Why is this a sentence with a quantifier an open sentence?

Question

From these notes on Relational Logic, the following two sentences are given as examples of 1) open and 2) closed sentences. The definition of an open sentence is one with at least free variables.

Isn't $x$ bound in (1)?

\begin{align} 1) & \qquad p(y) \Rightarrow \exists x.q(x,y) \\ 2) & \forall y.( p(y) \Rightarrow \exists x.q(x,y) ) \end{align}

Update: Updated my question to reflect the answers. An open sentence must have at least one free variable, not only free variables.

Thanks everyone for your quick responses.

Thank you.

Answer 1

An open "sentence" is one with at least one free variable: (with the occurrence of a variable which is not bound, it is not a "sentence" at all, but a formula). So in: $$(1) \quad p({\bf y}) \implies \exists x.q(x, {\bf y})$$

...while it is true that $x$ occurs strictly as a bounded variable, $\;\bf y$ is an unbounded variable, so the formula is open, since $y$ occurs unbound (twice).

ADDED for clarification:

From you link you have included, the following definition is written:

"A sentence is open if and only if it has free variables. Otherwise, it is closed."

The if and only if connective applies to the existence of one or more free variables. That is, is should be read as

"A sentence is open if and only if it has any free variable(s). Otherwise it is closed".

...or ...

"A sentence is open if and only it has one or more free variables. Otherwise it is closed."

And so is not meant to say: "A sentence is open if and only if it has [only] free variables."

Answer 2

A formula is open if there is a variable with at least one free occurrence in the formula. A formula is closed if it is not open: if there is no free occurrence of any variable in the formula.

In formula (1) of the question, both occurrences of $y$ are free occurences.

Closed formulas are important because each of them is true or false in each first-order structure. An open formula, on the other hand, is only true or false relative to a variable assignment, and the truth value can differ for different variable assignments.