Estimable Function Definition: Why $\forall \beta?$ I am reading Linear Models with R, 2nd Ed., by Julian J. Faraway, and on page 22, section 2.8 on the Gauss-Markov Theorem, he defines an estimable linear combination as follows:

A linear combination of the parameters $\psi=c^T\beta$ is estimable if and only if there exists a linear combination $a^Ty$ such that:
$$Ea^Ty=c^T\beta\qquad \forall\beta.$$

Note that the author uses a plain $E$ for expectation.
Now I understand why we need this definition, but I am hung up on those last two symbols: $\forall\beta.$ Why are those included in the definition? I've looked around a bit, and other authors do the same thing. It looks sloppy to me, though, because haven't we already said which particular $\beta$ we're talking about by writing $\psi=c^T\beta?$
Another way of looking at it is that order matters in quantifiers. If I say $(\forall\varepsilon)(\exists\delta)$ such that..., it's very different from saying $(\exists\delta)(\forall\varepsilon)$ such that...
A third way of looking at it is this: once you find a linear combination $a^Ty$ that works for $\psi=c^T\beta,$ if you were to change $\beta$ (because, apparently, we can!), how could you possibly still have $Ea^Ty=c^T\beta?$
Why isn't it phrased this way (which makes MUCH more sense to me):

A linear combination of the parameters $\psi=c^T\beta$ is estimable if and only if, for every $\beta,$ there exists a linear combination $a^Ty$ such that $E(a^Ty)=c^T\beta.$

A little further on, in the proof of the Gauss-Markov Theorem, the author reasons from $a^TX\beta=c^T\beta\quad\forall\beta$ to $a^TX=c^T,$ which would not necessarily be true if $\beta$ is a particular choice as opposed to arbitrary. So that would be an argument in favor of the $\forall\beta$ being tacked on at the end like that. Maybe if you could give me a simple example where we vary the $\beta$ but $Ea^Ty$ still equals $c^T\beta?$

Upon reflecting more about this in comments, I have come to see how poor this construction is.  About the only way to resolve the issue is to determine what "estimable" is really intended to mean.
Rather than go into all the ramifications, I intend only to offer a clear restatement.  The correct language to use is that of dual vector spaces.  The dual of any vector space $V$ is the space of all linear functions from $V$ into its ground field, written $V^{*}.$  These are called "forms."
When $V$ is finite-dimensional, $V^{*}$ consists of all functions of the form $x\to c^\prime x$ for $c\in V.$  In other words, left multiplication by vector transposes is just one particular way of naming (representing) a form.  But this does permit us to think of vectors as columns and forms as rows.
In the setting, $y$ is a random variable with values in the vector space $V=\mathbb{R}^n$ and the parameters $\beta$ are considered elements of the vector space $W=\mathbb{R}^k.$  Both are (column) vectors.
Before going on, let me remark that $\beta$ parameterizes a set of models.  It helps to think of this set very abstractly: that is, rather than considering $\beta$ to be an array of numbers, think of it as naming a model that is under consideration as an explanation of the data.  Now the underlying idea begins to gel: a function defined on the set of models is a property.  That is, the forms in $W^{*}$ are particular kinds of scalar properties of the models--namely, linear properties.
The models determine the distributions of the random variables $y.$  Thus, to be fully explicit, we ought to write $y$ as $y_\beta= (y_{1\beta}, y_{2\beta}, \ldots, y_{n\beta}).$
Consider a form $a\in V^*.$  Composing it with $y$ yields a scalar-valued random variable.  Thus, for any model represented by $\beta\in W,$ $E[a(y_\beta)]$ is a property of that model.
This construction has defined a map $f$ from $V^{*}$ into $W^{*}:$ the expectation of the form $a$ (composed with $y_\beta$) happens to be a linear property of $\beta,$ whence it is in $W^{*}.$  That is, $f(a):W \to \mathbb R$ is given by
$$(f(a))(\beta) = E[a(y_\beta)].$$
Now, despite whatever qualms one might have about the level of abstraction, the relationship explicitly references the key objects $a$ and $\beta$ on both sides!

The estimable properties are the image of $f:$ namely, the subspace $f(V^{*})\subset W^{*}.$

If you still prefer, you can write out the definition of the image along with the definition of $f.$

A form $c\in W^{*}$ is estimable if and only if there exists a form $a \in V^{*}$ for which $E[a(y_\beta)] = c(\beta).$

Translation into more meaningful language:

Estimable properties of a linear model are those that arise as expectations of linear combinations of the responses.

We have arrived at a good place, because we have been able to describe a concept without relying on a particular form of mathematical representation of the objects involved.  There is no reference to column or row vectors anywhere in this description.
Concrete example
Model three observations $y=(y_1,y_2,y_3) \in \mathbb{R}^3=V$ as iid draws from a Normal$(\beta_1+\beta_2, 1)$ distribution.  Although the space of models is one-dimensional (corresponding to the common mean $\beta_1+\beta_2$), the parameters $\beta_i$ are vectors in $W=\mathbb{R}^2.$
The forms $a\in V^{*} = \mathbb{R}^{3*}$ are all functions that can be expressed as
$$a(x_1,x_2,x_3)=a_1x_1 + a_2 x_2 + a_3 x_3$$
for three numbers $a_i.$  Compute expectations, using the fact that the expectation of each of the $y_{i\beta}$ is the expectation of a Normal$(\beta_1+\beta_2,1)$ variable, equal to $\beta_1+\beta_2:$
$$E[a(y_\beta)] = E[a_1y_{1\beta} + a_2 y_{2\beta} + a_3 y_{3\beta}] = a_1(\beta_1+\beta_2) + a_2(\beta_1+\beta_2) + a_3(\beta_1+\beta_2).$$
This is the map
$$\mathbb{R}^2 \ni (\beta_1, \beta_2) \to (a_1+a_2+a_3)(\beta_1+\beta_2),$$
considered as an element of $W^* = \mathbb{R}^{2*}.$  (As a row vector, it is $(1,1)$ multiplied by $a_1+a_2+a_3.$)  The set of all such maps is the subspace of forms
$$(\beta_1,\beta_2) \to A(\beta_1 + \beta_2)$$
for $A\in \mathbb R,$ generated by the form represented by $(1,1).$  It is a strict subspace, because it is (obviously) just one-dimensional.  For instance, the map $(\beta_1,\beta_2) \to \beta_1,$ represented by the row vector $(1,0),$ is not in this subspace -- and therefore, by definition, is not estimable.
For this parameterization of this model, we would say "only multiples of the sum of the two parameters are estimable -- no other functions of the parameters are estimable."
Resolution
How to interpret the quantifiers in the original quotation?  It should now be evident that the form $a$ is intended to work for all $\beta:$ it cannot vary with $\beta.$
Equivalently, the quotation refers to two forms, $\beta\to E[a^\prime y]$ and $\beta\to c^\prime \beta.$  As such the symbol "$\beta$" serves only as a placeholder for an arbitrary element of the common domain $W$ of those forms. Since the authors saw no problem not mentioning $\beta$ on the left hand side, they could with equal accuracy--and perhaps better clarity--simply write "$E[a^\prime y]=c^\prime,$" assuming both sides are understood as functions defined on $W.$
