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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.
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    The simple answer is that under very basic assumptions, the Liar is equivalent to a direct contradiction.
    – PW_246
    Jul 3, 2023 at 5:43
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    See Dialetheism and the Liar "A dialetheia is a sentence, A , such that both it and its negation, ¬A, are true. Dialetheism is the view that there are dialetheias. Paradoxes [like the Liar] have been known since antiquity. But they were thrown into prominence by developments in the foundations of mathematics around the turn of the twentieth century. In the case of each paradox, there appears to be a perfectly sound argument ending in a contradiction. If the arguments are sound, then dialetheism is true." Jul 3, 2023 at 10:24
  • @PW_246 "the Liar is equivalent to a direct contradiction" Contradictions are not usually characterised as paradoxical. They are just false. Asserting a contradiction would be regarded as just absurd, or even idiotic, not paradoxical. Jul 3, 2023 at 14:54
  • The reason the Liar is seemingly paradoxical is that the reason for its inconsistency is not obvious; self-referential statements like “this statement is true” are not inherently paradoxical, so we can’t ban self-reference outright. IMO, this whole thing boils down to either 1. defining a proposition as its negation at a meta-level, or 2. using something like diagonalization to reproduce the same result in a sufficiently expressive theory. The former is obviously incorrect, and Tarski showed that the latter issue prevents global truth predicates from being definable in a first-order theory.
    – PW_246
    Jul 3, 2023 at 16:31
  • @PW_246 "The reason the Liar is seemingly paradoxical" This is not the question. The question is what is the justification of philosophers for taking (L) as making the Liar paradoxical. Jul 4, 2023 at 9:18

3 Answers 3

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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:

  1. If L is true, then L is false. This is a given.
  2. If L is true, then L is true and L is false. Follows from 6 together with 4.
  3. If L is false, then L is true. This is a given.
  4. If L is false, then L is true and L is false. Follows from 8 together with 4.
  5. L is true or L is false. From 1 and 3.
  6. 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.

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  • It's hard to imagine a better answer than this. Jul 4, 2023 at 20:05
  • @Bumble Excellent answer. - 2. "so they conclude that L is simply false and not paradoxical, but this still involves rejecting assumption 1." They conclude that L is false, so why does it require to reject the assumption that it is either true or false. Jul 10, 2023 at 15:51
  • Isn’t it the case that you only get an unconditional contradiction if L is actually a legit proposition? The only way I could see that is if we either define L:=~L at a semantic level, or use a system that can formalize arithmetic. But, in the latter case, the Liar just shows that no truth predicate exists, and I think that the former is just an example of making an inconsistent definition.
    – PW_246
    Jul 10, 2023 at 16:09
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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...
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  • So we cannot decide whether it is true or false, but there are lots of things like that which are nonetheless not called "paradoxes". I may be missing something but don't how this justifies that (L) shows that the Liar is paradoxical. Jul 3, 2023 at 14:50
  • @Speakpigeon Graham Priest thinks that the liar sentence at least "proves"/"shows" that natural language is implicitly committed to the possibility of inescapable contradictions, but I think most analysts think that this just shows that natural language is deficient, then. It's supposed to be less that we can't decide a disjunction of true or false, and more that we can decide a conjunction of both. Besides "paradox," other names for such a sentence have included "insolubilia". Jul 3, 2023 at 14:56
  • I'm actually sympathetic to your intuition that the liar "problem" is not as obviously established as the typical logic on the liar sentence would have us believe; or then I think that the meaning of "true" and "false" simply cannot be the same for the liar sentence as the meaning of those words must be for (most) other sentences. However, showing that would mean trying to resolve the "problem." Jul 3, 2023 at 14:59
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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.

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  • "If you (...) aren’t convinced that the same thing can’t be both true and false" Where on Earth did you get that? 2. The question is: How do philosophers justify the idea that (L) shows that the Liar is paradoxical? you're not trying to answer that. Jul 3, 2023 at 9:30
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    I’ve answered exactly that, but obviously not convincingly. The problem is you’ve vetoed the field where the work is done, leaving an impoverished language with which to make the attempt. Alas.
    – Paul Ross
    Jul 3, 2023 at 19:04
  • @PauRoss "you’ve vetoed the field where the work is done" This is not true for two reasons. (a) I "vetoed" mathematical "logic" because it is irrelevant to human logic and so with the logic of the Liar; (b) Philosophers didn't wait for mathematical "logic" to discuss the Liar. The reasoning (L) has been standard for centuries before Boole was even born, and the Liar was already called a paradox by the time of Aristotle. Jul 4, 2023 at 9:32
  • @PauRoss "an impoverished language" You think natural languages are "impoverished"?! Clearly not. We can express everything we want using a natural language, including mathematical reasonings, but you couldn't express mathematically everything we say using a natural language. Plus, mathematical logic cannot even express human logic, whereas natural languages can. Jul 4, 2023 at 9:33
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    @Speakpigeon Ooooohhhh you're a solipsist, and for you human reasoning is the phenomenon. I think I understand why we will never be able to agree on this.
    – Paul Ross
    Jul 5, 2023 at 3:47

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