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In philosophy and logic, the 'liar paradox' encompasses paradoxical statements such as:
★ "I am lying now."
★ "This statement is false."
or
★ "The sentence below is false."
★ "The sentence above is true."
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."
However, it is arguable that this reformulation is little more than a syntactic expansion. The idea is that neither sentence accomplishes the paradox without precisely its counterpart.
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 equivalent or interchangeable with the "liar paradox", but they are not the same. The liar paradox is a statement that cannot consistently be true or false, while Epimenides' statement is simply false, as long as there exists at least one Cretan who sometimes tells the truth.
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. 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?''
The Apostle Paul's letter to Titus in the New Testament refers to this quote in the first century AD.
:''One of them, a prophet of their own, said, "Cretans are always liars, evil beasts, lazy gluttons." This testimony is true. '' (Titus 1:12, 13a NKJV)
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 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:
#Sentence 2 is true.
#Sentence 1 is false.
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 the 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.
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.
A further version is phrased as a question:
:"Are you lying when you answer this question?"
If the answer is affirmative, then it is the liar paradox. If the answer is negative, then either interpreting the answer as true or false is consistent.
Alfred Tarski resolved his "liar cycle" (see above) 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.
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. Eugene Mills[1] and Neil Lefebvre and Melissa Schelein[2] present similar answers.
But Prior never made clear how his approach would apply to the more complex versions of the paradox, such as the two sentence version: "The next sentence is false", "The preceding sentence is true". Moreover, if all sentences are really hidden conjunctions, then some rules of propositional logic, such as the rule that one can derive any conjunct immediately and the rule that from any two propositions one can immediately derive their conjunction, are called into question. If we can derive ''this statement is false'' from ''This statement is true and this statement is false'', then the paradox is back. And if we are not allowed to make such a derivation, then Prior has, in effect, invented a new kind of conjunction whose truth value characteristics are so mysterious, we cannot really say with any confidence that the paradox has been dissolved.
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.
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.
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").
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 falso quodlibet''. Such logics are called ''paraconsistent''.
★ Quine's paradox
★ List of paradoxes
★ Opposite Day
1. Mills, Eugene (1998) ‘A simple solution to the Liar’, Philosophical Studies 89: 197-212.
2. Lefebvre, N. and Schelein, M., "The Liar Lied," in Philosophy Now issue 51
★ Barwise, Jon and John Etchemendy 1987: ''The Liar''. Oxford University Press.
★ Greenough, P.M., 2001. American Philosophical Quarterly 38
★ Hughes, G.E., 1992. ''John Buridan on Self-Reference : Chapter Eight of Buridan's Sophismata, with a Translation, and Introduction, and a Philosophical Commentary'', Cambridge University Press, ISBN 0-521-28864-9 (Buridan's detailed solution to a number of such paradoxes).
★ Kirkham, Richard 1992: ''Theories of Truth''. Bradford Books. Chapter 9 is a very good discussion of the paradox.
★ Kripke, Saul 1975: "An Outline of a Theory of Truth" ''Journal of Philosophy'' 72:690-716.
★ Priest, Graham 1984: "The Logic of Paradox Revisited" ''Journal of Philosophical Logic'' 13:153-179.
★ Prior, A. N. 1976: ''Papers in Logic and Ethics''. Duckworth.
★ Lefebvre , Neil and Schelein, Melissa. "The Liar Lied". ''Philosophy Now'' issue 51, 2005.
★ Liar Paradox — at the Internet Encyclopedia of Philosophy
★ Smullyan, Raymond: ''What is the Name of this Book?'', ISBN 0-671-62832-1 (a collection of logic puzzles exploring this theme).
★ "I am lying now."
★ "This statement is false."
or
★ "The sentence below is false."
★ "The sentence above is true."
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."
However, it is arguable that this reformulation is little more than a syntactic expansion. The idea is that neither sentence accomplishes the paradox without precisely its counterpart.
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 equivalent or interchangeable with the "liar paradox", but they are not the same. The liar paradox is a statement that cannot consistently be true or false, while Epimenides' statement is simply false, as long as there exists at least one Cretan who sometimes tells the truth.
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. 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?''
The Apostle Paul's letter to Titus in the New Testament refers to this quote in the first century AD.
:''One of them, a prophet of their own, said, "Cretans are always liars, evil beasts, lazy gluttons." This testimony is true. '' (Titus 1:12, 13a NKJV)
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:
#Sentence 2 is true.
#Sentence 1 is false.
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 the 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.
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.
A further version is phrased as a question:
:"Are you lying when you answer this question?"
If the answer is affirmative, then it is the liar paradox. If the answer is negative, then either interpreting the answer as true or false is consistent.
Possible resolutions
Alfred Tarski
Alfred Tarski resolved his "liar cycle" (see above) 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.
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. Eugene Mills[1] and Neil Lefebvre and Melissa Schelein[2] present similar answers.
But Prior never made clear how his approach would apply to the more complex versions of the paradox, such as the two sentence version: "The next sentence is false", "The preceding sentence is true". Moreover, if all sentences are really hidden conjunctions, then some rules of propositional logic, such as the rule that one can derive any conjunct immediately and the rule that from any two propositions one can immediately derive their conjunction, are called into question. If we can derive ''this statement is false'' from ''This statement is true and this statement is false'', then the paradox is back. And if we are not allowed to make such a derivation, then Prior has, in effect, invented a new kind of conjunction whose truth value characteristics are so mysterious, we cannot really say with any confidence that the paradox has been dissolved.
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 falso quodlibet''. Such logics are called ''paraconsistent''.
See also
★ Quine's paradox
★ List of paradoxes
★ Opposite Day
References
1. Mills, Eugene (1998) ‘A simple solution to the Liar’, Philosophical Studies 89: 197-212.
2. Lefebvre, N. and Schelein, M., "The Liar Lied," in Philosophy Now issue 51
★ Barwise, Jon and John Etchemendy 1987: ''The Liar''. Oxford University Press.
★ Greenough, P.M., 2001. American Philosophical Quarterly 38
★ Hughes, G.E., 1992. ''John Buridan on Self-Reference : Chapter Eight of Buridan's Sophismata, with a Translation, and Introduction, and a Philosophical Commentary'', Cambridge University Press, ISBN 0-521-28864-9 (Buridan's detailed solution to a number of such paradoxes).
★ Kirkham, Richard 1992: ''Theories of Truth''. Bradford Books. Chapter 9 is a very good discussion of the paradox.
★ Kripke, Saul 1975: "An Outline of a Theory of Truth" ''Journal of Philosophy'' 72:690-716.
★ Priest, Graham 1984: "The Logic of Paradox Revisited" ''Journal of Philosophical Logic'' 13:153-179.
★ Prior, A. N. 1976: ''Papers in Logic and Ethics''. Duckworth.
★ Lefebvre , Neil and Schelein, Melissa. "The Liar Lied". ''Philosophy Now'' issue 51, 2005.
★ Liar Paradox — at the Internet Encyclopedia of Philosophy
★ Smullyan, Raymond: ''What is the Name of this Book?'', ISBN 0-671-62832-1 (a collection of logic puzzles exploring this theme).
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