Write ‘There is exactly 1 person…’ without the uniqueness quantifier

Question

During a lecture today the prof. posed the question of how we could write "There is exactly one person whom everybody loves." without using the uniqueness quantifier.

The first part we wrote as a logical expression was "There is one person whom everybody loves.", ignoring the 'exactly one' part of the question initially. From this he wrote

$L(x,y): x$ loves $y$; domain for $x$ and $y$: $\{\text{people}\}$

$\exists x\forall y: L(y,x)$

Which I understand to mean 'There is a person $x$ such that for all $y$, $x$ is loved by $y$' AKA 'There is a person who is loved by everyone'. I get that part.

The part I don't get is how the expression of 'exactly one'.

$\forall z(\forall y(L(y,z))\to x =z)$

which then creates the joint expression

$\exists x\forall y(L(y,x))\land \forall z(\forall y(L(y,z))\to x=z)$

I just can't seem to understand how $\forall z(\forall y(L(y,z))\to x =z)$ means exactly one here. I suppose you can take $\forall z$ here to mean 'for any given person', which means the $\forall z$ is considering every person in the world. This would translate the second expression block to something like, "For any given person $z$, if everybody loves $z$ then $z$ is the same person as $x$".

To me though $\forall z$ generally means for every element in the domain which I see as meaning every person in the world simultaneously, as it seems to for $y$. Is that just plain wrong? How can I tell when $\forall$ means 'all' and when it means 'for any (one)'? In the previous English translation the only reason I was able to translate it (if it's even right) is because I already knew what the statement was suppose to mean.

Is it just that $\forall z$ means that this statement could be true for any element, and if so what's the difference between $\forall z$ and $\exists z$? Someone told me that $\exists z$ would be redundant because the expression says $x=z$ but how do I know that $x$ and $z$ are automatically the same person if $\exists$ is used for both?

Sorry if this is a bit long with too many questions. I just wanted to try to make the cause of my confusion as clear as possible so you can help me figure this out.

Answer 1

Is it just that $\forall z$ means that this statement could be true for any element,

Yes, exactly.

...and if so what's the difference between $\forall z$ and $\exists z$?

If we only asserted the existence of some particular $z$ such that if $y$ loves $z$, then that particular person $z$ would thus be $x$. But then we are not ruling out that there might be another person, different from $x$, that is also loved by everyone. And we have already asserted the existence of someone (namely, x) who is loved by all. So in that sense, $\exists z$ such that... is reduntant.

We need the universal quantifier for $z$ (to assert a statement $\forall z$) in the second clause to indicate that if there is any $z$ (which means that the claim that follows - as it relates to $z$, is true for every z) $z$ such that $L(y, z)$, then any/every such $z$ must be $x$, since there is exactly one person, namely $x$, who is loved by all $y$. I.e., for every $z,$ if every y loves z, then z must be x: i.e., that $z$ is not anyone other than $x$. This gives us that $x$ (whose existence we asserted at the start) is therefore the one unique person loved by all.

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Now, just one oversight to "clean up" your expression, which you state as:

$$∃x(\forall y L(y,x))\land ∀z(∀y(L(y,z))⟹x=z)$$

But here we have two independent clauses that creates a problem, because in your second clause, you have $x$ appear outside the scope of its quantifier. I.e., it is a free, unquantified varible.

So we want the scope of $\exists x$ to persist over the entire statement, hence the square brackets below.

That is, $$\exists x \big[\forall y(L(y, x))\land \forall z(\forall y(L(y, z)) \rightarrow z = x)\big]$$

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Answer 2

The simplest definition for $\langle \exists! x :: P(x) \rangle$ I know, which I've probably learned from Dijkstra et al.'s works, is $$\langle \exists y :: \langle \forall x :: P(x) \equiv x = y \rangle \rangle$$ which does only use $P(x)$ once.

Note how we have the following nice symmetry: $$ \begin{array} \\ \langle \exists! x :: P(x) \rangle & \;\equiv\; & \langle \exists y :: \langle \forall x :: P(x) & \equiv & x = y \rangle \rangle \\ \langle \exists\phantom! x :: P(x) \rangle & \;\equiv\; & \langle \exists y :: \langle \forall x :: P(x) & \Leftarrow & x = y \rangle \rangle \\ \langle \phantom\exists! x :: P(x) \rangle & \;\equiv\; & \langle \exists y :: \langle \forall x :: P(x) & \Rightarrow & x = y \rangle \rangle \\ \end{array} $$ where $\langle ! x :: P(x) \rangle$ means "there is at most one $x$ such that $P(x)$".

Answer 3

The idea behind the formula is that, to say exactly one person is loved by everybody, you first say that there is such a person $x$ (this is the part you said you understood), and then you say that there isn't a second such person. That second part is expressed here as saying, if any other $z$ shows up who is also (like $x$) loved by everybody, then this wasn't really another $z$ but just the same $x$.

Answer 4

$$ \begin{aligned} \exists!_x\forall_yL(y,x)\equiv & \exists_x\forall_y(L(y,x)\land \forall_w(L(y,w)\rightarrow w=x))\\ \equiv & \exists_x\forall_y(L(y,x)\land \forall_w(L(y,w)\leftrightarrow w=x))\\ \equiv & \exists_x\forall_y\forall_w(L(y,w)\leftrightarrow w=x)\\ \end{aligned} $$ Their equivalence can be proved by expanding $\exists$ into $\lor$ and $\forall$ into $\land$. Try it :)