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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?

Then how about “Every shirt belongs to a unique boy”?

ryang
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    I would interpret the sentence in the second way. But in natural language it can mean either one. – Cheerful Parsnip Aug 04 '12 at 10:49
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    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! – Dario Aug 04 '12 at 10:49
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    If I meant that no shirt belongs to two boys, I would say "every boy has a distinct shirt". – MJD Aug 04 '12 at 11:21
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    Or: There exists a unique shirt, and every boy has it … – Hagen von Eitzen Apr 11 '22 at 19:52
  • @HagenvonEitzen Hehe. Then you must mean that the shirt's design is unique, not that there's only a single shirt or single ownership. – ryang Apr 12 '22 at 08:52
  • Closely related: Confusion on understanding a proposition on equivalence classes – MJD Apr 12 '22 at 14:34
  • Says "particularly in technical communication" and proceeds to give a non-technical example... – jjagmath Nov 04 '24 at 14:55
  • @jjagmath Heh. Just treat "boy" and "shirt" as free variables. – ryang Nov 04 '24 at 15:01
  • No, you also need to replace "has" by a relation. But then you need to fix the grammar for it to make sense, which force you to introduce quantifiers, since now you can't include the word "unique" in the expression "has(boy, shirt)". So, when you make it technical there's no ambiguity. – jjagmath Nov 04 '24 at 17:20
  • @jjagmath By "technical communication", I meant maths/science writing, not formal logic; of course formalising the four sentences reduces all ambiguity, and this has been analysed in the last section of my answer below, which observed that the four statements above turn out to have different meanings from one another. Anyway, as reflected below, my current position is that the given sentence should mean "every boy has exactly one shirt". – ryang Nov 04 '24 at 17:41

2 Answers2

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Further to MJD's, Dario's and Cheerful Parsnip's comments and answers, the sentence

  • Every $A$ has a unique $B$   (❔)

is arguably ambiguous between these clearer constructions:

  1. Every $\boldsymbol A$ has exactly one $\boldsymbol B$

    $\boldsymbol B$ is uniquely determined by $\boldsymbol A$

    (This reading corresponds to the common technical usage of ‘unique’ to mean sole.

    Notice that with this reading:

    • $\text“x-3=0$ has a unique root, $3\text”$ implies neither “The number $3$ is a unique root” nor “The root $3$ is unique to the equation $x-3=0\text”$ !
    • $\text“-2$ and $2$ are unique roots of $x^2=4\text”$ doesn't imply $\text“x^2=4$ has a unique root” !)

  2. Every $\boldsymbol A$ has a $\boldsymbol B$ that has no duplicate

    (In a technical setting, this reading is a bit liberal, and corresponds to the everyday meaning of ‘unique’ as one of a kind.)

To wit, consider the sentence “Every bin has a unique score” (bins are separated by indentation and scores by commas). \begin{gather} 7 &7 &8 &9\tag{1}\\ 6,0 &7,0 &8,0 &9,0\tag{2} \end{gather}

On the other hand, the conjunction of Readings 1 and 2 is logically equivalent to this sentence:

  1. Every $\boldsymbol A$ has a distinct $\boldsymbol B$

    Each $\boldsymbol A$ has exactly one $\boldsymbol {B,}$ which has no duplicate \begin{gather} 6&&7&&8&&9\tag{1,2,3}\\ \end{gather}

It turns out that the Question's four statements have four distinct meanings!

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

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

  • $(S_3)\quad$ “No two boys own 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 owns 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 $(S_2),$ based on Meaning 1 (alternatively: Meaning 2) in the previous section, 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.

ryang
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    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. – Eric Wofsey Nov 29 '20 at 14:50
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    @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. – ryang Nov 30 '20 at 04:00
  • You edited so much, you triggered the automatic "too many edits" flag. Please keep edits to a minimum. – Shaun Nov 06 '24 at 15:19
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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."

Cheerful Parsnip
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  • By the way this last sentence is the same as yours, but we are taking them to mean different things. – Cheerful Parsnip Aug 04 '12 at 11:05
  • 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.) – ryang Aug 04 '12 at 11:10
  • You might say something like "Every linear transformation is represented by a unique matrix." (Even though this is false. :)) – Cheerful Parsnip Aug 04 '12 at 11:20
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  • 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’.
    • – ryang Apr 11 '22 at 16:26
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    The "no shirt belongs to two boys" part seems a bit muddled and/or incorrect. "no shirt belongs to two boys" allows the possibility that some shirt belongs to no boys, but the formula (and subsequent phrase) that you say is equivalent to that phrase does not allow that possibility. – Don Hatch Nov 07 '24 at 11:07
  • @DonHatch agreed. – Cheerful Parsnip Mar 19 '25 at 00:29