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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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    $\begingroup$ $\Rightarrow\Leftarrow\quad$ $\endgroup$ Jun 18, 2012 at 20:02
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    $\begingroup$ 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. $\endgroup$ Jun 18, 2012 at 20:15
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    $\begingroup$ I usually prefer the symbol: $$\text{Contradiction}$$ $\endgroup$
    – Asaf Karagila
    Jun 18, 2012 at 20:18
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    $\begingroup$ 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... $\endgroup$
    – user1729
    Jun 19, 2012 at 9:19
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    $\begingroup$ My algebra professor used ?!. I found that quite amusing. $\endgroup$
    – tskuzzy
    Jun 25, 2012 at 16:46

12 Answers 12

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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.

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    $\begingroup$ 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. $\endgroup$ Oct 7, 2016 at 1:13
  • $\begingroup$ 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. $\endgroup$ Oct 29, 2017 at 12:32
  • $\begingroup$ My favorite for a completely silly reason: "You bot!" is a common youth insult, meaning "Stupid!". Even sillier given that it's the youth group of my chess club, where a bot would spank them even if set at half ELO... $\endgroup$ May 12, 2020 at 8:05
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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.

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    $\begingroup$ 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. $\endgroup$ Jun 18, 2012 at 20:42
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    $\begingroup$ Cute, @Andres. I hadn't seen that and I like its name. +1 $\endgroup$ Jun 19, 2012 at 2:07
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    $\begingroup$ I always thought that was Harry Potter's scar... $\endgroup$ Jun 19, 2012 at 3:43
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    $\begingroup$ Yes, I really like ↯ as my contradiction symbol. $\endgroup$
    – davidlowryduda
    Jun 19, 2012 at 6:58
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    $\begingroup$ The ↯ symbol seems to be very common : in France, my first teacher after high school, used this symbol too (since I use it unconsciously). $\endgroup$
    – JBC
    Jun 19, 2012 at 21:24
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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

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    $\begingroup$ How do you generate this symbol? $\endgroup$
    – Anthony
    Dec 17, 2015 at 23:51
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    $\begingroup$ @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);} $\endgroup$ Sep 25, 2016 at 0:05
  • $\begingroup$ This is in Unicode as U+2A33 ("smash product"), and the stix and boisik packages have it as \smashtimes (according to the CLSL). $\endgroup$
    – benrg
    Apr 22 at 18:03
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Some of my teachers and I use someone like (Harry Potter's scar) this $\unicode{x21af}$ (LaTex: \unicode{x21af})

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    $\begingroup$ Also had a professor who always used this. $\endgroup$
    – dreamer
    Jul 8, 2014 at 17:49
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    $\begingroup$ 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. $\endgroup$
    – Kitalda
    Mar 5, 2017 at 9:04
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    $\begingroup$ \unicode{x21af} is giving me an undefined control sequence error. $\endgroup$
    – qwertz
    Jan 14, 2018 at 7:45
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    $\begingroup$ @qwertz I believe \unicode{} is a XeLaTeX specific command. The package marvosym has a \Lightning command, that works with PDFLateX and LuaLaTeX. $\endgroup$
    – SeSodesa
    Mar 11, 2019 at 17:52
  • $\begingroup$ Also \blitz is suggested in the Comprehensive LaTeX symbols list ctan.org/pkg/comprehensive $\endgroup$
    – Andrestand
    Mar 11, 2021 at 10:50
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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

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    $\begingroup$ This one is my favourite. I've been looking for it, thanks! $\endgroup$ Feb 2, 2020 at 4:15
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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.

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    $\begingroup$ 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. $\endgroup$ Oct 13, 2018 at 7:20
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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 $\sqrt{2}\in\mathbb{Q}\to\bot$.

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  • $\begingroup$ Thanks for letting me know that. +1 $\endgroup$
    – Mikasa
    Feb 9, 2016 at 7:39
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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.

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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.

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  • $\begingroup$ Thanks very much for your nice survey. :-) $\endgroup$
    – Mikasa
    May 3, 2017 at 7:32
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One that all of my professors back in my college days used was "X" with each stroke looking like an axe.

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  • $\begingroup$ I don't know if it's the case, but I've seen people use strange/wrong/non-standard symbols in slides and or lecture notes just due to poor IT skills, lack of fonts in PowerPoint/printer, etc. $\endgroup$
    – Andrestand
    Mar 11, 2021 at 10:37
  • $\begingroup$ PS: may this be the case reported by user Stefan Octavian? $\endgroup$
    – Andrestand
    Mar 11, 2021 at 10:42
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I'm 8 years late, but I've seen that this answer is not on the list. In Romania we use a symbol similar to a pair of scissors, that I have learned from both my teachers I've studied with. Unfortunately, I was unable to find such a symbol in MathJax, but I will try reproducing it: $\require{HTML} \style{display: inline-block; transform: rotate(30deg) scaleX(-1)}{\flat} \style{display: inline-block; transform: rotate(-30deg) scaleX(1)}{\flat}$. Yes, something like that, but with smaller "ears".

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  • $\begingroup$ This may be what user drum suggests by axes-like X. $\endgroup$
    – Andrestand
    Mar 11, 2021 at 10:41
  • $\begingroup$ There are several scissors symbols in the Comprehensive LaTeX symbols list ctan.org/pkg/comprehensive They can be rotated with the command \rotatebox. $\endgroup$
    – Andrestand
    Mar 11, 2021 at 10:51
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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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  • $\begingroup$ As also mentioned... typical for tautology and always false, respectively, too. $\endgroup$
    – Andrestand
    Mar 11, 2021 at 10:40

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