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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?

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

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  • $\begingroup$ I don't know what "universal quantifiers are written in terms of conditionals, while existential quantifiers are written in terms of conjunctions" means. Could you clarify what you mean by that, please? $\endgroup$ – Arthur Aug 14 at 4:45
  • $\begingroup$ @Arthur The OP is referring to how the statements "Every A is B" and "There is some A which is B" are expressed. $\endgroup$ – Noah Schweber Aug 14 at 6:36
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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 "$\neg p\vee q$.")

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