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When using words like “unique” and “any”, particularly in technical communication, I sometimes find myself deliberating over which definition and tenor is the most natural, or which alternative phrasing might be clearer even if less succinct or accessible.

Does “Every boy has a unique shirt” mean that

  • no two boys have the same shirt,

or does it mean that

  • no two shirts belong to the same boy?

I suppose the former; if so, then does the latter mean “Every shirt belongs to a unique boy”?

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    $\begingroup$ I would interpret the sentence in the second way. But in natural language it can mean either one. $\endgroup$ Aug 4, 2012 at 10:49
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    $\begingroup$ It could as well mean "Every boy has at least one shirt that no other boy has". That's really the difference between mathematical/logic formulations and everyday speech. The latter is often ambiguous! $\endgroup$
    – Dario
    Aug 4, 2012 at 10:49
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    $\begingroup$ If I meant that no shirt belongs to two boys, I would say "every boy has a distinct shirt". $\endgroup$
    – MJD
    Aug 4, 2012 at 11:21
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    $\begingroup$ Or: There exists a unique shirt, and every boy has it … $\endgroup$ Apr 11 at 19:52
  • $\begingroup$ @HagenvonEitzen Hehe. Then you must mean that the shirt's design is unique, not that there's only a single shirt or single ownership. $\endgroup$
    – ryang
    Apr 12 at 8:52

2 Answers 2

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Closely approximating the English is the following logical formula $$\forall b \exists!s P(s,b)$$ where $b$ is a boy and $s$ is a shirt, and $P(s,b)$ means that s belongs to b. This means that for each boy there is one and only one shirt that belongs to him. If you want to say that no shirt belongs to two boys you would say $$\forall s\exists! b P(s,b),$$ and the natural language approximation would be "Every shirt belongs to a unique boy."

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  • $\begingroup$ By the way this last sentence is the same as yours, but we are taking them to mean different things. $\endgroup$ Aug 4, 2012 at 11:05
  • $\begingroup$ In natural language, how would you use the word "unique" to write that "Every linear transformation has exactly one matrix representation?" (A matrix may represent an infinite number of linear transformations, but leave this information or its implications out.) $\endgroup$
    – ryang
    Aug 4, 2012 at 11:10
  • $\begingroup$ You might say something like "Every linear transformation is represented by a unique matrix." (Even though this is false. :)) $\endgroup$ Aug 4, 2012 at 11:20
  • $\begingroup$ Yes, my first instinct was to write that too. ("Every LT, with respect to chosen bases in the domain and codomain, has a unique matrix representation.") Then I wavered and decided that the reader would as well interpret that to mean no two LT (with respect to the same bases) share the same representation, which is patently false! $\endgroup$
    – ryang
    Aug 4, 2012 at 13:06
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    $\begingroup$ 1. Actually, “no shirt belongs to two boys” (i.e., “no two boys share the same shirt”) means that every shirt belongs to at most one boy instead of that every shirt belongs to exactly one boy; as such, its formalisation is $$∀s\;∃b_1\;∀b_2\;\big(P(s,b_2){\implies}b_2{=}b_1\Big)$$ (equivalently: $∀s,b_1,b_2\;\big(P(s,b_1)∧P(s,b_2){\implies}b_2{=}b_1\Big)$) instead of $$\forall s\exists! b P(s,b).$$ 2. Summarising your Answer: your preferred interpretation of ‘unique’ is ‘exactly one’. $\endgroup$
    – ryang
    Apr 11 at 16:26
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Further to MJD's, Dario's and Cheerful Parsnip's comments and answers:

  1. Every $A$ has a unique $B$” has multiple interpretations:
    $(1)$ Each $A$ has at least one $B$ that no other $A$ has;
    $(2)$ Each $A$ has exactly one $B,$ which no other $A$ has;
    $(3)$ Each $A$ has exactly one $B.$

  2. To illustrate these interpretations, read each of the following lines (their labels correspond to the interpretation numbers above) as “Every bin has a unique score” (bins are separated by indentation and scores by commas): \begin{gather}6,4\quad 7,5\quad 8,0\quad 9,0\quad \tag{1}\\ 6\quad\quad 7\quad\quad 8\quad\quad 9\quad\quad \tag{1,2,3}\\ 6\quad\quad 7\quad\quad 8\quad\quad 8\quad\quad \tag{3}\end{gather}

  3. Interpretation $(1)$ corresponds to the most common definition of ‘unique’ as having no duplicate.

    Interpretation $(2)$ corresponds more accurately to “Each $A$ has a distinct $B$”.

    Interpretation $(3)$ corresponds to the common technical usage of ‘unique’ to connote having no alternative possibility.

  4. All in all, due to its ambiguity and frequent conflation with the word ‘distinct’, I would just avoid using the word ‘unique’.


It turns out that all the four statements in the Question have different meanings!

  • $(S_1)\quad$ “Every boy has a unique shirt.”

  • $(S_2)\quad$ “Every shirt belongs to a unique boy.”

  • $(S_3)\quad$ “No two boys have the same shirt.”

        Every shirt belongs to at most one boy. $$∀s,b_1,b_2\;\Big(P(s,b_1)∧P(s,b_2)\implies b_1=b_2\Big).$$

  • $(S_4)\quad$ “No two shirts belong to the same boy.”

        Every boy has at most one shirt. $$∀b,s_1,s_2\;\Big(P(s_1,b)∧P(s_2,b)\implies s_1=s_2\Big).$$

$(S_1)$ and $(S_2)$ are discussed in the previous section, and are clearly inequivalent;

since $(S_3)$ and $(S_4)$ each allows some boy to own no shirt, neither is equivalent to $(S_1);$

since $(S_3)$ and $(S_4)$ each allows some shirt to have no owner, neither is equivalent to $(S_2);$

$(S_3)$ and $(S_4)$ are clearly inequivalent.

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    $\begingroup$ I disagree--technical communication is exactly the context where you can use "unique". In mathematical writing, "Every A has a unique B" pretty much always means (1). It is only in a non-technical context (dealing with an audience who is not familiar with mathematical language) that the ambiguity arises. $\endgroup$ Nov 29, 2020 at 14:50
  • $\begingroup$ @EricWofsey While "she has a unique talent" is always unambiguous, I prefer to describe $[7\quad7\quad7\quad7]$ with "every bin has $\require{cancel}\cancel{\text{a unique}}$ exactly one$\,$ score" in both non-technical and technical contexts. $\endgroup$
    – ryang
    Nov 30, 2020 at 4:00
  • $\begingroup$ Please note that due to my Answer having been revised, the "(1)" in Eric's comment now actually refers to Interpretation (3) instead. $\endgroup$
    – ryang
    Apr 11 at 17:24

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