Question: Consider the following statement:

> 
> The king of France is bald.
> 
> 
> 

We know that France does not currently have a king, so what is the truth value of this statement? If it is true, then its negation must be false, so the king of France is **not** bald is false, But when France does not have a king, does it make sense to determine the truth and falsity of this statement? What should be done with these types of statements?

Answer: 
Insofar as this is not just a question about linguistics, the way this sort of issue is handled in mathematics is that to even use the word "the" to refer to a mathematical object is to make an implicit claim that that object both exists and is unique, e.g. when we speak of "*the* cyclic group of order $n$" and so forth. So there is an implicit existential quantifier there (one might render it more explicitly as "there exists a unique person who is king of France, and..."), and if the domain of discourse over which that quantifier ranges is empty (in this case, if France does not have a king) then the statement is false.

One needs to be a bit careful about this in mathematics, though, because sometimes the implicit quantifier is a universal quantifier and then if the domain of discourse being quantified over is empty such statements are [vacuously true](https://en.wikipedia.org/wiki/Vacuous_truth).
(Updated screenshot from proof checker)

[![enter image description here](https://i.stack.imgur.com/vUyI7.png)](https://i.stack.imgur.com/vUyI7.png)
[![enter image description here](https://i.stack.imgur.com/6FKEd.png)](https://i.stack.imgur.com/6FKEd.png)