Exclusive Or/Examples/One of Five Statements is True

Example of Exclusive Or
Which of the following $5$ statements is true?


 * $(1): \quad$ Exactly one of these statements is false.


 * $(2): \quad$ Exactly two of these statements are false.


 * $(3): \quad$ Exactly three of these statements are false.


 * $(4): \quad$ Exactly four of these statements are false.


 * $(5): \quad$ Exactly five of these statements are false.

Solution
more than $1$ of these statements is true.

That would mean both that:


 * $\text{(a)}: \quad$ Exactly $n$ of these statements is false.

and:


 * $\text{(b)}: \quad$ Exactly $m$ of these statements is false.

where $m \ne n$.

But that would mean that $\size {n - m}$ of the statements is both true and false at the same time.

That would contradict the Principle of Non-Contradiction.

Hence no more than $1$ of these statements can be true.

That would mean either $(4)$ or $(5)$ must be true.

$(5)$ is true.

But then that would mean that $(5)$ itself must be false.

Again, that would contradict the Principle of Non-Contradiction.

Hence it follows that $(4)$ must be true, and all the others false.