Liar paradox

From Wikipedia

In philosophy and logic, the liar paradox encompasses paradoxical statements such as:

• I am lying now.
• This statement is false.

These statements are paradoxical because there is no way to assign them a consistent truth value. Consider that if This statement is false. is true, then what it says is the case; but what it says is that it is false, hence it is false. On the other hand, if it is false, then what it says is not the case; thus, since it says that it is false, it must be true.

To avoid having a sentence directly refer to its own truth value, one can also construct the paradox as follows:

The following sentence is true. The preceding sentence is false.

Versions through history

Epimenides and Eubulides

In the sixth century BC the philosopher-poet Epimenides, himself a Cretan, reportedly wrote:

The Cretans are always liars.

The Epimenides paradox is often considered an equivalent or interchangeable term for the "liar paradox" but they are not the same. It is unlikely that Epimenides intended his words to be understood as a kind of liar paradox and they were probably only understood as such much later in history. Moreover this statement is not a paradox when false, because no proof exists that all Cretans really are liars.

The oldest known version of the liar paradox is instead attributed to the Greek philosopher Eubulides of Miletus who lived in the fourth century BC. It is very unlikely that he knew of Epimenides's words, even if they were intended as a paradox. Eubulides reportedly said:

A man says that he is lying. Is what he says true or false?

Bertrand Russell

Bertrand Russell formulated the liar paradox in terms of set theory. He discovered this form of the paradox, known as Russell's paradox, in 1901. First, he conceived of a set that included other sets. An example of this is the set of all sets. By definition, all sets, including this set, are members of the set of all sets. He then conceived of the set of all sets that do not include themselves. He pondered if this set included itself, and realized that it does if it does not, and it does not if it does.

Alfred Tarski

Alfred Tarski discussed the possibility of a combination of sentences, none of which are self-referential, but become self-referential and paradoxical when combined. As an example:

1. Sentence 2 is true.
2. Sentence 1 is false.

He resolved this "liar cycle" by arguing that when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the 'object language,' while the referring sentence is considered to be a part of a 'meta-language' with respect to the object language. It is legitimate for sentences in 'languages' higher on the semantic hierarchy to refer to sentences lower in the 'language' hierarchy, but not the other way around. This prevents a system from becoming self-referential.

Variants of the paradox

The problem of the liar paradox is that it seems to show that common beliefs about truth and falsity actually lead to a contradiction. Sentences can be constructed that cannot consistently be assigned a truth value even though they are completely in accord with grammar and semantic

rules.

Consider the simplest version of the paradox, the sentence:

This statement is false. (A)

If we suppose that the statement is true, everything asserted in it must be true. However, because the statement asserts that it is itself false, it must be false. So the hypothesis that it is true leads to the contradiction that it is true and false. Yet we cannot conclude that the sentence is false for that hypothesis also leads to contradiction. If the statement is false, then what it says about itself is not true. It says that it is false, so that must not be true. Hence, it is true. Under either hypothesis, we end up concluding that the statement is both true and false. But it has to be either true or false (or so our common intuitions lead us to think), hence there seems to be a contradiction at the heart of our beliefs about truth and falsity.

However, that the liar sentence can be shown to be true if it is false and false if it is true has led some to conclude that it is neither true nor false. This response to the paradox is, in effect, to reject a common beliefs about truth and falsity: the claim that every statement has to abide by the principle of bivalence, a concept related to the law of the excluded middle.

The proposal that the statement is neither true nor false has given rise to the following, strengthened version of the paradox:

This statement is not true. (B)

If (B) is neither true nor false, then it must be not true. Since this is what (B) itself states, it means that (B) must be true and so one is led to another paradox.

This result has led some, notably Graham Priest, to posit that the statement follows paraconsistent logic and is both true and false. Nevertheless, even Priest's analysis is susceptible to the following version of the liar:

This statement is only false. (C)

If (C) is both true and false then it must be true. This means that (C) is only false, since that's what it says, but then it can't be true, and so one is led to another paradox.

Another interesting variation is:

The writer of this statement cannot verify it to be true

In this version, the writer of the statement cannot verify it to be true, because in doing so he makes it obviously false, but at the same time cannot verify it to be false, as this would make it obviously true. Anybody else except the writer, however, can easily see and verify the statement's truth.

Not a paradox

The statement "I always lie" is often considered to be a version of the liar paradox, but is not actually paradoxical. It could be the case that the statement itself is a lie, because the speaker sometimes tells the truth, and this interpretation does not lead to a contradiction. The belief that this is a paradox results from a false dichotomy - that either the speaker always lies, or always tells the truth - when it is possible that the speaker occasionally does both.

Possible resolutions

A.N. Prior

A. N. Prior asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to Charles S. Peirce and John Buridan) is that every statement includes an implicit assertion of its own truth. Thus, for example, the statement "It is true that two plus two equals four" contains no more information than the statement "two plus two is four," because the phrase "it is true that..." is always implicitly there. And in the self-referential spirit of the Liar Paradox, the phrase "it is true that..." is equivalent to "this whole statement is true and ...".

Thus the following two statements are equivalent:

This statement is false

This statement is true and this statement is false.

The latter is a simple contradiction of the form "A and not A", and hence is false. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction. However, this argument is fallacious due to a semantic subtlety: The referents of the statements changed when they were conjoined. Neil Lefebvre and Melissa Schelein present a similar answer^[1].

Saul Kripke

Saul Kripke points out that whether a sentence is paradoxical or not can depend upon contingent facts. Suppose that the only thing Smith says about Jones is

A majority of what Jones says about me is false.

Now suppose that Jones says only these three things about Smith:

Smith is a big spender.

Smith is soft on crime.

Everything Smith says about me is true.

If the empirical facts are that Smith is a big spender but he is not soft on crime, then both Smith's remark about Jones and Jones's last remark about Smith are paradoxical.

Kripke proposes a solution in the following manner. If a statement's truth value is ultimately tied up in some evaluable fact about the world, call that statement "grounded". If not, call that statement "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.

Barwise and Etchemendy

Jon Barwise and John Etchemendy propose that the liar sentence (which they interpret as synonymous with the Strengthened Liar) is ambiguous. They base this conclusion on a distinction they make between a "denial" and a "negation". If the liar means "It is not the case that this statement is true" then it is denying itself. If it means This statement is not true then it is negating itself. They go on to argue, based on their theory of "situational semantics" that the "denial liar" can be true without contradiction while the "negation liar" can be false without contradiction.

Gödel's theorem

The proof of Gödel's incompleteness theorem uses self-referential statements that are similar to the statements at work in the Liar paradox.

In the context of a sufficiently strong axiomatic system A of arithmetic:

This statement is not provable in A. (1)

The statement (1) does not mention truth at all (only provability) but the parallel is clear. Suppose (1) is provable, then what it says of itself, that it is not provable, is not true. But this conclusion is contrary to our supposition, so our supposition that (1) is provable must be false. Suppose the contrary that (1) is not provable, then what it says of itself is true, although we cannot prove it. Therefore, there is no proof that (1) is provable, and there is also no proof that its negation is provable (i.e., there is no proof that it is also unprovable). Whence, A is incomplete because it cannot prove all truths, namely, (1) and its negation. Statements like (1) are called undecidable. We take for granted that all the provable statements of logic and arithmetic are true; Gödel showed that the converse, that all the true statements of a system are provable in that system, is not the case. (This does not mean that all true statements are not provable in some system or other. Additionally, there are systems, such as first-order logic, in which all true statements of the system are provable.)

Tarski's indefinability theorem, closely related to Gödel's Theorem, is a more direct application of the Liar Paradox, though there is no actual paradox involved; instead, the "paradox" simply demonstrates that all the true sentences of arithmetic are not arithmetically definable (or that arithmetic cannot define its own truth predicate; or that arithmetic is not "semantically closed").

Dialetheism

Graham Priest and other logicians have proposed that the liar sentence should be considered to be both true and false, a point of view known as dialetheism. In a dialetheic logic, all statements must be either true, or false, or both. Dialetheism raises its own problems. Chief among these is that since dialetheism recognizes the liar paradox, an intrinsic contradiction, as being true, it must discard the long-recognized principle of ex falso quodlibet. This principle asserts that any sentence whatsoever can be deduced from a true contradiction. Thus, dialetheism only makes sense in systems that reject ex false quodlibet. Such logics are called paraconsistent.

In popular culture

In the episode "I, Mudd" (Episode #41) of the original Star Trek series, Captain Kirk and Harry Mudd use the liar paradox to confuse and thus incapacitate an android who is holding the landing party captive.

In the 1973 Doctor Who serial The Green Death, The Doctor temporarily stumps the super-computer BOSS using the paradox.

A similar event to the above occurs in the anime Ghost in the Shell: Stand Alone Complex, when a mischievous Tachikoma think tank fools an admin drone using the paradox. The admin drone, which has a much simpler AI, is utterly confused and left stymied, allowing the Tachikomas to steal a piece of equipment left in the drone's care.

Another anime that features this paradox is Full Metal Panic!: The Second Raid, in which the paradox is briefly mentioned during the mission briefing. The Commanding Officer references the paradox again during the mission once he realizes that their communications were being intercepted and uses it to subtly tell his crew to do the opposite of his instructions.

Gregory House from House frequently says "Everybody lies". However, in the season one finale, he remarked that he was lying when he said that. Of course, there is no paradox here since he doesn't assert anything the first time except that everyone lies at least once in their lives. The second time, he says he was lying but he doesn't really have any way of knowing one way or another and neither do we. Is he lying if he believes he's lying, or is he lying if what he says is in reality false? Is he lying if he isn't one hundred percent certain that what he says is true? If the first statement was "Everybody lies all the time", then it by itself would constitute a liar paradox. The second statement would be another paradox. Actually, everything he said after the first statement would be a paradox if you accept that there is an implicit assertion of truth with every statement made.

A character from Disney's Timon and Pumbaa television series is called the "no good lying Toucan Dan", who never tells the truth. In the episode he first appeared in, Timon briefly probes into the liar paradox saying that if Toucan Dan never tells the truth and he's saying he did not steal anything, then he did steal it so to make him confess his crime, they'd have to trick him into thinking he didn't steal it, because he would lie and say he did. Toucan Dan hears all the muttering, so it doesn't work anyway.

In the book The Giver, the main character is given permission to lie upon becoming of age. He wonders about asking other adults if they received the same instruction. He then reasons that if they didn't, they'd be obligated to say "no"; yet if they did, they could always lie and say "no", so he'd never know even if he asked them.

In Douglas Adams' Hitchhiker's Guide to the Galaxy series, there is a passing reference to an old man who "claimed repeatedly that nothing was true, although he was later discovered to be lying."

On the George Carlin album A Place for My Stuff, Carlin says "The following statement is true. The preceding statement was false." His 2004 book When Will Jesus Bring The Pork Chops? also included the sentence "This statement is untrue," as well as "Ignore these four words."

A similar version of this phrase forms the title of a 2001 album by Sheila Chandra: "This Sentence Is True (The Previous Sentence Is False)". Note that this isn't actually a version of the liar paradox in that the two sentences merely need to have opposite truth values for the pair to be consistent.

In the animated television show Family Guy, star Peter Griffin tells his son, "Chris, everything I say is a lie. Except that. And that. And that. And that. And that. And that. And that.... And that."

In the video game Knights of the Old Republic, a liar's paradox is used as a riddle.

In the Deltora Quest book, The Lake of Tears, Lief is asked a riddle by a guard crossing him and his companions' path. The guard asks him to say one thing, and one thing only. If what Lief says is true, the guard would strangle him. If what Lief says is false, the guard would cut off his head. To this Lief says, "My head will be cut off". The guard does nothing, knowing that if the statement was true, he would have to strangle Lief, thus making the statement false. However, if the statement was false, he would have to cut off Lief's head, thus making the statement true. (Though he could strangle Lief and then cut his head off.)

In The Simpsons episode 'Itchy & Scratchy Land' (2F01), one of the deleted scenes (contained in the Sixth Season Box Set) shows Lisa trying to incapacitate a group of approaching homicidal robots by proposing to them the Liar Paradox. The ploy succeeds only in confusing Homer.

In the book Sideways Arithmetic from Wayside School, (part of the Sideways Stories From Wayside School series), one of the students, as punishment, must answer the two question version of the liar paradox.

In Werner Herzog's "The Enigma of Kaspar Hauser" the logician asks the idiot for a single clever question that can distinguish a liar from the truth teller. The idiot says to ask the liar/truth teller "Are you a tree frog?" which drives the logician into a state of apoplexy trying to prove that that is not the right approach.

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