(I don't know whether this can be classified as an actual mathematics question. Feel free to take this down if it doesn't qualify.)
Given the following statements, \begin{align} 1. &\text{ At least five of the statements listed below are true.}\\ 2. &\text{ At least four of the statements listed below are false.}\\ 3. &\text{ At least three of the statements listed below are true.}\\ 4. &\text{ At least two of the statements listed below are false.}\\ 5. &\text{ At least one of the statements listed below is true.}\\ 6. &\text{ At least one of the statements listed above is false.}\\ 7. &\text{ At least two of the statements listed above are true.}\\ 8. &\text{ At least three of the statements listed above are false.}\\ 9. &\text{ At least four of the statements listed above are true.}\\ 10. &\text{ At least five of the statements listed above are false.}\\ \end{align} Providing that true statements provide information that is supported by evidence and false statements provide information that contradicts evidence, what is the maximum number of true statements?
Right, first of all, suppose there are exactly four false statements between $(3)$ and $(10)$. (I should clarify that whenever "between $(a)$ and $(b)$" is mentioned, it applies to all statements $(n)$ where $a \le n \le b$.) The other four statements between $(3)$ and $(10)$ are true and additionally, statement $(2)$ is correct. This means there are exactly five true statements between $(2)$ and $(10)$, and statement $(1)$ is therefore correct.
Having identified the conditions given in statements $(1)$ and $(2)$ are met, we can also infer that statement $(7)$ is true, and from which infer that statement $(5)$ is also true. Refer to this logic again and the two next correct statements to be found are $(9)$ and $(3)$ in that order.
So far, six statements have to proven to be true - $(1) - (2) - (3) - (5) - (7) - (9)$ (which consist of all statements relating to the number of true statements and one statement on which we base our hypothesis). This eliminates the possibility of there being five or more correct statements between $(1)$ and $(9)$, thus, statement $(10)$ is false, implying statement $(5)$ is true. Continuing with this logic, conditions presented in statements $(7)$ and $(9)$ are not fulfilled, from which we can confirm their falsehood.
In the end, there are three false statements - $(6) - (8) - (10)$, which contradicts the presupposed hypothesis of there being "exactly four true statements between $(3)$ and $(10)$". (Why does it have to end this way? It was going so well~)
Let's start the other way around. Suppose there are exactly four true statements between $(1)$ and $(8)$, the other four statements between $(1)$ and $(8)$ are false. However, according to this hypothesis, statement $(9)$ is not incorrect, meaning that statement $(10)$ is false. Since we're looking for the maximum number of true statements, we'll assume that there are a minimum of four true statements between $(1)$ and $(8)$, reaffirming that statement $(9)$ is correct and statement $(10)$ is incorrect.
From this starting point, we can deduce that statement $(5)$ is true, (doesn't it start to look like our first case? Statements $(5) - (9) - (10)$ are factually the same), and actually, that's all of it. We can technically go further if we come up with more presumptions, but I'll stop here for now.
How about we revisit our first hypothesis? Since we're looking for the maximum number of true statements, we'll assume that statement $(2)$ is false, meaning that statement $(1)$ is true. Actually, let's skip the tedious reasoning. I've made a table detailing the process it takes to decide whether each statement is true according to previous claims. This is one of the configurations which provide supporting information to true statements and contradicting information to false ones.
But is this the case which maximises the number of true statements? Is there a way to prove that a combination with more than seven true statements is not logically attainable? What about one with the most false statements possible? I'd surmise that it's basically the opposite to the configuration above - $\text{T} - \text{F} - \text{T} - \text{F} - \text{F} - \text{F} - \text{F} - \text{F} - \text{F} - \text{T}$.