio ad absurdum process, someone would be sure to point out that, on such a premise, one who has sleeping sickness and sleeps for months on end is really in the best of health. The term also refers to a type of reductive-deductive syllogism:
    Major premise: Either A or B is true.
    Minor premise: A is not true.
    Conclusion: B is true." (A Handbook to Literature, 10th ed. Pearson, 2006)

James Jasinksi

  • - "This strategy is illustrated in a Dilbert cartoon from April 1995. The pointy-haired boss announces a plan to rank all of the engineers 'from best to worst' so as 'to get rid of the bottom 10%.' Dilbert's co-worker Wally, included in the bottom 10%, responds that the plan is 'logically flawed' and proceeds to extend the range of his boss's argument. Wally asserts that the boss's plan, if made permanent, will mean continual dismissals (there will always be a bottom 10%) until there are fewer than 10 engineers and the boss will 'have to fire body parts instead of whole people.' The boss's logic will, Wally maintains (with a touch of hyperbole), lead to 'torsos and glands wandering around unable to use keyboards . . ., blood and bile everywhere!' These horrendous results will be the consequence of extending the boss's line of argument; hence, the boss's position should be rejected."
    (Sourcebook on Rhetoric: Key Concepts in Contemporary Rhetorical Studies. Sage, 2001)

Walter Sinnott-Armstrong and Robert Fogelin

  • "[A] reductio ad absurdum argument tries to show that one claim, X, is false because it implies another claim Y, that is absurd. To evaluate such an argument, the following questions should be asked:
    1. Is Y really absurd?
    2. Does X really imply Y?
    3. Can X be modified in some minor way so that it no longer implies Y? If either of the first two questions is answered in the negative, then the reductio fails; if the third question receives an affirmative answer, then the reductio is shallow. Otherwise, the reductio ad absurdum argument is both successful and deep."
    (Understanding Arguments: An Introduction to Informal Logic, 8th ed. Wadsworth, 2010)

Adams Sherman Hill

  • "An argument which can be answered by reductio ad absurdum is said to prove too much--that is, too much for its force as an argument; since, if the conclusion is true, a general proposition which lies behind it and includes it is also true. To show this general proposition in its absurdity is to overthrow the conclusion. The argument carries in itself the means of its own destruction. For example:
    (1) Skill in public speaking is liable to great abuse; it should, therefore, not be cultivated.
    (2) Skill in public speaking is liable to great abuse; but so are the best things in the world--as health, wealth, power, military skill; the best things in the world should, therefore, not be cultivated. In this example, the indirect argument under (2) overthrows the direct argument under (1) by bringing into view the general proposition omitted from (1) but implied in it--namely, that nothing which is liable to great abuse should be cultivated. The absurdity of this general proposition is made apparent by the specific instances cited.
    "The argument that games of football should be given up because players sometimes sustain severe injuries may be disposed of in a similar way; for horseback-riders and boating-men are not exempt from danger.
    "In Plato's dialogues, Socrates often applies reductio ad absurdum to the argument of an opponent. Thus, in 'The Republic,' Thrasymachus lays down the principle that justice is the interest of the stronger. This principle he explains by saying that the power in each State is vested in the rulers, and that, therefore, justice demands that which is for the interest of the rulers. Whereupon Socrates makes him admit that it is just for subjects to obey their rulers, and also that rulers, not being infallible, may unintentionally command that which is to their own injury. 'Then justice, according to your argument,' concludes Socrates, 'is not only the interest of the stronger but the reverse.'
    "Another example of reductio ad absurdum is furnished by the reply to the arguments which attempt to prove by means of an alleged cipher that Bacon wrote the plays attributed to Shakespeare. All the arguments adduced in favor of this proposition may, as its opponents contend, be used to prove that anybody wrote anything."
    (Adams Sherman Hill, The Principles of Rhetoric, rev. edition. American Book Company, 1895)

Religion, Philosophy, and Popular Culture

Reductio ad absurdum has also been used in a variety of areas, from the teachings of Jesus, the foundations of philosophy, and even popular TV shows, as these excepts show.

Joe Carter and John Coleman

  • - "Reductio ad absurdum is a good and necessary way to work through the logical implications of a position. Most of Plato's Republic is an account of Socrates' attempts to guide listeners to the logical conclusions of their beliefs about justice, democracy, and friendship, among other concepts, through extended bouts of reductio ad absurdum. The United States Supreme Court also used this technique when it handed down its ruling in the famous 1954 case of Brown v. Board of Education. . . . While reductio ad absurdum can lead to long and complex arguments, it is often quite simple and practically useful. Take the following conversation as an example:
    Mother (seeing her child take a rock from the Acropolis): You shouldn't do that!
    Child: Why not? It is just one rock!
    Mother: Yes, but if everyone took a rock, it would ruin the site! . . . As you can see, reductio ad absurdum can be remarkably effective, whether in complex judicial arguments or in everyday conversations.
    "However, it is easy to move from reductio ad absurdum to what some people call the slippery slope fallacy. The slippery slope fallacy uses a logic chain similar to that employed in reductio ad absurdum that makes unreasonable logical jumps, many of which involve so-called 'psychological continuums' that are highly unlikely."
    (How to Argue Like Jesus: Learning Persuasion from History's Greatest Communicator. Crossway Books, 2009)

Leonard, Penny, and Sheldon

  • Leonard: Penny, if you promise not to chew the flesh off our bones while we sleep, you can stay.
    Penny: What?
    Sheldon: He's engaging in reductio ad absurdum. It's the logical fallacy of extending someone's argument to ridiculous proportions and then criticizing the result. And I do not appreciate it.
    ("The Dumpling Paradox." The Big Bang Theory, 2007)

Christopher Biffle

  • "The basic idea of theargumentum ad absurdum isthat if one can show that a belief leads to an obvious absurdity, then the belief is false. Thus, assume someone believed that being outside with wet hair caused sore throats. You could attack this belief by showing that if it were true that being outside with wet hair caused sore throats, then it would also be true that swimming, which involves getting wet hair, caused sore throats. But since it is absurd to say that swimming causes sore throats, it is false to say that being outside with wet hair causes sore throats."
    (Landscape of Wisdom:A Guided Tour of Western Philosophy. Mayfield, 1998)

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Nordquist, Richard. "Reductio Ad Absurdum in Argument." ThoughtCo, Apr. 5, 2023, thoughtco.com/reductio-ad-absurdum-argument-1691903.Nordquist, Richard. (2023, April 5). Reductio Ad Absurdum in Argument. Retrieved from https://www.thoughtco.com/reductio-ad-absurdum-argument-1691903Nordquist, Richard. "Reductio Ad Absurdum in Argument." ThoughtCo. https://www.thoughtco.com/reductio-ad-absurdum-argument-1691903 (accessed June 8, 2024).

When You Prove Something Is True by Showing the Opposite Is Not (2024)

FAQs

What is proving something by showing the opposite? ›

Proof by contradiction (also known as indirect proof or the method of reductio ad absurdum) is a common proof technique that is based on a very simple principle: something that leads to a contradiction can not be true, and if so, the opposite must be true.

What is the meaning of proof by contradiction? ›

Certain types of proof come up again and again in all areas of mathematics, one of which is proof by contradiction. To prove something by contradiction, we assume that what we want to prove is not true, and then show that the consequences of this are not possible.

What is the rule of contradiction? ›

This method is based on the fact that a statement X can only be true or false (and not both). The idea is to prove that the statement X is true by showing that it cannot be false. This is done by assuming that X is false and proving that this leads to a contradiction.

What is the difference between proof by contradiction and counterexample? ›

A counterexample disproves a statement by giving a situation where the statement is false; in proof by contradiction, you prove a statement by assuming its negation and obtaining a contradiction.

What is the word for saying or showing the opposite? ›

Antiphrasis is the rhetorical device of saying the opposite of what is actually meant in such a way that it is obvious what the true intention is. Some authors treat and use antiphrasis just as irony, euphemism or litotes.

What is a word for proving false? ›

disprove. expose. invalidate. knock the props out (from under) negate.

How powerful is proof by contradiction? ›

In classical logic, proof by contradiction is completely valid, and just as strong as any other method. However, in some non-classical logics, it doesn't work at all. Two particular properties of classical logic may be absent: Every statement must be either True or False; no statement can be neither True nor False.

What is an indirect proof? ›

With an indirect proof, instead of proving that something must be true, you prove it indirectly by showing that it cannot be false. Note the not. When your task in a proof is to prove that things are not congruent, not perpendicular, and so on, it's a dead giveaway that you're dealing with an indirect proof.

How to master proof by contradiction? ›

Proof By Contradiction
  1. Assume the opposite of your conclusion. ...
  2. Use the assumption to derive new consequences until one is the opposite of your premise. ...
  3. Conclude that the assumption must be false and that its opposite (your original conclusion) must be true.

What are the three laws of contradiction? ›

(1) For all propositions p, it is impossible for both p and not p to be true, or: ∼(p · ∼p), in which ∼ means “not” and · means “and.” (2) Either p or ∼p must be true, there being no third or middle true proposition between them, or: p ∨ ∼p, in which ∨ means “or.” (3) If a propositional function F is true of an ...

What is the rule of paradox? ›

Wittgenstein stated his famous rule-following paradox as follows: “this was our para- dox: no course of action could be determined by a rule, because every course of action can be made out to accord with the rule.” This is the paradox that Kripke develops in this essay via the example of plus and quus.

What is the paradox of contradiction? ›

b) A contradiction occurs when both A (some truth claim) and NOT A (some false claim) are asserted. Both arguments are incorrect, resulting in a paradox. In other words, a paradox occurs when two statements cannot both be valid at the same time.

What is meant by proof by contradiction? ›

In logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction.

Can a proof be proven by counterexample? ›

In fact, a theorem that has a counterexample can never be proven to be true. A proof by counterexample is then used to prove the theorem false.

What is the problem with proof by contradiction? ›

Another general reason to avoid a proof by contradiction is that it is often not explicit. For example, if you want to prove that something exists by contradiction, you can show that the assumption that it doesn't exist leads to a contradiction.

What is inverse proof? ›

For a statement p → q , the inverse is ¬ p → ¬ q , where the symbol means "not." The order of the hypothesis and the conclusion remains the same, but they are both negated. For example, the inverse of the statement "if the sky is blue, then it is sunny" is the statement "If the sky is not blue, then it is not sunny."

What is the opposite of proving something? ›

Opposite of to establish a fact to be true. refute. disprove. contradict.

What is it called when you describe something as the opposite? ›

Some common synonyms of opposite are antithetical, contradictory, and contrary. While all these words mean "being so far apart as to be or seem irreconcilable," opposite applies to things in sharp contrast or in conflict. opposite views on foreign aid.

What is the word for proving something to be wrong? ›

The verb refute is to prove that something is wrong.

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