Question about existential quantifiers

Question

I know that universal quantifiers are written in terms of conditionals, while existential quantifiers are written in terms of conjunctions. However, what does it mean when an existential quantifier uses a conditional?

For example, how would you translate the following into English?

$$\exists x \left( \text{Cube} (x) \to \text{Small} (x) \right)$$

Answer 1

An existential quantifier around an implication doesn't really correspond to a common phrase structure; we just translate the logical symbols directly. The example you give is just

There is something which, if it's a cube, is small.

We can rewrite this a little more clearly using the definition of (material) implication:

There is something which either is not a cube or is small.

(That is, "$p\rightarrow q$" is just "$(\neg p)\vee q$." Note that $\neg$ binds more tightly than $\vee$, so this can be written unambiguously as just "$\neg p\vee q$.")