Are There Any Symbols for Contradictions?

Question

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 (i.e. as shown the tombstone symbol ∎, Halmos).

Answer 1

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.

Answer 2

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.

Answer 3

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

Answer 4

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).

Answer 5

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.

Answer 6

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.

Answer 7

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 $\sqrt{2}\in\mathbb{Q}\to\bot$.

Answer 8

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.

Answer 9

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

Answer 10

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.

Answer 11

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".

Answer 12

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 $\bot$.