The indeterminacy of 0/0 and vacuous truth?

Question

Today, my roommate and I picked up our friend from the airport. We were supposed to pick him up yesterday, but he missed his flight. We joked that he misses flights a lot, and that only catches 70% of his flights.

I have never missed a flight, so I said "I have made 100% of my flights." My roommate has never flown/booked before. So I said "you've caught all of your flights too, so I guess you catch 100% of your flights. Of course, you've also missed all of your flights, so you catch 0% of your flights."

I can assign any percent of flight success to my roommate because he has had no scheduled flights. It is vacuously true. In a sense, because of the fact that any assigned percent is vacuously true, his percent rate is "indeterminant."

Similarly, if one were to calculate his success rate percent using simple arithmetic, with $s$ meaning "caught flights" and $f$ meaning "scheduled flights", we would have $100 \frac{s}{f} = 100 \frac{0}{0}$, - indeterminant.

Is this a reasonable demonstration of how the two ideas, vacuous truth and indeterminant form, agree with one another in their application? Are there any more profound connections?

Answer 1

When I have seen percentages counted in practice (like win percentages in video games) counted as 0 if there is no data to calculate. So I think if we were to make a function that would calculate the percentage it would be something like: $$p = \begin{cases} 100 \frac{s}{f}, &f\neq 0 \\ 0, &f=0\end{cases}$$

That way there is no ambiguity and I am sure that $f \neq 0$ in the formula you provided. :)

Answer 2

If we want to "handle" indeterminate operations like $0/0$ with the logical "machinery", I think that the theory of Definite descriptions - due to B.Russell - is more appropriate than the "model" of conditionals with false antecedent.

In a nutshell, Russell's analysis consider a denoting phrase in the form of "the $X$" where $X$ is e.g. a singular common noun.

The definite description is proper if $X$ applies to a unique individual or object. For example: "the first person in space" and "the 42nd President of the United States of America", are proper. The definite descriptions "the person in space" and "the Senator from Ohio" are improper because the noun phrase $X$ applies to more than one thing, and the definite descriptions "the first man on Mars" and "the Senator from Washington D.C." are improper because $X$ applies to nothing.

Thus, we can say that "the result of the operation $0/0$" is an improper definite description, because the operation $0/0$ lacks of a result.

If so, we can handle a statement referring to this description in the way proposed by Russell :

"The present King of France is bald" says that some $x$ is such that $x$ is currently King of France, and that any $y$ is currently King of France only if $y = x$, and that $x$ is bald:

$∃x[PKoF(x) \land ∀y[PKoF(y) \to y=x] \land B(x)]$

This is false, since it is not the case that some $x$ is currently King of France.

In the same way, we can try to formalize :

"You catched 100% of your flights"

as follows, where $PoFC_Y$ stands for "Your roommate Percentage of Flights Catched" :

$∃x[PoFC_Y(x) \land ∀y[PoFC_Y(y) \to y=x] \land PoFC_Y(x)=100]$

and again it is false because, due to the fact that your roommate has never flown, the term $PoFC_Y(x)$ has no denotation (because : the operation $0/0$ is undefined).