I was looking at the following question: which is the stronger logic statement
The original poster provides two statements with the only difference being the order of the quantifiers as follows:
$\forall a \exists b \forall c \; Sport(a,b,c)$ (1)
$\forall a \forall c \exists b \;Sport(a,b,c)$ (2) where a,b represent people.
My question is, why exactly is there a difference in the "strength" of a statement due to the order of the quantifiers? I have read the answer on the aforementioned question, but I am looking for something which is a bit more concrete.
My interpretation of (1) in English is as follows:
For each person $a$, there is a sport $b$, which is shared by every other person $c$
Similarly, my interpretation of (2) in English is as follows:
For every person $a$ and $c$, there is a sport $b$ which they share
Are these two statements not equivalent? Could someone prove this?