1. Statement 2 in this list is true
2. Statement 1 in this list is false
3. Statement 3 in this list is false
4. There are two misstakes in statement 4 in this list.
I can't help feeling that logic puzzles like this aren't legitimate at all. I think it's because they all involve some kind of self-reference. They are also difficult to solve because there is no conclusion, so it's difficult to know where to start. Any tips?
The concepts of true and false are not ordinary predicates. Propositions involving them have to have some meaning beyond them before they can apply.
Statements 1 and 2 refer to each other. 1 says 2 is true, 2 says 1 is false. Therefore, 2 belies 1; and so doing belies itself. This is therefore equivalent to a self-contradiction, and 2 should read that 1 is true (instead of false). In that case, 1 becomes acceptable as true.
The problem here is of course that this correction changes 2 - which means it is no longer the given 2. And that is a problem due to the use of true/false as ordinary predicates. They are not so, if the sentences involved have no other content besides truth/falsehood.
Statement 3 is likewise a straight self-contradiction. And similar comments apply.
Statement 4 is different - presumably the two mistakes are that there is one misspelled term (misstakes); and that this is just one, and not as it claims two, mistakes. So, the statement is therefore somewhat self-contradictory and therefore false - but here the issue is not the use of true or false as predicates.
In general, any use of true or false has to be the product of an inductive or deductive process - the use of these quasi-predicates without any justification is meaningless and valueless.
See more here: http://avisionparadoxes.bravesites.com/entries/general/6-the-liar-paradox-redux
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