$(\forall y \exists x Text(x,y)) \Rightarrow (\exists x \forall y Text(x,y))$ where Text($x$,$y$) means "$x$ sends a text to $y$"
Why is this statement invalid? My interpretation of the two statements is as follows:
Left:
Everybody gets a text from somebody
Right:
There is somebody that texts everybody
Now, it is pretty obvious that the first doesn't imply the second, but how do I actually prove this with a counterexample?
With some help from the chat rooms, I came up with the following answer:
Consider a universe with 3 people A,B,C. A gets a text from B, B gets a text from C and C gets a text from B. This satisfies the first statement, but definitely doesn't imply the second because there is no single person who texts everybody.