Can the generalized continuum hypothesis be disguised as a principle of logic? A cool way to formulate the axiom of choice (AC) is:

AC. For all sets $X$ and $Y$ and all predicates $P : X \times Y \rightarrow \rm\{True,False\}$, we have: $$(\forall x:X)(\exists y:Y)P(x,y) \rightarrow (\exists f : X \rightarrow Y)(\forall x:X)P(x,f(x))$$

Note the that converse is a theorem of ZF, modulo certain notational issues.
Anyway, what I find cool about formulating AC this way is that, okay, maybe its just me, but this formulation really "feels" like a principle of logic, as opposed to set theory. I mean, its just saying that we can commute an existential quantifier $(\exists y:Y)$ left across a universal quantifier $(\forall x:X)$ so long as we replace existential quantification over the elements of $Y$ with existential quantification over functions from $X$ into $Y.$ And sure, "function" is a set-theoretic concept; nonetheless, this feels very logical to me.

Question. Can the generalized continuum hypothesis also be formulated in a similar way, so that it too feels like a principle of logic?

Obviously this is pretty subjective, so lets lay down some ground rules.


*

*The axiom should be equivalent to the generalized continuum hypothesis over ZFC.

*It has to be two lines or fewer. No sprawling 4-page axioms, thank you very much!

*Use of concepts like "function" and "predicate" is good and desirable.

*Use of the concepts injection/surjection/bijection is moderately frowned upon.

*Completely disallowed: use of cardinality and/or cardinal numbers; use of ordinal numbers; mentioning $\mathbb{N}$ or $\omega$ or the words "finite" or "infinite."

 A: Certainly, the Continuum Hypothesis (simple or generalised) can be formulated in the language of pure second-order logic. 
This is done explicitly in Stewart Shapiro's wonderful book on second-order logic, Foundations without Foundationalism: see pp. 105-106.
The full story is a bit too long to give here. But as a taster, you start off by essentially using Dedekind's definition of infinity, so that you define Inf(X), for $X$ a property, as
$$\exists f[\forall x\forall y(fx = fy \to x = y) \land \forall x(Xx \to Xfx) \land \exists y(Xy \land \forall x(Xx \to fx \neq y))].$$
Ok, the first clause of this imposes the requirement that $f$ is injective: but we have done this in purely logical terms, so that shouldn't look worrying to someone looking for logical principles! 
Now we can carry on through a sequence of such definitions until you get to a sentence, still in the pure language of second-order logic, which formulates CH. Then with a bit more work we can go on to construct a sentence which expresses GCH. Further details are spelt out in Shapiro.
However, as Asaf rightly points out below, it is one thing to say that a statement can be formulated in the language of pure second-order logic, and another thing to say that it is a logical principle. And it is indeed a nice question what that actually means. Still, we can say this much: as Shapiro notes, on a standard account of the semantics of full second-order logic it turns out that either CH or NCH (a certain natural formulation of the falsity of the continuum hypothesis) must be a logical truth of full second-order logic. But here's the snag: we don't know which!
