Is there a standard for the priority of logical connectives and quantifiers?

Question

Examples of priority issues are as follows:

• $\lnot P\wedge Q$ is $(\lnot P)\wedge Q$ or $\lnot (P\wedge Q)$?
• $P \wedge Q\rightarrow R$ is $(P \wedge Q)\rightarrow R$ or $P \wedge (Q\rightarrow R)$ ?
• $\forall x \in A, P\rightarrow Q$ is $(\forall x \in A, P)\rightarrow Q$ or $\forall x \in A, (P\rightarrow Q)$ ?

Answer 1

The usual precedence convention is:

1. quantification
2. negation
3. conjunction, disjunction
4. conditional, biconditional

Among ∧ and ∨, or among →, or among → and ↔, always use parentheses.

And if the reader (or software) may not be familiar with the above precedence convention—or is suspected to be using a different one—then using parentheses helps; however, too many parentheses does decrease human-readability.

---

• $\lnot P\wedge Q$

  is $(\lnot P)\wedge Q\quad$ or $\quad\lnot (P\wedge Q)$ ?

The former.

• $P \wedge Q\rightarrow R$

  is $(P \wedge Q)\rightarrow R$ or $P \wedge (Q\rightarrow R)$ ?

The former.

• $\forall x \in A, P\rightarrow Q$

  is $(\forall x \in A, P)\rightarrow Q$ or $\forall x \in A, (P\rightarrow Q)$ ?

Ambiguous: it's unclear whether the comma is being used as a delimiter, in which case $$\forall x{\in} A\;(P\rightarrow Q)$$ is the intended meaning, or whether the comma is being logically superfluous (i.e., doesn't alter meaning and can be ignored), in which case $$(\forall x{\in} A\;P)\rightarrow Q$$ is the intended meaning.