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

Represent these two statement in first order logic:

A) Only Alligators eat humans

B) Every Alligator eats humans

Is Every represents ≡∃

and Only represents ≡∀ ??

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

• You can rephrase A) as "If $X$ eats humans then $X$ is an alligator" and B) as "If $X$ is an alligator then $X$ eats humans". This is rather a case of "$\Leftarrow$" vs. "$\Rightarrow$" (aka. "necessary" vs. "sufficient" condition) than of "exists" vs. "for all". -- Both my rephrasings can be (or perhaps should be) preceded with "$\forall X$". -- Also, the singular vs. plural differences is an artefact of natural language. "Every alligator eats .." is essentially the same as "All alligators eat ..." – Hagen von Eitzen May 18 at 10:57
• I read the "only" as $\forall$ and the "every" to imply that Alligators is a subset of things that eats humans. – CyclotomicField May 18 at 11:02
• Can someone write both the statements with first order logic?? – Srestha May 18 at 12:50
• ∀x(Human(x)->∀y(Eat(y,x)->Alligator(y))) ...........where Eat(x,y) means x eat y. Which sentence it represents? – Srestha May 18 at 13:00
• The sentence you've written is (A). If (A) is true, and you've just learned that a human has been eaten, then you can infer that the eater was an alligator; this should be a hint that there's a conditional in play. – Malice Vidrine May 18 at 14:18

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': $$\forall x (E(x) \to A(x))$$