“Unique” doesn't have a unique meaning

Question

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”?

Answer 1

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}\\ 7\quad\quad 7\quad\quad 7\quad\quad 7\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’ in the above sense.

   (On the other hand, “uniquely determined” isn't ambiguous and corresponds squarely to interpretation $(3)\,).$

---

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.

Answer 2

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."