What is the order of precedence for a statement containing the universal quantifier and an implication?

Question

In the statement $\forall x:X \bullet p(x) \Rightarrow q(x)$, does the universal quantifier apply over the predicate q?

i.e. it is equivalent to $$\forall x:X \bullet (p(x) \Rightarrow q(x))$$ or does the implication take precedence, i.e. $$(\forall x:X \bullet p(x)) \Rightarrow q(x)$$

Answer 1

The non-standard notation used is to be deprecated, exactly because it isn't clear how to scope it.

Suppressing for a moment the restriction on the quantifier, we need to distinguish in standard notation:

$\forall x(Px \to Qx)$

and

$(\forall xPx \to Qx)$

The usual convention for dropping outermost brackets would mean that

$\forall xPx \to Qx$

is allowed slang for the second wff.

Now, there is an alternative old-school system [old-school in standard first-order logic texts, at any rate] for replacing parentheses with dots (as required to fix scope), where we would write instead

$\forall x.Px \to Qx$

and (again, but now not slang)

$\forall xPx \to Qx$

But it is bad practice to mix dots and brackets (two different scoping systems), and we shouldn't write your first formula -- dots and brackets -- nor the second.

For restricted quantifiers - with $X$ giving the restricted domain - we could in some syntaxes replace $\forall x$ with $(\forall x \in X)$ [or perhaps $(\forall x \colon X)$] thus:

$(\forall x \in X)(Px \to Qx)$

vs [dropping outer brackets]

$(\forall x \in X)Px \to Qx$

A lot of work went into perfecting standard notation from Peano and Russell, through to Hilbert, Church and others: there are good reasons not to deviate!