Talk:Reductio ad absurdum/Archive 1

Merge with Proof by contradiction?
Discuss. --128.62.37.246 (talk) 01:22, 19 January 2010 (UTC)

Reductio de absurdum?
One night while watching the Colbert Report, Colbert made a comment for comedic effect that seems to hold logical water, and I was wondering if there have been any formal definitions of such an argument. He and his guest were discussing child labor in China. The guest was saying that forced child labor is morally wrong and should not be tolerated. Colbert compared this to parents forcing their children to do chores around the house, saying that it is essentially the same thing but most would not claim it to be morally wrong. This specific example is not necessarily a good argument; while it does disprove the assertion as stated (that any child labor is intolerable), it is attacking a strawman in the process of fighting the intended assertion that forced child labor of this extreme is intolerable. However, the idea behind the debate tactic is sound: one can disprove certain arguments by applying their premise to another situation which yields a less absurd result. Is there any documented acknowledgement of this sort of argument? If so, what is it called? Should it be included in this article as the opposite of the common reductio ad absurdum? --Daniel Draco (talk) 17:22, 25 December 2009 (UTC)


 * Daniel, in my opinion that would be a classic use of the straw man fallacy. Basically it's where you modify a line of argument in a way that is ridiculous and then show how ridiculous your modified argument is, and from that conclude that the original argument is ridiculous. I couldn't think of any good examples, but the name is pretty self-explanatory: rather than attacking your target, you make a weak imitation stuffed with straw, and you attack your imitation. 203.217.150.69 (talk) 00:43, 1 February 2010 (UTC)

The Definition and the description of "Proof by Contradiction"
The first paragraph and the second paragraph were nearly redundant; and the first paragraph was not satisfactory, since it did not describe a genus of the 'particular kind of RAA' that is detailed in the second paragraph, but rather described a vaguer wording of the same. I adjusted the definition, but in the process made it nearly identical to the second paragraph.

I thence undid my modification and made a minor edit to the definition, changing "implications to a logical but absurd consequence" to "implications logically to an absurd consequence" in order to emphasize that the process of arriving at the conclusion is logical and not emphasize that the conclusion itself is logical (though it is). This is because, to the layman, the adjective 'logical' has implications of truth value while in the field of Logic, it has no such connotation since a conclusion can be logically derived from a false premise.

I also made a nonminor modification to the second paragraph, since it is not entirely accurate. Proof by contradiction refers specifically to the proof of a proposition by proving that it is impossible for it to be false, therefore it is true.

Thus I have corrected the redundancy.

As is stated in the first cited text, Reductio ad Absurdum is not the same as Proof by Contradiction but is rather a genus of Proof by Contradiction. My amendment to the first two paragraphs should serve to make that distinction clear.

I have also added a Pop Culture section to refer to the episode of the Big Bang Theory wherein Sheldon misuses and misdefines Reductio Ad AbsurdumSpezied (talk) 07:45, 7 July 2010 (UTC)

The Addition of a New Example
A more substantial, logical, clear, nonspeculative, and especially unoriginal example was needed than the current one. I have added a prop from Euclid, his elements being a fine source of Reductio arguments. I was sure to pick a prop that did not build too greatly on other props so that a reader unacquainted with the work would not be confused. I also did not fully describe the prop for simplicity sake, leaving out the construction and the process of deducing the inequality of the radii.

An illustration for the Prop would be nice, but I do not know of any royalty-free illustration.

I did not think it appropriate to completely remove the former example after the addition of my own, since it touches on other topics that mine does not. But the section does need improvement.

If the example of Ad Coelum needs explanation that it is also a strawman fallacy, then a new example is needed since explaining the flaws of an example destroys the clarity of the example. An example should be immediately patent if possible.Spezied (talk) 16:10, 7 July 2010 (UTC)

Addition of a simpler example
The examples currently on the page are somewhat confusing. The Neils Bohr example assumes that the idea that "the opposite of every great idea is another great idea" is in itself a great idea, which leads to reductio ad absurdum, but it isn't really clear why it's assumed that Neils Bohr's statement is a great idea, and even if there is valid reasoning behind that assumption, examples should be clear, and free of the need for background explanation of assumptions.

The ad coelum example seems to be one specifically addressing the legal use of reductio ad absurdum, so while I find it confusing, its presence makes more sense.

I added a more intuitive, plainer example of a reductio ad absurdum. While I would like to take out the Bohr example, I will leave that decision in the hands of others.125.54.141.237 (talk) 03:59, 3 February 2011 (UTC)


 * I decided to clean up the ad coelum example, which was written quite confusingly. However, in doing so, once I finally figured out what it was actually saying, I realized that it wasn't an example of where legal use of reductio ad absurdum differs from conventional use, but an example of where reductio ad absurdum can be used (in both legal and conventional senses) on the literal meaning of ad coelum, but not (in either a legal or conventional sense) on the legal meaning of ad coelum. This would be a great example to put on the ad coelum page, but it has nothing to do with reductio ad absurdum itself.
 * Specifically, the example said that:


 * 1) Literally, reductio ad absurdum can only be used to argue against self-contradiction.
 * 2) In legal and everyday use, though, it is used to argue against things which are merely undesirable.
 * 3) It presented the concept of ad coelum, saying that used literally, it would imply that people, by buying a plot of land, would own everything which fell in a cone from the center of the earth to infinitely deep in space, including all the planets.
 * 4) Both legally and in an everyday sense, that is undesirable, so you could use the non-literal sense of reductio ad absurdum against it.
 * 5) But that's not what ad coelum legally means.
 * 6) Ad coelum, legally, only extends as high as is necessary to reasonably enjoy ones land.
 * 7) Therefore, you can't use reductio ad absurdum, legally or in the everyday sense, on the legal usage of ad coelum.


 * Again, that's great, and interesting, but it sheds no light on reductio ad absurdum, nor on how its legally used, nor on whether its legal usage varies from the everday usage. It just educates people about the difference between the everyday definition of ad coelum, and the legal definition of ad coelum.


 * The only important point that it even brought up was that literally reductio ad absurdum can only be applied when propositions result in self-contradiction, but that legally and in everyday use it can also be applied when propositions result in "absurd" conclusions. If anyone can find some citations for this, and make a clear section which explains the difference between the literal and conventional/legal meanings of 'reductio ad absurdum', it would be appreciated. 125.54.141.237 (talk) 04:41, 3 February 2011 (UTC)

"The opposite of every great idea is another great idea."
This statement made by Niels Bohr has been used as an example here but does a great idea have to be true? Take the case of reductio ad absurdum itself. We use it to prove an assumption wrong but that doesn't mean that the assumption isn't important as it can facilitate better understanding of something.Krisinboots (talk) 09:26, 20 May 2010 (UTC)


 * I know what you mean, it just has to be taken cynically. The grammar police won't come take you away if you mistakenly label something great :)  It's a subjective contingency to claim that great ideas are true ideas, so this argument is not true for all instances in time and space, though it is for some.  —Preceding unsigned comment added by 67.221.84.189 (talk) 00:44, 4 April 2011 (UTC)

Niels Bohr Example Removed
The following example was removed, because it represents several fallacies committed by Sagan (grrrr), who clearly wished to misrepresent Bohr's unorthodox comments. The "opposite" does NOT mean a logical dichotomy where a great idea is or is not, or that only two discrete ideas exist in a logical set. The dichotomy is Sagan's deliberate strawman to create an easy target for his reductio.

What Bohr means is analogous to saying that "the opposite of heads is tails" for a coin. A great idea is entailed by appropriate premises which also (sometimes) generate a converse of equal greatness.

"Another example concerns the following statement, attributed to physicist Niels Bohr: "The opposite of every great idea is another great idea." Carl Sagan used a reductio ad absurdum argument to counter this claim. If this statement is true, he argued, then it would certainly qualify as a great idea - it would automatically lead to a corresponding great idea for every great idea already in existence. But if the statement itself is a great idea, its opposite ("It is not true that the opposite of every great idea is another great idea", provided "opposite" is a synonym of "negation" in Bohr's aphorism) must also be a great idea. The original statement is disproven because it leads to an absurd conclusion: that an idea can be great regardless of whether it is true or false.

BlueMist (talk) 19:50, 18 October 2011 (UTC)

Ontological argument
A new user removed the following section for being a "subjective and incorrect insult". The source given seems fairly legit, and in my opinion the idea that it's an insult is very POV. I'm not sure it adds anything to the article though, so I've not reverted it to see if anyone has an opinion either way first.

"The ontological argument for the existence of God, as it was originally stated by Anselm of Canterbury, is an example of an attempted reductio ad absurdum. "

BulbaThor (talk) 20:02, 3 April 2012 (UTC)


 * It seem to me, that the user's comment is uninformed and biased. He doesn't seem to like the point of the argument. The Ontological Argument is a famous and important deductive reductio. Whether it succeeds is controversial, so, the argument's mention adds much to the article's reader appeal. I'd hate to lose it. However, it is also so convoluted in detail that it may not be the ideal encyclopedic example.  http://plato.stanford.edu/entries/ontological-arguments/ or http://www.princeton.edu/~grosen/puc/phi203/ontological.html
 * BlueMist (talk) 02:00, 5 April 2012 (UTC)

Poor example
The argument against the "raising taxes raises tax revenue" proposition is rather poor. It contains this claim: "If taxes were raised to 100% of income, individuals would not work". This is a synthetic claim, and does not follow purely from logical reasoning. (Also, the implied factual claim is false, as there have existed social systems where individuals (e.g. slaves) received no monetary income but worked anyway for other reasons.) The second example is better because the claim "If taxes were lowered to 0%, no taxes at all would be collected" is true by the definition of terms and does not rely on any hypotheses about human behavior. Can someone modify the first example so that it does not rely on presumed facts? If this proves impossible, I suggest it be removed. Augurar (talk) 04:36, 13 June 2012 (UTC)


 * Actually raising taxes to 100% of income is a perfect reductio ad absurdum argument. None of the examples listed are actually reductio ad absurdum arguments because there's nothing absurd being done.  You have to take the thing to absurd proportions to demonstrate how the restrained (in comparison to your absurd measures) measures being advocated are false.  Also the effects must be related between the restrained use and the total or absolute (absurd) use.   — Preceding unsigned comment added by 173.238.208.230 (talk) 18:45, 18 February 2013 (UTC)


 * This article is incomplete. Reductio ad absurdum refers not only to the logical fallacy but also to the concept that a statement can be shown to be false if its truth would lead to a false or contradictory statement being accepted as true. See http://www.iep.utm.edu/reductio/ the Internet Encyclopedia of Philosophy by Nicholas Rescher and http://en.wikipedia.org/wiki/Proof_by_contradiction the wiki article referencing reductio ad absurdum as a more general case of proof by contradiction. 20 June 2012 — Preceding unsigned comment added by 69.209.201.216 (talk) 03:19, 21 June 2012 (UTC)


 * Yeah, I thought I remembered reading that "reductio ad absurdum" wasn't a logical fallacy, but just a method of demonstrating that an argument is false because you take it to a logical conclusion that is terrible/absurd. We've only one link here and it doesn't look that great. Hm... Byelf2007 (talk) 03:33, 21 June 2012 (UTC)

Better example
We seriously can't get a better example than the current one? I don't think a logical fallacy is a helpful example for people trying to understand the concept. It's hard to come up with a good example, but it might be better to use a claim that's patently absurd on its face, instead of a straw man of a legitimate argument. Ideally, of course, what we'd need is an example of the technique used in a real debate--only problem there is that we have an issue with neutrality, as some might argue, if its a topical issue, that the reductio argument is faulty, etc.

Here's a good example of argumentum ad absurdum. Smoking is bad for you. If it's not, try smoking fifty cigarettes a day, or a hundred or two hundred. can you imagine the consequences? You'd be coughing up stuff constantly and would probably get cancer very quickly. — Preceding unsigned comment added by 173.238.208.230 (talk) 18:48, 18 February 2013 (UTC)

In any case though, I think the one that stands isn't terribly helpful. —Preceding unsigned comment added by 99.255.195.165 (talk) 20:01, 20 April 2010 (UTC)


 * The current example regarding ad coelum is a mess, and the "great ideas" example is questionable. I find that beyond the first paragraph, the rest of the article just serves to confuse the reader, and if I hadn't known what a reductio was before coming here I'd probably just assume it's too complicated for me to bother and move on. Does anyone have an example that does not need to be followed by two paragraphs explaining how it isn't really a reductio, or that it might not be, or that it may be but only if X and Y but not Z, or whatever. - 83.233.147.12 (talk) 12:02, 22 January 2011 (UTC)

Text from previous article
Below copied from previous article. Only relevant to the common reductio ad absurdum, not the proof by contradiction:

However, Reductio ad absurdum is also often used to describe any argument where a conclusion is derived in the belief that everyone (or at least those being argued against) will accept that it is false or absurd. This is a comparatively weak form of reductio, as the decision to reject the premise requires that the conclusion is accepted as being absurd. Although a formal contradiction is by definition absurd (unacceptable), a weak reductio ad absurdum argument can be rejected simply by accepting the purportedly absurd conclusion. Such arguments also risk degenerating into strawman arguments, an informal fallacy caused when an argument or theory is twisted by the opposing side to appear ridiculous.

Humour The often humorous outcome of extending the simplification of a flawed statement to ridiculous proportions with the aim of criticising the result is frequently utilised in forms of humour. In fiction, seemingly simple and innocuous actions that are extended beyond reasonable circumstance to chaotic outcomes, typically by use of stereotype and literal interpretation, can also be categorised as reductio ad absurdum. See farce.

Example:
 * Prove: All positive integers are interesting.


 * Proof: Assume there exists an uninteresting positive integer. Then there must be a smallest uninteresting positive integer. However, being the smallest uninteresting positive integer is in itself interesting. Thus you have a contradiction and all positive integers are interesting.

Quotations In the words of G. H. Hardy (A Mathematician's Apology), "Reductio ad absurdum, which Euclid loved so much, is one of a mathematician's finest weapons. It is a far finer gambit than any chess gambit: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game."

In the first paragraph of the Quentin Section (Part 2: June Second, 1910) of William Faulkner's The Sound and the Fury, Quentin's father, Mr. Compson, gives his son a watch that has been in the family for many generations. His father explains, "It [the watch] was Grandfather's and when Father gave it to me he said I give you the mausoleum of all hope and desire; it's rather excruciating-ly apt that you will use it to gain the reducto absurdum of all human experience which can fit your individual needs no better that it fitted his or his father's". This example represents a corruption of the Latin phrase Reductio ad absurdum.

—Centrx→talk &bull; 19:19, 17 April 2009 (UTC)

Removed this due to irrelevance:

== Reductio ad ridiculum ==

Another everyday use of the term reductio ad absurdum even further removed from the pure logical sense simply asserts an absurd consequence without claiming any formal derivation from the proposition.

For example: "If X is true, then I am a monkey's uncle" Presumably we do not accept that the speaker is really a monkey's uncle, so the 'argument' presented invites us to reject the proposition "X" as equally untrue. This is not really a reductio ad absurdum at all any more, merely a reductio ad ridiculum - an expression of the speaker's opinion that takes a similar form to the strict logical argument.
 * Isn't this example more related to Ex falso sequitur quodlibet? BertSeghers (talk) 20:12, 13 January 2010 (UTC)
 * Yes: but only in the ablative, there.  Basket Feudalist 21:33, 31 March 2013 (UTC)

Mathematics Example in Intro
I'm curious about the mathematical example in the introduction. In my opinion that's not an example. It's more a statement of fact that's hard for people to understand because they struggle to grasp the concept of infinity. Sabre ball t c 14:31, 8 April 2013 (UTC)
 * To a person familiar with mathematics, I agree, this statement may seem self-evident. But formally, it is a theorem which can be proved from the definition of rational number as shown, using reductio ad absurdam.  -- Chetvorno TALK 19:14, 9 April 2013 (UTC)

Strawman vs RAA
I'm really not sure why the reference to the straw man argument is on this page because it seems to me (and I'm no expert by the way) that reductio ad absurdum is not at all the same as the straw man...


 * A "straw man" argument is always used to attack a position or course of action, by ostensibly showing the bad consequences if that position is upheld; that makes it a reductio ad absurdam argument. The ostensible form of the argument is: "Don't support A, because A will have this terrible result; support not-A instead."  It's a reductio because it "...demonstrates that a statement is false by showing that a false, untenable, or absurd result follows from its acceptance".


 * The difference is that the "straw man" argument is a false reductio, because the premise is misrepresented. The consequence given is not a consequence of the original premise, but of a slightly different premise.   The actual form of a straw man argument is: "Don't support A, because B will have this terrible result."   The argument is nonsense, a non sequitur.  The author of the argument hopes you won't notice this. -- Chetvorno TALK 05:18, 13 January 2013 (UTC)


 * Not really, a strawman is any time A says X and B responds to X'. It's possible that X is a premise in an argument that B attempts to prove false by constructing a reductio against X', but it's not necessary for this to be the case. — Preceding unsigned comment added by 182.48.140.193 (talk) 10:27, 23 March 2013 (UTC)
 * No, a strawman is always used to attack a position, not support it, check the defs. B argues that X' should not be supported because a "false, untenable, or absurd result" follows from it.  This makes it a reductio. -- Chetvorno TALK 13:01, 11 April 2013 (UTC)

New world Encyclopedia
My understanding was that the New World Encyclopedia is a mirror (or semi-mirror) of wikipedia, and so not acceptable as a source. Yet over half of this article uses that as its source. Unless someone objects, I'll delete these citations in a few days. Original Position (talk) 17:28, 16 February 2016 (UTC)

Removal of example from introduction
Rstrug has repeatedly removed this well-known sentence as an example from the introduction:
 * Society must have laws, otherwise there would be chaos.

Rstrug has given these reasons in his edit comments:
 * "Chaos" is subjective. There are a lot of ideas in Game Theory that argue some "laws" are actually natural outcomes.
 * This is not reductio ad absurdum. You begin with an "absurd" statement that leads to an contradictory conclusion. Chaos is not contradictory, it's deductive.

I think the example is fine and should be put back. Rstrug seems to be arguing about the content of the example. It seems to me he's missing the point, it is the form of the argument, not its content, that makes it a reductio. Whether or not anyone agrees with the sentence, or thinks it is a good argument, the form of the argument is still a reductio ad absurdam.

From the definition, a reductio argument "seeks to demonstrate that a statement is true by showing that a false, untenable, or absurd result follows from its denial". The statement that this example is trying to demonstrate is "Society must have laws". In support of this the sentence argues that if this is denied and society didn't have laws, "there would be chaos". The meaning of "chaos" in this sentence is easy to understand: crime, disagreements or fights between people, auto accidents because people don't know which side of the road to drive on, no prohibitions against offensive or injurious behavior,  no defense of contracts or private property. It is saying laws prevent conflict; society without laws would be in a continual state of conflict. This comes under the heading of an "untenable" result; the word "untenable" means unsustainable, unworkable, flimsy, weak, or shaky. It is clear that this sentence is arguing that the result of denying the premise would be an "untenable" society.

Many people, including me, might disagree with this argument. Not the point; it's still a reductio.-- Chetvorno TALK 23:08, 26 January 2016 (UTC)


 * http://scholarship.law.marquette.edu/cgi/viewcontent.cgi?article=3570&context=mulr
 * Chaos exists with laws. Order can exist without laws. This is not a reductio ad absurdum because it is not logical. Additionally, there is no citation. Rstrug (talk) 00:11, 27 January 2016 (UTC) rstrug
 * Chaos exists with laws. Order can exist without laws. This is not a reductio ad absurdum because it is not logical. Additionally, there is no citation. Rstrug (talk) 00:11, 27 January 2016 (UTC) rstrug


 * Again, the question is whether or not the form of the argument is a reductio ad absurdum, not who is for or against it. Thomas Aquinas doesn't address that issue in your reference.     And "...falling off the edge of the Earth" (first example) is logical?  None of the other examples in the introduction have citations.  Sounds like the only real argument you have against this example is that you don't personally like it. -- Chetvorno TALK 03:56, 27 January 2016 (UTC)


 * Chaos means different things in mathematics than in politics. In the context of the example it is the opposite of laws.  Bringing in complexity or chaos theory in this context is not sensible as we are using the common sense meaning of the term in that example.  It would be useful to have a citation however Snowded  TALK 07:20, 27 January 2016 (UTC)
 * Various forms of the sentence appear widely in writings about the philosophy of law:, , , ,  -- Chetvorno TALK 15:03, 27 January 2016 (UTC)


 * Here is a citation: https://en.wikipedia.org/wiki/Superdeterminism70.126.133.47 (talk) 04:07, 18 March 2016 (UTC)

Suggestion: Why not make it "Society must have laws, otherwise there would be social disorder."

The problem lies in the ambiguity of the words law and chaos, which makes the example elliptical ... Mathematicians overloaded the already perfectly unclear theological and philosophical uses of chaos with their own 'chaos' theories, where the word was specially intended to confuse all non-chaos theorists.

BlueMist (talk) 17:14, 27 January 2016 (UTC)
 * That would be fine with me. What do you think, ? -- Chetvorno TALK</i> 19:42, 27 January 2016 (UTC)


 * In Game Theory, cooperation can result in non-cooperative games (Nash Equilibrium). Not every participant will be in disorder; not every participant will be in order. Depending on arbitrary sizes, each system could be categorized as its own society. A society of two people cooperating in the face of adversity does not necessarily have to be dysfunctional. There is an extremely rare possibility that everyone on the entire planet finds a way to cooperate, but it's not impossible.


 * Additionally, technology and other forces can take the place of "laws." You don't need to pass a law to prevent most people from walking across a minefield naked.


 * Even in just philosophy, is free will considered chaos and disorder? Rstrug (talk) 21:54, 27 January 2016 (UTC)


 * Sorry, Rstrug, with your latest remarks you lost me. You now appear to lack the background in any of logic, philosophy, or mathematics to edit this topic. You're giving probabilistic, empirical, and deterministic jargon as your justification.
 * The way I see it, Reductio is a special case arising from application of binary logic. For a coin that is either heads or tails, if it is not heads then it must necessarily be tails. Absurdum then says that the only other option is impossible (in math) or absurdly wrong (in philosophy). BlueMist (talk) 23:52, 27 January 2016 (UTC)


 * And your ad hominem defines your appearance. It's not impossible. Take the corollary. You can pretend people are not breaking the law all you want in your "binary" world, but they still exist. Back to the original criticism. Watch something like the Hunger Games -- even in a free for all, people still cooperated. There was no "chaos" between those groups of people. Rstrug (talk) 15:47, 28 January 2016 (UTC)


 * "Honor among theives." "Nearly 97 percent of those bets will be wagered illegally." Rstrug (talk) 16:28, 28 January 2016 (UTC)


 * No one seems to agree with you. And again, you're missing the point.  The question being discussed here is not whether "Society must have laws, or there would be chaos", but whether this sentence is a reductio ad absurdam.  It doesn't matter what you think of the argument, much less what is in the Hunger Games.  It is still a reductio because it "seeks to demonstrate that a statement is true by showing that an untenable result follows from its denial".  The fact that you disagree with the argument has nothing to do with what form of argument it is. -- Chetvorno <i style="color:purple; font-size:smaller;">TALK</i> 16:47, 28 January 2016 (UTC)


 * This is a logical argument that has to make logical sense for it to be logically true. This is not an English class where you can just pop any P and any Q into a sentence and expect it to always work. I don't actually care if people disagree with me. Go ahead and keep living in your fallacious paradox. I did my good deed for the day. If you MUST put this ridiculous assertion back into the article, it MUST have a [citation needed] right next to it. I want to see a complete deductive proof that leads to an explicit contradiction. Rstrug (talk) 17:56, 28 January 2016 (UTC)


 * The Roman Empire had laws. And collapsed into chaotic anarchy. Pretend that didn't happen. I dare you.Rstrug (talk) 18:00, 28 January 2016 (UTC)


 * Reductio ad absurdum isn't common sense. Reduction to the average is common sense.Rstrug (talk) 18:36, 28 January 2016 (UTC)


 * If you had read the part of the introduction which you are editing, you would have seen that a proof by contradiction, in which it is shown deductively that denying the premise results in a logical contradiction, is not the only type of reductio. That was the whole point of the paragraph.   The word "absurd" is not synonymous with "contradictory", it can also mean "unworkable" or "impractical" as it does here, and rhetorical arguments don't have "proofs".  And did you notice the word "seeks" in the phrase "seeks to demonstrate" in the definition?  -- Chetvorno <i style="color:purple; font-size:smaller;">TALK</i> 07:19, 29 January 2016 (UTC)


 * Define the size of society. An individual can function without the rule of law. A married couple can as well. You can extrapolate form there. Stop pretending this entire school of thought does not exist: https://en.wikipedia.org/wiki/Anarchism. Additionally, here is an entire field of study about people behaving in a predictable manner WITHOUT LAWS: https://en.wikipedia.org/wiki/Game_theory. Rstrug (talk) 22:33, 16 June 2016 (UTC)

This is not an article about the rule of law or civilization or anarchy, this article is about logic. Discussions of these subjects should be made on Rule of law or Anarchy. The article is not pushing the idea that "Society must have laws, otherwise there would be chaos"; it is merely using this common sentence as an example of an argument technique. This is made clear in the introduction. Like you, I don't happen to agree with the sentence, but it is a perfectly good argument which is widely used, and a good example of a reductio. Are you really afraid that just because someone reads that common cliche here he will be inspired to change his entire attitude toward laws? Do you really believe that the way to promote ideas we support is to censor all opposing views out of Wikipedia, even innocuous examples like this in unrelated articles? Come on. -- Chetvorno <i style="color:purple; font-size:smaller;">TALK</i> 00:13, 17 June 2016 (UTC)

You're already ad homineming. There is a list of anarchist communities that existed: https://en.wikipedia.org/wiki/Anarchy. Reductio ad absurdum is not a syllogism. Rstrug (talk) 12:30, 1 July 2016 (UTC)

I have cited other articles from Wikipedia like 10 times. All of those citations have been reviewed by more Wikipedia editors than this piece. Why are you guys pretending you don't see the contradictions? Like seriously. Rstrug (talk) 19:29, 1 July 2016 (UTC)

`We've never seen aliens before therefore reductio ad absurdum aliens are impossible.` That's your argument. It's not an argument. And don't you dare say I started this edit war. It was removed for quite some time. Rstrug (talk) 19:47, 1 July 2016 (UTC)

A statement is not an argument. Reductio ad absurdum is a type of argument. Ergo, the statement cannot be a reductio ad absurdum. BabyJonas (talk) 01:33, 14 July 2016 (UTC)

I don't see how this is even supposed to be an example of a reductio. A reductio is an argument of the form "if [some thing], then [absurdity]; (obviously) not [absurdity], therefore not [the thing]." The example given isn't even a conditional, much less a modus tollens syllogism like that (I'd be OK with omitting the second premise as obvious and the conclusion as tacitly implied from the premises). Something like "if there were no laws, then there would be chaos (and obviously we can't have chaos, therefore we must have laws)" might count as a reductio, except that whether "chaos" is an absurdity or not is a matter of some debate. If we allow this kind of argument to count as a reductio, then "if [policy I don't want] then [thing I don't like]", or "if [thing I don't believe] then [other thing I don't believe]", or basically any modus tollens, would be a reductio too; and obviously, that would be absurd, therefore we can't allow this kind of argument to count as a reductio. --Pfhorrest (talk) 22:44, 14 July 2016 (UTC)
 * I also find it unconvincing as reductio ad absurdum because "chaos" is not absurd, it's just a state of affairs. The premise that chaos is acceptable could be taken as absurd (that's a point of view, of course, but that's not the point), so "society must have laws, because otherwise it means that chaos is acceptable" would be a reductio.  But that is not the same argument, at least not exactly. --Trovatore (talk) 20:20, 17 July 2016 (UTC)

Straw man argument
Why is this here? Just because it is someone's favorite fallacy and they want it on this page as well as it's own? There are any number of ways to reject a hypothesis. Reductio ad absurdum focuses on the fact that *good* arguments based on the hypothesis lead to something obviously false. "Straw man" really has nothing to do with it. 50.202.216.74 (talk) 20:25, 18 November 2014 (UTC)


 * I've just made a brief attempt to clean up the "Straw man" section. The two sources cited in this version do bring up "straw man" in connection with reduction ad absurdum but they seem like very mediocre sources. Without some better sources and a stronger connection between "straw man" and the topic, then we should probably remove the section. —Ben Kovitz (talk) 02:09, 17 August 2016 (UTC)

Recent changes to introduction
An editor has been repeatedly adding this to the introduction as a general definition of reductio ad absurdam:
 * "if A then both B and not-B, so not-A" and "if not-A then both B and not-B, so A".

I don't think this belongs in the introduction. First, many general readers are not going to understand symbolic logic. Second, the addition falsely implies that this is a general example of the form of all reductio arguments. It's not. This is a particular type of reductio called proof by contradiction, which is already illustrated by the "positive integer" example in the introduction. This example is a better way to introduce the concept than the above abstract sentence, which is going to be meaningless to many readers. -- Chetvorno <font color="Purple">TALK  18:56, 19 November 2014 (UTC)


 * Anyone else have an opinion about this?-- Chetvorno <font color="Purple">TALK  18:09, 6 December 2014 (UTC)


 * I agree: that example was unnecessarily abstract and obscure. The concept of proof by contradiction should be mentioned, maybe even covered in the body of the article, but a plain example is much better. Even an abstract, mathematical example could work if introduced properly, but right after the opening definition is not the place for it. —Ben Kovitz (talk) 02:24, 17 August 2016 (UTC)

Supposedly
I'm not trying to make it clear that that statement is hypothetical; I'm making it clear that there really isn't a smallest positive rational number, and that's because, as I made clear on Paul August's talk page, the stated result would not follow from the premise "if there were" (i.e. if there really were). If there were, it would not be able to be divided by two to get a smaller one. Esszet (talk) 13:06, 5 September 2016 (UTC)
 * As Paul explained to you, adding "supposedly" in addition to "if" is redundant; it changes the meaning, giving the implication that the entire proof is uncertain or in doubt. The existing sentence follows the standard form for this type of reductio: "A is not true, because if A were true then [contradiction]".  Look at the other examples on this page, and on Proof by contradiction.  They only have one conditional, "if", not two. -- Chetvorno  <font color="Purple">TALK  17:35, 5 September 2016 (UTC)
 * Sorry about the very long delay, but I'm not adding another "if" and I'm not casting doubt on the entire proof, I'm casting doubt on the hypothetical condition because it could never really be satisfied. The form is exactly the same; as it is, it would have to read "There is no smallest positive rational number, because if there were, it could not be divided by two to get a smaller one". As it is, the given conclusion does not follow from the premise. Esszet (talk) 19:53, 2 October 2016 (UTC)
 * Oh wait, I think I get what you're saying, the "supposedly" isn't part of the standard form, and that inference is thus out of place. The issue I have with the correct form (i.e. with the "not") is that it may not be entirely clear to people who don't know much about math; a different example altogether may be best in this case. Esszet (talk) 20:11, 2 October 2016 (UTC)

Conflated with proof by contradiction
The recent addition by 173.67.38.174 struck me as being somewhat contrary to the meaning of reductio ad absurdum and I see that confusion occurring again and again in this article. In an earlier discussion (archived) about whether to merge the two articles, I found this:

While I recognize that proof by contradiction is seen by many as related to or even a sub-species of reductio ad absurdum, I agree that there are good reasons to have separate articles for them and to have examples for this article that are not simply examples of proof by contradition. Since the article discusses some of Socrates's arguments as recorded by Plato as early examplars of this technique, perhaps we can choose one to use as the primary example here. The flat earth example is terrible (more straw man than reductio} and the mathematical one is more proof by contradiction. The main drawback I see with this approach is that Socrates's arguments can get pretty drawn out and digesting them may leave out something important, all the while confusing the reader.

Do we still have a considerable population of editors who see no difference between the two? Anyone want to help?  &#8212;jmcgnh  (talk) (contribs)  06:29, 11 January 2017 (UTC)
 * I don't suppose that many of the repeated contributors to this article are confused about the distinction. Did you have a concrete change in mind? &mdash; Charles Stewart (talk) 14:56, 11 January 2017 (UTC)


 * On the recent unsourced addition by 173.67.38.174
 * "Greek mathematicians used the technique to prove fundamental propositions which did not have any preceding propositions to prove the antithetical or superposition arguments."
 * I understand what he's trying to say, that proof by contradiction was used when the Greeks couldn't find supporting propositions to build a direct proof, but it is certainly confusingly worded. What is a 'superposition argument'?   If it is to be kept, this needs to be sourced and rewritten so it is comprehensible. -- Chetvorno  <font color="Purple">TALK  09:33, 12 January 2017 (UTC)
 * To support the above statement you cited Miller, p.20 I don't see any reference to reductio ad absurdum on p.20; besides, this page is simply an analysis of Euclid's Proposition 4 and says nothing about Greek mathematics in general.  In fact, I don't see anything about reductio in the entire paper. Chetvorno<i style="color: Purple;">TALK</i> 04:20, 24 December 2017 (UTC)

(talk)


 * Hello I am the IP in question. In an addition today, I tried to clarify my meaning and updated the sources. First of all, I suppose the only citation you would have required would be one that MENTIONED reductio ad absurdum, which I gave, in an online article detail a Euclidean proposition for which it is used. There is that, and also the book by Nathaniel Miller. Obviously nowhere in the book is reductio ad absurdum mentioned, but I made an addendum on the wikipedia which was almost all but deleted today (go ahead and check it in the history) for which was said how superposition relates to reductio ad absurdum. I will type it here again in the hope the article gets amended
 * Greek mathematicians proved fundamental propositions, which did not have any preceding propositions, to prove the reductio ad absurdum and superposition arguments; this is done through proving that a line or angle cannot be in any other length or relation in the first case and in the second case that all lines and angles cannot be in any other position at all
 * This strikes me as an important addition because it does a few things A). It defines a term which really SHOULD have a page on wikipedia, the mathematical term of superposition B). It helps the reader understand this is NOT a 'proof by contradiction', which as Openfuture referred to is utilized far too much in this page (my vote goes to changing the numerical example at the beginning) C). It actually clarifies the entire entry, which is what one of the current footnotes is in regards to -- athenaswill <font color="Purple">TALK '' 11:11, 23 December 2017 (UTC)


 * On the subject of whether proof by contradiction is the same as reductio ad absurdum; they are not the same because contradiction is not synonymous with absurdity. OpenFuture's unsourced argument above is erroneous, although his conclusion is correct.    According to these sources, ,  proof by contradiction is a specific type of reductio limited to mathematics and symbolic logic, in which it is shown that the premise implies a logical contradiction: two statements that contradict each other: $$Q \land \lnot Q\;$$.  However, in rhetorical logic the conclusion of a reductio can take a broader range of forms.  As these sources , ,  say, a reductio can take at least two other forms:
 * the premise implies a patently ridiculous or absurd conclusion (reductio ad ridiculum), i.e. "The Earth cannot be flat, otherwise we would find people falling off the edge."
 * the premise implies an untenable (impractical, unworkable, or intolerable) conclusion (reductio ad incommodum), i.e. "Society must have laws, otherwise social order would break down and there would be chaos." or "Motor vehicles must have brakes; leaving them off would result in disaster on the roads."
 * The introduction originally had three examples illustrating this range of types, and my preference would be to see three examples restored, although I am flexible on which examples are used. -- Chetvorno  <font color="Purple">TALK  09:33, 12 January 2017 (UTC)


 * It would have helped the clarity of this discussion if the new additions had been neatly added at the bottom instead of somewhere in the middle. I still think the flat earth example is not an example of RAA. I don't see how bringing up the superposition argument in any way relates to RAA. I like the Rescher example and think we could use something like it to add to this topic.  &mdash; jmcgnh  (talk) (contribs)  05:52, 24 December 2017 (UTC)


 * A few things. First of all, I apologize for not adding the entry at the bottom of the page. Second of all, you are completely right about the flat earth theory, but oddly enough, the second is also not an example of RAA. In this way, both jmcgnh and Openfuture have made great points about the two bullet points at the top of the page. The page moderator should remove these soon. This is besides the main point at hand though, superposition is by its very nature Reductio ad absurdum. We're lucky this detail is at the beginning of Euclid's elements, therefore it is easily explicated by many. The example posted from the 1996 webpage detailing Euclid's proposition shows how one side of a triangle cannot be any other length than the one defined by the subtending angle on a triangle with two congruent angles. Based on this, no matter what the position of the triangle, that side has to be the same. Thus it is a reductio ad absurdum. Much like this, superposition posits that because three sides are the same length or, in the case of proposition at hand, two sides of the triangle and one angle define the remaining sides and angles of a triangle. Euclid posits that the side could be in another position (another length), which is impossible due to the definitions and a common notion. In this way, I tried to illustrate how similar these two concepts were and link them. Hopefully we can bring back the original edit to the article, as I thought it a simple way to illustrate the concept. At the very least we could drop off the superposition altogether, but I thought this a nice way to show multiple methods of Reductio ad absurdum in both Euclid and Archimedes. And if you would like a source of Archimedes utilizing the concept I can easily do that as well. Thank you for the consideration. Regardless, those two bullets at the beginning need to be dropped. -- athenaswill TALK   6:26, 24 December 2017 (UTC)  — Preceding unsigned comment added by 72.71.132.54 (talk)


 * We're all "page moderators" in the sense I think you intended, but moving boldly on this topic tends to be contentious so we try to first assemble consensus (sometimes over months or years of discussion) here on the talk page. In neither the Clark nor the Miller source is there any sense of this method being a reductio. Miller comments, in particular, that the bracketed interpolation (which does resemble a reductio) is considered a later addition and not part of the method of superposition. So saying Thus it is a reductio ad absurdum., when the sources don't, constitutes an unallowed WP:synthesis.


 * What I like about the Rescher example is that a proposition is revealed to be false by way an obviously absurd concocted example of something that would follow if the proposition were true. What I don't like about the flat earth example is that it imports "edges" and "something that would cause people to fall", when neither is absolutely part of the concept of the earth being flat. Some additional argument is needed for the example to make sense as an RAA. Maybe there is no edge? Maybe leaping off an edge is no more effective than leaping straight up? It would be good to have valid examples of RAA that would help show that not just any argument ending in an absurdity is an RAA.  &mdash; jmcgnh  (talk) (contribs)  07:58, 24 December 2017 (UTC)


 * Both the Rescher and the Internet Encyclopedia of Logic make clear that reductio ad absurdum covers a wider range of arguments than just logical or mathematical proof by contradiction, it covers "rhetorical logic". The latter source defines it as a "mode of argumentation".  The definition says the conclusion can be "ridiculous, absurd, or impractical", it's not limited to the logically impossible.  This range of types is why we need multiple examples, and they can't be limited to geometrical or mathematical examples like Euclid.  jmcgnh, you can always pick holes in any rhetorical argument.  The example of people falling off the edge is a common "mode of argumentation" used to ridicule the idea of a flat earth.  I think both the examples in the intro are good; the "flat earth" is clearly an example of a "reductio ad ridiculum", and the second is clearly an example of a proof by contradiction.  We could even have a third that demonstrates a conclusion which is "untenable" or "impractical"; something like "Society must have laws, or civilization would descend into violence", or "Government regulations requiring brakes on vehicles are necessary to prevent carnage on our roads." --Chetvorno<i style="color: Purple;">TALK</i> 10:46, 24 December 2017 (UTC)


 * I think this recently added sentence should be removed:
 * "Greek mathematicians used the technique to prove fundamental propositions which include the preceding propositions to prove the antithetical or superposition arguments."
 * This confusing run-on sentence does not make clear to readers whether "preceding propositions" refers to the Greek geometry proofs or propositions in this article. The term "antithetical or superposition arguments" is not explained, and anyway superposition proofs do not use reductio.  The sentence is redundant as the following section already discusses the use of reductio in Greek mathematics.   I agree with jmcgnh; I don't see that superposition proofs, for example Euclid, Book 1, Prop. 4, have anything to do with reductio ad absurdum at all.  No propositions are being assumed false; superposition just consists in using a correspondence between the angles and sides of two figures, to deduce the equality of other sides and angles.--Chetvorno<i style="color: Purple;">TALK</i> 12:08, 24 December 2017 (UTC)


 * If the bracketed part is, as you said, questioned as an interpolation by Miller, then no need to include superposition at all. Lets remove that source, superposition, and keep the reductio ad absurdum comment. I've amended the sentence so there is no superposition, and changed the word antithetical because that makes no sense at all regarding geometry. If you would still like to remove the sentence because Aristotle is metioned in the next section, that is a different matter entirely --athenaswillTALK 4:14, 24 December 2017 (UTC)
 * Reductio arguments in mathematics and geometry are called proof by contradiction. A proposition is proved true by showing that its negation leads to a contradiction.  These are discussed in the next section, so Euclid should be mentioned there. Chetvorno<i style="color: Purple;">TALK</i> 17:49, 24 December 2017 (UTC)

Postmodernism Reductio Ad Absurdum
I don't know why this text has been removed, but it was on the page for quite some time without a problem. I'm being blamed for edit warring, but I've been told in the past that removing content is the edit war. Which is it? This sounds incredibly contrived.

The following text is 100% a Reductio Ad Absurdum:

"Postmodernism says there are no absolute truths, meaning that there is always an exception to any assertion. This in itself is an assertion or an absolute truth. The only possible exception, since there is always an exception, to there's always an exception, is except when there's no exception." Rstrug (talk) 19:46, 11 November 2019 (UTC)


 * That's a cute tongue twister, but it's not going to be understood by the general readers who come to this page for a simple explanation of the subject. If you want to know why the other 3 editors have reverted it, read their edit comments.  You have been repeatedly adding this against the consensus of 4 editors, that's edit warring. --Chetvorno<i style="color: Purple;">TALK</i> 20:50, 11 November 2019 (UTC)


 * For more information please see BOLD, revert, discuss cycle and Edit warring. I also put a welcome post on your talk page with many useful links about what kind of material to include in articles and how to include it.—Anita5192 (talk) 21:00, 11 November 2019 (UTC)


 * I would actually call that a rather extreme example of circulus in probando. A tongue twister to be sure, but it's not an example of argument to absurdity, where the conclusion is well beyond the scope of reality. This tongue twister may in fact provide a valid conclusion, but the conclusion is the same as the premise; it's a long-winded way of talking in circles. (ie: "Every rule in English has an exception, except the rule that says every rule has an exception." This seemingly presents a paradox where none truly exists, and you can just keep going 'round and 'round.) Zaereth (talk) 22:50, 11 November 2019 (UTC)


 * That's 5 editors against it. --Chetvorno<i style="color: Purple;">TALK</i> 16:28, 13 November 2019 (UTC)

on Logical Fallacy
This section suggests that a fallacious RAU argument is shown when we reduce the working hours to zero. But the formula for productivity then requires a division by zero.

I suggest, this is an introduced absurdity, not essential to the argument, which should not just be assumed to require it. The arguer should be allowed to make an argument, that reducing working hours increases productivity, without requiring thim to explicitly state, Except for zero hours.

Actually, the section is ambiguous. It might be already saying what I just said. Davrids (talk) 21:23, 3 December 2019 (UTC)


 * That section is entirely nonsensical and should be deleted in my opinion. Most of it is unsourced and, what is sourced is not supported by the sources. The first source is used to cherry pick three words, "In common speech..." What the source actually says is "In common speech the term reductio ad absurdum refers to anything pushed to absurd extremes". The seond source, to finish the sentence, also says nothing of the kind. What is says is that reductio ad absurdum is used to show the fallaciousness of an argument, and then gives a similar example (if more sleep equals greater health, then by reductio ad absurdum one could point out that someone in a coma has the best health of all).


 * In the example within our article, it's reductio ad absurdum if someone uses the second part of the sentence to show the fallacious nature of the first part, but as written here, like a complete argument, it's a syllogistic fallacy and a type of false dichotomy. While the second part is absurd, the first part may be true in whole or in part. (ie: a happy worker is a productive worker, thus a reduction in hours may in fact increase productivity to a certain extent.) Zaereth (talk) 00:11, 4 December 2019 (UTC)


 * You know, upon further examination and looking deeper at the ThoughtCo source, I think what our author here is trying to say is that it's possible for an argument to be logically constructed yet still be fallacious in other ways, or that an argument can contain more than one fallacy at a time. But that goes for any argument that is poorly conceived, like the examples above. According to the source, when reductio ad absurdum consists of a long chain of events to reach the absurdity, one can easily stray off track and into slippery slope territory, although the two concepts are distinctly different things. Or, when poorly thought through, reductio ad absurdum can contain other fallacies that invalidate the argument, such as formal and syllogistic fallacies, etc... But that's a concept that is really true of any logic, so perhaps this is better suited to a more general article. Zaereth (talk) 02:44, 4 December 2019 (UTC)

Nothing about being nonconstructive?
Is it worth mentioning that an argument that ends in a contradiction is not constructive? (I suppose one could call it destructive though I've never seen it actually called that in any logic text.) Vaughan Pratt (talk) 03:11, 11 August 2020 (UTC)

Example is actually a straw man argument
The example given in the text "The Earth cannot be flat, otherwise we would find people falling off the edge." is a straw man. There is no reason why the earth cannot be flat without people falling of. According to some flat earth theories the edge is so far behind Antarctica that nobody has ever seen it. So the fact that we have no report of people falling of does not mean that it is not possible. And when it would be possible it does not mean that it must happen. It all depends on the location of the edge. The idea that people will start falling off the edge just because the earth is round is as true to the flat earth theory as saying that it is proven that the earth is flat because else people in Australia will start dropping of the surface is to believing in a round earth. (Note: I do not say anything about the actual shape of the earth, just that this argument is not a good example.) --195.240.98.246 (talk) 18:50, 14 June 2016 (UTC)
 * I agree. Both current examples are not very good in my opinion. The German Wiki has a decent (classic) example:
 * Not all people can be greek. Proof per Reductio ad absurdum: Lets assume all people would be greek. Then Cicero would also be greek. We know as a fact that he was roman.  As a person can not be greek and roman at the same time, the assumption must be wrong and the opposite must be true.     Simple, but has all needed elements in it. Please note that the reductio ad absurdum lies in the "can not be greek and roman at the same time" part. THAT is the absurd outcome of the opposite assumption (not the part where you just present one non greek person).--Soulman (talk) 13:17, 21 January 2017 (UTC)

Since Reductio ad absurdum is sound (unless you're an intuitionist or a constructivist), examples of it need to be sound. The problem with these examples is not so much that they're examples of one or another kind of fallacy as that they are simply unsound. Vaughan Pratt (talk) 03:25, 11 August 2020 (UTC)


 * That is too narrow a definition. If you look at the sources,  the term is not just used for arguments that denying the premise results in a falsehood, but also where it results in a ridiculous, untenable, or absurd conclusion.  The term is not just used in logic, but also for rhetorical arguments.  Both examples are clearly reductios. --Chetvorno<i style="color: Purple;">TALK</i> 04:46, 11 August 2020 (UTC)

I do not like the flat earth example. I do not like the "no non-Greeks" one even more.

Something clearly unrefutable, please?

Verba docent, exempla trahunt, and such.

Zezen (talk) 17:24, 19 August 2020 (UTC)

PM's formal definition of a reductio ad absurdum + Reichenbach + Tarski
After reading the confusions above it seemed time to do some research. What (historically) is a "reductio ad absurdum"?

This comes from PM (Principia Mathematica) 2nd edition (1927) page 100 paperback edition to *56:


 * “∗2.01 ⊦: p ⊃ ~p . ⊃ . ~p


 * "This proposition states that, if p implies its own falsehood, then p is false. It is called the “principle of the reductio ad absurdum,” and will be referred to as ‘Abs.”* [the proof follows . . . ]

We would write this today in the following way (the turnstile symbolizing “I Assert” or "It is true that")
 * ∗2.01 ⊦: (p → ~p ) → ~p

But Reichenbach 1947:38 (formula 1g) presents the reductio ad absurdum in the form of a logical equivalence, where the symbol ≡ is logical equivalence ←→ ; thus if we want to we can substitute (a ⊃ ā) for ā [he puts a bar over the a to indicate NOT-a], or vice versa:


 * a ⊃ ā ≡ ā

A truth table of the above reveals this equivalence. A truth table also reveals the equivalence of:
 * ā ⊃ a ≡ a

By itself these are not a very interesting notions. We need a syllogism (a chain of reasoning) to make them fun. At *3.35, after introduction of the logical AND (&), PM (re-)defines the first of two forms of “the syllogism” as [in modern symbols]:


 * ∗3.33 ⊦ ((p →q ) & (q →r)) → (p →r)

Going backward to a previous proof (in a note to *2.15 ) PM discusses the notion of a sorites, and defines it as a syllogism-structure [Syll] of any length:


 * “[Syll] ⊦ . (a) . (b) . (c) . ⊃ ⊦ . (d)


 * “where (a) is of the form p1 ⊃ p2, (b) of the form p2 ⊃ p3, (c) of the form p3 ⊃ p4, and (d) of the form p1 ⊃ p4. The same abbreviation will be applied to a sorites of any length.”

We now have the tools to construct an interesting reductio, keeping in mind that the chain on the left is really logically equivalent to the assertion ~p on the right:


 * ⊦. (p →a) & (a → b) & . . . & (k → ~p) ≡⊦. ~p

Alternately:


 * ⊦. (~p →a) & (a → b) & . . . & (k → p) ≡⊦. p

Here is an example (it wasn’t easy to produce BTW): We begin with a statement that we assert to be true “there is no need for laws” (~L) but end up via the sorites with the negation of the statement “there is need for laws” (L). From the fact that our argument consists of a sorites-chain that starts with ~L and ends with L, we can substitute L for the entire chain:


 * I assert that “There is no need for laws" (~L). However, If “there is no need for laws” (~L)” then “murder will become the norm” (M). But if “murder becomes the norm” (M) then “citizens will form a posse (aka police force) to protect themselves from murderers” (P). If “citizens form a posse to protect themselves from murderers (P)” then “there is a need for laws (L)”. As this chain is logically equivalent to “there is a need for laws (L), the original assumption of no laws (L) is false.


 * ⊦. (~L → M) & (M → P) & (P → L) → ⊦. L

So there we go. For the argument to work we need to believe or to demonstrate the truth of all the implications in the chain. Tarksi (1946 2nd ed:158; caps in the original) refers to the “so-called LAW OF REDUCTIO AD ABSURDUM of sentential calculus” in two forms:
 * [p → (~p)] → (~p)

and in the footnote to the above the alternate form
 * [(~p) → p] → p

He references G Vailati (1863-1909) [who] devoted a special monograph to its history” (ibid). [A reference to Vailati also appears in PM in a footnote on p. 100.] On the next page he notes that a proof that proceeds by use of the reductio ad absurdum form is called


 * “an INDIRECT PROOF or a PROOF BY REDUCTIO AD ABSURDUM. Proofs of this kind may quite generally be characterized as follows: in order to prove a theorem, we assume the theorem to be false, and then derive from that certain consequences which compel us to reject the original assumption.” (page 159).

Tarski seems to be saying that sometimes we can stop before the end because of the obvious falsehood of the consequences. But for a proper reductio we have to demonstrate the entire chain of logic and then prove by extra-logical means (evidence) that all the implications in the chain are in fact truths. Bill Wvbailey (talk) 19:28, 18 July 2016 (UTC)


 * Thanks for that interesting piece of research, Bill Wvbailey.
 * Thanks for that interesting piece of research, Bill Wvbailey.


 * We seem to have here another case of the elephant and the blind men. Each person insists that only their understanding is right and everyone else is wrong.


 * Broadly, there are formal applications, and there are informal, empirical applications. Reductio goes back at least as far as the Presocratics. Zeno, Socrates, and Plato employed a number of forms of the argument. This was all well before what is today's formal logic. BlueMist (talk) 20:23, 18 July 2016 (UTC)


 * I concur with Blue Mist, though I don't think I understand Wvbailey's point. Both the concept of reductio ad absurdum and the term are much, much older than Principia Mathematica. For a start, see this Google Books search. Russelll & Whitehead were invoking a well-known phrase when they named p ⊃ ~p . ⊃ . ~p "reductio ad absurdum". Is there a change to the article being proposed here? —Ben Kovitz (talk) 01:34, 17 August 2016 (UTC)

It is an elegant construction. I posit that we include it here, e.g. in the lead.

Zezen (talk) 17:29, 19 August 2020 (UTC)


 * Are you talking about putting the formal theorem above in the lead? I think it's inclusion would be WP:undue weight, too abstract and peripheral to the article.  It certainly should not be in the lead, as general readers won't be able to understand symbolic logic.   The appropriate place for this is Proof by contradiction, the article covering the formal use of reductio in logic.  This article is a broader article covering all uses of reductio including in rhetoric and informal argument.  --Chetvorno<i style="color: Purple;">TALK</i> 19:26, 19 August 2020 (UTC)

Straw man argument example

 * "For example, in a 1977 appeal of a U.S. bank robbery conviction, a prosecuting attorney said in his closing argument[6]
 * I submit to you that if you can't take this evidence and find these defendants guilty on this evidence then we might as well open all the banks and say, 'Come on and get the money, boys', because we'll never be able to convict them.
 * The prosecutor was tacitly equating the failure to convict the defendants in one particular trial with the inability to convict any bank robbers, a situation with self-evident unpleasant consequences but very little connection with the outcome of the trial."

In the USA, isn't it used the doctrine of precedents in a trial to determine a verdict on a similar case? If that is the case, the outcome of that trial in the example COULD (even though not as absurd as the attorney mentions) infer in the inability to convict defendants in future trials, right? 160.83.30.185 (talk) 17:20, 18 December 2013 (UTC)
 * To say that the precedent from one case would make it impossible to convict any bank robbers, is absurd enough to fall into the category of a "straw man". I think many "straw man" arguments could also be described as exaggeration or hyperbole.  If you look at the citation, I took this quote from the beginning of the "Straw man" chapter in the American Bar Association's litigation guide, so the bar association seems to feel it is a good example. -- Chetvorno  TALK   19:15, 18 December 2013 (UTC)


 * So, in order for it to be considered a straw man argument, it doesn't have to be actually false, just an exaggeration? That makes sense, thanks! Sekkuar (talk) 00:23, 25 December 2013 (UTC)


 * As I understand it, the consequence has to be false, or perhaps irrelevant to the discussion, to be a straw man. The source text for the above example implies that the point is that whatever consequence the verdict has as a precedent for future bank robbery cases is irrelevant to the current case, and should not be considered by the judges when deciding on their verdict. -- Chetvorno  TALK   08:47, 25 December 2013 (UTC)


 * The ordinary meaning of "straw-man argument" is an argument that attempts to refute a proposition but actually refutes a different proposition, which is trivially easy to refute. It's a form of equivocation. The example of setting a precedent that would make it impossible to convict bank robbers is not a straw-man argument. Maybe it's a slippery-slope argument, but we can't tell from the example alone; we'd have to know the facts of the case. I deleted the example. —Ben Kovitz (talk) 02:16, 17 August 2016 (UTC)


 * The "open the bank doors" argument is false, if only because difficulty in convicting robbers doesn't mean there's no point in putting security inside the bank. Indeed, the harder it is to convict, the more important it becomes to have effective security measures to stop the robbery from succeeding. So it's incorrect to say that "we might as well open all the banks". If the attorney had argued, "Banks will have to massively increase their security budgets, because we'll never be able to convict them after the fact," then that would be a more sound argument. The way it was actually phrased was presenting an absurd conclusion that didn't necessarily follow from the starting point. — Preceding unsigned comment added by Carl.antuar (talk • contribs) 03:40, 19 January 2021 (UTC)
 * The "open the bank doors" statement is not meant literally, it is just a hyperbolic way of saying that if the judge doesn't convict in this case the precedent will make it impossible to convict any bank robbers, so there will be no deterrent for bank robbery. That is the straw man. --Chetvorno<i style="color: Purple;">TALK</i> 04:44, 19 January 2021 (UTC)