What reasons philosophers give to justify the claim that the Liar is paradoxical?

Question

Most philosophers seem to see the Liar as paradoxical. Typically, they would say:

(L) -- If the Liar is true, then it is false; if it is false, then it is true.

According to what I have read, (L) seems sufficient in itself to motivate the characterisation of the Liar as a paradox. However, this is essentially like pointing your finger at something, saying "See?", without explanation.

I haven't found any philosopher articulating the reason that (L) proves that the Liar is paradoxical.

My question is therefore as follows:

How do philosophers justify the idea that (L) shows that the Liar is paradoxical?

Thank you for scholarly references.

I'm not interested in:

1. resolutions of the paradox.
2. explanations involving the principle of explosion, the horseshoe or mathematical logic at large.

Answer 1

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.

1. A proposition is always either true or false.
2. A proposition is never both true and false.
3. L is a proposition.
4. From "if A then B" it follows that "if A then (A and B)".
5. From "A or B", "if A then C", "if B then C" it follows that C.

Now we can proceed as follows:

6. If L is true, then L is false. This is a given.
7. If L is true, then L is true and L is false. Follows from 6 together with 4.
8. If L is false, then L is true. This is a given.
9. If L is false, then L is true and L is false. Follows from 8 together with 4.
10. L is true or L is false. From 1 and 3.
11. 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:

1. 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.

2. 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.

3. 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.

4. There are some logics that dispense with assumption 4, but 4 seems highly plausible if we are just dealing with simple truth valued propositions.

5. 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.

Answer 2

The reasoning involves taking both initial options and seeing where that goes. First we assume that the liar sentence is true. Then it "is what it says it is," so it must be false. But if it's false, it's not true, ergo...

Then, going the other way, if we assume that it's false, then again, it "is what it says it is," which in general means it's true. But then if it's true, it's still "what it says it is," and the cycle continues.

So in a way, it's like a back-and-forth of the following:

1. Assume that it's true in general; then it's false in particular.
2. But if it's false in general, then it's true in particular.
3. Assume that it's true in general and in particular...
4. But if it's false in general and in particular...

Answer 3

If you’re not interested in Mathematical logic, but aren’t convinced that the same thing can’t be both true and false, it’s going to be hard to explain why you should be.

Roughly speaking, it comes down to the consistency of the truth. The thing about truth and logic is that when something is true, other statements related to that thing also gain value related to it being true. Consequence and Implication are both highly connected to truth in this way.

So the combination of truth holding statements in this kind of structured way and a statement being both true and having impossible consequences would a paradoxical situation where the impossible things should be true. Since this is what the Liar does, it needs to be addressed.