Precedence rules

Question

From my understanding

1. $P \land Q \lor R$ should be understood as $((P \land Q) \lor R)$ due to the precedence-rules and
2. $\exists x Px \land Qx$ should be understood as $(\exists x Px \land Qx)$ and not $\exists x (Px \land Qx)$ as the scope of $\exists$ is just $x$.

But how do you express this meaningful if you combine the two rules in an example? I would think that $\exists x P(x) \land Q(x) \lor R(x)$ is $(\exists x ((P(x) \land Q(x)) \lor R(x)))$, but that conflicts "rule 2" above.

Answer 1

The, in my impression, most common convention (but your textbook may define it differently, so check the syntax definitions there) is that quantifiers have precedence over all connectives:

1. $\forall$, $\exists$
2. $\neg$
3. $\land$
4. $\lor$
5. $\to$
6. $\leftrightarrow$

From this your second bullet point ($\exists$ having precedence over $\land$) would automatically follow, and in addition that $\exists$ precedes $\lor$.

So
$\exists x P(x) \land Q(x) \lor R(x)\\ = (\exists x P(x)) \land Q(x) \lor R(x) \quad (\exists > \land, \lor)\\ = ((\exists x P(x)) \land Q(x)) \lor R(x) \quad (\land > \lor).$

In general, combining two precedence rules amounts to simply merging the orders they prescribe into one:
(1) "$\land$" > "$\lor$" + (2) "$\exists$ > "$\land$"
= "$\exists$" > "$\land$" > "$\lor$"