Syllogism question using set theory notation

Question

i'm stuck on a question where I have to explain the set theory notation of the syllogism below;

boats are vessles $(A ⊆ B)$

boats operate in water $(A ∈ C)$ or $(A ⊆ C)$

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Some vessles operate in water $(B ∩ C ≠ \emptyset)$

The question my friends and I have had is surrounding the set notation to use for 'boats operate in water' - Some are of the assumption that we should use A ⊆ C, but i'm really stumped because I was under the assumption of using A ∈ C. We've all been back and forth about it and was just hoping to get some clarity on here for anyone that can help speculate, thank you.

Answer 1

"Boat are vessels" is $\forall x (\text {Boat}(x) \to \text {Vessel}(x))$ and this is equivalent, in set language, to :

$\text {Boat} \subseteq \text {Vessel}$.

The same for "Boats operate in water" : $\text {Boat} \subseteq \text {Water}$, because we can derive it as the conclusion of a Barbara-type syllogism whose second premise is : "Vessels operate in water", i.e. $\forall x (\text {Vessel}(x) \to \text {Water}(x))$.

Regarding "Some vessels operate in water", we have : $\exists x (\text {Boat}(x) \land \text {Water}(x))$.

This means that: $\text {for some } x : x \in \text {Boat} \text { and } x \in \text {Water}$ that, translated into set language, is as you said :

$\text {Boat} \cap \text {Water} \ne \emptyset$.

Answer 2

It depends on what you call $A$, $B$ and $C$.

If $A$ is the set of all boats (that is $x\in A\iff x\textrm{ is a boat}$), $B$ is the set of all vessels and $C$ is the set of all the means of transport which operate in water (that is $x\in C\iff x\textrm{ operates in water})$, then:

Boats are vessels: $x\in A\implies x\in B$, that is $A\subset B$;

Boats operate in water: $x\in A\implies x\in C$, that is $A\subset C$;

Some vessels operate in water: there exists $x\in B$ such that $X\in C$, that is $B\cap C\neq\emptyset$.

But you can change the definition of $C$, for instance you can decide that $C$ is the set of all categories of means of transportation which operate on water, that is $x\in C$ iff $x$ is a category of means of transportation which operate on water. Then you would have $A\in C$. However, you would not have $B\cap C\neq\emptyset$ anymore. Hence the first definition of $C$ is more natural because more compatible with the definitions of $A$ and $B$.