Translation into predicate calculus

Question

I want to translate the following sentence into predicate calculus:

"Anything taller than something Alice is taller than is taller than Alice."

Let $a$ denote Alice and $T(x,y)$ the predicate asserting that $x$ is taller than $y$. I believe the tranlation is $$(\forall x)(\forall y)[(T(x,y)\land T(a,y))\implies T(x,a)].$$ But I also think it might be $$(\forall x)[(\exists y)(T(x,y)\land T(a,y))\implies T(x,a)].$$

Which is the correct one, and why?

Answer 1

In classical logic, $(\forall y)[\phi(y, x) \Rightarrow \psi(x)]$ is equivalent to $[(\exists y)\phi(y, x)] \Rightarrow \psi(x)$, so both of your proposed answers are correct, if classical logic is the right logic to use in Wonderland.

Answer 2

The direct translation would be (with $>$ instead of $T$): $$ \forall x( \exists y(x > y \land a > y) \to x > a ) $$ where the $\exists y$ is inside the premise of the implication.

This is equivalent to $$ \forall x \forall y( (x>y \land a>y) \to x > a ) $$ where $\forall y$ ranges over the entire implication. So arguably both of your proposals are right, but the one with an $\exists$ can be said to be "more verbatim".

Note, though, that $$ \forall x \exists y( (x>y \land a>y) \to x > a ) $$ is something different. This is not even particularly suited to be rendered in English; the meaning of $\exists x(\cdots\to\cdots)$ is not very intuitive.