Open statement of the implication and the convention

Question

Let $p(x)$ and $q(x)$ be two open sentences over a set $M$. If $x$ is replaced by some element from $M$, then $p(x)$ and $q(x)$ are closed sentences, and so is, say, $p(x)\wedge q(x)$. The statement $$\forall x\in M (p(x)\wedge q(x))$$ is clearly true if both $p(x)$ and $q(x)$ are true for each $x\in M$. There is a problem that I'd like to understand from some books. Consider the following statement $$\forall x\in M (p(x)\implies q(x))$$ This is false if $p(x)$ is true and $q(x)$ is false for each $x\in M$, otherwise the statement is true. However, some books used a convention that the following statement $$p(x)\implies q(x)$$ is not an open sentence, even though it is in reality. This is because that that statement should be read as $$\forall x(p(x)\implies q(x))$$ where $x$ runs over a set, say $M$. One of the books says that the interpretation $p(x)\implies q(x)$ means "$q(x)$ is true for each $x$ for which $p(x)$ is true". I do not agree it completely, because the statement could be true if $p(x)$ is false and $q(x)$ is true for each $x$. Is it because that we are not interested if $p(x)$ is false or not when it comes to writing proofs?

Questions: Why should we use this kind of convention, and is my understanding correct about the truth value of the last statement?

Answer 1

Correct.

There are two basic approaches in order to "give meaning" to open formulas.

According to the first one, the meaning (and the truth-value) of an open formula with respect to an interpretation $\mathfrak A$ is defined for specific "instances" of the formula.

In this case, we do not consider $p(x) \to q(x)$ but the corersponding instance obtained replacing the variable $x$ with a "name" or considering a variable assignment function $s$ that assign an object $a$ of the domain of $\mathfrak A$ to $x$.

In this case, the satisfaction relation holds for "instances":

$\mathfrak A \vDash \varphi [s]$.

See: Herbert Enderton, A Mathematical Introduction to Logic, Academic Press (2nd ed. 2001), page 83.

The second case, limits the definition of meaning and truth value to sentences, i.e. "closed" formulas.

For open one, it adopts the convention that:

$\mathfrak A \vDash \varphi$ iff $\mathfrak A \vDash Cl(\varphi)$,

where $Cl(\varphi)$ is the universal closure of $\varphi$.

See: Dirk van Dalen, Logic and Structure, Springer (5th ed. 2013), page 67.