First-order logic mindset when checking if A fulfills B

Question

Task

I'm supposed to check if the following is correct:

$\forall x(P(x) \lor Q(x)) \vDash \forall xP(x) \lor \forall xQ(x)$

Questions

If I remember the definition correctly of $\vDash$ it means entails/fulfills. I would definite it this way:

$T \vDash \psi \iff $ For every model $M \vDash T$ we also have $M \vDash \psi$.

1. What is the recommended way to reason to see if the task is correct? Am I supposed to sit and try different models or is there a better way to solve it?

2. I'm getting a bit confused with what logical consequence means because isn't that the same as $\vDash$ or am I wrong?

I'm thankful for any tips but please don't solve the task for me, I just want some pointers to the right direction. Thank you!

Answer 1

1. What is the recommended way to reason to see if the task is correct? Am I supposed to sit and try different models or is there a better way to solve it?

If you believe the statement to be false, there is no real other way than try models until you find one where it is indeed false. However, in general if you believe a statement to be false, you have a reason to believe so, and therefore you can build a model that reflects your reason.

If you believe the statement to be true, then obviously just trying models does not tell you anything -- you could never exhaust all possible models. In that case, you fix an (arbitrary) model, assume that the antecedent (in this case $\forall x(P(x) \lor Q(x))$) holds in the model, and then try to prove that the conclusion holds in the model (in this case $\forall x(P(x)) \lor \forall x(Q(x))$).

2. I'm getting a bit confused with what logical consequence means because isn't that the same as $\models$ or am I wrong?

I don't think there is strict consistency in the usage of this phrase -- although I would be happy to be corrected -- but I usually take logical consequence to mean any kind of consequence relation in logic. This relation could be syntactic (i.e. $\phi$ is a logical consequence of $\psi$ means that you can prove $\phi$ from $\psi$) or the relation could be semantic (the way you described, with reference to models). This makes $\models$ a kind of logical consequence, but not the only kind.

Answer 2

Your understanding of "logical consequence" is correct. So for this problem, you should be thinking about models $M$ that satisfy $\forall x\,(P(x)\lor Q(x))$, and I recommend that you start by getting a clear intuitive picture of what such a model looks like. Then ask whether every such model satisfies $(\forall x\,P(x))\lor(\forall x\,Q(x))$. If your picture of models of $\forall x\,(P(x)\lor Q(x))$ is clear enough, it will be obvious whether they all satisfy $(\forall x\,P(x))\lor(\forall x\,Q(x))$.