Logic determine whether a set is consistent

Question

I got a question regarding defining whether a set of formulas is consistent in predicate logic.

For example if we have the following sets:

{∀x(P(x) ∨ Q(x)), ∃x ¬P(x), ∃x ¬Q(x)}

How can we determine whether this set is consistent or not. I know that the theory says that:

Γ is consistent if there is some model M and some environment L such that M ⊧ L ψ holds for all ψ ∈ Γ

But I am having trouble understanding this. Can anyone help / guide me by explaining it or by referencing to some good documentation? Thank you!

Answer 1

The set is consistent. Let $M$ be the model with domain $\{a,b\}$. $M$ interprets $P$ as $\{a\}$ and $Q$ as $\{b\}$. Now you can check each sentence is true in $M$.