Yeh sorry, it's definitely not the clearest working out but it's a fast method and would be how I'd actually solve it in rough working out, hence the scrappiness of the method itself =/
So starting with I, set up the given details with a diagram like this. This is saying that statement 1 is saying statement 2 is a lie (-), and statement 2 is saying statement 1 is true (+).
Now, start with the assumption that statement 1 is true. i.e. It is saying that statement 2 is actually a lie. I'll write this as follows:
Now with truth and lies in this situation, given that there's only two options (a statement is either true or not regarding the other statement), it's actually similar to other positive and negative systems. If you think about it, if a statement which says another statement is true, is true itself, then that other statement is obviously true. If a statement which says another statement is a lie, but is a lie itself, then the other statement has to be true as that's the only other viable option. So you can see that as with Maths, multiplying a positive and positive, or a negative and negative will always give you a positive. The converse applies with multiplying a positive and negative, meaning that a lying statement saying the other is true, or a true statement saying the other is false, obviously mean the other statement is false. Hence, using your typical mathematical positive-negative multiplication laws to simplify things from here on:
This diagram is saying that if you assume 1 to be true, and 1 is saying 2 is false, then 2 must be false. Hence the negative sign underneath it. However, from here you can already see a problem. 2 is false, but it's saying 1 is true, so hence 1 must be false. But we've already said 1 is true. So there's a contradiction here.
If you then follow suit and let 1 be false as the system is indicating (draw this diagram yourself and see what happens, I cbf drawing another), 2 becomes true, but then this causes 1 to become true again. Hence, the system never stays constant on its signs and hence will continue to contradict itself indefinitely.
However if you take say III:
Multiplying the + and - makes 2 negative as shown by:
And then multiplying the - and - makes a positive, which was what we started off with. Hence, the entire system just stays constant and there's no contradictions arising. You'll see that this particular system actually works whether you assume 1 to be true or false to begin with.
If you still don't get it, look, copying someone else's methods probably isn't the best idea anyway. I work in a far more mathematically based system than most so if you're not mathematically minded, there's no point copying my methods. Just work out what's best for you because there's plenty of ways to figure out these problems. The point I was trying to get across initially is that these truth and lies questions tend to be based upon a system where you assume one of the statements to initially be either true or false, and then follow the system of logic through to reach your answer, whether that's figuring out whether any contradictions arise, or shifting through combinations of statements until none do etc.
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