Predicate logic: $(\forall x\varphi \rightarrow \forall x\psi ) \nRightarrow (\forall x(\varphi \rightarrow \psi))$

Question

Given $L$ language and $\varphi$ and $\psi$ are formulas.

Needs to show that is happening in general: $$(\forall x\varphi \rightarrow \forall x\psi ) \nRightarrow (\forall x(\varphi \rightarrow \psi))$$

I do not how to separate the right part and how to make it to the other part. I'd be happy to suggest how to start working on the problem.

Answer 1

One good way to think about this is in terms of sets. Think about $\varphi$ and $\psi$ as sets they define in some arbitrary $L$-structure.

Then $\forall x\varphi\rightarrow\forall x\psi$ is essentially saying "If $\varphi$ is everything, then $\psi$ is everything". On the other hand, $\forall x(\varphi\rightarrow\psi)$ says that "$\varphi$ is a subset of $\psi$".

Now find a structure where $\varphi$ is not a subset of $\psi$, but "if $\varphi$ is everything, then $\psi$ is everything" (e.g. neither is everything, but neither is a subset of the other).

Answer 2

Suppose I tell you, "If everyone cheats on the exam, then everyone will not get caught."

Can you conclude, "Everyone who cheats on the exam will not get caught" ?

No way, maybe you cheat but no one else does. Then you might get caught.