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I find the word "iff" for "if and only if" quite helpful for brief statements, but is there a similar one meaning "one and only one"?

edit In light of the ambiguities some of the answers so far hint at, here's an example I'd like to shorten:

Of the statements x, y, z, one and only one statement is true

should become

xorne of the statements x, y, z is true

where xorne is the sought word.

Asaf Karagila
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Tobias Kienzler
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    If not, due to the relationship with xor I propose xone or xorne – Tobias Kienzler Mar 10 '15 at 12:43
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    What's wrong with "exactly one"? Or perhaps "a unique"? – Rory Daulton Mar 10 '15 at 12:48
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    onee?${}{}{}{}$ – GFauxPas Mar 10 '15 at 12:50
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    $\exists !$ is often used for "there exists a unique..." though I wouldn't say it was standard – Matthew Towers Mar 10 '15 at 12:54
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    "Exactly one" is the term I've heard. – apnorton Mar 10 '15 at 14:10
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    Should this be on [english.SE]? – Cole Tobin Mar 10 '15 at 14:54
  • @ColeJohnson I don't think they would even accept iff as a word there... – Tobias Kienzler Mar 10 '15 at 14:57
  • @TobiasKienzler True. It does look like a typo. But AFAIK, it's just as much a word as "xor" (which is short for "exclusive or") – Cole Tobin Mar 10 '15 at 15:00
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    I think in most context not involving truth of statements using "one" rather than "a" usually already means "exactly one". For instance "this equation has one solution" or "this polynomial has one root" would already exclude multiple (distinct) solutions/roots. In context where you cannot use "a" one could use "some" or "at least one", so that "one" would still mean "exactly one". – Marc van Leeuwen Mar 10 '15 at 17:14
  • Coincidentally I find the term "iff" to be redundant; if already means if, and it already is a conditional meaning "only when the following criteria are met". – TylerH Mar 10 '15 at 18:48
  • @MarcvanLeeuwen that's a very good point,you should propose visa as answer,I might even accept it – Tobias Kienzler Mar 10 '15 at 18:51
  • @TylerH indeed, but unfortunately everyday usage tends to mean "if,but not restricted by" – Tobias Kienzler Mar 10 '15 at 18:53
  • @TobiasKienzler I don't find that to be the case at all, but now we're just derailing the comments section. Oops – TylerH Mar 10 '15 at 18:54
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    @TylerH what do you find "iff" redundant for? It means "if and only if", which is important, because "if" and "only if" don't mean the same thing. "A if B" means that "B implies A". "A only if B" means that "A implies B". That's why "A iff (if and only if) B" means "A implies B and B implies A", or "A and B are logically equivalent". – Joshua Taylor Mar 10 '15 at 21:26
  • I don't know why you'd want one - a phrase works perfectly well, flows better in most all cases, and will be familiar to a wider audience. (I feel the same way about "iff" and don't use it myself) Unless you need to use this idea so often that saying "one and only one" would start to sound repetitious and artificial (i.e. when the reader starts seeing it and parsing it as one chunk rather than a phrase), I would stick with using a phrase - but that'd be hard to do with "one and only one", since you need a list of at least two things follows each use of such a phrase. – Milo Brandt Mar 11 '15 at 02:12
  • @JoshuaTaylor I find it redundant because, outside of math's peculiar use of "if"/"iff", it means something different. – TylerH Mar 11 '15 at 05:49
  • Hey, where did that new users' answer about "onne" go? They gave a source for Conway saying that, which actually sounded reasonable? @Jyrki can you undelete it or what was wrong with that? – Tobias Kienzler Mar 11 '15 at 07:52
  • @Tobias: The user deleted their answer themselves. In a comment they state that the answer is included in Blue's as the reason for the deletion. – Jyrki Lahtonen Mar 11 '15 at 07:58
  • @JyrkiLahtonen That's a shame, complementary answers aren't that bad iff truly complementary... Could you quote their named source as a comment on Blue's answer then? – Tobias Kienzler Mar 11 '15 at 08:00

10 Answers10

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I have never come across such a word. My best ideas for replacements are: Precisely one, Exactly one, just one.

Sidenote: Some Authors prefers "if and only if" over "iff", since it can be easy to forget to show both ways. So maybe it is a good thing that there is no shorthand, if indeed there is none.

William Kurdahl
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    I guess "precisely one" is the best solution. Oh, shouldn't your answers' end read "iff indeed there is one"? :P – Tobias Kienzler Mar 10 '15 at 13:30
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    Can you clarify the last sentence? Do you mean "a good thing that there is no shorthand"? – Ypnypn Mar 10 '15 at 18:11
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    Also let's not forget that if vs. iff seems very sensitive to typos. – Arthur Mar 10 '15 at 20:26
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    @Arthur: meh. $12$ vs $122$ seems equally sensitive to typos, but mathematicians struggle on regardless ;-) – Steve Jessop Mar 11 '15 at 01:13
  • @SteveJessop No, it is not equally sensiitive. It's much easier to overlook a typo in a word than in a number. – yo' Mar 11 '15 at 21:41
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    @yo': I wasn't talking about whether it is typoed, Arthur only claimed that it seems sensitive to typos and I was responding to that. In fact, my experience is that if vs iff is not typoed. I speculate that the two words are so important to mathematicians that the difference simply is not overlooked, at least not frequently enough to deprecate "iff". Those who find they are typoing it, of course can address that if they choose by ceasing to use "iff" :-) – Steve Jessop Mar 12 '15 at 05:21
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John H. Conway has said that whenever he writes "iff", he finds himself compelled to also write "thenn" (ie, "then and only then!") for parallel emphasis in both parts of a statement. (It's only fair, after all, given that the statement asserts the parts' logical equivalence.)

I see no reason that we can't follow his lead (as well as consonant-doubling precedent) and introduce, say, "onne" into the mathematical vernacular.

Edit. According to @AndreasBlass, Conway (unsurprisingly) beat me to it.

Edit. User @Someone else posted a sourced answer linking Conway to "onne", then deleted that answer for being a "duplicate" of mine. OP and I find the reference(s) helpful, so I'll quote the post here. (If @Someone has an objection ---or if this is a violation of some StackExchange policy--- I can delete it.)

J H Conway of Princeton introduced "onne" for "one and only one". (Reference: Margie Hale: "Essentials of Mathematics: Introduction to Theory, Proof, and the Professional Culture")

I have seen this a few times in books and scholarly papers, although it is obviously much less common than "iff".

In a follow-up comment, @Someone wrote:

Here's a second (although possibly not very reliable) source for the same thing - MathForum post.

Community
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Blue
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  • Good suggestion, though I wonder how to properly pronounce that. Hm, "One-Nh" perhaps. Then again, that problem already arises for iff and thenn... – Tobias Kienzler Mar 10 '15 at 13:27
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    @TobiasKienzler: Why, of course, you pronounce it as "one, and only one!" ... with a suitably dramatic flourish. :) – Blue Mar 10 '15 at 13:31
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    I've heard Conway propose not only "onne" but also "whenn" (for "when and only when") and "threee". – Andreas Blass Mar 10 '15 at 14:08
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    Shall we just take it back to Middle English? Whan that Aprille with his shoures soote the droghte of Marche hath perced to the roote ... ;) – Neal Mar 10 '15 at 14:25
  • A new user also added "onne" as answer and provided a reference in a comment, but they deleted their answer. @Blue Since you're ≥10k, can you maybe find and copy that into your answer? – Tobias Kienzler Mar 12 '15 at 07:44
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    FWIW, the Wolfram page has "onee", not "onne"... and "onee" is in the comments-on-the-question, too. – TallTed Feb 15 '17 at 06:08
  • I heard Conway lecture once, where he used "unlesss" for "unless and only unless". – GEdgar Oct 26 '22 at 01:22
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You can see it written symbolicallly: $$ \exists ! $$ So like this:

$(\exists! x \in \mathbb N)( x^2=4)$

GEdgar
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7

A unique. ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$

Neal
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    Oh come on, instead of that "creative" abuse of MathJax you could simply quote Mirriam-Webster or similar... – Tobias Kienzler Mar 10 '15 at 12:58
  • ...however I cannot fully agree: "7 is unique in X" would, to me at least, merely mean that X does contain 7 only once, but not that there are no other (potentially repeated) elements – Tobias Kienzler Mar 10 '15 at 12:59
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    Unfortunately, "unique" is ambiguous. Some take it to mean "precisely one", others take it to mean "at most one." – goblin GONE Mar 10 '15 at 13:10
  • I've added an example phrase to my question to illustrate where I think "unique" doesn't fit. But maybe my English is just too bad and you can correct me ;) – Tobias Kienzler Mar 10 '15 at 13:14
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    (Just to clarify, I didn't downvote, but I am not yet convinced of upvoting ;) – Tobias Kienzler Mar 10 '15 at 13:28
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    This is grammatically wrong; it has to be two words: "a unique". Incidentally, while "a unique" has as many letters and words as "just one", the latter is preferable in having only two syllables. – John Bentin Mar 10 '15 at 14:02
  • @JohnBentin Thanks! – Neal Mar 10 '15 at 14:23
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    @goblin Really? Seems to me the natural interpretation of that is "... iff for all $x\in X$, there is exactly one $y\in Y$ satisfying $(x,y)\in R$." – Neal Mar 10 '15 at 14:27
  • @goblin: Please could you quote chapter and verse for any text where "a unique $y\in Y$ satisfies [ ... ]" allows the possibility that no $y\in Y$ satisfies [ ... ]? – John Bentin Mar 10 '15 at 14:40
  • @JohnBentin, not off the top of my head, but see Allen Knutson's comment here. – goblin GONE Mar 11 '15 at 05:45
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    @JohnBentin, on second thoughts, perhaps it is not often used this way. I've had a look around and I can't see anyone doing it, anywhere. But I'm sure I've heard it once or twice. I'll let you know if I can think of a realistic sentence where it might realistically be used. – goblin GONE Mar 11 '15 at 07:55
  • If your answer is too short without Mathjax abuse, your answer is too short. Try quoting a definition. – TRiG Mar 11 '15 at 14:25
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    This is correct, and exactly what "unique" means and how it is used. Unique is correct. – David Boshton Mar 11 '15 at 15:04
  • You are not answering the question. (S)he is asking for an abbreviation of "one and only one", not a word/expression. – Rubem Pacelli Nov 29 '22 at 13:08
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"$\exists$ a unique" would be the shortest common way to say what you're asking for.

Christopher
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    If you must use symbols, then $\exists!$ is fairly common. But I'd stick to words. – lhf Mar 10 '15 at 13:18
  • Indeed, I'd prefer something pronounceable (while still short) – Tobias Kienzler Mar 10 '15 at 13:18
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    @TobiasKienzler: Saunders MacLane taught me to pronounce $\exists!$ as "E-shriek", and I have done so ever since. – WillO Mar 10 '15 at 17:45
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    @WillO good idea, but then why not simply "eek"? – Tobias Kienzler Mar 10 '15 at 18:49
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    I'm not sure that $\exists !$ would cover the example asked for of "one and only one of $x,y,z$ is true", at least not in any efficient way: would you say "$\exists !$ letter such that one of the following is true", or how would it be paraphrased? – Geoff Robinson Mar 10 '15 at 20:04
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Although there are many answers at this point, I'm compelled to answer because the original question did not use a quantifier, so depending on the logic it may be difficult to use $\exists!$ directly. For example, the statement "exactly one of the sets $x,y,z$ is blue" can be written "$\exists!n\in\{x,y,z\}\, n$ is blue", but "exactly one of the following propositions is true: the moon is made of cheese, I know how to swim, there is an elephant in the White House" is much harder to write in this manner, because there are no sets available to quantify over (propositions can not necessarily be treated as sets; the difference is what lead to Russell's paradox).

If $P,Q,R$ are propositional variables, then "exactly one of $P,Q,R$ is true" can be faithfully represented as:

$$(P\land\lnot Q\land\lnot R)\lor(\lnot P\land Q\land \lnot R)\lor(\lnot P\land\lnot Q\land R)$$ $$(P\oplus Q\oplus R)\land\lnot(P\land Q\land R)$$ $$((P\oplus Q)\land\lnot R)\lor(\lnot P\land\lnot Q\land R)$$

where $P\oplus Q=(P\land\lnot Q)\lor (\lnot P\land Q)$ is the exclusive or.

The third example actually generalizes well; if $\phi_k^n$ means "exactly $k$ of $P_1,P_2,\dots,P_n$ is true" then these formulas can be defined inductively as

$$\phi_0^1=\lnot P_1\quad\phi_1^1=P_1\qquad\phi_0^n=\phi_0^{n-1}\land\lnot P_n\quad\phi_n^n=\phi_{n-1}^{n-1}\land P_n\qquad$$ $$\phi_k^n=(\phi_{k-1}^{n-1}\land P_n)\lor (\phi_k^{n-1}\land\lnot P_n)$$

although the length of the formula is still quite long, ${n+2\choose k+1}-2$ in general, and I don't know if there is an asymptotically shorter formula.

One last special case: The number one case (in fact, the only case) where I've seen this propositional "exactly one" occur in math, and which may be the prototype from which you abstracted this question, is in the statement of the trichotomy law for real numbers:

If $x,y$ are real numbers, then exactly one of $x<y$, $x=y$, $x>y$ is true.

In this particular case (in the presence of antisymmetry), it turns out that you can rewrite this law into a simpler one using only the biconditional:

If $x,y$ are real numbers, then $\lnot\, x<y\leftrightarrow(x=y\lor y<x)$.

It is a good exercise to show that this axiom implies both trichotomy and antisymmetry.

Mario Carneiro
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  • Note that expressing "exactly one of $P$, $Q$ and $R$" as $(P\oplus Q\oplus R)\land\lnot(P\land Q\land R)$ does not generalize well to higher numbers of propositions; for, say, four propositions, it would look something like $(P\oplus Q\oplus R\oplus S)\land\lnot(P\land Q\land R)\land\lnot(P\land Q\land S)\land\lnot(P\land R\land S)\land\lnot(Q\land R\land S)$, which isn't nearly so convenient any more. – Ilmari Karonen Mar 11 '15 at 11:04
  • @IlmariKaronen True. It might be interesting to investigate whether there is a way to write "exactly $k$ of $n$ propositions are true" with a formula of less than $O(n{n\choose k})$ if you allow combintations of arbitrary two-argument boolean functions (this bound is what you get from the DNF form of the expression). – Mario Carneiro Mar 11 '15 at 11:16
  • I hope I'll never meet a person willing to write "exists a unique" as $(P\oplus Q\oplus R)\land\lnot(P\land Q\land R)$. (I don't speak about computer algebra, but neither the OP is.) – yo' Mar 11 '15 at 21:45
  • @yo' I realize this, but as the tenth or so answer I wanted to put a different spin on the question for other readers, as I feel the linguistic question is quite well settled by "exactly one of ... is true", as I use throughout the answer. For some people the question of how to state this in propositional logic is not obvious, and indeed as I show it's not as straightforward as other logical constructions, so I provide some insights on this front. – Mario Carneiro Mar 12 '15 at 00:13
  • @MarioCarneiro The question doesn't need more than 2 answers. But that may be only my point of view. – yo' Mar 12 '15 at 00:19
  • @yo' If my answer was some tongue-in-cheek "onne" or "thenn" (which I personally find to be a non-answer as you would never be able to use it in a real proof), then yes I wouldn't have bothered. But there is a nonzero probability that someone looking for ways to say this in propositional logic will be led to this popular question, and I hope that this answer will help them. – Mario Carneiro Mar 12 '15 at 00:23
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"xorne" would usually be written as "precisely one" or "exactly one." I don't think there is an accepted abbreviation; if you really want to find something, your best bet is probably to research the family of functions $f_n : \mathbb{B}^n \rightarrow \mathbb{B}$ defined by asserting that $f_n(x)=1$ iff there is a unique $i \in \{1,\ldots,n\}$ such that $x_i=1$.

goblin GONE
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Ok, for the sake of it, I propose

xorne

which portmanteaus xor and one as an actual answer.

Tobias Kienzler
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$\exists !$ won't really work by itself as the OP is potentially restricting to a subset of a larger space, i.e. there could be many objects that satisfy the property in question, but only one out of a specific collection which satisfies the property: $\exists ! x\in A\subset E$ such that $\Phi(x)$, but $\exists y\in E$, such that $y\neq x$, and $\Phi(y)$.

So we need something more along the lines of $$``\text{there is only one element of}" = \exists ! (\text{fill-in-the-blank})\in.$$

Any word would work, you just have to define it as there isn't any standard. "xorne" isn't quite as nice as "iff" or "thenn" as the latter are very natural sounding in speech, and are almost a natural slur of the statements they substitute for. In light of that, I would suggest "olnel" for "only one element of" (pronounced \ōl'nel\).

Olnel $A$ $\Phi$ = "only one element of set $A$ satisfies statement $\Phi$."

Edit: On second thought... How about "onelf"? Pronounced one elf. Or " alonelf" pronounced a lone elf for "a lone element of".

jdods
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  • Note that $\exists! x\in A,\varphi$ means exactly that there is a unique element of $A$ such that $\varphi$ holds. There could be more elements outside of $A$ for which $\varphi$ holds, these don't matter for the evaluation. It can be defined in terms of unrestricted existential uniqueness as $\exists! x,(x\in A\land\varphi)$. – Mario Carneiro Mar 10 '15 at 21:31
  • That was my point. That you can't overlook the containment in / restriction to a particular set. Many here were just commenting "\exists!". – jdods Mar 10 '15 at 21:33
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Of course one could say "exactly one", "an unique", "exactly one" instead of what is being proposed here, in the same way one could say "if and only if" instead "iff". No one wonders that. So I do not think useful to say it...

The term xorne seems reasonable since it is the portmanteau of xor+one. I was going to propose "oone", but I notice that others had similar-but-different propositions (onee, onne...). xorne, on the other hand, seems to be less prompt to divergence.

Rubem Pacelli
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