I have a follow-up to a previous question: True or false or not-defined statements.
(In the following I might use the word statement incorrectly.)
In that question/answer I learned that the "statement" that $\frac{1}{0} =1$ is not true or false because the expression isn't a well-formed formula (i.e. the left have side is not well defined). I do understand that one might make the "symbol" $\frac{1}{0}$ in certain areas of mathematics. For this question, though, I am just interested in the math we teach in say an undergraduate calculus class.
My follow-up question is what about the statement
$\frac{1}{x} = 1$ for all real numbers $x$
Here the statement makes fine sense if we had said "for all values of $x\neq 0$".
But would this statement also be not-true and not-false because the expression is not defined/well-formed since $x=0$ is part of the statement?
(Even though I am interested in how one might answer this in a calculus class, I would like the strict logic answer. I know that we often say and write things that technically/strictly speaking not true, but when say teaching about true and false statements, I would like to keep things as precise as possible. Even though we can sometimes overlook some things in an undergraduate class, I also think it is wrong to teach stuff that is outright wrong.)
Best Answer
The general framework to handle logical systems in which terms can be undefined is called "free logic". However, free logic has been studied mostly in the context of philosophy, and is conspicuous only in its absence in mathematical logic textbooks. This is because the way we handle things in mathematics is just to rephrase the question to avoid the undefined values, and to use vacuous quantification in other settings to work around them.
In free logic there are several semantics, but all of the ones that people usually consider do give a truth value to "$(\forall x \in \mathbb{R})[ 1/x = 1]$". All the usual semantics would say that it is false because it is false for $x = 2$. I spent a while looking into this a little bit ago, and what I learned is that the usual trend in free logic is to assign a truth value to as many quantified sentences as possible. Of course one could define a semantics in which a quantified statement has no truth value if there is a substitution instance that has no truth value (e.g. "1/0 = 1" has no truth value in free logic because $1/0$ is an undefined term; but $(\forall x \in \mathbb{R})[1/x = 1]$ is nonetheless false in all the usual semantics).
My interest in this came from looking at statements such as "$(\forall X \subseteq \mathbb{R})(\forall z \in X) [ z \leq \sup X]$". My opinion is that this statement is erroneous, and has no truth value, because it has no truth value when we take $X = \emptyset$. However, by looking at some references I realized that all of the usual semantics for free logic make this statement true, because they make $(\forall z \in \emptyset)[z \leq \sup \emptyset]$ true as a vacuous quantification. Nevertheless my personal opinion, as yours may be, is that the formula at the beginning of this paragraph cannot be asserted in ordinary mathematics because to do so implies that $\sup\emptyset$ has to be defined. It would be possible, I believe, to write a short paper developing a semantics for free logic that mirrors this, although I have not spent any significant time on it.