How should I read the statement: $(\forall x)(Sx \to (\forall y)(Sy \to x=y))$?

Question

"There is at most one student" is symbolized as $(\forall x)(Sx \to (\forall y)(Sy \to x=y)$

Earlier I asked this question $(\forall x)[Mx -> (\forall y)(My -> Kxy)]$, can $x\;\&\; y$ be the same thing?

and they told me that x and y can be the same thing. Now in confused because I would read the statement above as "for everything, if it's a student, then, for everything else if it's a student, then they have to be the same student" which it would make sense. But if x and y can be the same I don't understand how I should interpret it. Could you guys help please?

Answer 1

Not every FOL formula has a "natural" natural language counterpart...

But we can apply some useful transformations. First of all, we may convert the formula in the equivalent prenex form:

$(∀x)(∀y)(Sx → (Sy → x=y))$,

followed by the tautological equivalence called Exportation to get:

$(∀x)(∀y)((Sx \land Sy)→x=y)$.

Now the reading is much more easy:

"Every two students are identical".

Answer 2

Should $(\forall x)(Sx\to(\forall y)(Sy\to x=y))$ be true, then the statement $Sa\to(Sb\to a=b)$ will be true for any term $a$ and any term $b$ in the domain — whether these terms have the same value or not.

Now, when those terms have the same value, the statement holds whether a term of that value satisfies $S$ or not.   However, the statement cannot be satisfied by two distinct terms that satisfy $S$.

Thus the claim of $(\forall x)(Sx\to(\forall y)(Sy\to x=y))$ is that predicate $S$ can be satisfied by at most one entity in the domain.

Answer 3

I don't know what exactly your question is, but let me say

"for everything, if it's a student, then, for everything else if it's a student, then they have to be the same student"

is not a good interpretation (not specifying the variables after quantifiers, and 'for everything else' means $\forall y $ such that $\lnot x=y$, which is not provided.).

"for any $x$ if it's a student, then, for any $y$ if it's a student, then $x$ is $y$"

will be more precise. And also as Mauro said, not every logical formula can be translated into natural language. For example, 'is' used in the statement just above, is used for equal ($=$). However, in natural language 'is' can mean several things such as Josh is student- here 'is' is used for relation, namely $\in$. (see $Sj\leftrightarrow j\in S$.)