1. Sentence 2 is true.
2. An odd number of prime-numbered sentences are true.
3. A prime number of these sentences are true.
4. Sentence 1 is true.
5. A prime number of these sentences are false.
Which sentences are true and which sentences are false?
A statement may be true, false, or a contradiction. I assert that a contradiction is neither true nor false.
The prime-numbered sentences referred to in statement 2 are statements 2, 3, and 5. Because a contradiction is neither true nor false, it is possible that 3 and 5 are both true or both false. Let's consider three cases.
Case 1: Assume that between 3 and 5, only one is true.
If statement 2 is true, then an even number of prime-numbered sentences are true, so statement 2 cannot be true. If statement 2 is false, then an odd number of prime-numbered sentences are true, so statement 2 cannot be false. Therefore, statement 2 is a contradiction.
Then sentence 1 is false.
Then sentence 4 is false.
So there are 3 false statements, 1 true statement, and one contradiction (neither true nor false). That makes statement 3 false and statement 5 true, which is consistent with our assumption.
So, one solution is:
1. False
2. Contradiction
3. False
4. False
5. True
Case 2: Assume that neither 3 nor 5 are true.
If statement 2 is true, then an odd number of prime-numbered sentences are true, so statement 2 could be true. If statement 2 is false, then an even number of prime-numbered sentences are false, so statement 2 could be false.
Case 2a: Assume Statement 2 is true.
Then 1 and 4 are also true.
Then we have 3 true statements, so statement 3 is true, which violates our case 2 assumption.
Case 2b: Assume Statement 2 is false.
Then 1 and 4 are also false.
Then we have 3 false statements, so statement 5 is true, which violates our case 2 assumption.
So, case 2 yields no valid solutions.
Case 3: Assume that both 3 and 5 are true.
If statement 2 is true, then an odd number of prime-numbered sentences are true, so statement 2 could be true. If statement 2 is false, then an even number of prime-numbered sentences are false, so statement 2 could be false.
Case 3a: Assume Statement 2 is true.
Then 1 and 4 are also true.
Then we have 5 true statements and 0 false statements, so statement 5 is false, which violates our case 3 assumption.
Case 2b: Assume Statement 2 is false.
Then 1 and 4 are also false.
Then we have 3 false statements and 2 true statements, so statements 3 and 5 are both true.
Second solution (which some may assert is the only solution)
1. False
2. False
3. True
4. False
5. True
Edited on February 9, 2022, 11:21 am
an alternative method
