The nth statement in a list of 100 statements is “Exactly n of the > statements in this list are false.” I am trying to solve a question from "Discrete Mathematics and its Applications - 7th edition" by Konneth H Rosen which I don't quite understand:



*The nth statement in a list of 100 statements is “Exactly n of the
statements in this list are false.”
a) What conclusions can you draw from these statements?
b) Answer part (a) if the nth statement is “At least n of the
statements in this list are false.”
c) Answer part (b) assuming that the list contains 99 statements.

I understand (a) and (b), but why is (c) not possible?
I have found a few explanations for this but still don't understand;
Why can't it be that 1 through 49 are all true and statements 50 through 99 are all false?
 A: Given an English statement $eng$, the statement $\ulcorner eng \urcorner$ is an encoding of $eng$ into propositional logic (in a presumably obvious way). This is so I can avoid having to write out very long and convoluted formulas in propositional logic, but you should make sure that you can see how the English statement could be encoded into propositional logic.
You should consider the $n$th statement to be an atomic proposition $P_n$. We then have a bunch of logical formulas of the form $\phi_n :\equiv (P_n \iff \ulcorner$exactly $n$ of the $P_i$ are false$\urcorner)$. The question is: is there any truth assignment to the atomic propositions such that all the $\phi_n$ are satisfied? In other words, is the statement $\phi_1 \land \cdots \land \phi_{100}$ satisfiable? This is how you should approach part (a). Hint: if all the $\phi_n$ are satisfied by some truth assignment, then let $m$ be the number of statements $P_n$ which are false in the truth assignment. What can you say about $m$?
For part (b), you should do the same, but instead define $\psi_n :\equiv (P_n \iff \ulcorner$at least $n$ of the $P_i$ are false$\urcorner)$. Again, you should consider some truth assignment which might satisfy all of the $\psi_n$, and let $\ell$ be the number of statements which are false in the truth assignment. What can you say about $\ell$?
Part (c) is just part (b) with a different number of formulas, but you should check your logic carefully, as there is a key difference between 99 and 100 which will make itself plain.
As for your example, if statements 50 through 99 are false, then there are at least fifty false statements. But then $P_{50}$ is false.
A: The only difference between your question (b) and (c) is there're odd number of sentences in question (c). Since you said you've understood question (a) and (b) already, you basically apply the same reasoning which is examining from the tail of the whole list recursively and forced to assert an opposite sentence from the head as true. But when you get to the middle of the list there'll be only 1 sentence without a distinct paired sentence with a lower $n$ (keep in mind sentence with a lower $n$ is truth-functionally weaker than sentence with a higher $n$ in question (b) and (c) and thus more likely to be true relatively speaking). The falsity of this middle sentence by the above same reasoning will be forced to imply its own truth since no other sentences whose truth-condition are left to be evaluated, thus it becomes a paradoxical liar sentence which is no longer a proposition...
