There are different ways of writing the liar sentence, L. It is usually written as "This statement is false". It can also be made somewhat more indirect by using more than one sentence, e.g.
The statement immediately below this one is true.
The statement immediately above this one is false.
The consequences of the liar statement may be expressed conditionally, as you have written it.
If L is true, then L is false; and if L is false, then L is true.
To see why this is paradoxical, let us help ourselves to a few assumptions. We can challenge them later.
- A proposition is always either true or false.
- A proposition is never both true and false.
- L is a proposition.
- From "if A then B" it follows that "if A then (A and B)".
- From "A or B", "if A then C", "if B then C" it follows that C.
Now we can proceed as follows:
- If L is true, then L is false. This is a given.
- If L is true, then L is true and L is false. Follows from 6 together with 4.
- If L is false, then L is true. This is a given.
- If L is false, then L is true and L is false. Follows from 8 together with 4.
- L is true or L is false. From 1 and 3.
- L is true and L is false. Follows from 5, 10, 7, 9.
Hence we have proved a contradiction with 11. Note that this is not merely a case of assuming L, showing that it leads to a contradiction and then concluding that L is false. What this proves is that if L is true then a contradiction follows and also if L is false a contradiction follows. We do not need to assume a truth value for L: irrespective of whether it is true or false we can prove a contradiction. This is what gives L its paradoxical flavour. It is not merely equivalent to a contradiction, which would make it false but not paradoxical. It appears to show that we can prove a contradiction unconditionally, whether L is true or not.
Let's examine the assumptions:
We can drop assumption 1 by supposing that the liar sentence does not have a truth value or has a value other than true or false. We still have to deal with the weakened version of the liar paradox, "This sentence is either false or does not have a truth value". Some have claimed that the basic liar sentence does not entail "if L is false, then L is true", so they conclude that L is simply false and not paradoxical, but this still involves rejecting assumption 1.
We can drop assumption 2 and go along with the dialetheists and suppose that some propositions are both true and false and L is one of them.
We could reject 3 and say that L fails to qualify as a proposition for some reason. Many reasons have been offered, e.g. that it is not legitimate for a proposition to be self-referential, that it lacks a stable truth value, that it violates the object language/metalanguage boundary, that there is no satisfactory way to define a truth predicate, etc.
There are some logics that dispense with assumption 4, but 4 seems highly plausible if we are just dealing with simple truth valued propositions.
Assumption 5 is pretty hard to argue with. Offhand I cannot think of a logic that does not include it.
The upshot is that the paradox is highly problematic and there is a substantial literature on the options for resolving it. The SEP article is a good start. I also recommend Hartry Field's book, Saving Truth From Paradox.