This whole line of reasoning is predicated upon 10 being absolute truth, because no dog can be both dead and alive. If that is not the case and we have a quantum dog that can be in a superposition of states, none of this is valid.
1 is a simple statement.
2 is a simple statement.
3 says that only one of 6 or 9 is true. If 3 is false, then either both 6 and 9 are true, or both 6 and 9 are false. Since statement 10 is true, if 9 is true, then 6 must be false. Therefore, they cannot both be true. IF 9 is false, then 6 must be true, otherwise 9 would be true. Therefore, they cannot both be false. Therefore, statement 3 must be true. Therefore, either 6 is true, or 9 is true, but not both.
9 states that only one of 6 or 10 is true. 10 tells us something we already knew to be true; therefore, if 9 is true, 10 is true and 6 is false. (We can also derive this from 3 being true, which makes 6 false if 9 is true). If 6 is false, then either 1 and 10 are both true, or they are both false. 10 is true, therefore 1 must be true; therefore, if 6 is false, rocky is alive.
If 6 is the one that is true, either 1 is true or 10 is true. 10 must be true, as 1 and 2 are logically opposed. Therefore, if 6 is true, 1 is false, making 2 true. Therefore, if 6 is true, Rocky is Dead.
On to the next statement. 4 states that one of 2 and 6 is false. If 2 is false, Rocky is Alive. However, if 2 is false, 6 is true, and if 6 is true, as we have shown, 2 is true, and Rocky is Dead. Therefore, if 2 is false, 2 is true, which is a logical contradiction. Therefore, 2 is true, and 6 is false. Therefore, if 4 is true, Rocky is Dead.
If statement 4 is false, then either 2 and 6 are both true or 2 and 6 are both false. If 2 is false (therefore Rocky is Alive) and 6 is false (therefore making statement 1 true, since 10 cannot be false for them both to be false), then Rocky is alive.
To recap the state of Rocky:
if 1: alive; if !1: dead
if 2: dead; if !2: alive
3, therefore either dead or alive.
if 4: Dead; if !4: alive
if 6: dead; if !6: alive.
If Statement 5 is true, then statements 4, 5, and 10 are all false. Since statement 5 being true precludes statement 5 being false, then statement 5 is false. This lets us know that statements 4, 5, and 10 are not all false. Statement 5 is false, so both of statements 4 and 10 cannot be false. Therefore, one ore both of statements 4 and 10 is true. However, we know that statement 10 is true. Therefore, either statement 4 is false or statement 4 is true. This tells us nothing of interest about statement 4; we already knew it was either true or false.
Statement 7 claims that five statements are true. This is not useful at this time, so we set it aside for now.
Statement 8 claims that exactly one of 3 and 10 is false. We know that 10 is empirically true. Therefore, if 8 is true 3 is false. We know, however, that 3 must be true. Therefore, 8 is false. You can reason out here to show that both statements 3 and 10 are true, but we knew that already.
To recap the state of Rocky:
if 1: alive
if 2: dead
since 3, either Dead or alive
if 4: Dead; if !4: alive
if 6: dead
since !8, either dead or alive
Statement 11 claims that 1, 8, and 11 are all false. As with statement 5, 11 being true leads to a logical contradiction, so we know that 11 is false. Since we know that 11 is false and 8 is false, 1 must be true (otherwise 11 would be true). Therefore, 1 is true and rocky is ALIVE. Since he is alive, he cannot be dead, so 2 is false.
So far:
1: true
2: false
3: true
5: false
8: false
10: true
11: false
We deduced earlier that if statement 4 is true, then rocky is dead. Since rocky is not dead, statement 4 must be false. We deduced earlier that that means 2 and 6 must both be true or both be false; since 2 is true, 6 must be false.
So far:
1: true
2: false
3: true
4: false
5: false
6: false
8: false
10: true
11: false
Statement 9 claims that exactly one of 6 and 10 is true. This is obviously true.
1: true
2: false
3: true
4: false
5: false
6: false
8: false
9: true
10: true
11: false
We now have four true statements. Statement 7 states that exactly 5 statements are true. If 7 is true, then five statements are true, as 7 is the fifth. If 7 is false, then ther are only four true statements, making 7 false. So 7 is impossible to determine. I will mark this as "mu".
As a side note, I am ridiculously proud that I managed to avoid confusing myself, but please let me know if I mixed something up in the middle!
Edited: I had mixed up statement 8 somehow. Expanded and fixed.
