# Talk:Proof by contradiction

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This is a decent article, though it is lacks any source.. Luckily, there seem to be several good web sites which can be cited for the majority of this article's content. Here's what a quick google search yielded: [1], [2], [3] Jellocube27 20:25, 11 March 2007 (UTC)

## Use of a straw man as an Example

I'm not attemping to argue for creationism here, but the Pastafarianism reference is definitely a straw man and not an reductio ad absurdum. It does not imply formal contradiction, and seems to be a gratuitious stab at creationists in the midst of the article for no reason. The fact that the example itself references its inability to be a formal argument ad reductio underscores this point. Creationism, while it may be absurd, would be backed up by its followers with 2000 years of theological argumentation, and more recently by some cavalier 'scientists' attempting to prove it as well. Pastafarianism is an obvious straw man argument, as it does not adequately represent the creationist's views.

More to the point, the entire argument about creation is a theological, and not philosophical one. It belongs to the realm of theology, because it pertains to the meaning of creation, especially when we get into creation of new religions (Pastafarianism...). Again, I'm not here to argue creationism or intelligent design or darwinism or evolutionary topics at all, I'm just pointing out that this example does not belong in a philosophical definition of a reductio ad absurdum, mainly because it does not give example of an actual argument reducing to the absurd (it is far closer to argument by analogy), and secondarily because even then it is a vast reduction of the creationist's argumentation to a pathetic appeal to authority, which they would obviously deny, and as such it is a straw man.

I've removed the example, feeling that this is a pretty clear case. If you disagree things can obviously be debated and changed.

71.58.54.96 19:30, 5 March 2007 (UTC) Jim Brown

## Comment by UserRyguasu

Can someone please explain this:

basically: if
S union { ¬ t } |-- F
then
S |-- t

I suspect the equivalent point can be made just as well in more widely known notation, perhaps even in plain-old English. --Ryguasu

seems that the english version of the above is in the first paragraph. -- Tarquin 10:34 Jan 10, 2003 (UTC)
could you at least explain what S and t are? It probably doesn't add clarity that A, used in the English version, does not figure into the symbols. --Ryguasu
Ok, thanks for the in-line explanation. I still think the TeX graphic and the textual description should use the same letter to represent "the statement". Changing the text might be easier, but I think, Tarquin, your choice of "t" for the statement in the TeX graphic is unfortunate, as it reminds the untrained eye of "true". Is this a routine convention for this sort of formal symbol representation? --Ryguasu
I would prefer A instead of t, for the sake of consistency within the article. After all, proposition F already has the form of a capital letter. --FvdP 20:35 Jan 10, 2003 (UTC)

F is not a proposition, it represents logical False. Propositions are small letters. small a could be ok. -- Tarquin 20:37 Jan 10, 2003 (UTC)

I should have known (and did realize, but a bit late.) OK for a. I've had the idea to replace F with a more common sign for "false", like $\bot$, but that would annoy non-mathematically readers more than it would be useful, IMO. --FvdP
Does it make sense to replace A with a (or pehaps a) in the body of the text as well? This would save us from including an overly philosophical sentence along the lines of "Take a to be the same as A." Also, if we put italic a in the text, can we make it italic in the graphic as well? --Ryguasu
I (boldly!) jumped in and relaced occurrences of "A", a, and t with p (for proposition), and relaced "B" with q. Apologies if the term "hypothesis" is being used in a technical sense here - h just seemed less obvious as a choice.
Ah, p ! That had become too obvious to be seen (by me). Thanks Chas for the change. --FvdP
I agree that $\bot$ seems unneccesarily obtuse here - I'd be just as happy replacing F with "false", i.e.:
$S \cup \{ \neg p \} \vdash false$
unless that's problematic. Also, the law of the excluded middle is being invoked here - do we need to add a link to intuitionism or the like here? Chas zzz brown 01:01 Jan 11, 2003 (UTC)
I have no definitive opinion on this. The interested reader can get the link through law of the excluded middle. Perhaps we should interest him more by telling a few words on intuitionism ? --FvdP
Even in intuitionistic logic, if $S\cup \{\neg p\} \vdash false$, one can still conclude that $S\vdash \neg\neg p$. The real difference is that in classical logic, $\neg\neg p \supset p$ but not in intuitionistic logic. So I don't think it's a major difference in this case, or particularly worth mentioning. Dominus

I don't like too much the given in

and S is a set of statements which are given as true

because it reads (to me) like we're just giving them the status of being true, out of the blue, which can't be. But I don't know what to write instead:

• and S is a set of true statements
• and S is a set of statements which are known to be true
• and S is a set of statements which have been proven to be true
• ...

--FvdP 01:05 Jan 11, 2003 (UTC)

But that ultimately is all we can say; that they are given (or is it taken?) as true "out of the blue". For example (loosely speaking), "parallel lines meet at infinity" is not inherently true - but if we take it as true then we get euclidean geometry. Alternatively, we could just as easily say that "parallel lines always intersect at exactly two points"; this would then imply a different geometry.
The statements in S are either a given set of axioms which we postulate as true, or theorems following from axioms which essentially comes to the same thing (in that we assume that these theorems really do follow from some axiom set A; i.e., that p can be deduced directly from A as well as from S). From this point of view, we don't so much prove that p is "True", as that p is true assuming that (i.e., given that) the statements in S are true (which generally has been previously proven if needed).
By my phrasing, I was hoping to distinguish between "Truth" and "truth". Philosophically, one can argue that certain axioms are "really" true, e.g. platonism; but it is not required that one hold this philosophical view to use reductio ad absurdum. Cheers Chas zzz brown 01:50 Jan 11, 2003 (UTC)
We can get rid of this "given as true" confusion by adding a second part to the if in the formal notation. It can now read
if
it is not the case that S |-- F
and
S union { ¬ p } |-- F
then
S |-- t
Does this deal with both side's objections? --Ryguasu
I have no idea what this is saying. -- Zoe
Do you understand the formal logic notation used elsewhere on this page? My comment was aimed at those who did. --Ryguasu

The formal notation is useless and should be relegated to a footnote. Why do I push this heresy? Because anyone who can read it already knows what a reductio ad absurdum is, and anyone who doesn't know what it is has got Buckley's chance of figuring out how to read :::$S \cup \{ \neg p \} \vdash false$ or any similar formula. A plain-text example in simple English is badly needed. This should come first, so that the non-expert reader can find it immediately, and be followed by the more formal explanation. Tannin 02:15 Jan 11, 2003 (UTC)

Yes, much improved now Chas. It would be better still if it were a non-mathematical example, but (to my shame) I can't think of one that's clear enough! Tannin 09:44 Jan 11, 2003 (UTC)

It's always nice when an article dramatically improves overnight! :-) The "smallest rational number" example is great. -- Tarquin 10:43 Jan 11, 2003 (UTC)~

Text from Reductio Ad Absurdum:
A method of disproving a proposition, by demonstrating that it leads to a contradiction.

An example would be the proof that the square root of 2 is irrational. Begin with the proposition that there exist integers M and N with no factors in common, where M*M = 2*N*N. If they exist, then the square root of 2 is the rational number M/N.

2*N*N is even, and therefore M*M is even. This implies that M is even, since an odd number cannot have an even square. Therefore K = M/2 is an integer. Then 2*K*K = N*N, and it follows that N is even.

The equation M*M = 2*N*N cannot hold unless both M and N are even. This contradicts the initial assumption that M and N have no factors in common.

I'd propose moving this article to Proof by contradiction and redirecting there. Reductio ad absurdum is the historical name used in philosophy and formal logic, but proof by contradiction is the more common English name and the one preferred in most modern mathematics articles. --Delirium 22:51, Oct 15, 2003 (UTC)

Interesting idea, but isn't proof by contradiction just one type of reductio argument? Granted a contradiction is an absurdity, but many reductio arguments work by exhibiting conclusions that are counter-intuitive or otherwise philosophically expensive (eg they come with heavy ontological commitments). For example, Zeno's paradoxes are reductio arguments but they aren't really proofs by contradiction. For that reason I personally would tend not to call the classical proof of the irrationality of root 2 a reductio ad absurdum but would use the more specific term. In general I think "proof by contradiction" appeals to formal inconsistency (ie deriving p and not-p), whereas reductio arguments can be less rigorous and still be effective. You could almost say that the conclusion can be "strongly absurd" (contradiction) or "weakly absurd" (unattractive). You can respond to a weak reductio argument by accepting it, if you wish to, and you are not commited to a contradiction by so doing, although you may have work to do to make a convincing case. I'm making some of this up so take it with a pinch of salt, but that's the way it seems to me. Ornette 15:45, 2 November 2005 (UTC)

I used to be appalled at seeing people (non-mathematicians, mostly) use the term to refer simply to an absurd argument, seemingly translating the Latin to "absurd reduction" instead of "reduction to the absurd". However, I've noticed that Merriam-Webster's lists a second (presumably less formal) meaning of the term as "the carrying of something to an absurd extreme". I'm curious to find out whether this is an instance of a sloppy translation gaining status as an acceptable meaning simply through widespread usage. If anyone can shed some light on the non-specialist use (or abuse) of the term, it could be an interesting aside to this article. rajneesh 04:15, 19 Aug 2004 (UTC)

I've never heard it misused like that. Note, however, that the mathematical use is not the primary one. It is mainly used in philosophy. Chameleon 09:06, 17 Nov 2004 (UTC)

I arrived at this page to find out if there were non-mathematical uses of this term. Specifically, where one tries to kill a project, by "improving" it to the point where it must fail. For instance, take a product that breaks even with costs of \$0.90, and a sale price of \$1, and improve it so that it now costs \$1.05 to make, has to sell for \$2, and now is a failure. Perhaps there is another term for that kind of effort. Mcr314 02:37, 4 May 2005 (UTC)

## Nazism

I'm out of my depth, so just wondered if anyone else felt that using Nazism in the article seems a bit odd, especially against the flat earth thing. --[[User:Bodnotbod|bodnotbod » .....TALKQuietly)]] 10:01, Dec 5, 2004 (UTC)

How is it odd? Whether you happen to agree with the Nazis or not, it is the typical example used in such reductions. So much so that Godwin's law has been humorously invented as a remark upon the phenomenon. Chamaeleon 01:43, 19 Jan 2005 (UTC)
I think what he means is that the Earth has been proven to be round, and thus the "world is flat" argument is itself absurd or illogical. Nazism may be morally reprehensible, but that doesn't equate to it being illogical. The illogical part is when A contradicts himself. Dforest 15:28, 30 October 2005 (UTC)
Even if it's a fact, it is still not a logical tautology that the Earth is round; the point is that the other speaker accepts it (third line). To assert "all beliefs are of equal validity and must not be denied" and "it is right to deny Nazism" is illogical. -Dan 16:24, 31 October 2005 (UTC)

I agree, I was struck the by inappropriateness of the Nazi reference as well. This is actually an NPOV issue; revolting though Nazism may be, some people believe in it and I don't think that how we feel about other people's beliefs has any place in an explanation of a logical argument. The flat earth example is less inflamatory and just as illustrative on its own. Ornette 15:25, 2 November 2005 (UTC)

It has nothing to do about how we feel about Nazism. Reductions are quite often made to fascism, socialism, communism, anarchy, genocide, and other inflammatory conclusions. By contrast, a reductio argument to the Earth being flat is rather unusual. -Dan 18:40, 2 November 2005 (UTC)

## Law of excluded middle

At least some references in the article to this law are inaccurate. The law of non-contradiction might be what was intended in those cases. To go from something like "If A, then something absurd; therefore not A." is perfectly constructive. Indeed "If A, then something absurd" is a common constructive reading of "not-A". The law of the excluded middle is only required for "Unless A, then something absurd; therefore A", and is non-constructive. -Dan 19:58, 22 August 2005 (UTC)

## Reductio creep

Would an article on "Reductio creep", a neologism coined by Julian Sanchez, be suitable for wikipedia?

The term is used to describe the following process: During the debate about doing X, opponent B says that if X is accepted, then Y would also be accepted, and isn't Y ridiculous? Subsequently, X is indeed accepted, and now serious debate about doing Y occurs. Opponent C then says that if Y is accepted, then Z would also be accepted, and Z is ridiculous. Then Y gets accepted, and serious debate occurs about Z, and so on... Thus there is a general erosion about what is absurd.

Julian coined it when he heard about someone suing the fast food industry. In his article, he said that when there was debate over suing tobacco companies, suggesting that people sue fast food companies for getting fat was considered a reductio ad absurdum, but that it was now a reality. Andjam 08:45, 25 October 2005 (UTC)

I've never heard of "reductio creep" or Julian Sanchez, but if you feel the term is commonly used, I say go for it. Even if not, it would be good to have here something in this article to the effect of "on the other hand, one might simply accept the logical consequences of one's ideas despite the absurdity" with the examples of the "all beliefs are equally valid" fellow accepting that Nazism is ok, and the fast food example. -Dan 16:50, 31 October 2005 (UTC)
"Reductio creep" sounds to be just another name for the "slippery slope" fallacy. 193.122.47.170 08:50, 22 May 2007 (UTC)
Or, from the political sphere, "defining deviancy down." I like that term "reductio creep", though. ~ CZeke 16:25, 2 August 2007 (UTC)
I like that idea too. Its a very good anecdote. I have never taken a philosophy class, so don't know if the term is 'scholarly' but its not slippery slope because, the form is prepositional. 76.4.141.131 (talk) 08:04, 29 June 2008 (UTC)

## “Reductio ad absurdum”, “material implication”, “contraposition”, and “self-contradiction”

Self-contradiction in reductio ad absurdum argument (“reduction to absurdity” --- in its strictest form, “reduction to self-contradiction” [please refer to Nicholas Rescher, “Reductio ad Absurdum” in Stanford Encyclopedia of Philosophy @Internet]) is inherent with the very definition of material implication --- with P true and ~P --> Q as well as ~P --> ~Q being both true at the same time so that ~P --> P (by contraposition and the transitive property of material implication) or with P false and P --> Q as well as P --> ~Q being both true at the same time so that P --> ~P (by contraposition and the transitive property of material implication). Contraposition (~Q --> ~P) is definitionally equivalent to material implication (P --> Q) --- their truth tables are identical. Moreover, contraposition checks infinite regress of reasoning — that is, one needs to justify P in P --> Q with O --> P, O with N --> O, N with M --> N, and so on ad infinitum but contraposition prevents the necessity for this infinite justifications so contraposition must be a “first principle” (not merely a “theorem”) just like the first principles of identity (P --> P), excluded middle (P OR ~P), and non-contradiction [~(P AND ~P)] (Aristotle’s 3 “laws of thought”) all of which are in fact embodied in the very definition of truth-functional logic (that is, Boolean or 2-valued logic wherein the truth-value of a compound formula is determined by the truth-values of its prime constituents).

A reductio ad absurdum (“reduction to self-contradiction”) proof goes either (~P --> P) --> P or (P --> ~P) --> ~P.

The first reductio ad absurdum tautologous scheme simply says that if one assumes P to be false and establish by logical reasoning (that is, some valid argument) that in fact P is true then P is actually true. By the very definition of material implication, the assumption that P is false means P --> ~P is true so that together with the “proven” antecedent ~P --> P they are equivalent to P <--> ~P which is equivalent to P AND ~P (a self-contradiction) from which (being false) any conclusion immediately follows (again, from the very definition of material implication).
The second reductio ad absurdum tautologous scheme simply says that if one assumes P to be true and establish by logical reasoning (some valid argument) that in fact P is false then P is actually false. By the very definition of material implication, the assumption that P is true means ~P --> P is true so that together with the “proven” antecedent P --> ~P they are equivalent to ~P <--> P which is equivalent to ~P AND P (a self-contradiction) from which (being false) any conclusion immediately follows (again, from the very definition of material implication).
What must be emphasized here is that ---
(1) reductio ad absurdum (“reduction to self-contradiction”) is a self-contradictory argument [that is, it is absurd to derive a self-contradictory proposition P inasmuch as this does not truly establish the truth or falsity of proposition P];
(2) P is a self-contradictory proposition or formula scheme [that is, P is not only false or true it is also true or false, respectively, at the same time]; and (3) P AND ~P is a contradiction [that is, it is always false for any truth-value for P] from which, used as an antedent, any consequent follows.
It is reiterated: in a self-contradiction --- P AND ~P --- it is the conjunction which is false while the proposition P itself is both true and false which violates the very definition of a proposition in truth-functional logic as being single-valued.

What this means is simply that reductio ad absurdum (“reduction to self-contradiction”) is ridiculous and absurd while the other cases of reductio (that is, not factually “reduction to self-contradiction”) may well be “valid reasoning” if: (1) in the case of the first reductio ad absurdum tautologous scheme, with the assumption that P is false, it is not _factually_ true that P --> Q and P --> ~Q (for any proposition or formula scheme Q) are both true at the same time; or, (2) in the case of the second reductio ad absurdum tautologous scheme, with the assumption that P is true, it is not _factually_ true that ~P --> Q and ~P --> ~Q (for any proposition or formula scheme Q) are both true at the same time.
First, consider the widely accepted “Euclid’s” proof of the proposition P about the infinitude of the prime natural numbers. An assumption is made, ~P, that there are only finitely many prime natural numbers. Now, there is no proposition Q such that both P --> Q and P --> ~Q are factually true at the same time --- thus, this is actually not a true reductio ad absurdum (reduction to “self-contreadiction” ) argument because there is no “self-contradiction” involved here. The initial supposition ~P (which references “infinity”) simply need not be stated --- this is called finitary argument: it simply asserts that given any finite list of prime natural numbers one could always find another prime natural number that is not in the list (this “no last element” scenario is to be taken as the meaning of “infinite”).
Next, consider the standard argument proving that 1/0 is not an element of the field of real numbers. An assumption is made, P, that 1/0 = c is a real number and it is argued that this leads to 1 = 0 [since 0 ٠ c = 0 for all real number c (this is easily derived from the field axioms)]. There is no “reduction to self-contradiction” here but only “reduction to falsehood” or “reduction to contradiction” [that is, the argument is merely reduced to the result 1 = 0 which is a contradiction --- it is false] so this argument is valid.
Next, consider Cantor’s diagonal argument “proving” the “uncountability” of the real numbers by first assuming that all the fractional real numbers are countable and they could be row-listed in the standard enumeration form x1, x2, x3, … from which a fractional real number not in the row-listing could be formed from the anti-diagonal digits. Now, this so-called “proof” is replete with so many self-contradictions and, thus, is an untenable reasoning ---
(1) the assumption that all the fractional real numbers could be row-listed uniquely and exhaustively in the standard enumeration form x1, x2, x3, … presupposes some list inclusion and imposition of order condition; but any of this specification is tantamount to the prescription that the nth-row number must have some particular digit at its nth-column position --- that is, the row-listing could be specified as such that the diagonal digits are all 0s, or all 1s, or alternating 0s and 1s, etc. Clearly, with the adoption of a prefixed fractional expansion point before the diagonal digits, the “real number” thus formed by the diagonal digits must be an “irrational number” because one can easily find an excluded fractional real number from the row-listing if it was a “rational number” (that is, one with a discernible pattern in its digits); hence, the anti-diagonal number must also be an “irrational number” that could not possibly satisfy ab initio its own row-list inclusion and imposition of order condition being different digit-for-digit to it;
(2) the diagonal-digits-with-prefixed-fractional-expansion-point “real number” is a variable --- it is not a true real number which is a constant;
(3) the row-listed fractional real numbers are mostly _intervals_ and not true real number _points_ (so Georg Cantor himself had to posit ordinal numbers --- in particular, omega as the first transfinite number, followed by omega + 1, omega + 2, and so on);
(4) the standard enumeration form x1, x2, x3, … and the fact that each fractional real number is an infinite sequence of place-value base digits (at least 2 for binary system) is a self-contradiction --- the non-standard enumeration (still countable!) form a1, a2, a3, …, b1, b2, b3, … is more appropriate for the row-listing;
(5) etc., etc., etc. . . .
Likewise, Kurt Godel’s argument which invokes the self-contradiction “This assertion cannot be proved”, Alan Turing’s argument which invokes the self-contradiction “a computer program halts if and only if it does not halt”, and many others that involve self-contradictions are untenable “proofs”.

Please read my Wikipedia discussion notes on “Cantor’s diagonal argument”, “Cantor’s theorem”, “Cantor’s first uncountability proof”, “Ackermann’s function”, “Boolean satisfiability problem”, “Entscheidungsproblem”, “Definable number”, and “Computable number”. (BenCawaling@Yahoo.com [14 December 2005])

One of the “absorption laws” of propositional logic or statement calculus is: (P --> (Q AND R)) <--> ((P --> Q) AND (P --> R)). Thus, (P --> (Q AND ~Q)) <--> ((P --> Q) AND (P --> ~Q)).

The formal argument in the “proof” of general Cantor’s theorem can be summarized as follows ---

If there is a 1:1 correspondence between S and P(S), then the generator of T is in T. [1]
If there is a 1:1 correspondence between S and P(S), then the generator of T is not in T. [2]
Therefore, there is no 1:1 correspondence between S and P(S).

The conclusion follows from the “belief” that propositions [1] and [2] are contradictory. It might be argued that the conclusion follows from the contradiction “the generator of T is in T” AND “the generator of T is not in T”. However, the “absorption law” equivalence could not be discounted — that is, the latter contradiction claim also asserts the former “contradiction” claim. Moreover, it is the claims P --> Q and P --> ~Q that are separately demonstrated in this type of “proof by contradiction” and not the claim P --> (Q AND ~Q).

The following discussion by Alice Ambrose and Morris Lazerowitz in their book entitled “Fundamentals of Symbolic Logic” (New York: Holt, Rinehart and Winston; 1962) is enlightening in pointing out the logical defect in the preceding Cantor’s reasoning ---

Any two propositions of ordinary discourse are related in one of the seven ways described (pages 85 – 92) [equivalence, superimplication, subalteration, subcontrariety, contrariety, contradiction, and logical independence]. Failure to understand their relationships is responsible for many of the fallacies in reasoning. For example, contradictories and contraries, contraries and subcontraries, are frequently confused, and propositions are sometimes supposed to be equivalent when they are not.
As an illustration of a further logical relation commonly confused, take the two propositions ---
If it rains, the crops will be good. [1]
If it rains, the crops will not be good. [2]
It might be supposed that these two propositions could not both be true, and that, hence, a person who made both these statements would be uttering an inconsistency. One needs merely to note that both propositions are true under the condition that it does not rain, to see that they are consistent with each other, and that therefore the supposition of their inconsistency is a mistake. These propositions are subcontraries.

In symbolic logic, both Georg Cantor’s argument as well as Alice Ambrose and Morris Lazerowitz’s example are of the form: P --> Q and P --> ~Q. Moreover, ((P --> Q) AND (P --> ~Q)) --> ~P is a tautology --- it is a flawed (particularly when there is no relevant relation between P and Q) variant of “proof by contradiction”.

By the very definition of material implication, both P --> Q and P --> ~Q are true at the same time when P is false (regardless of the presence or absence of material relevance of the antecedent P to the consequent Q or ~Q) — so, invoking that these 2 propositional forms are inconsistent with each other is indeed preposterous. It might be argued (in fact, as guaranteed by the tautology scheme cited above) that the simultaneous truth of these two propositions _implies_ the falsity of the antecedent P. What I am counter-arguing is that they are _defined_ to be so — that is, it is a gross self-contradiction to call upon a definition to rationalize an argument. However, I reiterate that, as presented, the flaw in Cantor’s argument is in the false belief that [1] and [2] are contradictories (that is, when 2 propositions cannot both be true or both be false at the same time or that their conjunction is always false for all truth-value assignments to their atomic formulas or prime constituents).
Georg Cantor inferred his conclusion without regard for the material or factual truth of his two implication premises. In the simpler-to-analyze example by Ambrose and Lazerowitz on the “rain” and “good crops” relation, we can easily see that both the given implication-premises lack material truth:
There are countless of actual true-to-life circumstances whereby either “the crops will be good” or “the crops will not be good” is true without their truth being a direct consequence of the truth or falsity of “it rains”.
On the other hand, if “it rains” adequately only then “the crops will be good” is true while if “it rains” exceedingly hard so that flooding occurs then “the crops will not be good” is also true.

Ambrose and Lazerowitz expounded on the issue of escaping commitment to the conclusion of an inference which is also particularly relevant in pointing out the flaw in Cantor’s line of reasoning ---

It is to be noted that whenever an inference is made, not only is an implication asserted to hold between premises and conclusion, but both premises and the conclusion are asserted to be true [it is emphasized that modus ponens ((P --> Q) AND P) --> Q is a tautology]. Both these facts are relevant to a consideration of the means of escaping commitment to the truth of an inferred conclusion. There are, in general, two ways of doing this. One way is to deny that the implication holds. This amounts to pointing out that the argument is formally invalid. The second way is to take exception to the material truth of the asserted premises; that is, either to refuse to agree to the initial assumptions or to point out their actual falsity. The relevance of denying the truth of the premises depends upon a logical fact about the relation between the antecedent and consequent of any implication when the antecedent is false.
Consider the argument ---
If it rains, it pours.
It is raining.
Therefore, it is pouring.
This asserts, in part, that if the first two propositions are true then “It is pouring” must be true. Suppose now that either the implicative proposition “If it rains, it pours” is false or that “It is raining” is false. The conjunction of the two is in either case false. . . . the falsity of the antecedent (P --> Q) AND P is consistent with the falsity of Q as well as with its truth; hence, the truth of Q does not follow from the falsity of (P --> Q) AND P.
The function ~((P --> Q) AND P) --> Q is not tautologous --- the truth-value of Q is not uniquely determined by ~((P --> Q) AND P). In general, if one denies the material truth of the premises or refuses to assent to it, there is no logical necessity of assenting to the truth of the conclusion.

Applying Ambrose and Lazerowitz’s well-informed logical declarations to Georg Cantor’s alleged ”proof” of his hierarchy-of-infinite-power-sets theorem, it is easily seen that Georg Cantor’s argument is not a valid application of “proof by contradiction” deduction — we firmly deny the material truth of its implication premises or we refuse to assent to them on the ground that T [the set of all the elements of the infinite set S which are not contained in their respective images for the presumed one-to-one correspondence between S and its power set P(S)] is not really a completely constructible set (as defined, T is an indeterminate infinite set) or the contradiction with regard to the generator of T occurs only if we look at T as a completed totality of an infinite set.

The simultaneous truth of P --> Q and P --> ~Q when P is false are embodied in the definition of modern Fregean logic’s material implication (regardless of the presence or absence of material relevance of the antecedent P to the consequent Q or ~Q) --- so, it is a gross self-contradiction to call upon some definition (which cannot be contradicted) to rationalize a faulty alleged “proof by contradiction” argument. Furthermore, because both P --> Q and P --> ~Q are defined true when P is false, then they do not form a contradiction. The self-contradiction is in invoking that they form a contradiction in spite of the concerted efforts by present-day logicians justifying the modern meaning of material implication that they do not.

It might be argued that the defect is merely in the nomenclature “proof by contradiction” which could be immediately remedied by just dropping the “contradiction” reference to the argumentation. However, it is stressed that this is not simply the case --- rather, it is the abandonment by modern Fregean logic of the existential import of the universal quantifier that jettisoned such relations as subcontraries and left only contradiction relations in the traditional so-called Aristotle’s “square of opposition” (relating “All S is P”, “No S is P”, “Some S is P”, “Some S is not P”) --- the sides (contrariety, subcontrariety, superimplication, and subalternation) are discarded while the diagonal (contradiction) is retained.
In other words, in the classical Aristotlean logic, “All S is P” (also expressible as “No S is non-P”) and “No S is P” (also expressible as “All S is non-P”) implies the existence of S. With the Fregean logic interpretation dropping the existential import of a universal quantifier (cajoled by the seeming simplification offered by adhering to a Boolean algebra implementation), comes the definition of material implication with P --> Q and P --> ~Q being both true when P is false without any regard for any factual relevance relating the antecedent P and the consequent Q or ~Q. As a consequence, in modern Fregean logic, “it will be necessary to accept what at first sight is paradoxical” --- for example, “both ‘All leprechauns are bearded’ [which can also be stated as “No leprechauns are not bearded”] and ‘No leprechauns are bearded’ [which can also be expressed as “All leprechauns are not bearded”] will be counted true, given the circumstance that there are no leprechauns” [Ambrose and Lazerowitz].
Scientific theories rigorously observe the Aristotlean logic’s implied existential import of the universal quantifier so they are successfully applied in practice. In Fregean predicate logic, the formula (For all)vP --> (There exist)vP is a generally accepted theorem which makes explicit what was implicit in Aristotlean logic. However, the rationalization for defining material implication to be true whenever the antecedent is false had already been forgotten --- hence, the hidden self-contradiction (in the example cited about leprechauns, the apparent contradiction is easily seen when the statements with the same quantifier are compared).
It is noted that every model for a first-order theory is prescribed to have a non-empty domain. It is also stressed that any specification of a self-contradiction serves to define an empty set.

Related as “vacuous truths” to logic’s material implication “paradox” is the inherent “paradox” in set theory --- if the empty set is an element of any set’s power set (or a subset of any set), then the empty set is also an element of the power set of the given set’s complement set (or a subset of the given set’s complement set) --- thus, the set of all subsets of a given set and the set of all subsets of its complement set are not mutually exclusive despite the fact that their intersection set contains the empty set (this means a hierarchical level of interpretation for the supposedly unique empty set). In the present case, the self-contradiction is in discarding the existential import of the universal quantifier while giving existential attribute to the empty set — that is, the empty set has cardinal number 0 while the set that contains the empty set has cardinal number 1.

This engenders positive motivation for formulating some set theory based on the primitive concepts “set” and “subset” by considering power sets or sets of subsets instead of sets of individual elements (that is, {a} instead of just a) — the isomorphism with the incompletable set of all natural numbers whose every element has a successor is evident from the fact that every subset implies a superset. Thus, paradoxes involving the comparison of the sizes of two distinct sets with respect to “one being a subset of the other” and “one-to-one correspondence of their respective individual elements” would be mooted ab initio — that is, there will be no “uncountable set” unless the latter signifies “every subset (or, element) always has a superset (or, successor element)” or “not a completeable set”.

Every semantic paradox has its analog in set theory, and every set theory paradox has its semantic analog --- that is, every truth-value statement can be rephrased as a statement about sets, and vice versa. The liar-paradox assertion --- “This statement is false” --- can be translated into --- “This statement is a member of the set of all false statements”. The correlation with the completed infinite set self-contradiction is evident — that is, the countably infinite set of all false statements cannot be truly completed. Also, the more basic association with the general inexpressibility of the negation of a countably infinite whole (that is, “the set of all false statements” as the negative of “the set of all true statements”) in terms of its elements and their negatives (that is, the truth or falsity of every statement) is equally manifest. It is further stressed that the assertion “This statement is an element of the set of all true statements” is not a self-contradiction.

The preceding discussion is simply stated in symbolic logic. The truth of a countably infinite disjunction P1 OR P2 OR P3 OR … could be simply established by the truth of just one of its variables even though the latter are countably infinite; however, the truth of a negated countably infinite disjunction ~( P1 OR P2  OR P3 …) = ~P1 AND ~P2 AND ~P3 … could only be ascertained from the truth of all of its negated variables which might be impossible to establish for a domain with countably infinite number of elements.

This provides a very simple proof for Godel’s incompleteness theorem (the relevance of the encompassed natural number system is clear). Please read my discussion text in the Wikipedia article “Godel’s Incompleteness Theorems”. [BenCawaling@Yahoo.com --- 10 February 2006]BenCawaling 06:24, 28 March 2006 (UTC)

## Poor example...

The following example is given:

Example: Disproving statement 1 by reductio ad absurdum:

1. Living by no moral rules is just as correct as living by any given set of moral rules.

1->2

2. No one should have to live by any moral rules.

2->3

3. Society has no right to punish those who choose not to live by moral rules (murdering, stealing, etc).

4. Since 3 is an absurd supposition, 1 is incorrect.

Three is not an absurd supposition because murdering, stealing, etc. are outlawed because the acts infringe upon the victim's legal right to living and owning property, not because they're morally incorrect. --Berserk798 07:00, 5 March 2006 (UTC)

If we accept 1, society has no moral right to write those laws.Loodog 04:03, 24 March 2006 (UTC)

I guess I didn't make it clear that I mean laws, in most countries, aren't morals. They aren't saying what's right and wrong, they exist to keep the country under control so it can function properly. You could easily have a law-abiding nation (i.e. one in which murder, stealing, etc. are punished) with no real morals. --Berserk798 21:56, 28 March 2006 (UTC)

That societies should be kept under control so they can function properly is a moral statement, rendered unenforcable by 1.Loodog 18:06, 29 March 2006 (UTC)

But I'm trying to say that punishment to keep things under control and manageable is different from punishment because of right and wrong, which would be for moral reasons. It is not a moral statement. --Berserk798 22:41, 29 March 2006 (UTC)

It's meant in the context of having the moral right to do it. Anyway, suffice it to say, if the issue requires this much sorting out, I'll concede it's not the best example for someone just wanting to know what a reductio ad absurdum argument is, but I don't think the others are as clear.
Yeah, don't even mind me. I can't believe it wasn't clear to me to begin with. --Berserk798 00:15, 31 March 2006 (UTC)

By the by, the current first example regarding the flat earth belief is nonsensical from an everyday standpoint. If the statements "the earth is round" and "the earth is flat" are of equal validity, how can either statement be denied? MrWallet 01:05, 7 September 2006 (UTC)

## Mathematical logic

The following reductio ad absurdum method should be included:

(p&¬q-->c)<-->(p-->q)

with c being the argument to be disproved.Ciacchi 23:09, 12 March 2006 (UTC)

The second example in the mathematical logic section should be corrected, to read S |- ~~P. As it reads now you're conflating this conclusion with that of double negation, and there's no reason to force classical logic. — Preceding unsigned comment added by 99.31.15.148 (talk) 18:02, 21 March 2012 (UTC)

## Ad Impossibile not a Synonym

As this article shows, "reductio ad impossibile" is often used as a synonym for "reductio ad absurdum." But when the two are used more strictly, they have diferent meanings. --Christofurio 20:38, 31 March 2006 (UTC)

## Slippery slope

Someone has added some nonsense about slippery slopes. This is not a reductio ad absurdum. It is simply a straw man. It is a fallacy, not a logical argument. — Gulliver 08:12, 1 April 2006 (UTC)

I agree. We should remove it. --Berserk798 20:26, 2 April 2006 (UTC)

## Fallacy?

Any reason this is in the logical fallacies category? --W0lfie 22:19, 19 May 2006 (UTC)

## What sets apart appeal to ridicule and reductio ad absurdum?

Mother — Why did you start smoking? Son — All my friends were doing it. Mother — You're saying that if all your friends jumped off a cliff, you would do that too?

Seems to me like an exaggerated example, one of the reason why someone would want to start smoking in the first place would be to fit in a social group. But telling someone that he would suicide if his friends would do it sounds rather like a straw man to the premise. What exactly would set apart her reply from this : Mother — You're saying that if all your friends started eating s*** , you would do that too? Wouldn't that be a form of appeal to ridicule since they both sound as offensive?

I think the point was that the son did not use the argument that he was doing it to fit in; his argument was simply that he was doing it because everyone else was. A social group argument is valid, but never mentioned. Just because everyone else is doing it is an absurd argument, which is what is being shown. GSlicer 15:57, 13 December 2006 (UTC)

Although the "if all your friends did unsafe activity X" argument is essentially a straw-man argument—it is possible, albeit highly unlikely, that the son's friends could be ignorant of the risks of jumping off of bridges (or eating feces) yet possess some unknown knowledge that makes smoking safe)—this is entirely within the realm of inductive reasoning; one cannot argue definitively that the son's friends lack (or possess, for that matter) knowledge about smoking unknown to medical science. Reductio ad absurdum is deductive; it shows that assuming a given statement to be false leads to an absolute logical impossibility. 3.14 (talk) 03:53, 1 June 2012 (UTC)

## Second argument: revision?

A — You should respect C's belief, for all beliefs are of equal validity and cannot be denied.
B —
1. I deny that belief of yours and believe it to be invalid.
2. According to your statement, this belief of mine (1) is valid, like all other beliefs.
3. However, your statement also contradicts and invalidates mine, being the exact opposite of it.
4. The conclusions of 2 and 3 are incompatible and contradictory, so your statement is logically absurd.

Points 3/4 hide the rather obscure but quite definitely unassumed "fact" that two contradictory statements cannot be true at the same time

How about changing it to:

A — You should respect C's belief, for all beliefs are of equal validity and cannot be denied.
B —
1. I deny that belief of yours and believe it to be invalid.
2. I believe that contradictory statements cannot be true at the same time.
3. According to your statement, my beliefs 1 and 2 are valid, like all other beliefs.
4. However, statement 1 contradicts your statement and by 2 they are incompatible, therefore your statement is logically absurd

By explicitly making the two statements incompatible, this problem is removed. Obscurans 10:57, 23 October 2006 (UTC)

Ridiculous. If you don't accept that something can't be both true and false (in the same sense, at the same time) you're simply not in the business of being rational in the first place. PurplePlatypus 23:13, 13 December 2006 (UTC)
cf: paraconsistent logic Obscurans 22:40, 2 March 2007 (UTC)

## Holocaust?

I would really appreciate if we picked a different subject for question one... It seems like a cavalier, thoughtless reference to a mass murder. Obviously, if the article were "Examples of blatant insensitivity" this would be appropriate, but...

I know that Wikipedia is not censored, and I've always supported that policy. However, in this article, it just doesn't seem necessary at all. Of course, I can't think of anything else that works, but I'm sure y'all can. (Also, the logic is a bit stretched... for instance, taking showers is fine, but I don't think you'd ever want to step in a _gas chamber_, which does have a direct connection to why we consider the Holocaust to be, you know, bad.)

(That is, what I don't see as necessary and object to is the very glib way the subject of the Holocaust is brought up and then dismissed; it doesn't seem appropriate for an article on a logical fallacy, that does not deal with the Holocaust in any way. It's just an extreme example, I think...)Cherry Cotton 07:18, 3 March 2007 (UTC)

Wikipedia not censored ? This remark ranks among the most naive and untrue statements I have ever come across. I suggest you review a few articles in the history section and include the views of Historical Revisionism. Then you will see how quickly your contributions will be deleted by the political correct Wikipedia censors.

multo cum dubio

## Use of tilda (~) for logical NOT

Is the tilda symbol ~ in common use to mean logical NOT? I have never seen its usage before. I am more familiar with ¬, if others agree then I think it should be changed.

The ¬ symbol is used later on in the article, so either ~ or ¬ should be used throughout (not both) for consistency.Bah23 12:51, 5 March 2007 (UTC)

It is extremely common. It was introduced by Giuseppe Peano, and is widely used, for example in Principia Mathematica. I agree with you that the usage in the article should be consistent. -- Dominus 16:48, 5 March 2007 (UTC)

One will also find ` and ! through the literature, as well as the bar.

## Where can we use the "Reductio ad Absurdum" proof?

As far as I know, this kind of proof can be used only in a theory based on a consistent set of axioms. For example, it cannot be used in the naïve set theory, because of its well known inconsistency proven by Russell's paradox. Moreover, it is not proven that the axiom sets used for mathematics are consistent, even if no one has found a counter-example (see Axiom#Modern_development or Zermelo-Fraenkel_set_theory#Introduction for a general discussion). From this point of view the "Reductio ad Absurdum" proof is very risky because it assumes the consistency of the theory in which we are working, wich cannot be proven without using a stronger one.

Thus, why no one says this fact?

--patrick 22:58, 1 May 2007 (UTC)

I agree, it would be nice to include something about this in the article, as I've often wondered about it also. ffangs (talk) 21:51, 29 March 2009 (UTC)
Reductio ad absurdum does not apply only in a consistent theory. Really, it just relies on modus ponens and the following theorem of propositional calculus: (¬A →(B∧¬B)) → A. (It is easily checked that this formula is a tautology, so it follows that it is a theorem.) Any theory which has the propositional calculus as a subtheory will therefore have proof by contradiction. 130.239.235.34 (talk) 15:44, 27 March 2012 (UTC)

## Just added a more neutral example

Added a disproof that the "cubing the cube" puzzle is solvable. Perhaps the cite could be reworked to still show that this is not a copyright breach, whilst removing the POV. 193.122.47.170 09:19, 22 May 2007 (UTC)

## Flying Spaghetti Monster

Correct me if I'm wrong, but I'm pretty sure the argument for Pastafarianism is a Reductio argument, right? I think it would be an interesting addition to real-life examples. Electriceel [ə.lɛk.tʃɹɪk il] 12:49, 1 August 2007 (UTC)

Wow. That's a great example. Please do add. The Evil Spartan 19:10, 1 August 2007 (UTC)
That's been in the article before -- scroll up to "Use of a straw man as an example", which I think sums things up perfectly. The FSM is a parody, not an argument; it doesn't prove anything. I'm going to replace it with a better example. ~ CZeke 16:32, 2 August 2007 (UTC)
Just because it's a parody doesn't mean it's not an argument. It seems to me that it takes much the same slant as the first example (the smoking girl): she obviously wouldn't jump off a cliff just because her friends were, but it's reducing the notion of peer pressure to absurdity by providing a clearly false and ludicrous outcome. It "proves the point" in the same way that FSM does.
That said, I'm no logician, and it's possible if not likely that I don't understand it properly, but it just seems that if FSM is being dismissed for this reason, the other example should be too. That, or keep them both. Electriceel [ə.lɛk.tʃɹɪk il] 01:09, 3 August 2007 (UTC)
Oho! I missed that, but you're right -- that's not reductio either, it just sounds kind of like it. I'll see if I can think of a replacement in the same spirit. ~ CZeke 09:03, 5 August 2007 (UTC)
Actually, I'm still not convinced of the FSM example not being reductio. I definitely thinks it's not a straw man on the grounds that it's putting forward an identical claim to that of the creationists, definitely not misrepresenting them, as the author of "Use of a straw man as an example" thinks. Unless it's a contentious area in philosophy (which I can't imagine it being) there must be a clear-cut definition. I found (after a brief googling) this page that, as far as I can tell, differentiates uses like the current and FSM examples from strictly mathematical arguments, saying: Despite its departure from what is strictly speaking so construed - conditionals with self-contradictory - time to time conclusions – this sort of thing is also characterized as an attenuated mode of reductio. So, it appears to me to be a weakened, but nonetheless valid form of reductio. Electriceel [ə.lɛk.tʃɹɪk il] 06:39, 6 August 2007 (UTC)
Another one, even though it's from a blog. Electriceel [ə.lɛk.tʃɹɪk il] 06:48, 6 August 2007 (UTC)

## Deleted "daughter smoking" example

"If your friends all jumped off a bridge, would you jump off too?" is not a reductio ad absurdum, but a dubious use of slippery slope. The father has not refuted his daughter's argument in the least.

The daughter could say that her friends have influence over her because she trusts and respects them. This influence is enough to get her to start smoking, which is a choice some rational people make, but not enough to get her to do something so obviously dangerous as jumping off a cliff.

Please do not re-add this terrible example.

I beg to differ. However, I have not had the opportunity to read the original post. Still, I see the argument this way (please explain if I'm mistaken):

The daughter argues that she smokes because her friends do, because 'she trusts and respects them'. The father reduces this argument to the absolute basics by insisting that she respects and trusts their every action. His extension of this to the level of mass suicide is therefore an (extreme/classic) example of reductio ad absurdum. The fact that the outcome is ridiculous is just a symptom of the reductio ad absurdum technique.

Your comment appears to suggest that an absurd outcome is unacceptable in the application of reductio ad absurdum, which is a contradiction in terms. Also, that there is obligation to refute the original statement, where none exists. Granted though, it is a technique that can be used to highlight inconsistencies or flaws in an argument, but the technique itself can demonstrate how reductio ad absurdum is a flawed method of analysis. In this case, a simple concept undergoes scrutiny by reductio ad absurdum, producing an outcome that is so absurd that it can serve no possible contribution to any logical argument. This 'cliff-jumping' example is one of the latter.

PS: Please sign your comments in future. Nikie42 (talk) 03:23, 1 June 2008 (UTC)

Is the following dialogue an example of reductio ad absurdum where the father refutes the daughter's justification by showing the absurdity of its consequences?
Father: Why are you taking aspirin?
Daughter: The doctor told me to take it.
Father: You're saying that if the doctor told you to take arsenic, you would do that too?
131.211.113.1 22:46, 1 June 2008 (UTC)

In my opinion, yes. Just because his argument is flawed and appears to be irrelevant, doesn't mean that this is a poor example. For the record, reductio ad absurdum does not need to actually refute. It can be used as an attempt to refute, which subsequently backfires, or just for comic effect. A good example of reductio ad absurdum would be:

Person A: Democracy is good because decisions are made by the majority.
Person B: Are you saying that minorities should be expelled from the political process?

An extreme (but still valid) example would be:

Person A: I like science fiction.
Person B: Are you saying that you would embrace a dystopian environment?

Nikie42 (talk) 23:28, 7 June 2008 (UTC)

This sounds like Steven Colbert! Frunobulax (talk) 20:42, 26 June 2008 (UTC)

If this is relevant to the discussion, then I don't understand the reference. Otherwise: WP:NOT#CHAT (sorry). Nikie42 (talk) 23:50, 26 June 2008 (UTC)

## Euclid on primes: right and wrong information

I have deleted the Euclid example from this article.

Many eminent authorities assert in books and journal articles that Euclid's famous proof of the infinitude of primes is by contradiction. They're just repeating each others' assertions. It is not true. Wikipedia's article on prime numbers gets it right, and I repeat it here:

The oldest known proof for the statement that there are infinitely many prime numbers is given by the Greek mathematician Euclid in his Elements (Book IX, Proposition 20). Euclid states the result as "there are more than any given [finite] number of primes", and his proof is essentially the following:

Consider any finite set of primes. Multiply all of them together and add one (see Euclid number). The resulting number is not divisible by any of the primes in the finite set we considered, because dividing by any of these would give a remainder of one. Because all non-prime numbers can be decomposed into a product of underlying primes, then either this resultant number is prime itself, or there is a prime number or prime numbers which the resultant number could be decomposed into but are not in the original finite set of primes. Either way, there is at least one more prime that was not in the finite set we started with. This argument applies no matter what finite set we began with. So there are more primes than any given finite number.

Notice that it says "any finite set of primes". Not the set of all primes, assumed to be finite. Not the first n primes, for some n. Just any finite set of primes. This is not a translation of Euclid's original words in Greek, but it is faithful to what Euclid wrote; it is the same proof. (Catherine Woodgold and I have written a paper recently submitted to a scholarly journal examining the prevalent misunderstanding of the history. If anyone's interested I can email them a preprint.) Michael Hardy (talk) 13:54, 17 December 2007 (UTC)

## Proof by negation cannot be used

Proof by negation is argued to be valid by using closed systems as a demonstration and then saying it can be used anywhere.

However in reality we do not define what assumptions must be made, and in fact we do not even know what assumptions are being made. For example: If a person bumps into me on the subway I might assume that he did not pickpocket me and therefore my wallet should still be there.

If my wallet is not there, proof by negation, he did pickpocket me. This is wrong because in truth I have made an infinite number of other assumptions. Maybe someone else stole it, perhaps someone I did not even see. Maybe it fell out at an unknown location. Maybe I never even put it in my pocket this morning.

There are an infinite number of things that could have happened that sound plausible once someone points them out to me or it turns out to have been what happened, but that I would not have thought of on my own.

There are also an infinite number of things that could have happened that I would be very surprised to know occurred. Yet the label "trivial" assumptions does not apply because in short "you don't know what you don't know". At best trivial assumption could be applied to anything that would cause all human knowledge to break down if it turned out to be false.

Also, any designation between plausible and implausible is meaningless in any realm outside of normal human experience. For example, we have no clue what possibilities are plausible or not regarding what causes the behavior of sub-atomic particles. —Preceding unsigned comment added by 69.252.158.32 (talk) 13:38, 18 February 2008 (UTC)

But it's obviously not true that if he didn't steal your wallet, then it must be in your pocket. Obviously it could have left your pocket in others ways, including by being stolen by someone else besides that person. Michael Hardy (talk) 22:42, 23 May 2008 (UTC)
Is this in any way related to the content of the article?  --Lambiam 08:20, 20 February 2008 (UTC)

Yes very much.

Silently the writer of the above critique assumes we are all skilled enough at logic to do a proof by negation quickly. Let me clarify with some formal logic;

before we start := is a symbol that assigns meaning, so X := Y states that X symbolizes Y.

p := I got pick pocketed w := I have my wallet

-> := implies

1) not p -> w The writer assumes " I did not get pickpocketed implies that my wallet is there"

2) not w The writer observes "My wallet is not there"

Now the writer does a proof by negation, this means assume some fact X and then show that X leads to a contradiction. *assuming a correct logical model* then that model is not contradictory. So if X leads to a contradiction in the logical model, that must be correct, then X can not be true. X is either true or false, hence X must be false. We can now add not X to the model.

Applied;

suppose I did not get pickpocketed

| 3) not p ( some assumed fact similar to X )

| 4) w ( if not p -> w and not p then surely w must be true)

| 5) w and not w (!? how can my wallet be in my pocket and not be in my pocket at the same time? this is impossible therefore my assumption not p must be false)

6) not not p 7) p ( if it is false that p is false then p is true)

But... the writer states that there might be other reasons the wallet is missing. This does not attack the proof of negation, this attacks the model. And of course if the model is just plain contradictory and wrong, then logic cannot be applied. This holds for any and all logical proofs.

The situation with more reasons why a wallet could be missing could be modelled as follows;

f := forgot at home b := in my bag etc...

( p or f or b ( or etc...) )-> not w

without further derivation (I have other things to do) this can be rewritten as;

( not p and not f and not b ( and not etc ) ) -> w

Given it is an attack on logical modeling it is definitely a strong attack in terms of philophy of logics, but that holds no place in this part of wikipedia. That needs mentioning in the section on logic in general.

Personally I consider the statistical/empiric evidence of the power of logic overwhelming though ;)

Guidocalvano (talk) 21:58, 23 May 2008 (UTC)

## Humour

Apparently there is a source (the chapter "Comic Films" by Gerald Mast in Nancy A. Walker (ed.), What's So Funny: Humor in American Culture, Rowman & Littlefield, 1998) that uses the term reductio ad absurdum for the plot device of blowing up a position to ridiculous proportions. As far as I can see that is a somewhat idiosyncratic and not very notable use of the term.

The subsequent statement in the section "Humour" that this plot device is employed in an episode of the Big Bang Theory cannot be ascribed to a reliable source and is therefore original research. While the term is uttered in the episode, the utterance does not refer to a humoristic plot device but to a logical fallacy.  --Lambiam 04:28, 30 May 2008 (UTC)

Regarding your first point, effectively that the information in this publication is merely opinion without weight or experience and that it has little significance to the wider context of film. The introduction to that very chapter establishes the author's position in the realm of film. However, if this isn't enough to settle concerns regarding the relevance of the reference to the arena of film and cinematic research, then perhaps this article will:

http://query.nytimes.com/gst/fullpage.html?res=940DE5DE1731F931A3575AC0A96E948260

It records the numerous academic qualifications and publications of the author, Gerald Mast. He was apparently not only a well-qualified gentleman but the ex-Chairman of the English Department at the University of Chicago. In his article he declares reductio ad absurdum to be one of the eight basic comic structures and gives multiple examples in evidence.

To address your second point: The context of that 'utterance' is in reference to a humoristic plot device utilised by the other character (Leonard). He uses reductio ad absurdum to generate humour, as evidenced by the clip given in reference (http://www.youtube.com/watch?v=lYA0reSm6qw). After which the second character (Sheldon) calls him on it. To quote:

After Sheldon expresses concern for their emergency food rations in view of an overnight guest. Leonard: "Are you suggesting that if we let Penny stay, we might succumb to cannibalism?" Sheldon: "No-one ever thinks it'll happen until it does." 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."

The statement in the Wiki article concerning reductio ad absurdum does not require additional sources because the evidence of the clip alone confirms the use of the technique in humour. If reductio ad absurdum were not a humoristic plot device, then Leonard's request of Penny would not have generated laughter from the audience. Since it does, and since he has quite obviously used the technique here, then reductio ad absurdum must therefore be a humoristic technique. Argument to the contrary either denies the evidence of the reaction of the audience in response, or the fact that Leonard's statement qualifies as reduction ad absurdum in the first place. Nikie42 (talk) 02:57, 1 June 2008 (UTC)

I did not doubt the author's qualifications, but notwithstanding that, I see no evidence contradicting my suspicion that his use of this term for the hyperbolic plot type he describes is somewhat idiosyncratic and not very notable. Many people have contributed to the theory of comedy, and as far as I can tell Mast is the only author using this term.
My main objection, however, is to your ascription of this to an episode of a sitcom. It is a jump to extend Mast's concept of a plot type that gives the structural principle of a film to a mere humoristic technique. What I see and hear in the sitcom episode is a far cry from what Mast describes: "The typical progression of such a plot—rhythmically—is from one to infinity. Perfect for revealing the ridiculousness of social or human attitudes, such a plot frequently serves a didactic function." The audience laughs on cue, as required, but it is impossible to deduce from this how to classify and name the humoristic device involved, and it is your interpretation that the utterances of the Leonard character are an instance of the use of a technique named reductio ad absurdum.  --Lambiam 19:28, 1 June 2008 (UTC)

Then you certainly have not made the effort to dig deep enough. Even a cursory glance at a selection of works on both comedy and literary technique have revealed plenty of sources that quote or reference reductio ad absurdum as a humoristic technique or plot type. These include Paulos, J.A. (1980) Mathematics and Humor. University of Chicago Press, p11-12; New, W.H. (2003) A History of Canadian Literature. McGill-Queen's Press, p127 concerning Stephen Leacock; and Costa, R. (2000) Humor in Borges. Wayne State University Press, p128-9.

Your second point is simply a problem of scale. Just as the creationist takes issue with natural selection being extended to large-scale evolution, your argument is that a plot type and humoristic structure is not applicable on the smaller scale. From Paulos (1980): Humor can easily be contrived in this manner. An odd premise is accepted, and the joke or story develops the premise to the point of absurdity. Note here that scale is not an issue and that the technique is applicable to the plot and individual jokes, as in this episode of The Big Bang Theory.

The extract you have selected from Mast omits his qualification of the above statement: But the reductio ad absurdum need not serve didactic purposes exclusively. Even given your quotation, I cannot see how it supports your argument. Surely, the purpose of Leonard's question: Are you suggesting that if we let Penny stay, we might succumb to cannibalism? is through its design and execution revealing the ridiculousness of [Sheldon's] ...attitudes. It seems to me that you have not applied the basis of your own argument to the context.

Your comment about the audience laughing on cue, as required. Indicates that you would like to enter into a debate about the nature of comedy, and pose the question as to whether or not something can be labelled as funny just because people laugh at it. I will not be drawn into such a discussion, seeing as it would be as unproductive as an attempt to define art. If you are unprepared to accept a laughing audience as evidence of a humorous act, then I would ask you to please reduce your levels of cynicism.

I would also like to very strongly point out that the statement is not wholly based upon my interpretation of the episode. May I remind you that the character Sheldon himself identifies the technique, and defends it by giving a definition of reductio ad absurdum that, in turn, agrees with the definition laid down by this Wikipedia article. It is a flawed argument to pronounce that it is simply my interpretation when you are presented with two separate definitions both of which are in agreement. I am merely reporting their agreement. The fact that I happen to understand, appreciate and utilise this technique is a coincidence. Nikie42 (talk) 23:06, 7 June 2008 (UTC)

This seems quite confused to me.
First, when Aristotle uses reductio ad absurdum to argue a point, he is not engaging in a humoristic plot device, and if someone were to claim in the article that Aristotle used humoristic plot devices, citing Mast, Paulos, New, and Costa, then it would be entirely correct to label these citations as dubious and not supporting the statement. The Sheldon character (to whose judgement you appeal) does not suggest at all that the Leonard character's technique he identifies as reductio ad absurdum is a humoristic technique.
Second, since when do the characters in a skit count as reliable sources? They say just what the text writers make them say, for which the correctness of what is being said is not a consideration.
Third, while the definition given by the Sheldon character for reduction ad absurdum ("extending someone's argument to ridiculous proportions and then criticizing the result") agrees with the meaning given in the disputed section, it conforms so well because that is the wording you put in. Indeed, this agreement is not a coincidence, and using it as an argument for defending the section is incestuous.
Fourth, some writers use a character's lack of understanding to make the audience laugh. In such cases the character's lack of understanding is employed for its humoristic effect; it is not itself a humoristic technique. Likewise, the character's use of reductio ad absurdum in this skit is employed for its humoristic effect; it is not itself a humoristic technique.
Fifth, in the sense of the plot device described by Mast, it is not a character who blows up an argument to ridiculous proportions, but it is the script writer who lets the situation run out of control by magnification; the characters are helplessly carried along in the maelstrom.
Finally, the issue is not whether the Big Bang is funny or not, and so the laughter of the studio audience is irrelevant.  --Lambiam 09:43, 8 June 2008 (UTC)

Firstly, I've made no direct reference to Aristotle and I'm confused as to why you've included him in this discussion. While he certainly did use reductio ad absurdum in his work, I'm pretty certain that it wasn't for humour. Just as other literary techniques can be used for a variety of purposes, you have shown nothing for your view by identifying this as another. If you acknowledge that the audience laughed at Leonard's question, then also accept that Sheldon's definition is correct, why are you so unwilling to combine the evidence and conclude that his use of reductio ad absurdum was funny to the audience?

Secondly and thirdly, you misunderstand. I am referring to the audience reaction as evidence that what was occuring on stage was humorous. The fact that reductio ad absurdum was then defined in a way that agreed with this article merely strengthens this assertion. Please refer to the introductory paragraph and the explanation section of this article (which were not written by myself). My argument was never, nor ever intended to be, incestuous. If you'd like me to trawl through the comedy archives and find more examples, then please get on and do so. This example happened to be convenient because of the definition therein.

Fourthly, you appear to have proven my point. Since when is a technique employed for its humoristic effect not a humoristic technique? Please refer to the references I have provided before replying with your opinion about the nature of humour.

Fifthly, again you appear to be bogged down by scale and have ignored my supporting reference. Please look again at Paulos (1980). Alternatively, please consider the definition of reductio ad absurdum and convince me that it does not apply to Leonard's question.

Lastly; incorrect, the issue is whether or not reductio ad absurdum is a technique that can produce humour. Hence, the evidence of a laughing audience is entirely valid. I would not attempt to call myself a critic by discussing the success or not of The Big Bang Theory. —Preceding unsigned comment added by Nikie42 (talkcontribs) 14:11, 8 June 2008 (UTC)

I introduced Aristotle to make a point that seemed (and seems) necessary, in such a way that I was sure you would agree: the fact that something can be categorized as reductio ad absurdum is by itself insufficient to establish that it is a humoristic technique. I do not accept Sheldon's definition as being correct. The introductory paragraph does mention "ridiculous", but does so to characterize a straw man argument – and then combined with appeal to ridicule. Categorizing such degenerated arguments as reductio ad absurdum is your doing (as it is above, on this talk page, to the father criticizing the daughter for taking aspirin). Although people use straw man arguments and appeal to ridicule, they do so to win the argument, and not because they want to be funny. You argue: "if X is employed for its humoristic effect, then X is a humoristic technique". Hopefully you agree, however, that lack of understanding is not by itself a humoristic technique and should not be classified as such. The humoristic technique is to create a situation in which characters, through lack of understanding, act in ways that have entirely foreseeable consequences – that is, entirely foreseeable to the audience, but totally unknown or unexpected to the acting character. I don't buy your scale argument. In the examples given by Paulos, it is the author who applies the technique, and not a character using it by way of argument. No amount of scaling turns the author into a character. In the sitcom, the audience laughs, not because the script writers employ the reductio ad absurdum technique, but because the characters show themselves for what they are: pompous selfish dolts. You don't have to be a critic to see that.  --Lambiam 21:10, 8 June 2008 (UTC)

I'm getting very tired of your obstinacy. You accuse me of original research in my assertion that reductio ad absurdum is a humoristic technique. I present multiple, published references that demonstrate that it has long been considered so. Yet you maintain that it isn't. If you're an expert yourself in comedy and humour then please announce it now. Otherwise, please accept these sources as evidence. Your refusal in doing so smells of arrogance and the belief that your opinion on the subject should somehow be held in higher regard than these authors.

However, if you're simply looking for a less convenient example of the technique, then please say so. I've sure I could find a collection of clips for you.

[Nb: Paulo (1980) states that the "joke or story develops..." - thus there is no requirement for it to be an author/writer over a character. I ask again that you please consult my references before throwing them back at me.]

You have replied in what appears to be a stream of consciousness. If you'd like me to address your points then please rewrite them in a more structured format. While you're there: convince me that the phrase: Are you suggesting that if we let Penny stay, we might succumb to cannibalism? in the given context does not qualify as reductio ad absurdum. Then you might have a leg to stand on.

Oh, and another thing. Please don't insult the characters like that. Perhaps you don't find them funny, but there's no need to show such bias towards my sources. It undermines your argument in my view. Nikie42 (talk) 00:27, 9 June 2008 (UTC)

My complaint of original research is partly based on your formulation of the meaning of reduction ad absurdum as a plot type (or humoristic technique), whose source is the utterance of a character in a sitcom – which character, however, does not identify the use covered by the formulation as a humoristic technique. The formulation given by Sheldon and you does not match those supplied by your written sources, in which the aim of criticizing the result plays no role. The main basis of the complaint is the identification of the humour in the skit as an application of reductio ad absurdum as a humoristic technique. I disagree with your judgement that it is, but actually my or your judgement in this matter is of no importance. Just give us a reliable source that states that the "The Dumpling Paradox" episode of The Big Bang Theory offers an instance of reductio ad absurdum used and analyzed as a humoristic technique, and I'll shut up.
In my previous reply above I have treated your comments point by point in the order in which you gave them.
I don't want an edit war, which is why I try to discuss this on the talk page, rather than just deleting original research from the article, and I'm disenchanted with the stubbornness and the feeble arguments with which you defend this original research. If a claim made in the article is challenged, you should come up with reliable sources for it, otherwise it should go. It is not manifest, self-evident, or whatever, from the primary source, that the critical analysis given in the article is correct.
I'm truly sorry if you identify so much with the characters in the show that you are offended by my characterization of their narcissistic personalities. You should realize, however, that they can't help it; they have been endowed with these personalities because the script writers thought it would be funny.
To avoid another complaint of stream-of-consciousness writing, let me summarize the issues I have:
--Lambiam 06:01, 9 June 2008 (UTC)

Don't start putting words in mouth! At no point did I express offense at your treatment of the characters. I merely pointed out that your belittling them reduced the value of your argument in my eyes. I delight in pointing out that despite this request you are now being sarcastic towards me, and continue to treat them with disrespect. The value of your contributions to this talk page has lowered further in my eyes.

Another concern of mine is that you've failed to meet my challenge. The term reductio ad absurdum is adequately defined in this Wikipedia article, and I asked you to show how the line laid out above does not conform to it. If it is shown to be inaccurately ascribed to the technique, then I shall find another example. But I fundamentally maintain that it is a technique in humour.

Furthermore: you appear to be entirely focused upon the opinion that my given example is flawed, when I have quite openly offered to provide another. Instead of simply continuing your attacks upon my contribution, perhaps you should address this offer and decide whether or not you'd like to take me up on it. Wouldn't that be a better example of giving constructive criticism? Shouldn't we all strive in unison to produce a more accurate encyclopedia? From what I can tell, you're not playing it that way. Nikie42 (talk) 22:28, 12 June 2008 (UTC)

Please tread easy on me, this is my first time attempting to contribute to wikipedia in this sort of manner, but I enjoyed this topic and want to help. I think that Nikie42 has produced a selection of works that shows his point, and these works convince me of the accuracy of his claim. However, I would agree with Lambiam to the extent that when I read the transcript of the clip of the Big Bang Theory used as evidence (sadly I am unable to use speakers in this setting, however the evidence of reductio ad absurdum in humor should be able to be seem from a transcript), it is at best not evident, and at worst confusing, that this is an example of the use of reductio ad absurdum in humor. The use of the actual phrase reductio ad absurdum in the quotation listed confuses the point, creating a situation where we can reasonably conclude some may believe we are defining reductio ad absurdum as humor by citing the fact the phrase was uttered in a comedy, which holds no weight. I have a slight issue with the idea of using a TV sitcom as well, considering the fact that laugh tracks are added. I can only say for certain an example from a different medium would be stronger.

I think plenty of stage comedians have used reductio ad absurdum effectively with proof of humor coming from live sets. For a crude example, George Carlin's set "Rape Can Be Funny", a transcript of which can be found online, uses reductio ad absurdum in my opinion and I recall hearing this live and it was effective. I find the selection of works used by Nikie42 evidence enough of reductio ad absurdum in humor, however I would say the example used from the Big Bang Theory could be stronger or perhaps an unnecessary distraction from the point. Hope I added to this discussion. Kmvenne (talk) 13:01, 20 June 2008 (UTC)

In reply to Nikie42: My objections are, definitely, not solely directed towards the specific example. I object, in general, to any statement claiming that some specific (fragment of a) work exemplifies said humorous technique, unless we have a citation from a reliable source to that effect. Otherwise, the ascription to the technique is original research, regardless of whether my attempts to show that it is "inaccurately" thusly ascribed are successful (who will be the judge of that?) or not.
Let me recapitulate the meaning of reductio ad absurdum. This term refers to a form of argument, namely one that aims to establish that some proposition (or point of view, or position) P is untenable, by showing that P entails, necessarily, a consequence Q that is manifestly untenable. In the traditional form, Q has the form of the conjunction of some statement with the negation of that same statement, such as "God is merciful and God is not merciful". A manifestly untenable statement is called the absurdum.
As used in philosophy, logic, and mathematics, there is in general no identifiable proponent of the position P, let alone an "opponent". In the proof by reductio ad absurdum that the square root of 2 is irrational, the aim of the argument is merely to establish the result, by letting the assumption of the contrary lead to an absurd consequence, namely that a fraction witnessing the rationality is and is not reduced to lowest terms. A fortiori, there is no aim to make anything or anyone appear "ridiculous", or to "criticize" any result.
Presumably because the meaning of the term is not generally well understood and the word "absurd" is catchy, the term is often improperly applied to something that is not an argument, but a rhetorical technique intended to floor an opponent. The technique consists of seizing an opponent's ideas and carrying them further in an absurdly consequent and rectilinear fashion, until they assume a form that is manifestly inconsistent with common sense. For example, a proponent of animal rights may be attacked by suggesting that they must be in favour of extending suffrage to jellyfish. In the classical approach, that outrageous conclusion is reached in a careful build-up, starting from a quite reasonable assumed consequence and proceeding stepwise to increasingly nonsensical alleged consequences, presented in an accordingly increasingly ironic way. This rhetorical technique fits the pattern of the "straw man" fallacy, combined with the "continuum fallacy" and often – but not necessarily – with the fallacy of "appeal to ridicule". It rests on a deliberate distortion, in this case by hyperbole, of an opponent's views. A true reductio ad absurdum argument is not a fallacy and does not involve such distortion.
Even beyond that, latching on to the word "absurd", some people have applied the term to any hyperbole that exaggerates and distorts something (no longer a position, but any situation) to extreme proportions. For example, The Architectural Guidebook to New York City calls the former AT&T building at 195 Broadway "the reductio ad absurdum of the classical skyscraper". It would, in my opinion, not be a good idea to add a section "In architecture" to the article, describing reductio ad absurdum as an architectural technique, even with this citation. The row of contexts in which people have abused the term reductio ad absurdum to name any grotesque form of hyperbole is endless. Really, this should have been called magnificatio ad absurdum.
Now to the use of the term in the context of humour. You have mentioned a number of sources that supposedly use the term in the way described in the section "Humour". But they all use the term in significantly different ways! The starting point need not be a "flawed statement"; it can be a "simple human mistake" (Mast), or a "social question" (also Mast), or an "odd premise" (Paulos). The aim may be to criticize, not so much the result of the grotesque exaggeration, but rather the starting point: an attitude, or an approach to a social question. But not all cases are satire and aim to criticize; for example, Mast calls the Laurel and Hardy films the perfect example of "pure fun". For Mast and Paulos the term reductio ad absurdum as a humoristic technique relates to the development of the plot or story, and for all sources it refers to a technique applied by the author, not by a character. So even granting the fact that the term has been used – improperly – for a humoristic technique, the description given in the section "Humour" is inadequate.  --Lambiam 11:33, 22 June 2008 (UTC)

The Humor section makes no sense, when are we removing that? Beam 14:08, 4 July 2008 (UTC)

I'm afraid that the foregoing discussion was simply too long and I did not read it. The analysis of the quoted exchange from the television episode was original research, and thus I have removed it. Nikie42, if you wish to illustrate the points you are making here in the article, please find a quotable source to do it with. — Hex (❝?!❞) 23:06, 6 July 2008 (UTC)

Many thanks for taking decisive action in simplifying the issue. I agree with the deletion of controversial examples and await the inclusion of more suitable ones. As long as the references connecting reductio ad absurdum to humour are recognised, then with due respect to all previous authors, I opt to stand down from this discussion. Nikie42 —Preceding undated comment was added on 12:08, 25 February 2009 (UTC).

## Lacks an easy-to-follow example

Why doesn't the article have an educational, everyday, poignant example of the use if this form of argument? __meco (talk) 12:43, 9 September 2008 (UTC)

## contraposition

I notice that there is a detailed discussion of contraposition in one of the comments above, but none of it seems to have made it into the article Reductio ad absurdum itself. It may be helpful to have a simple-minded comparison of the two as part of the article. Also, if anybody cares to explain the difference between "proof by contradiction" and law of excluded middle", I would be much obliged. Katzmik (talk) 12:43, 26 October 2008 (UTC)

## In mathematics section

This section has an ambiguity. The discussion is about about first and second "kind" of arguments. At the outset the discussion states the 'first kind' is to disprove, but then latter sites the 'first kind' as an example to "prove" a statement. Moroever, at the bottom of the discussion both "kinds" are explicitly stated as means to "prove" something. The ambiguity arises from if we take a statement q that is equivalent to not p, then both "kinds" of arguments can be used to be "proofs". That is, the same argument that "prooves" q, "disproves" p. I would drop the use of the word "kind" throughout this discussion. —Preceding unsigned comment added by 98.150.186.198 (talk) 18:14, 17 December 2008 (UTC)

Would a link to Russell's Paradox be appropriate here? It gives an example of needing to know that one of the options is consistent. In brief, the paradox posits the existence of a set S, which contains all sets that do not contain themselves. If we let "x < Y" mean x is contained in set Y and "x <> Y" mean x is not contained in Y, then by looking at one half of the paradox, we get:

1. "S < S" -> "S <> S" 2. 1 is a contradiction, therefore "S <> S"

This ignores the second half, which shows that a similar result is obtained for "S <> S". Iain marcuson (talk) 20:15, 9 November 2009 (UTC)

I would not insert a link to Russell's Paradox. It has (had) great importance, but the interested reader wants to know what a Proof by contradiction is, not something a bit related to it. BertSeghers (talk) 18:58, 25 January 2010 (UTC)

Well, actually, it is very much related just like Goedels theorems. Now, I'm not an expert in the field, but what about the possible undecidable statements? —Preceding unsigned comment added by 88.207.15.189 (talk) 06:13, 10 April 2011 (UTC)

## Merge with Reductio ad absurdum?

They're the same thing. Discuss. --128.62.37.246 (talk) 01:21, 19 January 2010 (UTC)

Strictly speaking, they are not identical, reductio ad absurdum, is a more general concept. Paul August 12:40, 1 March 2010 (UTC)
I don't see what you have in mind. Please cite an example of a reductio ad absurdum that's not a proof by contradiction. Michael Hardy (talk) 17:44, 1 March 2010 (UTC)
See this IEP article: Reductio ad Absurdum. Paul August 18:15, 1 March 2010 (UTC)
The only thing I feel I should add is I came here specifically looking for Proof by Contradiction for a math class. If I landed in Reductio ad Absurdum I would have been thoroughly confused. —Preceding unsigned comment added by 24.61.197.189 (talk) 02:30, 31 August 2010 (UTC)

No, they're not the "same thing". There need to be separate articles for classical and modern logical usage. The classical usage, since Plato and Aristotle, never got discrete, polychotomy, and excluded middle quite sorted out, which resulted in a literature of fallacies based on false dichotomies. BlueMist (talk) 15:08, 18 October 2011 (UTC)

They are absolutely not the same thing at all. Proof by contradiction is when you show that a statement is true by showing that the contrary statement is false. For example, the statement "Not all swans are white" is shown to be true, because the statement "All swans are white" can be shown to be false by presenting a black swan. Reductio ad absurdum is showing that assuming a statement to be correct has unintended, absurd conclusions. For example, the statement "We should sterilize those person who has genetic defects" can be shown to be absurd by pointing out that since everyone has some genetics that can be branded a "defect" we should sterilize all of humanity, a clearly absurd position that the person stating the original position most likely did not intend. --OpenFuture (talk) 16:11, 14 July 2012 (UTC)

## right-triangle proof

I edited the proof that the sum of the length of the legs is greater than the length of the hypotenuse. The main change is that I ignored the degenerate case by just stating the claim for non-degenerate right triangles. There are three reasons why I thought that change would improve the article. The first is just that discussion of the triangle inequality holding in equality for certain degenerate right triangles distracts from the article. For the reader who wants to know about proof by contradiction, it would be better not to end the example with "the result is that a + b is greater than or equal to c. It is entirely possible for the latter possibility to hold true, if we include a straight line in our definition of a right triangle." Also, in the proof before my edit(s sorry kept noticing small mistakes I made) it was assumed that a and b were positive which implies that the triangle is non-degenerate. It also doesn't seem like non-degenerate right triangles are widely considered. A google search for "degenerate right triangle" showed only 15 hits (though it said 120) with a few duplicates. "nondegenerate right triangle" had only 4 and "non-degenerate right triangle" just one.

This last point applies to my edit as well as I specified non-degenerate instead of just not mentioning it. I'm not sure whether it's better to leave it in there where it could serve as a distraction or take it out, in which case the claim is technically false.

I also shifted things around, waiting until the middle and end of the proof to refer to the Pythagorean theorem and the assumption that a and b are positive. I feel that this makes the method more clear; it's more instructive to start the proof explicitly with the negation of the theorem rather than state results and assumptions to be used later. I did reference the Pythagorean theorem before starting the proof so that one could get some sense what was coming and the reasoning behind squaring both sides and so on. I also changed the wording in a couple spots both to make it more clear and emphasize the process. Jdl22 (talk) 05:49, 25 February 2010 (UTC)

## non-mathematical examples

I note that there used to be some non-mathematical examples that had problems of dealing with socially problematic topics. I think its still important an attempt should be made to find an example of this argument that doesn't include formal logic or maths. That kind of thing is very useful to people initially about logic. If anyone could think of one... romnempire (talk) 21:49, 10 April 2011 (UTC)

## Formulating examples -- WP:NOR

Anyone with training in logic can make up examples of proof by contradiction. But doesn't this constitute original work, commonly referred to at wikipedia as original research (WP:NOR)? Shouldn't we, instead, select examples from WP:SOURCES? Surely, there are plenty of textbooks available, chock full of textbook examples. --SV Resolution(Talk) 16:08, 21 October 2011 (UTC)

## Should a proof not using excluded middle be called "by contradiction"?

A apologize if this has already been discussed, but I didn't find much trace of that (notably there is no reply in this section of this talk page). I my opinion one should only call a proof "by contradiction" if it conceivably could have been otherwise, i.e., direct. If the statement to be proved is negative, then there is no real option to prove if but to assume the negated statement an to prove it false (the only other option would be the application of a previously proved result that has exactly that negative statement as conclusion, in which case a "proof by contraction" will be hidden in the proof of that result). The very first example given, proving the irrationality of √2, is exactly of that nature, and therefore in my eyes a poor example. I know no other description of being irrational than as "not being rational", so how would one go out to prove the irrationality of √2 otherwise than to disprove its supposed rationality? Once it is shown that √2 cannot be rational, one need not say "therefore by the law of the excluded middle √2 is irrational" but one could simply say "therefore √2 satisfies the definition of being irrational". This proof is as direct as it could possibly be. I don't want to dispute about this, maybe for some it is easier to just say "a proof is by contradiction if at some point before concluding it reaches (under assumptions) a contradiction; however I think it would be mathematically more interesting to distinguish proofs that do truly need to apply the law of the excluded middle, and to reserve the terms "indirect" and "by contradiction" to them. By the way, a statement can be inherently negative even without have a negation at its outer level; for instance "all positive powers of π are irrational". In that case too, no proof without attaining a contradiction can be imagined. Marc van Leeuwen (talk) 08:04, 19 February 2012 (UTC)

I believe it's proof by contradiction because it starts with the assumption that there are integers a and b such that a/b = √2—the opposite of what is desired—and then derives a contradiction from that. A direct proof is at least conceivable, although I can't think of one offhand. But suppose for example that one were to show that some numerical property P was possessed only by irrational numbers, and then that √2 had property P.
For comparison, consider the proof that Liouville's number L is transcendental. One could prove it by contradiction, assuming that there is some polynomial that has L as a root, and then deriving a contradiction, but this is not what was done. Instead, Liouville use a direct proof: he showed that there was an upper bound on how closely an irrational algebraic number could be approximated by rationals, and then that L could be approximated much more closely than that. —Mark Dominus (talk) 18:55, 19 February 2012 (UTC)
The example you give is an instance of my parenthesized remark: "the only other option would be ... a "proof by contraction" will be hidden in the proof of that result". To wit, the fact that "there was an upper bound on how closely an irrational algebraic number could be approximated by rationals" is the Thue–Siegel–Roth theorem and its proof proceeds by assuming the existence of an irrational algebraic number with many good rational approximations, and obtaining a contradiction. (The fact that Roth's result is non-effective, in the sense of not excluding any single (very) good approximation but just a certain type of infinite collection of good approximations, is interesting but beside the point I want to make here. One could very well imagine a proof of the Thue–Siegel–Roth theorem that is effective in this sense, but not one that avoids obtaining any form of contradiction at some point.) Marc van Leeuwen (talk) 08:36, 20 February 2012 (UTC)