is this formula semantically entailed from the empty set of premises?

Question

I did the truth table of the below logic:

((p ∨ q) → r) → ((p → r) ∨ (q → r))

However I didn't quite understand what semantically entailed form the empty set of premises? What that mean exactly?

As far as i understand, Whatever P I pick, the conclusion should always be true. So in this case, is it semantically entailed form the empty set of premises?

I think it is not because

((p ∨ q) → r) <> p

in case p is T q is T and r is false

Answer 1

In this setting, semantic entailment $S \models Q$ simply means that if you write down the truth table and throw away the rows where any of the statements in S are false (i.e. you keep only the rows where every statement in $S$ is true), then $Q$ is identically true in the remaining rows.

Thus, your truth table does indeed prove

$$ \models ((p \vee q) \to r) \to ((p \to r) \vee (q \to r)) $$

Answer 2

However I didn't quite understand what semantically entailed form the empty set of premises? What that mean exactly?

Is it entailed by only the rules of inference? That is: is it a tautology?

$$\models((p\lor q)\to r)\to((p\to r)\lor (q\to r))$$