Relationship between domain and quantifiers in predicate logic I'm confused about the domain of discourse and their predicates. 
Lets say we have Domain: {1,2,3} and predicates P: {1,2} S: {1,2}
If we were to say ∀x(P(x) ∧ S(x)) would this mean, that for all entities that hold for P, they also hold for S? Or would this mean "Every entity in the domain holds for both P and S" (which would obviously not hold because 3 is in neither P nor is it in S.
 A: I assume you meant $\forall x(P(x) \land S(x))$ (with $S(x)$ inside the scope of the quantifier $\forall x$), not $\forall x(P(x)) \land S(x)$.
If you say $\forall x(P(x) \land S(x))$, this means "For all entities in the domain, $P$ holds and $S$ holds". This statement is false in the domain $\{1,2,3\}$, since neither $P$ nor $S$ hold of $3$.  
If you want to say "For all entities for which $P$ holds, $S$ holds", this would be formalized as $\forall x (P(x) \to S(x))$: "For all entities, if $P$ holds of that entity, then $S$ holds of that entity". This statement is true in the domain $\{1,2,3\}$, since for all the entities for which $P$ holds, namely $1$ and $2$, $S$ holds as well.
Here it doesn't matter that $P$ doesn't hold of $3$, since with $\to$ all we're saying is that if it is true that $P$ holds of $3$, then $S$ holds of $3$, and since $P(x)$ is false for $x=3$, we have by the semantics of $\to$ that $P(x) \to S(x)$ is true as well.
But this is not a question on how quantifiers work on domains that's special to predicate calculus, it's simply the meaning of the connectives $\land$ and $\to$.
