Difference between 'Only' and 'Every' Keyword in Mathematical logic

Question

Represent these two statement in first order logic:

A) Only Alligators eat humans

B) Every Alligator eats humans

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Is Every represents ≡∃

and Only represents ≡∀ ??

Can we differentiate it with verb ‘eat’ and ‘eats’??

Answer 1

Is Every represents ≡∃

and Only represents ≡∀ ??

No. Both 'Every alligator eats humans' and 'Only alligators eat humans' are general statements, and so both require a $\forall$

Can we differentiate it with verb ‘eat’ and ‘eats’??

No ... the fact that one statement in English ends up with 'east' and the other with 'eat' is just a fluke of the English language. But since you mean the same action of one thing eating another, you'll have to use the same predicate for logic.

The correct translations, where $A(x)$ mreans '$x$ is an alligator' and $E(x)$ means '$x$ eats humans':

'Every alligator eats humans': $\forall x (A(x) \to E(x))$

'Only alligators eat humans': $\forall x (E(x) \to A(x))$