Is $x$ a free variable in sentences like "$x\lt 7\Rightarrow x\lt 5$"?

Question

Is $x$ a free variable in sentences like "$x\lt 7\Rightarrow x\lt 5$"? I consulted an advanced calculus book and it says that this is a false sentence, which indicates that $x$ is a bound variable.

However, $\Rightarrow$ is a logical operator which connects two "sentences" involving free varibales, namely $P(x): x\lt 7$ and $Q(x):x\lt 5$. The truth values of P(x) and Q(x) cannot be determined since $x$ is free there. If we take $x=1$, then $P(x)\Rightarrow Q(x)$ becomes $1\lt 7\Rightarrow 1\lt 5$, which is a true sentence. If we take $x=6$, then $P(x)\Rightarrow Q(x)$ becomes $6\lt 7\Rightarrow 6\lt 5$, which is a false sentence.

So I think the truth value of $x\lt 7\Rightarrow x\lt 5$ isn't determined either. In order to turn $x$ into a bound variable, we should instead say $\forall x\in\mathbb{R}, x\lt 7\Rightarrow x\lt 5$. Am I understanding it correctly?

Answer 1

It depends on the context. If your statement here was a line in a formal proof, the $x$ would be a free variable in that statement since it is not quantified.

For readability in informal presentations such as found in most math textbooks, a universal quantifier is often assumed if there is no possibility of confusion. If the author was talking about a single value of $x$, he or she might say something like, "Let $x$ be a real number such that...".

Answer 2

"$x<7\to x<5$" is not always TRUE. Because the logic behind it is $x<7$ implies $x\epsilon(-\infty,7)$ and $x<5$ implies $x\epsilon(-\infty,5)$ . Now according to your question "$x<7\to x<5$" is not true because $x<7$ never implies that $x$ have to be always $<5$ . $x$ can be $>5$ . This implies $x\epsilon (5,7)$ . I think you have understood my point.