Given some function involving two variables, $\mathit p(x,y)$, is the formula $$\mathit \exists!x\exists yp(x,y)$$ equivalent to $$\mathit\exists y\exists!xp(x,y)$$ I have tried writing out the formal definition for the unique existential quantifier, but I get stuck simplifying the statement when the unique quantifier is first. I have also tried giving $\mathit p(x,y)$ some meaning and talking through it, but I can't decide if they mean the same thing. Any help is appreciated!
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In the second instance, do you perhaps mean a formula and not a function? – Chubby Chef Jan 31 '20 at 02:50
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Yes that is what I meant. – yungtoad Jan 31 '20 at 03:07
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No, they do not commute.
Consider for example the nonnegative reals as a linear order. Then $$\exists x\exists !y(y\le x)$$ is true (take $x=0$), but $$\exists!y \exists x(y\le x)$$ is false since for every $y$ there is some $x$ with $y\le x$.
This becomes less surprising when we "unpack" $\exists!$. There are a few ways to do this, but I think the simplest is to view "$\exists !x\varphi(x,...)$" as an abbreviation for "$\exists x\forall z(\varphi(z,...)\leftrightarrow x=z)$." Then we have:
"$\exists x\exists!y\varphi(x,y)$" is an abbreviation for $$\exists x\exists y\forall z(\varphi(x,z)\leftrightarrow y=z),$$
while "$\exists!y\exists x\varphi(x,y)$" is an abbreviation for $$\exists y\forall z[(\exists x\varphi(x,z))\leftrightarrow y=z].$$
This has a clear instance of swapping $\forall$ and $\exists$ (in fact, it's even messier than that - unpack "$\leftrightarrow$" ...), so we should expect them to be inequivalent in general.
(There are of course other ways to unpack "$\exists!$," but they'll all yield the same general picture: despite the symbol suggesting only existentiality, it's hiding a very important universal quantifier, and so the usual dangers of $\forall/\exists$-switching carry over to $\exists!/\exists$-switching.)
Noah Schweber
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9To make this (and other similar statements involving two quantifiers) intuitive: imagine a table where the rows represent the possible values of $x$, the columns represent the possible values of $y$, and we put a tick mark in a cell if the corresponding pair $(x, y)$ satisfies $p(x,y)$. Then the first formula means "there is exactly one non-empty row", while the second formula means "some column has exactly one marked cell" - a weaker statement. – Ilia Smilga Jan 31 '20 at 18:15
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