$P=(\forall x)(\exists y) GTOE(x,y)$
$Q=(\exists y)(\forall x) GTOE(x,y)$
And I want to know whether Q is an logical consequence of P. I know P is a logical consequence of Q. But I cannot identify whether Q is logical consequence of P.
GTOE(x,y) - x is geater than or equal y.
Consider the following structure $M$. The underlying set of $M$ is the set of integers, positive, negative, and $0$. The relation $GTOE$ is interpreted as ordinary "greater than or equal," usually denoted by $\ge$.
Then sentence $P$ is true in $M$, for it is true that for any $a\in M$ there is a $b$ such that $a\ge b$. Choose, for example, $b=a$.
But sentence $Q$ is not true in $M$. This is because there is no $b\in M$ such that for all $a$ in $M$ we have $a\ge b$. For example, given $b$, we can let $a=b-1$.
Since there is a structure $M$ in which $P$ is true but $Q$ is false, $Q$ cannot be a logical consequence of $P$.