# Quantifiers bind tightly?

Is this true that it is a commonly agreed rule that $\forall x\in A:P(x) \wedge Q$ and $\forall x\in A:P(x) \Rightarrow Q$ should be interpreted correspondingly as $(\forall x\in A:P(x)) \wedge Q$ and $(\forall x\in A:P(x)) \Rightarrow Q$?

The question is about implied parentheses. Are the other interpretations $\forall x\in A:(P(x) \wedge Q)$ and $\forall x\in A:(P(x) \Rightarrow Q)$ common?

• I would read this as $\forall x \in A : (P(x) \wedge Q)$. But one should add parentheses wherever there's ambiguity (e.g., here.)
– user98602
Jun 10, 2014 at 20:05
• I would have said that in the absence of any parentheses, quantifiers always extend as far as possible to the right, agreeing with Mike Miller's reading.
– MJD
Jun 10, 2014 at 20:22
• @MJD Exactly the opposite happens. The convention is that if there are no parentheses, then the scope of the quantifier is only the closest predicate. Jun 10, 2014 at 20:26
• The rule is that there is no rule that can be counted on. Jun 10, 2014 at 20:47