8

As "there is at least one" and "there is exactly one" both have their symbols, I wonder what is the common notation for "there is at most one"? By "common" I mean the desired notation can be used without re-defining it.

Yes
  • 20,910
  • 3
    There are no such notation, that can be safely used without first defining it. – vadim123 Mar 25 '15 at 04:46
  • 1
    Why is there a close vote for "unclear what you are asking"? I think it's quite clear what you are asking, but it does not seem at all likely that there actually is a standard symbol to indicate what you want. – Daniel W. Farlow Mar 25 '15 at 04:50
  • 1
    To the close-voter: Please specify your problem or add additional details to highlight exactly why you voted to close this question. Certainly I am glad to see my question gets clearer if it is not that clear at the present stage for most people. – Yes Mar 25 '15 at 04:53
  • 1
    As others said, there is no common such notation: this quantification is simply not often used. However, the one I like best for reasons of symmetry is $;!;$ (as in $;\langle ! n :: P(n) \rangle;$); see my answer http://math.stackexchange.com/a/398539/11994. – MarnixKlooster ReinstateMonica Mar 25 '15 at 05:15
  • @MarnixKlooster: Thanks. In actuality I considered that. However, I feel that it may be not suitable in a context involving factorial, and factorial is more and more ubiquitous :) – Yes Mar 25 '15 at 05:24

2 Answers2

9

The quantifier you are looking for is a special case of a counting quantifier. Wikipedia's link Counting quantifier only mentions quantifiers of the form "there exists at least k elements that satisfy property X", but a more general definition can be given, see for instance Majority logic, p. 60.

These quantifiers are used in various domains of theoretical computer science (circuit complexity [3,4], constraint satisfaction problems [2], complexity [1], etc.). There is no agreement on notation yet, but $\exists^{\leqslant 1}$ seems to be a reasonable suggestion.

[1] K. Etessami, Counting Quantifiers, Successor Relations, and Logarithmic Space, Journal of Computer and System Sciences 54, (1997) 400–411.

[2] F. Madelaine, B. Martin, J. Stacho, Constraint Satisfaction with Counting Quantifiers, LNCS 7353, 2012, pp 253-265.

[3] N. Schweikardt Arithmetic, First-Order Logic, and Counting Quantifiers, ACM Transactions on Computational Logic (2002)

[4] H. Straubing, Finite automata, formal logic, and circuit complexity. Progress in Theoretical Computer Science. Birkhäuser Boston Inc., Boston, MA, 1994.

J.-E. Pin
  • 42,871
5

$\exists_{\leq 1}$

$(\nexists\vee\exists!)$

Warning: to my knowledge, neither notation is a well-established standard. I think, at any rate, that the first one has the advantage of being quite intuitive, while the second one consists of a combination of already familiar symbols.

triple_sec
  • 23,935
  • 2
    That seems fairly ad hoc, do you have a reference for a text that actually uses that? (i.e. the op seems to be asking for whether there is a standard use symbol, to which the answer is likely "no.") – Adam Hughes Mar 25 '15 at 04:48
  • @AdamHughes Admittedly, the first one is non-standard. The second one, however, is a combination of well-established symbols. The particular combination may be uncommon as well, but the constituents are not. – triple_sec Mar 25 '15 at 04:51
  • 3
    Certainly I agree, I think it might be best to add into your answer "while technically unambiguous and using of standard symbols, you will probably never encounter anyone using it in the wild." It answers the question more fully that way, at least IMO. – Adam Hughes Mar 25 '15 at 04:53
  • @AdamHughes You're right, a clarifying comment was in order. I had gotten too carried away with the urge to post a short and witty answer. – triple_sec Mar 25 '15 at 04:56
  • Thanks for taking the trouble to write. Yes, to me the first is more appealing and more suggestive, so I would choose the first. The second one may be somewhat cumbersome, no disrespect, but I think writing "there is a.m.o", say, maybe more convenient than the second. I appreciate your work anyway. – Yes Mar 25 '15 at 04:57