I've encountered a working paper in which the author discusses two different notions about the legitimate definition of a certain functional. He expresses these notions using the universal and existential quantifiers:


Notion 1: $\forall \theta \in \Theta : f(\phi(\cdot), \theta) \equiv \int\_{\mathbb{R}}\ \phi(x)g(x,\theta)dx$


Notion 2: $\exists \theta^{\*} \in \Theta : f(\phi(\cdot), \theta^{\\*} ) \equiv \int\_{\mathbb{R}}\ \phi(x)g(x,\theta^{\\*})dx$


where $\theta^\*$ is to be thought of as the element of $\Theta$ which actually obtains in physical reality. 


Is this a legitimate use of the quantifiers?



Quantifiers in function definition -- is this legitimate?
It's not clear to me whether the line "where $\theta^\*$ ... physical reality" is intended to be part of Notion 2, perhaps even within the scope of the existential quantifier on $\theta^\*$. The formatting suggests that it is not. On that reading, both notions look like legitimate statements, but Notion 2 does not look like a "legitimate definition of a certain functional". There could be many functionals $f$ that satisfy Notion 2, for different values of $\theta^\*$. The terminology "*the* element of $\Theta$ which actually obtains" suggests that $\theta^\*$ is regarded as unique, which it might not be.


So let me consider the alternative reading, where the specification of $\theta^\*$ from physical reality is considered part of Notion 2. Of course, on this reading, Notion 2 is no longer a mathematical statement. Furthermore, it still doesn't define the functional $f$, because $f$ could have entirely arbitrary values whenever its second argument differs from the actual physical $\theta^\*$.


I'm inclined to conclude (until somebody shows me a better reading) that Notion 2, though it might be a well-formed mathematical statement, does not serve as a definition of a functional $f$. My guess as to the author's intention is that $\theta^\*$ should be fixed first, on physical grounds, and then Notion 2 should be stated for just this one value of $\theta^\*$, without an existential quantifier and without a second argument in $f$.


