"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?

  • $\begingroup$ For all $x$, if $x$ is a student, then every student is equal to $x$. $\endgroup$
    – GEdgar
    Jan 21, 2020 at 16:41
  • 2
    $\begingroup$ I don’t see your problem. Since “$x=y$” is a conclusion, it seems to me that of course $x$ may be $y$. $\endgroup$
    – Lubin
    Jan 21, 2020 at 16:45
  • $\begingroup$ My problem is that i dont understand how I should read it, nobody speaks like this and it's so confusing. It should be read "for everything if it's a student, then for everything if it's a student, then they are the same" the "everything" confuses me, once you say it you're talking about everything already so why say it again... Its weird that I don't even know how to explain it clearly $\endgroup$
    – Abcd
    Jan 21, 2020 at 16:51
  • $\begingroup$ Consider $x$ and $y$ ways to identify people. "If any person selected by process-x is a student, then: if any person selected by process-y is a student, then they must have selected the same person". That is of course, just along winded way of saying "There is at most one student to select." $\endgroup$ Jan 22, 2020 at 1:03
  • $\begingroup$ As per your previous post start from the simpler part: $(∀y)(Sy→x=y)$ means that "every student is equal to $x$". Then quantifiy over $x$. $\endgroup$ Jan 22, 2020 at 7:25

3 Answers 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 $Sj\leftrightarrow j\in S$.)


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".

  • $\begingroup$ Why do we even use all this it's so much easier in normal language $\endgroup$
    – Abcd
    Jan 21, 2020 at 18:23
  • 1
    $\begingroup$ Natural language can also be rather ambiguous at times. Formal symbolism is precise about what it means to say. $\endgroup$ Jan 22, 2020 at 0:31

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.


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