The Paradox of the Court is a very old problem in logic stemming from ancient Greece. It is said that the famous sophist Protagoras took on a pupil, Euathlus, on the understanding that the student pay Protagoras for his instruction after he had won his first case. Some accounts claim that Protagoras demanded his money as soon as Euathlus completed his education, others say that Protagoras waited until it was obvious that Euathlus was making no effort to take on clients[1] and still others [2] assert that Euathlus made a genuine attempt but that no clients ever came. In any case, Protagoras decided to sue Euathlus for the amount owed.
Protagoras argued that if he won the case he would be paid his money. If Euathlus won the case, Protagoras would still be paid according to the original contract, because Euathlus would have won his first case.
Euathlus, however, claimed that if he won then by the court's decision he would not have to pay Protagoras. If on the other hand Protagoras won then Euathlus would still not have won a case and therefore not be obliged to pay.
The question is: which of the two men is in the right?
1. Analysis
From a moral standpoint it may be that either party was right, or that both were, or weren't, due to the ambiguous nature of the scenario. However, as a matter of law, if the Court were to rule in favor of Protagoras, the conditions of the original contract between him and his pupil would be invalid and Euathlus would have to pay Protagoras. If, on the other hand, Euathlus were to win, the Court could also void Euathlus's obligation of payment.
How, from an objective standpoint, the way the Court could make its ruling is not necessarily a paradox either. The Court would either rule that Euathlus (as the defendant) had violated the terms of the contract, or had not. The subsequent conundrum would have no legal bearing on the court's decision.
In some civil cases the respondent, if he receives the favor of the court, is also shielded from payments associated with the act of going to court. The Court could indeed rule that Protagoras, as the unsuccessful plaintiff, pay Euathlus the amount which it cost to win. In this case, Euathlus would pay Protagoras only to have the money returned by order of the court. The original contract would have been fulfilled, and Euathlus would bear no further obligation to pay Protagoras for his instruction. The net outcome for Protagoras would be to lose his case, receive payment per the original contract, and then have to pay for the defendant's losses due to his failed suit (Which would be equal to, or exceeding, the cost of Euathlus's education.)
Additionally, Euathlus could hire a lawyer to take on the case, thus invalidating this case as a standard for payment.
1. 1. Another theory
Another means of viewing this case is as follows:
Euathlus would win his case because Protagoras sued him BEFORE Euathlus won his first case. Protagoras would lose that particular case because Euathlus has not yet won a case, and therefore Protagoras's cause of action had not yet manifested itself.
The new victory of Euathlus would qualify as new evidence for Protagoras, thus constituting grounds for a new trial.
2. Notes
1. ^ Peter Suber, Department of Philosophy, Earlham College, Richmond, Indiana, 47374, U.S.A.
2. ^ Eugene P. Northrop, "Riddles in Mathematics", Penguin Books
Friday, December 26, 2008
Quine's paradox
Quine's paradox is a paradox concerning truth values, attributed to W.V.O. Quine. It is related to the liar paradox as a problem, and it purports to show that a sentence can be paradoxical even if it is not self-referring and does not use demonstratives or indexicals (i.e. it does not explicitly refer to itself). The paradox can be expressed as follows:
“Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.
If the paradox is not clear, consider each part of the above description of the paradox incrementally:
it = yields falsehood when preceded by its quotation
its quotation = “yields falsehood when preceded by its quotation”
it preceded by its quotation = “yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.
With these tools, we may now reconsider the description of the paradox. It can be seen to assert the following:
The statement “‘yields falsehood when preceded by its quotation’ yields falsehood when preceded by its quotation” is false.
In other words, the sentence implies that it is false, which is paradoxical - for if it is false, what it states is in fact true. Note, however, the type error: what is quoted is a string of words, whereas what yields falsehood is a proposition.
Contents:
1. Motivation
2. Application
3. See also
4. Bibliography
1. Motivation
The liar paradox ("This sentence is false", or "The next sentence is true. The previous sentence is false") demonstrates essential difficulties in assigning a truth value even to simple sentences. Many philosophers, attempting to explain the liar paradox, concluded that the problem was with the word "this". Once we properly understand this sort of self-reference, they claimed, the paradox no longer arises.
Quine's construction demonstrates that paradox of this kind arises independently of such direct self-reference. In fact, there is no way to eliminate the paradoxes short of a severe crippling of the language. Any system, such as English, that contains entities such as words or sentences that can be used to apply to themselves, must contain this type of paradox.
2. Application
In Gödel, Escher, Bach: an Eternal Golden Braid, author Douglas Hofstadter suggests that the Quine sentence in fact uses an indirect type of self-reference. He then shows that indirect self-reference is crucial in the proofs of Gödel's incompleteness theorems.
3. See also
* Grelling paradox
* Russell paradox
4. Bibliography
* Hofstadter, Douglas. (1979) Gödel, Escher, Bach: an Eternal Golden Braid New York: Basic Books.
* Quine, W. V. O. (1962) "The Ways of Paradox" reprinted in Quine (1966) The Ways of Paradox and Other Essays Cambridge: Harvard Univ. Press. pp. 1-21.
“Yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.
If the paradox is not clear, consider each part of the above description of the paradox incrementally:
it = yields falsehood when preceded by its quotation
its quotation = “yields falsehood when preceded by its quotation”
it preceded by its quotation = “yields falsehood when preceded by its quotation” yields falsehood when preceded by its quotation.
With these tools, we may now reconsider the description of the paradox. It can be seen to assert the following:
The statement “‘yields falsehood when preceded by its quotation’ yields falsehood when preceded by its quotation” is false.
In other words, the sentence implies that it is false, which is paradoxical - for if it is false, what it states is in fact true. Note, however, the type error: what is quoted is a string of words, whereas what yields falsehood is a proposition.
Contents:
1. Motivation
2. Application
3. See also
4. Bibliography
1. Motivation
The liar paradox ("This sentence is false", or "The next sentence is true. The previous sentence is false") demonstrates essential difficulties in assigning a truth value even to simple sentences. Many philosophers, attempting to explain the liar paradox, concluded that the problem was with the word "this". Once we properly understand this sort of self-reference, they claimed, the paradox no longer arises.
Quine's construction demonstrates that paradox of this kind arises independently of such direct self-reference. In fact, there is no way to eliminate the paradoxes short of a severe crippling of the language. Any system, such as English, that contains entities such as words or sentences that can be used to apply to themselves, must contain this type of paradox.
2. Application
In Gödel, Escher, Bach: an Eternal Golden Braid, author Douglas Hofstadter suggests that the Quine sentence in fact uses an indirect type of self-reference. He then shows that indirect self-reference is crucial in the proofs of Gödel's incompleteness theorems.
3. See also
* Grelling paradox
* Russell paradox
4. Bibliography
* Hofstadter, Douglas. (1979) Gödel, Escher, Bach: an Eternal Golden Braid New York: Basic Books.
* Quine, W. V. O. (1962) "The Ways of Paradox" reprinted in Quine (1966) The Ways of Paradox and Other Essays Cambridge: Harvard Univ. Press. pp. 1-21.
Liar paradox
In philosophy and logic, the liar paradox, known to the ancients as the pseudomenon, encompasses paradoxical statements such as "This sentence is false." or "The next sentence is false. The previous sentence is true." These statements are paradoxical because there is no way to assign them a consistent truth value. 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.
Contents:
1. History
2. Explanation of the paradox and variants
3. Possible resolutions
4. Logical structure of the liar paradox
5. See also
6. Notes
7. References
8. External links
1. History
Generally attributed to Epimenides, the original version of the Liar's Paradox was actually devised in the fourth century BC by Greek philosopher Eubulides. Epimenides is rather a fictional speaker in the dialogue in which the Liar's Paradox first makes its appearance. The fictional speaker Epimenides, a Cretan, reportedly stated:
The Cretans are always liars.
The 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?
2. Explanation of the paradox and variants
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.
The simplest version of the paradox is the sentence:
This statement is false. (A)
If 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 false. Yet the sentence cannot be 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, the statement is both true and false.
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.
Another reaction to the paradox of (A) is to posit, as Graham Priest has, 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 is what it says, but then it cannot be true, creating another paradox.
This statement is probably a lie. (D)
If the sentence is probably a lie, then it probably not probably a lie, whereas if it is probably the truth, it probably isn't probable.
2. 1. Non-paradoxes
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.
3. Possible resolutions
3. 1. Alfred Tarski
Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed" by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, 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.
3. 2. 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 equals 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.
3. 3. Saul Kripke
Saul Kripke points out that whether a sentence is paradoxical or not can depend upon contingent facts. If the only thing Smith says about Jones is
A majority of what Jones says about me is false.
and 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.
and Smith really is a big spender but 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, that statement is "grounded". If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.
3. 4. 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.
3. 5. 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, which 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.
3. 6. Four-state logic
The manual for the computer language INTERCAL refers to "Warmenhovian" logic gates which have four states, "low, high, undefined (value of an uninitialized flip-flop), and oscillating (output of a NOR gate with one input low and the other input connected to the output)". The NOR gate, with one input low, is effectively a NOT gate, which negates its input ("the input is false"). By linking its output to its input (the input is assigned the value of this statement, hence "this statement is false") the paradox is induced. The electronic effect of this is oscillation, hence the name.
In this case the value of the liar paradox is "oscillating"; "low" and "high" correspond to the more usual values of "false" and "true".
In terms of the "undefined" value, many people will really claim not to be liars, but some of them will be lying; while this is not a paradox, no truth value can be assigned to the statement "I am not a liar" without further information, and, like paradoxes, this is an important class of statement. This "solution" can be seen to beg the question; higher orders of paradox can be generated which are ill-defined in this system.
4. Logical structure of the liar paradox
For a better understanding of the liar paradox, it is useful to write it down in a more formal way. If "this statement is false" is denoted by A and its truth value is being sought, it is necessary to find a condition that restricts the choice of possible truth values of A. Because A is self-referential it is possible to give the condition by an equation.
If some statement, B, is assumed to be false, one writes B = false. The statement (C) that the statement B is false would be written as C = "B = false". Now, the liar paradox can be expressed as the statement A, that A is false:
A = "A = false"
This is an equation from which the truth value of A = "this statement is false" could hopefully be obtained. In the boolean domain "A = false" is equivalent to not A and therefore the equation is not solvable. This is the motivation for reinterpretation of A. The simplest logical approach to make the equation solvable is the dialetheistic approach, in which case the solution is a A being both "true" and "false". Other resolutions mostly include some modifications of the equation e.g. A. N. Prior claims that the equation should be A = "A = false" and "A = true" and therefore A is false.
5. See also
* Quine's paradox
* List of paradoxes
6. Notes
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
7. References
* Jon Barwise 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 Univ. Press, ISBN 0-521-28864-9. Buridan's detailed solution to a number of such paradoxes.
* Kirkham, Richard (1992) Theories of Truth. MIT Press. Especially chapter 9.
* Saul Kripke (1975) "An Outline of a Theory of Truth," Journal of Philosophy 72: 690-716.
* Lefebvre, Neil, and Schelein, Melissa (2005) "The Liar Lied," Philosophy Now issue 51.
* Graham Priest (1984) "The Logic of Paradox Revisited," Journal of Philosophical Logic 13: 153-179.
* A. N. Prior (1976) Papers in Logic and Ethics. Duckworth.
* Smullyan, Raymond (19nn) What is the Name of this Book?. ISBN 0-671-62832-1. A collection of logic puzzles exploring this theme.
8. External links
* Internet Encyclopedia of Philosophy: "Liar Paradox" -- by Bradley Dowden.
Contents:
1. History
2. Explanation of the paradox and variants
3. Possible resolutions
4. Logical structure of the liar paradox
5. See also
6. Notes
7. References
8. External links
1. History
Generally attributed to Epimenides, the original version of the Liar's Paradox was actually devised in the fourth century BC by Greek philosopher Eubulides. Epimenides is rather a fictional speaker in the dialogue in which the Liar's Paradox first makes its appearance. The fictional speaker Epimenides, a Cretan, reportedly stated:
The Cretans are always liars.
The 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?
2. Explanation of the paradox and variants
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.
The simplest version of the paradox is the sentence:
This statement is false. (A)
If 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 false. Yet the sentence cannot be 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, the statement is both true and false.
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.
Another reaction to the paradox of (A) is to posit, as Graham Priest has, 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 is what it says, but then it cannot be true, creating another paradox.
This statement is probably a lie. (D)
If the sentence is probably a lie, then it probably not probably a lie, whereas if it is probably the truth, it probably isn't probable.
2. 1. Non-paradoxes
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.
3. Possible resolutions
3. 1. Alfred Tarski
Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed" by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, 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.
3. 2. 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 equals 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.
3. 3. Saul Kripke
Saul Kripke points out that whether a sentence is paradoxical or not can depend upon contingent facts. If the only thing Smith says about Jones is
A majority of what Jones says about me is false.
and 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.
and Smith really is a big spender but 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, that statement is "grounded". If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.
3. 4. 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.
3. 5. 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, which 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.
3. 6. Four-state logic
The manual for the computer language INTERCAL refers to "Warmenhovian" logic gates which have four states, "low, high, undefined (value of an uninitialized flip-flop), and oscillating (output of a NOR gate with one input low and the other input connected to the output)". The NOR gate, with one input low, is effectively a NOT gate, which negates its input ("the input is false"). By linking its output to its input (the input is assigned the value of this statement, hence "this statement is false") the paradox is induced. The electronic effect of this is oscillation, hence the name.
In this case the value of the liar paradox is "oscillating"; "low" and "high" correspond to the more usual values of "false" and "true".
In terms of the "undefined" value, many people will really claim not to be liars, but some of them will be lying; while this is not a paradox, no truth value can be assigned to the statement "I am not a liar" without further information, and, like paradoxes, this is an important class of statement. This "solution" can be seen to beg the question; higher orders of paradox can be generated which are ill-defined in this system.
4. Logical structure of the liar paradox
For a better understanding of the liar paradox, it is useful to write it down in a more formal way. If "this statement is false" is denoted by A and its truth value is being sought, it is necessary to find a condition that restricts the choice of possible truth values of A. Because A is self-referential it is possible to give the condition by an equation.
If some statement, B, is assumed to be false, one writes B = false. The statement (C) that the statement B is false would be written as C = "B = false". Now, the liar paradox can be expressed as the statement A, that A is false:
A = "A = false"
This is an equation from which the truth value of A = "this statement is false" could hopefully be obtained. In the boolean domain "A = false" is equivalent to not A and therefore the equation is not solvable. This is the motivation for reinterpretation of A. The simplest logical approach to make the equation solvable is the dialetheistic approach, in which case the solution is a A being both "true" and "false". Other resolutions mostly include some modifications of the equation e.g. A. N. Prior claims that the equation should be A = "A = false" and "A = true" and therefore A is false.
5. See also
* Quine's paradox
* List of paradoxes
6. Notes
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
7. References
* Jon Barwise 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 Univ. Press, ISBN 0-521-28864-9. Buridan's detailed solution to a number of such paradoxes.
* Kirkham, Richard (1992) Theories of Truth. MIT Press. Especially chapter 9.
* Saul Kripke (1975) "An Outline of a Theory of Truth," Journal of Philosophy 72: 690-716.
* Lefebvre, Neil, and Schelein, Melissa (2005) "The Liar Lied," Philosophy Now issue 51.
* Graham Priest (1984) "The Logic of Paradox Revisited," Journal of Philosophical Logic 13: 153-179.
* A. N. Prior (1976) Papers in Logic and Ethics. Duckworth.
* Smullyan, Raymond (19nn) What is the Name of this Book?. ISBN 0-671-62832-1. A collection of logic puzzles exploring this theme.
8. External links
* Internet Encyclopedia of Philosophy: "Liar Paradox" -- by Bradley Dowden.
Intentionally blank page paradox
Intentionally blank page: Many documents contain pages on which the text "This page is intentionally left blank" is printed, thereby making the page not blank.
Grelling-Nelson paradox
The Grelling-Nelson paradox is a semantic self-referential paradox formulated in 1908 by Kurt Grelling and Leonard Nelson and sometimes mistakenly attributed to the German philosopher and mathematician Hermann Weyl. It is thus occasionally called Weyl's paradox as well as Grelling's paradox. It is closely analogous to several other well-known paradoxes, in particular the Barber paradox and Russell's paradox.
Contents:
1. The Paradox
2. Similarities with Russell's paradox
3. See also
4. External links
1. The Paradox
Suppose one interprets the adjectives "autological" and "heterological" as follows:
1. An adjective is autological (sometimes homological) if and only if it describes itself. For example "short" is autological, since the word "short" is short. "English," "unhyphenated" and "pentasyllabic" are also autological.
2. An adjective is heterological if and only if it does not describe itself. Hence "long" is a heterological word, as are "abbreviated" and "monosyllabic."
All adjectives, it would seem, must be either autological or heterological, for each adjective either describes itself, or it doesn't. The Grelling-Nelson paradox arises when we consider the adjective "heterological". To test if the (imaginary) word "'foo" is autological one can ask: Is "foo" a foo word? If the answer is 'yes', "foo" is autological. If the answer is 'no', "foo" is heterological.
By comparison, one can ask: Is "heterological" a heterological word? If the answer is 'yes', "heterological" is autological (leading to a contradiction). If the answer is 'no', "heterological" is heterological (again leading to a contradiction).
But, then, we cannot say what is logical is not logical, and so on.
The paradox can be eliminated, without changing the meaning of "heterological" where it was previously well-defined, by modifying the definition of "heterological" slightly to hold of all nonautological words except "heterological." But "nonautological" is subject to the same paradox, for which this evasion is not applicable because the rules of English uniquely determine its meaning from that of "autological." A similar slight modification to the definition of "autological" (such as declaring it false of "nonautological" and its synonyms) might seem to fix that, but the paradox still obtains for synonyms of "autological" and "heterological" such as "selfdescriptive" and "nonselfdescriptive," whose meanings also would need adjusting, and the consequences of those adjustments would then need to be pursued, and so on. Freeing English of the Grelling-Nelson paradox entails considerably more modification to the language than mere refinements of the definitions of "autological" and "heterological," which need not even be in the language for the paradox to arise. The scope of these obstacles for English is comparable to that of Russell's paradox for mathematics founded on sets, argued as follows.
1. 1. Is "Autological" autological?
One may also ask if "autological" is autological. It can be chosen consistently to be either:
* if we say that "autological" is autological, and then ask if it applies to itself, then yes, it does, and thus is autological;
* if we say that "autological" is not autological, and then ask if it applies to itself, then no, it does not, and thus is not autological.
This is the opposite of the situation for heterological: while "heterological" logically cannot be autological or heterological, "autological" can be either. (It cannot be both, as the category of autological and heterological cannot overlap.)
In logical terms, the situation for "autological" is:
"autological" is autological if and only if "autological" is autological
A if and only if A, a tautology
while the situation for "heterological" is:
"heterological" is autological if and only if "heterological" is heterological
A if and only if not A, a contradiction.
2. Similarities with Russell's paradox
The Grelling-Nelson paradox can be translated into Bertrand Russell's famous paradox in the following way. First one must identify each adjective with the set of objects to which that adjective applies. So, for example, the adjective "red" is equated with the set of all red objects. In this way, the adjective "pronounceable" is equated with the set of all pronounceable things, one of which is the word "pronounceable" itself. Thus, an autological word is understood as a set, one of whose elements is the set itself. The question of whether the word "heterological" is heterological becomes the question of whether the set of all sets not containing themselves contains itself as an element.
3. See also
* List of autological words
* Metamagical Themas
4. External links
* Autological words
Contents:
1. The Paradox
2. Similarities with Russell's paradox
3. See also
4. External links
1. The Paradox
Suppose one interprets the adjectives "autological" and "heterological" as follows:
1. An adjective is autological (sometimes homological) if and only if it describes itself. For example "short" is autological, since the word "short" is short. "English," "unhyphenated" and "pentasyllabic" are also autological.
2. An adjective is heterological if and only if it does not describe itself. Hence "long" is a heterological word, as are "abbreviated" and "monosyllabic."
All adjectives, it would seem, must be either autological or heterological, for each adjective either describes itself, or it doesn't. The Grelling-Nelson paradox arises when we consider the adjective "heterological". To test if the (imaginary) word "'foo" is autological one can ask: Is "foo" a foo word? If the answer is 'yes', "foo" is autological. If the answer is 'no', "foo" is heterological.
By comparison, one can ask: Is "heterological" a heterological word? If the answer is 'yes', "heterological" is autological (leading to a contradiction). If the answer is 'no', "heterological" is heterological (again leading to a contradiction).
But, then, we cannot say what is logical is not logical, and so on.
The paradox can be eliminated, without changing the meaning of "heterological" where it was previously well-defined, by modifying the definition of "heterological" slightly to hold of all nonautological words except "heterological." But "nonautological" is subject to the same paradox, for which this evasion is not applicable because the rules of English uniquely determine its meaning from that of "autological." A similar slight modification to the definition of "autological" (such as declaring it false of "nonautological" and its synonyms) might seem to fix that, but the paradox still obtains for synonyms of "autological" and "heterological" such as "selfdescriptive" and "nonselfdescriptive," whose meanings also would need adjusting, and the consequences of those adjustments would then need to be pursued, and so on. Freeing English of the Grelling-Nelson paradox entails considerably more modification to the language than mere refinements of the definitions of "autological" and "heterological," which need not even be in the language for the paradox to arise. The scope of these obstacles for English is comparable to that of Russell's paradox for mathematics founded on sets, argued as follows.
1. 1. Is "Autological" autological?
One may also ask if "autological" is autological. It can be chosen consistently to be either:
* if we say that "autological" is autological, and then ask if it applies to itself, then yes, it does, and thus is autological;
* if we say that "autological" is not autological, and then ask if it applies to itself, then no, it does not, and thus is not autological.
This is the opposite of the situation for heterological: while "heterological" logically cannot be autological or heterological, "autological" can be either. (It cannot be both, as the category of autological and heterological cannot overlap.)
In logical terms, the situation for "autological" is:
"autological" is autological if and only if "autological" is autological
A if and only if A, a tautology
while the situation for "heterological" is:
"heterological" is autological if and only if "heterological" is heterological
A if and only if not A, a contradiction.
2. Similarities with Russell's paradox
The Grelling-Nelson paradox can be translated into Bertrand Russell's famous paradox in the following way. First one must identify each adjective with the set of objects to which that adjective applies. So, for example, the adjective "red" is equated with the set of all red objects. In this way, the adjective "pronounceable" is equated with the set of all pronounceable things, one of which is the word "pronounceable" itself. Thus, an autological word is understood as a set, one of whose elements is the set itself. The question of whether the word "heterological" is heterological becomes the question of whether the set of all sets not containing themselves contains itself as an element.
3. See also
* List of autological words
* Metamagical Themas
4. External links
* Autological words
Exception paradox
Exception paradox: if every rule has an exception, then there must be an exception to the rule that every rule has an exception.
From the logical point of view, this can be taken as a proof that the sentence "every rule has an exception" is false - a simple example of a proof technique known as reductio ad absurdum. However, if the rule to which there was no exception was the rule that all rules have an exception, then the sentence 'all rules have an exception' could remain true without paradox.
The Exception paradox is quite similar to the liar paradox, in which either answer creates an infinite loop of questions which again leads back to the beginning of each initial question being asked (or answered).
A similar idea is this: If everything is possible, then it is possible for anything to be impossible. Also: The only rule is that there are no rules.
From the logical point of view, this can be taken as a proof that the sentence "every rule has an exception" is false - a simple example of a proof technique known as reductio ad absurdum. However, if the rule to which there was no exception was the rule that all rules have an exception, then the sentence 'all rules have an exception' could remain true without paradox.
The Exception paradox is quite similar to the liar paradox, in which either answer creates an infinite loop of questions which again leads back to the beginning of each initial question being asked (or answered).
A similar idea is this: If everything is possible, then it is possible for anything to be impossible. Also: The only rule is that there are no rules.
Epimenides Paradox
The Epimenides paradox is a problem in logic. It is named after the Cretan philosopher Epimenides of Knossos (alive circa 600 BC), who stated Κρῆτες ἀεί ψεύσται (Kretes aei pseystai), "Cretans, always liars". There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach, by Douglas R. Hofstadter:
Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."
It is commonly supposed that self-referential paradox arises when one considers whether Epimenides spoke the truth. However, if Epimenides knew of one Cretan (other than himself) who is not a liar, his statement is a lie (because he asserts all) even though it correctly describes the speaker as a liar.
Contents:
1. History of the phrase
2. Logical analysis
3. References
1. History of the phrase
Epimenides was a philosopher and religious prophet who, against the general sentiment of Crete, proposed that Zeus was immortal, as in the following poem:
They fashioned a tomb for thee, O holy and high one The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being.
- Epimenides, Cretica
Denying the immortality of Zeus, then, is the lie of the Cretans. It appears that by "Cretans", Epimenides intended "Cretans other than myself". The phrase "Cretans, always liars" was quoted by the poet Callimachus in his Hymn to Zeus, with the same theological intent as Epimenides. The entire second line is quoted by the Apostle Paul in the Epistle to Titus.
One of Crete's own prophets has said it: 'Cretans are always liars, evil brutes, lazy gluttons'. He has surely told the truth.
- Epistle to Titus, 1:12
The logical inconsistency of a Cretan asserting all Cretans are always liars may not have occurred to Epimenides, nor to Callimachus. In the original context, Epimenides necessarily meant "Cretans other than myself", so there is no self-reference and thus no logical problem to speak of. The liar paradox was known in antiquity, but it was not associated with Epimenides and Saint Augustine restates the liar paradox, without mentioning Epimenides or Titus, in Against the Academicians (III.13.29). Many variations of the liar paradox (called insolubilia) were studied in the Middle Ages, but none of the extant medieval works on insolubilia refer to Epimenides, neither directly nor through the Epistle to Titus. The earliest appearance of Epimenides in the context of a logical problem dates only to the nineteenth century. Since that time, the Epimenides paradox has been commonly employed in discussions of logic.
2. Logical analysis
If one defines "liar" as someone who is never truthful, then the statement "All Cretans are liars," if uttered by a Cretan, Epimenides, leads inescapably to the conclusion that the speaker is a liar who knows that some Cretans are not liars.
Several interpretations and analyses are available, if the statement is considered false. It might be contended that the truth-value "false" can be consistently assigned to the simple proposition that "All Cretans are liars," so that this statement by itself, when deemed false, is not, strictly speaking, paradoxical. Thus, if there ever existed a Cretan (not Epimenides in this instance) who even once spoke the truth, the categorical statement "All Cretans are (always) liars," would be false, and Epimenides might be simply regarded as having made a false statement himself. But if Epimenides' statement is understood as in essence asserting its own falsehood, then the statement cannot consistently be false, either, because its falsehood would imply the truth of its self-asserted falsehood.
An interesting asymmetry is possible under one interpretation: the statement's truth clearly implies its falsehood, but, unless the statement is interpreted to refer specifically to itself (rather than referring categorically to all statements by Cretans), the statement could be contingently false without implying its own truth.
Paradoxical versions of the Epimenides problem are closely related to a class of more difficult logical problems, including the liar paradox, Russell's paradox, and the Burali-Forti paradox, all of which have self-reference in common with Epimenides. Indeed, the Epimenides paradox is usually classified as a variation on the liar paradox, and sometimes the two are not distinguished. The study of self-reference led to important developments in logic and mathematics in the twentieth century.
3. References
All of the works of Epimenides are now lost, and known only through quotations by other authors. The quotation from the Cretica of Epimenides is given by R.N. Longenecker, "Acts of the Apostles", in volume 9 of The Expositor's Bible Commentary, Frank E. Gaebelein, editor (Grand Rapids, Michigan: Zondervan Corporation, 1976-1984), page 476. Longenecker in turn cites M.D. Gibson, Horae Semiticae X (Cambridge: Cambridge University Press, 1913), page 40, "in Syriac". Longenecker states the following in a footnote:
The Syr. version of the quatrain comes to us from the Syr. church father Isho'dad of Mero (probably based on the work of Theodore of Mopsuestia), which J.R. Harris translated back into Gr. in Exp ["The Expositor"] 7 (1907), p 336.
An oblique reference to Epimenides in the context of logic appears in "The Logical Calculus" by W. E. Johnson, Mind (New Series), volume 1, number 2 (April, 1892), pages 235-250. Johnson writes in a footnote,
Compare, for example, such occasions for fallacy as are supplied by "Epimenides is a liar" or "That surface is red," which may be resolved into "All or some statements of Epimenides are false," "All or some of the surface is red."
The Epimenides paradox appears explicitly in "Mathematical Logic as Based on the Theory of Types", by Bertrand Russell, in the American Journal of Mathematics, volume 30, number 3 (July, 1908), pages 222-262, which opens with the following:
The oldest contradiction of the kind in question is the Epimenides. Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie?
In that article, Russell uses the Epimenides paradox as the point of departure for discussions of other problems, including the Burali-Forti paradox and the paradox now called Russell's paradox. Since Russell, the Epimenides paradox has been referenced repeatedly in logic. Typical of these references is Gödel, Escher, Bach by Douglas Hofstadter, which accords the paradox a prominent place in a discussion of self-reference.
Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."
It is commonly supposed that self-referential paradox arises when one considers whether Epimenides spoke the truth. However, if Epimenides knew of one Cretan (other than himself) who is not a liar, his statement is a lie (because he asserts all) even though it correctly describes the speaker as a liar.
Contents:
1. History of the phrase
2. Logical analysis
3. References
1. History of the phrase
Epimenides was a philosopher and religious prophet who, against the general sentiment of Crete, proposed that Zeus was immortal, as in the following poem:
They fashioned a tomb for thee, O holy and high one The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being.
- Epimenides, Cretica
Denying the immortality of Zeus, then, is the lie of the Cretans. It appears that by "Cretans", Epimenides intended "Cretans other than myself". The phrase "Cretans, always liars" was quoted by the poet Callimachus in his Hymn to Zeus, with the same theological intent as Epimenides. The entire second line is quoted by the Apostle Paul in the Epistle to Titus.
One of Crete's own prophets has said it: 'Cretans are always liars, evil brutes, lazy gluttons'. He has surely told the truth.
- Epistle to Titus, 1:12
The logical inconsistency of a Cretan asserting all Cretans are always liars may not have occurred to Epimenides, nor to Callimachus. In the original context, Epimenides necessarily meant "Cretans other than myself", so there is no self-reference and thus no logical problem to speak of. The liar paradox was known in antiquity, but it was not associated with Epimenides and Saint Augustine restates the liar paradox, without mentioning Epimenides or Titus, in Against the Academicians (III.13.29). Many variations of the liar paradox (called insolubilia) were studied in the Middle Ages, but none of the extant medieval works on insolubilia refer to Epimenides, neither directly nor through the Epistle to Titus. The earliest appearance of Epimenides in the context of a logical problem dates only to the nineteenth century. Since that time, the Epimenides paradox has been commonly employed in discussions of logic.
2. Logical analysis
If one defines "liar" as someone who is never truthful, then the statement "All Cretans are liars," if uttered by a Cretan, Epimenides, leads inescapably to the conclusion that the speaker is a liar who knows that some Cretans are not liars.
Several interpretations and analyses are available, if the statement is considered false. It might be contended that the truth-value "false" can be consistently assigned to the simple proposition that "All Cretans are liars," so that this statement by itself, when deemed false, is not, strictly speaking, paradoxical. Thus, if there ever existed a Cretan (not Epimenides in this instance) who even once spoke the truth, the categorical statement "All Cretans are (always) liars," would be false, and Epimenides might be simply regarded as having made a false statement himself. But if Epimenides' statement is understood as in essence asserting its own falsehood, then the statement cannot consistently be false, either, because its falsehood would imply the truth of its self-asserted falsehood.
An interesting asymmetry is possible under one interpretation: the statement's truth clearly implies its falsehood, but, unless the statement is interpreted to refer specifically to itself (rather than referring categorically to all statements by Cretans), the statement could be contingently false without implying its own truth.
Paradoxical versions of the Epimenides problem are closely related to a class of more difficult logical problems, including the liar paradox, Russell's paradox, and the Burali-Forti paradox, all of which have self-reference in common with Epimenides. Indeed, the Epimenides paradox is usually classified as a variation on the liar paradox, and sometimes the two are not distinguished. The study of self-reference led to important developments in logic and mathematics in the twentieth century.
3. References
All of the works of Epimenides are now lost, and known only through quotations by other authors. The quotation from the Cretica of Epimenides is given by R.N. Longenecker, "Acts of the Apostles", in volume 9 of The Expositor's Bible Commentary, Frank E. Gaebelein, editor (Grand Rapids, Michigan: Zondervan Corporation, 1976-1984), page 476. Longenecker in turn cites M.D. Gibson, Horae Semiticae X (Cambridge: Cambridge University Press, 1913), page 40, "in Syriac". Longenecker states the following in a footnote:
The Syr. version of the quatrain comes to us from the Syr. church father Isho'dad of Mero (probably based on the work of Theodore of Mopsuestia), which J.R. Harris translated back into Gr. in Exp ["The Expositor"] 7 (1907), p 336.
An oblique reference to Epimenides in the context of logic appears in "The Logical Calculus" by W. E. Johnson, Mind (New Series), volume 1, number 2 (April, 1892), pages 235-250. Johnson writes in a footnote,
Compare, for example, such occasions for fallacy as are supplied by "Epimenides is a liar" or "That surface is red," which may be resolved into "All or some statements of Epimenides are false," "All or some of the surface is red."
The Epimenides paradox appears explicitly in "Mathematical Logic as Based on the Theory of Types", by Bertrand Russell, in the American Journal of Mathematics, volume 30, number 3 (July, 1908), pages 222-262, which opens with the following:
The oldest contradiction of the kind in question is the Epimenides. Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie?
In that article, Russell uses the Epimenides paradox as the point of departure for discussions of other problems, including the Burali-Forti paradox and the paradox now called Russell's paradox. Since Russell, the Epimenides paradox has been referenced repeatedly in logic. Typical of these references is Gödel, Escher, Bach by Douglas Hofstadter, which accords the paradox a prominent place in a discussion of self-reference.
Curry's paradox
Curry's paradox is a paradox that occurs in naive set theory or naive logics, and allows the derivation of an arbitrary sentence from a self-referring sentence and some apparently innocuous logical deduction rules. It is named after the logician Haskell Curry.
It has also been called Löb's paradox after Martin Hugo Löb. [1]
Contents:
1. In natural language
2. In formal language
3. In naive set theory
4. Discussion
5. See also
6. References
7. External links
1. In natural language
Claims of the form "if A, then B" are called conditional claims. It is not necessary to believe the conclusion (B) to accept the conditional claim (if A, then B) as true. For instance, consider the following sentence:
If a man with flying reindeer has delivered presents to all the good children in the world in one night, then Santa Claus exists.
Imagine that a man with flying reindeer has, in fact, done this. Does Santa Claus exist, in that case? It would seem so. Therefore, without believing that Santa Claus exists, or that this scenario is even possible, it seems that we should agree that if a man with flying reindeer has delivered presents to all the good children in the world in one night, then Santa Claus exists, and so the above sentence is true.
Now consider this other sentence:
If this sentence is true, then Santa Claus exists.
As before, imagine that the antecedent is true - in this case, "this sentence is true". Does Santa Claus exist, in that case? Well, if the sentence is true, then what it says is true: namely that if the sentence is true, then Santa Claus exists. Therefore, without necessarily believing that Santa Claus exists, or that the sentence is true, it seems we should agree that if the sentence is true, then Santa Claus exists.
But then this means the sentence is true. So Santa Claus does exist. Furthermore we could substitute any claim at all for "Santa Claus exists". This is Curry's paradox.
2. In formal language
In formal languages, we sometimes interpret "If X then Y" as a material conditional. On this reading, it simply means "Y, or else not X". Here we would read the sentence as "Santa Claus exists, or this sentence is false". On this reading, Curry's paradox is simply a variant on the liar paradox. However, in natural language this is not usually what we mean by "If X then Y". For instance, "if 6*7=42, then the moon exists" is true as a material implication, but is generally not considered true in natural language, because the moon's existence does not seem to be related to this fact of arithmetic.
Nevertheless we arrived at paradox in natural language. In fact, not only did we arrive at a contradiction, but we were actually able to prove anything at all, without relying on such principles as the principle of explosion which are generally held to be false in accounts of natural language. Thus Curry's paradox poses an additional problem.
To arrive at this formally, let us denote by Y the proposition to prove, in this case "Santa Claus exists". Then, let X denote the statement that asserts that Y follows from the truth of X. Mathematically, this can be written as X = (X → Y), and we see that X is defined in terms of itself. The proof proceeds:
1. X → X
rule of assumption, also called restatement of premise or of hypothesis
2. X → (X → Y)
substitute right side of 1, since X = X → Y
3. X → Y
from 2 by contraction
4. X
substitute 3, since X = X → Y
5. Y
from 4 and 3 by modus ponens
3. In naive set theory
Even if the underlying mathematical logic does not admit any self-referential sentence, in set theories which allow unrestricted comprehension, we can nevertheless prove any logical statement Y from the set
The proof proceeds:
This can be seen as a variant on Russell's paradox, but is in an important way more general. Some proposals for set theory have attempted to deal with Russell's paradox not by restricting the rule of comprehension, but by restricting the rules of logic so that it tolerates the contradictory nature of the set of all sets that are not members of themselves. This reasoning shows that such a task is not so simple, because again, we have not only a contradiction, but we have in fact proved any statement whatsoever, without recourse to the full apparatus of the propositional calculus.
4. Discussion
Curry's paradox can be formulated in any language meeting certain conditions:
1. The language must contain an apparatus which lets it refer to, and talk about, its own sentences (such as quotation marks, names, or expressions like "this sentence");
2. The language must contain its own truth-predicate: that is, the language, call it "L", must contain a predicate meaning "true-in-L", and the ability to ascribe this predicate to any sentences;
3. The language must admit the rule of contraction, which roughly speaking means that a relevant hypothesis may be reused as many times as necessary; and
4. The language must of course admit the rules of identity (if A, then A) and modus ponens (from A, and if A then B, conclude B).
Various other sets of conditions are also possible. Natural languages nearly always contain all these features. Mathematical logic, on the other hand, generally does not countenance explicit reference to its own sentences, although the heart of Gödel's incompleteness theorems is the observation that usually this can be done anyway; see Gödel number. The truth-predicate is generally not available, but in naive set theory, this is arrived at through the unrestricted rule of comprehension. The rule of contraction is generally accepted, although linear logic (more precisely, linear logic without the exponential operators) does not admit the reasoning required for this paradox.
Note that unlike the liar paradox or Russell's paradox, this paradox does not depend on what model of negation is used, as it is completely negation-free. Thus paraconsistent logics can still be vulnerable to this, even if they are immune to the liar paradox.
The resolution of Curry's paradox is a contentious issue because resolutions (apart from trivial ones such as disallowing X directly) are difficult and not intuitive. Logicians are undecided whether such sentences are somehow impermissible (and if so, how to banish them), or meaningless, or whether they are correct and reveal problems with the concept of truth itself (and if so, whether we should reject the concept of truth, or change it), or whether they can be rendered benign by a suitable account of their meanings.
Linear logic disallows contraction and so does not admit this paradox directly, but one must remove its exponential operators, or else the paradox reappears in a modal form.
5. See also
* Richard's paradox
* Kleene-Rosser paradox
6. References
1. Barwise, Jon and John Etchemendy 1987: The Liar, p. 23. Oxford University Press.
7. External links
* The Stanford Encyclopedia of Philosophy: "Curry's Paradox" -- by J. C. Beall.
* Grossman, Jason, Australian National University: A Proof that Penguins Rule the Universe. A brief and entertaining discussion of Curry's paradox.
"[1]" "[2]" Relevant First-Order Logic LP# and Curry's Paradox http://front.math.ucdavis.edu/0804.4818
It has also been called Löb's paradox after Martin Hugo Löb. [1]
Contents:
1. In natural language
2. In formal language
3. In naive set theory
4. Discussion
5. See also
6. References
7. External links
1. In natural language
Claims of the form "if A, then B" are called conditional claims. It is not necessary to believe the conclusion (B) to accept the conditional claim (if A, then B) as true. For instance, consider the following sentence:
If a man with flying reindeer has delivered presents to all the good children in the world in one night, then Santa Claus exists.
Imagine that a man with flying reindeer has, in fact, done this. Does Santa Claus exist, in that case? It would seem so. Therefore, without believing that Santa Claus exists, or that this scenario is even possible, it seems that we should agree that if a man with flying reindeer has delivered presents to all the good children in the world in one night, then Santa Claus exists, and so the above sentence is true.
Now consider this other sentence:
If this sentence is true, then Santa Claus exists.
As before, imagine that the antecedent is true - in this case, "this sentence is true". Does Santa Claus exist, in that case? Well, if the sentence is true, then what it says is true: namely that if the sentence is true, then Santa Claus exists. Therefore, without necessarily believing that Santa Claus exists, or that the sentence is true, it seems we should agree that if the sentence is true, then Santa Claus exists.
But then this means the sentence is true. So Santa Claus does exist. Furthermore we could substitute any claim at all for "Santa Claus exists". This is Curry's paradox.
2. In formal language
In formal languages, we sometimes interpret "If X then Y" as a material conditional. On this reading, it simply means "Y, or else not X". Here we would read the sentence as "Santa Claus exists, or this sentence is false". On this reading, Curry's paradox is simply a variant on the liar paradox. However, in natural language this is not usually what we mean by "If X then Y". For instance, "if 6*7=42, then the moon exists" is true as a material implication, but is generally not considered true in natural language, because the moon's existence does not seem to be related to this fact of arithmetic.
Nevertheless we arrived at paradox in natural language. In fact, not only did we arrive at a contradiction, but we were actually able to prove anything at all, without relying on such principles as the principle of explosion which are generally held to be false in accounts of natural language. Thus Curry's paradox poses an additional problem.
To arrive at this formally, let us denote by Y the proposition to prove, in this case "Santa Claus exists". Then, let X denote the statement that asserts that Y follows from the truth of X. Mathematically, this can be written as X = (X → Y), and we see that X is defined in terms of itself. The proof proceeds:
1. X → X
rule of assumption, also called restatement of premise or of hypothesis
2. X → (X → Y)
substitute right side of 1, since X = X → Y
3. X → Y
from 2 by contraction
4. X
substitute 3, since X = X → Y
5. Y
from 4 and 3 by modus ponens
3. In naive set theory
Even if the underlying mathematical logic does not admit any self-referential sentence, in set theories which allow unrestricted comprehension, we can nevertheless prove any logical statement Y from the set
The proof proceeds:
This can be seen as a variant on Russell's paradox, but is in an important way more general. Some proposals for set theory have attempted to deal with Russell's paradox not by restricting the rule of comprehension, but by restricting the rules of logic so that it tolerates the contradictory nature of the set of all sets that are not members of themselves. This reasoning shows that such a task is not so simple, because again, we have not only a contradiction, but we have in fact proved any statement whatsoever, without recourse to the full apparatus of the propositional calculus.
4. Discussion
Curry's paradox can be formulated in any language meeting certain conditions:
1. The language must contain an apparatus which lets it refer to, and talk about, its own sentences (such as quotation marks, names, or expressions like "this sentence");
2. The language must contain its own truth-predicate: that is, the language, call it "L", must contain a predicate meaning "true-in-L", and the ability to ascribe this predicate to any sentences;
3. The language must admit the rule of contraction, which roughly speaking means that a relevant hypothesis may be reused as many times as necessary; and
4. The language must of course admit the rules of identity (if A, then A) and modus ponens (from A, and if A then B, conclude B).
Various other sets of conditions are also possible. Natural languages nearly always contain all these features. Mathematical logic, on the other hand, generally does not countenance explicit reference to its own sentences, although the heart of Gödel's incompleteness theorems is the observation that usually this can be done anyway; see Gödel number. The truth-predicate is generally not available, but in naive set theory, this is arrived at through the unrestricted rule of comprehension. The rule of contraction is generally accepted, although linear logic (more precisely, linear logic without the exponential operators) does not admit the reasoning required for this paradox.
Note that unlike the liar paradox or Russell's paradox, this paradox does not depend on what model of negation is used, as it is completely negation-free. Thus paraconsistent logics can still be vulnerable to this, even if they are immune to the liar paradox.
The resolution of Curry's paradox is a contentious issue because resolutions (apart from trivial ones such as disallowing X directly) are difficult and not intuitive. Logicians are undecided whether such sentences are somehow impermissible (and if so, how to banish them), or meaningless, or whether they are correct and reveal problems with the concept of truth itself (and if so, whether we should reject the concept of truth, or change it), or whether they can be rendered benign by a suitable account of their meanings.
Linear logic disallows contraction and so does not admit this paradox directly, but one must remove its exponential operators, or else the paradox reappears in a modal form.
5. See also
* Richard's paradox
* Kleene-Rosser paradox
6. References
1. Barwise, Jon and John Etchemendy 1987: The Liar, p. 23. Oxford University Press.
7. External links
* The Stanford Encyclopedia of Philosophy: "Curry's Paradox" -- by J. C. Beall.
* Grossman, Jason, Australian National University: A Proof that Penguins Rule the Universe. A brief and entertaining discussion of Curry's paradox.
"[1]" "[2]" Relevant First-Order Logic LP# and Curry's Paradox http://front.math.ucdavis.edu/0804.4818
Berry paradox
The Berry paradox is a self-referential paradox arising from the expression "the smallest possible integer not definable by a given number of words." Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G. Berry, a librarian at Oxford's Bodleian library, who had suggested the more limited paradox arising from the expression "the first undefinable ordinal".
Contents:
1. The paradox
2. Resolution
3. Relationship with Kolmogorov complexity
4. Notes
5. See also
6. References
7. External links
1. The paradox
Consider the expression:
"The smallest positive integer not definable in under eleven words."
Since there are finitely many words, there are finitely many phrases of under eleven words, and hence finitely many positive integers that are defined by phrases of under eleven words by the pigeonhole principle. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words — that is, positive integers satisfying the property "not definable in under eleven words". By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under eleven words". This is the integer to which the above expression refers; that is, this integer is defined by the above expression. Note that the above expression is only ten words long; so, this integer is defined by an expression that is under eleven words long; so it is definable in under eleven words, and is not the smallest positive integer not definable in under eleven words, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is, clearly, definable in under eleven words), there cannot be any integer defined by it.
2. Resolution
The Berry paradox as formulated above arises because of systematic ambiguity in the word "definable." In other formulations of the Berry paradox, such as one that instead reads: "...not nameable in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious-circle fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal. [1]
One of the ways it is proposed that this family of paradoxes be resolved is by incorporating stratifications of meaning in language. Terms with systematic ambiguity may be written with subscripts denoting that one level of meaning is considered a higher priority than another in their interpretation. The number not nameable0 in less than eleven syllables may be nameable1 in less than eleven syllables under this scheme. [2]
The argument presented above that "Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words" assumes that "there must be an integer defined by this expression" which is counterfactual as most phrases "under eleven words" are ambiguous to their defining of an integer, with this ten word paradox being an example. Assuming one can match word phrases to numbers is a mistaken assumption. [3]
It is generally accepted that the Berry paradox results from interpreting sets of possibly self-referential expressions: it and similar paradoxes embody so-called "vicious-circle" fallacies. To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them.
Using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, as has been done by Gregory Chaitin. Though the formal analogue does not lead to a logical contradiction, it does prove certain impossibility results, including an incompleteness theorem similar in spirit to Gödel's incompleteness theorem.
George Boolos (1989) built on a formalized version of Berry's paradox to prove Gödel's Incompleteness Theorem in a new and much simpler way. The basic idea of his proof is that a proposition that holds of x if x = n for some natural number n can be called a definition for n, and that the set {(n, k): n has a definition that is k symbols long} can be shown to be representable (using Gödel numbers). Then the proposition "m is the first number not definable in less than k symbols" can be formalized and shown to be a definition in the sense just stated.
3. Relationship with Kolmogorov complexity
Main article: Kolmogorov complexity
It is possible to unambiguously define what is the minimal number of symbols required to describe a given string. In this context, the terms string and number may be used interchangeably, since a number is actually a string of symbols, i.e. an English word (like the word "eleven" used in the paradox) while, on the other hand, it is possible to refer to any word with a number, e.g. by the number of its position in a given dictionary, or by suitable encoding. Some long strings can be described exactly using fewer symbols than those required by their full representation, as is often experienced using data compression. The complexity of a given string is then defined as the minimal length that a description requires in order to (unambiguously) refer to the full representation of that string.
The Kolmogorov complexity is defined using formal languages, or Turing machines, that allow to avoid ambiguities about what string results from a given description. After defining that function, it can be proved that it cannot be computed. The proof by contradiction shows that if it were possible to compute the Kolmogorov complexity, then it would also be possible to systematically generate paradoxes similar to this one, i.e. descriptions shorter than what the complexity of the described string implies. That is to say, the definition of the Berry number is paradoxical because it is not actually possible to compute how many words are required to define a number, and we know that such computation is not feasible because of the paradox.
4. Notes
1. Russell and Whitehead (1927).
2. Willard Quine (1976) Ways of Paradox. Harvard Univ. Press
3. French (1988) demonstrated that an infinite number of numbers could be uniquely described in the exact same words.
5. See also
* Definable number
* Busy beaver
* Richard's paradox
* Interesting number paradox
6. References
* Charles H. Bennett (1979) "On Random and Hard-to-Describe Numbers." IBM Report RC7483.
* George Boolos (1989) "A new proof of the Gödel Incompleteness Theorem," Notices of the American Mathematical Society 36: 388-90; 676. Reprinted in his (1998) Logic, Logic, and Logic. Harvard Univ. Press: 383-88.
* Gregory Chaitin (1995) "The Berry Paradox,." Complexity 1: 26-30.
* French, James D. (1988) "The False Assumption Underlying Berry's Paradox," Journal of Symbolic Logic 53: 1220-1223.
* Bertrand Russell (19nn) "Les paradoxes de la logique," Revue de métaphysique et de morale 14: 627-650
* ------ and Alfred N. Whitehead (1927) Principia Mathematica. Cambridge University Press. 1962 partial paperback reissue goes up to *56.
7. External links
* Roosen-Runge, Peter H. (1997) "Berry's Paradox."
Eric W. Weisstein, Berry Paradox at MathWorld.
* Weisstein, Eric W. "Berry paradox," Wolfram Research's MathWorld
Contents:
1. The paradox
2. Resolution
3. Relationship with Kolmogorov complexity
4. Notes
5. See also
6. References
7. External links
1. The paradox
Consider the expression:
"The smallest positive integer not definable in under eleven words."
Since there are finitely many words, there are finitely many phrases of under eleven words, and hence finitely many positive integers that are defined by phrases of under eleven words by the pigeonhole principle. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words — that is, positive integers satisfying the property "not definable in under eleven words". By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under eleven words". This is the integer to which the above expression refers; that is, this integer is defined by the above expression. Note that the above expression is only ten words long; so, this integer is defined by an expression that is under eleven words long; so it is definable in under eleven words, and is not the smallest positive integer not definable in under eleven words, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is, clearly, definable in under eleven words), there cannot be any integer defined by it.
2. Resolution
The Berry paradox as formulated above arises because of systematic ambiguity in the word "definable." In other formulations of the Berry paradox, such as one that instead reads: "...not nameable in less..." the term "nameable" is also one that has this systematic ambiguity. Terms of this kind give rise to vicious-circle fallacies. Other terms with this type of ambiguity are: satisfiable, true, false, function, property, class, relation, cardinal, and ordinal. [1]
One of the ways it is proposed that this family of paradoxes be resolved is by incorporating stratifications of meaning in language. Terms with systematic ambiguity may be written with subscripts denoting that one level of meaning is considered a higher priority than another in their interpretation. The number not nameable0 in less than eleven syllables may be nameable1 in less than eleven syllables under this scheme. [2]
The argument presented above that "Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words" assumes that "there must be an integer defined by this expression" which is counterfactual as most phrases "under eleven words" are ambiguous to their defining of an integer, with this ten word paradox being an example. Assuming one can match word phrases to numbers is a mistaken assumption. [3]
It is generally accepted that the Berry paradox results from interpreting sets of possibly self-referential expressions: it and similar paradoxes embody so-called "vicious-circle" fallacies. To resolve one of these paradoxes means to pinpoint exactly where our use of language went wrong and to provide restrictions on the use of language which may avoid them.
Using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, as has been done by Gregory Chaitin. Though the formal analogue does not lead to a logical contradiction, it does prove certain impossibility results, including an incompleteness theorem similar in spirit to Gödel's incompleteness theorem.
George Boolos (1989) built on a formalized version of Berry's paradox to prove Gödel's Incompleteness Theorem in a new and much simpler way. The basic idea of his proof is that a proposition that holds of x if x = n for some natural number n can be called a definition for n, and that the set {(n, k): n has a definition that is k symbols long} can be shown to be representable (using Gödel numbers). Then the proposition "m is the first number not definable in less than k symbols" can be formalized and shown to be a definition in the sense just stated.
3. Relationship with Kolmogorov complexity
Main article: Kolmogorov complexity
It is possible to unambiguously define what is the minimal number of symbols required to describe a given string. In this context, the terms string and number may be used interchangeably, since a number is actually a string of symbols, i.e. an English word (like the word "eleven" used in the paradox) while, on the other hand, it is possible to refer to any word with a number, e.g. by the number of its position in a given dictionary, or by suitable encoding. Some long strings can be described exactly using fewer symbols than those required by their full representation, as is often experienced using data compression. The complexity of a given string is then defined as the minimal length that a description requires in order to (unambiguously) refer to the full representation of that string.
The Kolmogorov complexity is defined using formal languages, or Turing machines, that allow to avoid ambiguities about what string results from a given description. After defining that function, it can be proved that it cannot be computed. The proof by contradiction shows that if it were possible to compute the Kolmogorov complexity, then it would also be possible to systematically generate paradoxes similar to this one, i.e. descriptions shorter than what the complexity of the described string implies. That is to say, the definition of the Berry number is paradoxical because it is not actually possible to compute how many words are required to define a number, and we know that such computation is not feasible because of the paradox.
4. Notes
1. Russell and Whitehead (1927).
2. Willard Quine (1976) Ways of Paradox. Harvard Univ. Press
3. French (1988) demonstrated that an infinite number of numbers could be uniquely described in the exact same words.
5. See also
* Definable number
* Busy beaver
* Richard's paradox
* Interesting number paradox
6. References
* Charles H. Bennett (1979) "On Random and Hard-to-Describe Numbers." IBM Report RC7483.
* George Boolos (1989) "A new proof of the Gödel Incompleteness Theorem," Notices of the American Mathematical Society 36: 388-90; 676. Reprinted in his (1998) Logic, Logic, and Logic. Harvard Univ. Press: 383-88.
* Gregory Chaitin (1995) "The Berry Paradox,." Complexity 1: 26-30.
* French, James D. (1988) "The False Assumption Underlying Berry's Paradox," Journal of Symbolic Logic 53: 1220-1223.
* Bertrand Russell (19nn) "Les paradoxes de la logique," Revue de métaphysique et de morale 14: 627-650
* ------ and Alfred N. Whitehead (1927) Principia Mathematica. Cambridge University Press. 1962 partial paperback reissue goes up to *56.
7. External links
* Roosen-Runge, Peter H. (1997) "Berry's Paradox."
Eric W. Weisstein, Berry Paradox at MathWorld.
* Weisstein, Eric W. "Berry paradox," Wolfram Research's MathWorld
Subscribe to:
Posts (Atom)
