Written in English, does "the set S contains only members of set T" imply that S does contain some member of set T?
How would this relationship be written formally?
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My view is that vernacular would consider that S is not empty, i.e. $\emptyset \neq S\subseteq T$, while mathematical language would consider that S can be empty, i.e. $\emptyset\subseteq S\subseteq T$.
Lay people do not speak of empty sets, while mathematicians are aware of their role in their work.
That means that the sentence is not ambiguous, but the meaning depends on the community where it is used.
Of couse, people aware of the two readings can play games with it. But that is yet something else.
You may notice that I did not specifically mention English, because the same is probably true of many "languages".
To think that English (or German, or French) is a well defined language is unrealistic. Actually, I doubt you can find two people who agree on what is English and the meaning, or on the ambiguity, of its sentences,
It might be interesting to ask the linguists.
It's ambiguous whether "contains only elements of $T$" implies that it must contain at least one element. Formally, you could write $\emptyset\subset S\subseteq T$ or $\emptyset\neq S\subseteq T$ or say "the non-empty set $S$ contains only elements of $T$."