Perhaps, this question has been answered already but I am not aware of any existing answer. Is there any international icon or symbol for showing Contradiction or reaching a contradiction in Mathematical contexts? The same story can be seen for showing that someone reached to the end of the proof of a theorem (The tombstone symbol ∎ , Halmos).

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    $\Rightarrow\Leftarrow\quad$ – Bill Dubuque Jun 18 '12 at 20:02
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    I’m more familiar with a slight variant of Bill’s symbol, $\rightarrow\leftarrow$, which I’ve used for over 40 years and picked up from other people. I’ve also seen something similar to # but larger and rotated slightly clockwise, though not so often. – Brian M. Scott Jun 18 '12 at 20:15
  • 51
    I usually prefer the symbol: $$\text{Contradiction}$$ – Asaf Karagila Jun 18 '12 at 20:18
  • 11
    I use ↯, but I only ever use this in my working. If I am writing something up I will always use the word "contradiction" somewhere, and after reading this thread I believe all of you will understand why... – user1729 Jun 19 '12 at 9:19
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    @user1729 I also ↯ in my own writing and on a blackboard, but I often still add "Contradiction". Usually, it is not a mistake to give students the time it takes to write "contradiction" to ponder the fact that the proof is finished despite the fact that we got something wrong ... – Phira Jun 22 '12 at 12:54

11 Answers 11

up vote 20 down vote accepted

Different sources use different symbols (if they use symbols at all). I've seen $\Rightarrow\Leftarrow$ most often. For some others, see "Symbolic Representation" here.

  • 11
    I like the \blitza symbol (the first one listed at the link). Apparently, it is commonly used in Germany. I always think it describes the path of a small remote control plane that ends up falling. – Andrés E. Caicedo Jun 18 '12 at 20:42
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    Cute, @Andres. I hadn't seen that and I like its name. +1 – Rick Decker Jun 19 '12 at 2:07
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    I always thought that was Harry Potter's scar... – The Chaz 2.0 Jun 19 '12 at 3:43
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    Yes, I really like ↯ as my contradiction symbol. – davidlowryduda Jun 19 '12 at 6:58
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    The ↯ symbol seems to be very common : in France, my first teacher after high school, used this symbol too (since I use it unconsciously). – JBC Jun 19 '12 at 21:24

I am surprised to see that nobody has mentioned $\bot$. In logic, this is a standard symbol for a formula that is always false, and therefore represents a contradiction exactly.

In almost all logical formalisms, one has a rule of inference that allows one to deduce $p$ from $\bot$ for any $p$ at all, and it is usually possible to prove that $(p\land\lnot p)\to \bot$ and so forth.

  • 1
    This seems like a pretty standard choice, especially when doing proofs about a logical system. The symbols are \bot ($\bot$) and the corresponding \top ($\top$) to show tautology. – Benjamin Kuykendall Oct 7 '16 at 1:13
  • This is my favorite answer. What brought me to this post was an impromptu Rorschach test: I came across the symbol $\dashv$ placed at the end of the second-to-last sentence of this proof by contradiction. Having (a) never seen that symbol used to mark the end of a proof before, and (b) never even considered the possibility of - let alone known of - any symbol to mark contradiction, despite its position at the end of the proof of the claim in that answer, my brain's first theory about its meaning was that it must be there to mark contradiction. – mathematrucker Oct 29 '17 at 12:32

The symbol I've seen most commonly in mathematical logic statements is also the one which was taught to me in a class called "Discrete Mathematics;" it is something like a sideways number sign or "pound sign" (or "hashtag," as some might call it today). http://i.stack.imgur.com/KpPlS.png

  • 1
    How do you generate this symbol? – Anthony Dec 17 '15 at 23:51
  • @Anthony Try \def\contra{\tikz[baseline, x=0.22em, y=0.22em, line width=0.032em]\draw (0,2.83)--(2.83,0) (0.71,3.54)--(3.54,0.71) (0,0.71)--(2.83,3.54) (0.71,0)--(3.54,2.83);} – Klint Qinami Sep 25 '16 at 0:05

Some of my teachers and I use someone like (Harry Potter's scar) this $\unicode{x21af}$ (LaTex: \unicode{x21af})

  • Also had a professor who always used this. – dreamer Jul 8 '14 at 17:49
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    This is the only contradiction symbol I've been taught with through both gymnasium (high school) and university. So I would say that it is the most common symbol here in Denmark. Only place I've seen other symbols have been in english-language books and on the internet. – Kitalda Mar 5 '17 at 9:04
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    \unicode{x21af} is giving me an undefined control sequence error. – qwertz Jan 14 at 7:45

The bottom and top symbols $\bot,\,\top$ respectively denote contradictions and tautologies in model theory. For example, a proof by contradiction that $\sqrt{2}\notin\mathbb{Q}$ can be rewritten as a proof that $\sqrt{2}\in\mathbb{Q}\to\bot$.

  • Thanks for letting me know that. +1 – mrs Feb 9 '16 at 7:39

One that all of my professors back in my college days used was "X" with each stroke looking like an axe.

The symbol is use came from my professors at Emory University and Auburn University (all Moore Method practitioners) which is octothorp bang, #!

I use it and teach my students to use it.

To indicate contradiction, I use either of the following three Arial Unicode MS letter-like symbols: Ⓡ or Ⓟ or Ⓒ. For me, Ⓡ indicates Reduction to Absurdity; review, revise, redo. (The 3 R's); Ⓟ indicates premise issue; Ⓒ indicates contradiction.

I got the initial idea from RPC meaning 'Remote Procedure Call' See How RPC Works at https:/technet.microsoft.com, The purpose is to call in your brain (Remote Procedure) to review, revise and redo the premises in your logical proofs or electronic designs. That's the real job.

In philosophy and mathematics, a proof by contradiction, shows the logical revision of a premise. Proof by Contradiction ● A proof by contradiction is a proof that works as follows: ● To prove that P is true, assume that P is not true. ● Based on the assumption that P is not true, conclude something impossible. ● Assuming the logic is sound, the only option is that the assumption that P is not true is incorrect. ● Conclude, therefore, that P is true.

Some Proofs by Contradiction: MATH DIY :here are many mathematical proofs by contradiction on the Internet,

RELIGION: Ponder Anselm's Argument for Existence of God at http://web.nmsu.edu/~dscoccia/101web/101ONT.pdf

P versus NP Problem: SEE Wikipedia at https://en.wikipedia.org/wiki/P_versus_NP_problem

LEGAL: Discredit the opponent's argument by showing it is absurd. SEE: 'Recording and Proof of Contradictions and Omissions, Their Evidential Value and Appreciation of Evidence of Hostile Witnesses' at http://mja.gov.in/Site/Upload/GR/summary%20of%20second%20work%20shop%20criminal%20dated%2010-01-15.pdf.

  • Thanks very much for your nice survey. :-) – mrs May 3 '17 at 7:32

An equivalent to \blitza can be found in the package stmaryrd in math mode via \lightning. Here is another option for the rotated pound sign:

\def\contradict
{
\tikz[baseline, x=0.2em, y=0.2em, line width=0.04em]
\draw (0,0) -- ({4*cos(45)},{4*sin(45)})
    (-1,1) -- ({-1 + 4*cos(45)},{1 + 4*sin(45)})
    (-1,3) -- ({-1 + 4*cos(315)},{3 + 4*sin(315)})
    (0,4) -- ({0 + 4*cos(315)},{4 + 4*sin(315)});
}

And, although I have never seen that as a contradiction symbol, I have seen $\Rightarrow\Leftarrow$ more often, and use it in my teaching. I generally try to avoid double meaning of symbols so in a class not solely for propositional logic I prefer not to use a perpendicular symbol $\perp$ for contradiction.

  • 1
    I'm not very familiar with tikz but I guess your answer is downvoted not due to the content of your tikz macro but solely due to the format (because you didn't highlight the codes). I edited as such. Feel free to further improve your post and overwrite my edit. I don't know who downvoted and wish that person hadn't. Please don't feel disheartened. – Lee David Chung Lin Oct 13 at 7:20

I always had used the following notation. At least in my academic environment this one was suggested and used. You can also see these links

  • Wikipedia (The part "Symbolic representation").
  • TeX (The first page of the section "3 Mathematical symbols").

They both has brought this symbol among symbols that are common for contradiction. About how to type it in TeX with better size, see this link.

enter image description here

The symbols are: $\top$ for truth (example: $100 \in \mathbb{R} \to \top$)
and $\bot$ for false (example: $\sqrt{2} \in \mathbb{Q} \to \bot$)

In Latex, \top is $\top$ and \bot is $\bot$.

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