I've been learning about translating English to logic and vice versa and I have to say that I find this quite difficult.
On a paper that I'm examining , I've across the following logic formula:
$ \forall x (circle(x) \Rightarrow \exists y (circle(y) \land above(x,y)))$
Each circle has a circle somewhere below it
$ circle(x) $ - $x$ is a circle.
$ above(x, y)$ - $x$ is above $y$.
I started to wonder what the nuance is if the exists quantifier was moved outside of the entire bracket like this:
$ \forall x \exists y (circle(x) \Rightarrow circle(y) \land above(x,y))$
But my feeling is that this formula I says the same thing as the former sentence but I also think it doesn't as I'm unsure of the meaning in natural English.
Former formula: For all $x$ , if $x$ is a circle then there exists a $y$ such that $y$ is a circle and it is above $x$.
Latter formula: For all $x$ , there exists a $y$ such that if $x$ is a circle then $y$ is a circle and is above $x$.
My question is , what the difference between these two formulas in logic and natural English?
Translating from logic to English is quite hard I think and even harder from English to logic.
Are there any tips for translating from logic to English and vice versa? How do you work out the nuance if two formulas seem alike when you read them?