Chapter 1 - Section 1.1 - Propositional Logic - Exercises - Page 16: 49

a) The 99th statement is true, the remaining are false. b) The first 50 statements are true, the remaining 50 are false. c) The list of statements is inconsistent.

a) Since the 100 statements all state a different thing, at most one of them can be true. If the statement were all false, there would be 100 false statements: but that is what the $100$th statement asserts! So this particular statement would be true, contradicting the fact that they are all false. Then, what if there actually is a a true statement among the list? The only reasonable choice would be statement n° 99, which asserts that exactly 99 statements are false: that doesn't lead to any contradiction, and make perfect sense. More precisely, statements 1 to 98 and statement 100 are false, while the 99th is true. b) First of all, notice that if the $n$th statement is false, the all the following statements (the ($n+1$)th, the $(n+2)$th etc) are false too (otherwise we would have that ''less than $n$ statements are false", but "at least $m$ statements are false", with $m>n$: a contradiction). Now, the first statement cannot be false, because that would imply that all statements are false, but also that 'less than 1 statement is false': clearly a contradiction. So the first statement is true. The same thing applies to all statements until the 50th: they are all true. In particular, the 50th statement being true, there are at least 50 false statements: those are precisely statements 51 to 100. The 'falseness' of this statements does not rise any contradiction, as there are indeed $only$ 50 false statements. c) In this case, the list of statements is contradictory: proceeding as in part b) we deduce that statements 1 to 49 are true. Now, if statement n° 50 were to be false, so would be statements 51 to 99, resulting in exactly 50 false statements: but n° 50 being false means that there are $strictly$ less than 50 false statements: a contradiction. So statement n° 50 should be true : but once again, what it states is in contradiction with the fact that we are left with only 49 statements - all the remaining being true. Since it is not possible to assign a truth value to statement n° 50, the whole list is inconsistent.