Logical translation of the verb " to be " in " the cat is a mammal ". ( Inclusion or membership?) 
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*Standardly, the sentence " the cat is a mammal " would be translated as


for all x , if x is a cat, then x is a mammal,

meaning that the set of cats is included in the set of mammals : $C\subseteq M$.

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*There is another interpretation: the species " cat" is a mammals species. Under this alternative interpretation ( admitting that a " species" is some sort of set or collection), we would have : $ C \in M $*

with C = the set of cats and M* = the set of species of mammals.

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*The distinction I'm referring to is analogous to the medieval distinction between personal supposition and simple supposition of a term. In " horses are animals" the term " horse" supposes personnaly while in " horse is a species" , the term supposes simply.


*What are the drawbacks of the second interpretation ( where " is" is interpreted as membership relation holding between a species and a set of species)? Why has the first interpretation become standard?
 A: The two interpretations refer to two distinct terms. $M$ is the set of mammals, where each element of $M$ is an individual (a mammal); while $M^*$ is the set of species that are mammals, where each element of $M^*$ is a set (a species) of individuals (the mammals).
The meaning of the verb "to be" is an inclusion or a membership relation depending on whether the "predicate" you want to consider is $M$ or $M^*$.
In the former, $C$ is a subset of $M$, in the latter $C$ is an element of $M^*$.

In set theory, you can say that $C \subseteq M$ is equivalent to $C \in \mathcal{P}(M)$, for any sets $C$ and $M$, where $\mathcal{P}(M)$ is the powerset of $M$ (the set of all subsets of $M$).
Here, you are doing something similar. Your $M^*$ is not $\mathcal{P}(M)$ but a subset of it ($M^*$ "lives in the same realm" as $\mathcal{P}(M)$, if you like this ontological way of thinking), more precisely $M^*$ is the set of subsets of $M$ that are species.
From a psychological standpoint, I find more natural to interpret the sentence "cats are mammals" as $C \subseteq M$ because it requires less mathematical structures than $C \in M^*$. Anyway, the two interpretations are equivalent, so no one is preferable to the other one from a logical point of view (as soon as you are entitled to use powersets).

In first-order logic, it is quite natural and standard to interpret sentences in a domain of individuals to which quantifiers refer, and then to gather together individuals by means of predicates. So, according to Fregeian tradition, the sentence "the cat is a mammal" is translated in first-order logic as

for all $x$ , if $x$ is a cat, then $x$ is a mammal,

i.e., (for $C(x) =$ "$x$ is a cat", and $M(x) =$ "$x$ is a mammal")

$\forall x (C(x) \to M(x))$

which corresponds exactly the set-theoretic definition of $C \subseteq M$.
If you prefer to interpret the sentence "the cat is a mammal" as $C \in M^*$ where $C$ and $M$ are first-class elements in the domain of discourse, then you should define a way to interpret sentences which would require more mathematical structures  and where it would be clumsier to talk about single individuals.
I guess the domain should have many sorts of elements with a hierarchical structure: individuals (as me, you, my cat Zoe) and classes of these individuals (cats, mammals, human beings). It is not impossible, but technically more elaborated and less intuitive. See for instance order-sorted logic.
A: "For all x , if x is a cat, then x is a mammal" is:

$\forall x (\text {Cat}(x) \to \text {Mammal}(x))$

and this is exactly the definition of set inclusion:

$\text{Cat } \subseteq \text { Mammal}.$


Regarding the multiple uses of "is" into natural language, we can identify three different "contexts" :

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*"Plato is a philosopher"; this context is relative to an object (or individual) belonging to a set or class.

In symbols this is expressed with $\text {Plato} \in \text {Philospher}$, i.e. $\text {Philospher}(\text {Plato})$.

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*"a man is a male"; this context is relative to a concept subsumed under a more general" one.

In symbols it is expressed with inclusion (your example above).

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*"$2+2 \text { is } 4$"; this context is relative to identity, i.e. the relation between two names denoting the same object.

It is simply: $2+2 = 4$.
A: To see the problem with your second approach, consider the sentence "The Siamese cat is a mammal." The first approach translates simply as $S\subseteq M$; but your second approach requires us to define a new set $Q$, the set of sub-species of mammals, before we can say $C\in Q$. You can see how cumbersome this might get.
