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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?

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  • $\begingroup$ Without further context, both of those look like legitimate statements to me. "Actually obtains in physical reality" is not a mathematical concept, without making further modelling assumptions explicit. $\endgroup$
    – Yemon Choi
    Commented Jun 13, 2012 at 2:06
  • $\begingroup$ The context is statistics. Θ is the parameter space, and the functional is an expectation. In notion 2, the idea is to restrict consideration of the functional to just the physically real value of the parameter. I confess to being surprised that this is legitimate -- I thought such quantifiers could only be applied to a formula with a free variable, yielding a proposition with a definite truth value (or another formula with a free variable, of course). $\endgroup$
    – Cyan
    Commented Jun 13, 2012 at 2:20
  • $\begingroup$ As a guide to thinking about this, one might look at partial versus total functions in universal algebra and/or recursion theory. 2 looks to me like the definition of a nontrivial partial functional. Gerhard "Ask Me About System Design" Paseman, 2012.06.12 $\endgroup$
    – Gerhard Paseman
    Commented Jun 13, 2012 at 2:26
  • $\begingroup$ To answer the question two ways: the short answer is that I do not have enough context to know what laws he is breaking; the long answer is that I can imagine situations where such definitions are legitimate, but I do not know if they are correct or even appropriate for the situation used by this person. Gerhard "Looking For Medium Length Answers" Paseman, 2012.06.12 $\endgroup$
    – Gerhard Paseman
    Commented Jun 13, 2012 at 2:31

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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 $f$.

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