645:"everything I am saying is false") Bhartrhari identifies a hidden parameter that can change unproblematic situations in daily communication into a stubborn paradox. Bhartrhari's solution can be understood in terms of the solution proposed in 1992 by Julian Roberts: "Paradoxes consume themselves. But we can keep apart the warring sides of the contradiction by the simple expedient of temporal contextualisation: what is 'true' with respect to one point in time need not be so in another ... The overall force of the 'Austinian' argument is not merely that 'things change', but that rationality is essentially temporal in that we need time in order to reconcile and manage what would otherwise be mutually destructive states." According to Robert's suggestion, it is the factor "time" which allows us to reconcile the separated "parts of the world" that play a crucial role in the solution of Barwise and Etchemendy. The capacity of time to prevent a direct confrontation of the two "parts of the world" is here external to the "liar". In the light of Bhartrhari's analysis, however, the extension in time that separates two perspectives on the world or two "parts of the world" â the part before and the part after the function accomplishes its task â is inherent in any "function": also the function to signify which underlies each statement, including the "liar". The unsolvable paradox â a situation in which we have either contradiction (
738:. Then the theory at hand, which is assumed to prove certain facts about numbers, also proves facts about its own statements. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence states that no natural number exists with a certain, strange property. A number with this property would encode a proof of the inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to the consistency hypothesis. So, under the assumption that the theory is consistent, there is no such number.
480:
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.
815:. However, lacking the intelligence to realize the statement is a paradox, he simply responds, "Um, true. I'll go with true. There, that was easy." and is unaffected. Humorously, all other AIs present barring GLaDOS, all of which are significantly less sentient and lucid than both her and Wheatley, are still killed from hearing the paradox. However, GLaDOS later notes that she almost killed herself from her own attempt to kill Wheatley.
514:) 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 ...".
2387:
2377:
731:, Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable", called the "Gödel sentence G". His proof showed that for any sufficiently powerful theory T, G is true, but not provable in T. The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.
695:"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 being both "true" and "false". Other resolutions mostly include some modifications of the equation;
491:+1 which asserts that the first statement is false." This is a true, meaningful statement about the hierarchy that Tarski defines, but it refers to statements at every level of the hierarchy, so it must be above every level of the hierarchy, and is therefore not possible within the hierarchy (although bounded versions of the sentence are possible).
641:(late fifth century AD) dealt with paradoxes such as the liar in a section of one of the chapters of his magnum opus the VÄkyapadÄ«ya. Bhartrhari's solution fits into his general approach to language, thought and reality, which has been characterized by some as "relativistic", "non-committal" or "perspectivistic". With regard to the liar paradox (
838:, titled "Access Denied", the main character Jesse and their friends are captured by a supercomputer named PAMA. After PAMA controls two of Jesse's friends, Jesse learns that PAMA stalls when processing and uses a paradox to confuse him and escape with their last friend. One of the paradoxes the player can make Jesse say is the liar paradox.
126:(late fifth century AD) was well aware of a liar paradox which he formulated as "everything I am saying is false" (sarvam mithyÄ bravÄ«mi). He analyzes this statement together with the paradox of "unsignifiability" and explores the boundary between statements that are unproblematic in daily life and paradoxes.
798:, the Doctor temporarily stumps the insane computer BOSS by asking it "If I were to tell you that the next thing I say would be true, but that the last thing I said was a lie, would you believe me?" BOSS tries to figure it out but cannot and eventually decides the question is irrelevant and summons security.
579:
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
79:, reportedly stated that "All Cretans are liars." However, Epimenides' statement that all Cretans are liars can be resolved as false, given that he knows of at least one other Cretan who does not lie (alternatively, it can be taken as merely a statement that all Cretans tell lies, not that they tell
479:
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
299:
If F is assumed to bear a truth value, then it presents the problem of determining the object of that value. But, a simpler version is possible, by assuming that the single word 'true' bears a truth value. The analogue to the paradox is to assume that the single word 'false' likewise bears a truth
185:
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, the rejection of the claim that every statement has to be either true or false, also known as the
1421:
recognized that
Bhartrhari "wants to rise at once above all controversies by showing the conditions of possibility of any system of interpretation, rather than to prove the truth of a certain particular system" (Théorie de la connaissance et philosophie de la parole dans le brahmanisme classique,
117:
telling the truth or is he lying? If it is true that every man is a liar, and David's statement, "Every man is a liar" is true, then David also is lying; he, too, is a man. But if he, too, is lying, his statement that "Every man is a liar", consequently is not true. Whatever way you turn the
563:
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
274:
The multi-sentence version of the liar paradox generalizes to any circular sequence of such statements (wherein the last statement asserts the truth/falsity of the first statement), provided there are an odd number of statements asserting the falsity of their successor; the following is a
673:
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
300:
value, namely that it is false. This reveals that the paradox can be reduced to the mental act of assuming that the very idea of fallacy bears a truth value, namely that the very idea of fallacy is false: an act of misrepresentation. So, the symmetrical version of the paradox would be:
44:
is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary
327:, where the truth values may only be the integer values 0 or 1. In this system, the statement "This statement is false" is no longer paradoxical as it can be assigned a truth value of 0.5, making it precisely half true and half false. A simplified explanation is shown below.
628:
has argued in favour of a non-cognitivist solution to the paradox, suggesting that some apparently well-formed sentences will turn out to be neither true nor false and that "formal criteria alone will inevitably prove insufficient" for resolving the paradox.
603:. Dialetheism is the view that there are true contradictions. 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
936:
St. Jerome, Homily on Psalm 115 (116B), translated by Sr. Marie
Liguori Ewald, IHM, in The Homilies of Saint Jerome, Volume I (1-59 On the Psalms), The Fathers of the Church 48 (Washington, D.C.: The Catholic University of America Press, 1964),
181:
If (A) is false, then "This statement is false" is false. Therefore, (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.
526:
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 presents a similar answer.
39:
is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the
741:
It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as
703:" and therefore A is false. In computational verb logic, the liar paradox is extended to statements like, "I hear what he says; he says what I don't hear", where verb logic must be used to resolve the paradox.
851:
Robert Earl Keen's song "The Road Goes On and On" alludes to the paradox. The song is widely believed to be written as part of Keen's feud with Toby Keith, who is presumably the "liar" Keen refers to.
495:
is credited with identifying this incompleteness in Tarski's hierarchy in his highly cited paper "Outline of a theory of truth," and it is recognized as a general problem in hierarchical languages.
118:
proposition, the conclusion is a contradiction. Since David himself is a man, it follows that he also is lying; but if he is lying because every man is a liar, his lying is of a different sort.
271:
Assume (D1) is false. Then (D2) is false. This would mean that (D1) is true. Thus (D1) is both true and false. Either way, (D1) is both true and false â the same paradox as (A) above.
599:
and other logicians, including J. C. Beall and
Bradley Armour-Garb, have proposed that the liar sentence should be considered to be both true and false, a point of view known as
178:
If (A) is true, then "This statement is false" is true. Therefore, (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.
290:
Assume (E1) is false. Then (E2) is true, which means (E3) is false, and hence (E1) is true. Either way, (E1) is both true and false â the same paradox as with (A) and (D1).
916:
388:
466:
434:
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611:
propositions are true. Since trivialism is an intuitively false view, dialetheists nearly always reject the explosion principle. Logics that reject it are called
584:, that the "denial liar" can be true without contradiction while the "negation liar" can be false without contradiction. Their 1987 book makes heavy use of
768:
The liar paradox is occasionally used in fiction to shut down artificial intelligences, who are presented as being unable to process the sentence. In the
681:
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
1823:
293:
There are many other variants, and many complements, possible. In normal sentence construction, the simplest version of the complement is the sentence:
56:
If "this sentence is false" is true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on.
134:
130:
1971:
1568:
John
Buridan on Self-Reference : Chapter Eight of Buridan's Sophismata, with a Translation, and Introduction, and a Philosophical Commentary
607:, which asserts that any proposition can be deduced from a contradiction, unless the dialetheist is willing to accept trivialism â the view that
344:
1499:
1277:
1202:. Seventy-Second Annual Meeting American Philosophical Association, Eastern Division. Vol. 72. Journal of Philosophy. pp. 690â716.
716:
2411:
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has, that the statement is both true and false. Nevertheless, even Priest's analysis is susceptible to the following version of the liar:
1168:
2421:
1714:
1684:
1665:
1393:
2032:
743:
2001:
1838:
1644:
1575:
1379:
1307:
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538:
argued that whether a sentence is paradoxical or not can depend upon contingent facts. If the only thing Smith says about Jones is
1024:
2257:
161:. Sentences can be constructed that cannot consistently be assigned a truth value even though they are completely in accord with
197:
The proposal that the statement is neither true nor false has given rise to the following, strengthened version of the paradox:
848:" alluded to the paradox when the narrator ends the song by stating "I'll lie again and again and I'll keep lying, I promise".
912:
770:
133:
for at least five centuries, starting from late 9th century, and apparently without being influenced by any other tradition.
98:, who lived in the 4th century BC. Eubulides reportedly asked, "A man says that he is lying. Is what he says true or false?"
68:(c. 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The semi-mythical
2250:
723:
which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by
2074:
1534:
728:
287:
Assume (E1) is true. Then (E2) is false, which means (E3) is true, and hence (E1) is false, leading to a contradiction.
1904:
860:
483:
However, this system is incomplete. One would like to be able to make statements such as "For every statement in level
585:
691:
This is an equation from which the truth value of A = "this statement is false" could hopefully be obtained. In the
268:
Assume (D1) is true. Then (D2) is true. This would mean that (D1) is false. Therefore, (D1) is both true and false.
2026:
1966:
1025:"The Early Arabic Liar:The Liar Paradox in the Islamic World from the Mid-Ninth to the Mid-Thirteenth Centuries CE"
875:
191:
2094:
1909:
2069:
746:, was discovered independently by Gödel (when he was working on the proof of the incompleteness theorem) and by
2189:
845:
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1951:
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507:
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soft on crime, then both Smith's remark about Jones and Jones's last remark about Smith are paradoxical.
2235:
2179:
2064:
1899:
1793:
997:[Paradox and Perspectivism in Bhartrhari's Language Philosophy: Language, Thought and Reality].
834:
2352:
2337:
2312:
2307:
2159:
259:
There are also multi-sentence versions of the liar paradox. The following is the two-sentence version:
1264:. Logic, Epistemology, and the Unity of Science. Vol. 36. Dordrecht: Springer. pp. 211â243.
323:, the truth value of a statement can be any real number between 0 and 1 both inclusive, as opposed to
2357:
2342:
2332:
2297:
2245:
2174:
2089:
1961:
1956:
1879:
1874:
1803:
995:"Paradoxe et perspectivisme dans la philosophie de langage de Bhartrhari: langage, pensée et réalité"
95:
506:
asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to
2119:
2079:
2043:
1976:
1843:
1813:
1808:
1778:
1743:
1738:
625:
581:
899:
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2347:
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2240:
2038:
1991:
1981:
1894:
1758:
1700:
1612:
1468:
1418:
1340:
1211:
1134:
1126:
1091:
972:
964:
870:
720:
65:
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1939:
1934:
1924:
1889:
1798:
1235:
757:
735:
1064:
Zach Weber, Guillermo Badia and
Patrick Girard (2015). "What is an Inconsistent Truth Table?".
580:
means, "This statement is not true", then it is negating itself. They go on to argue, based on
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811:
attempts to use the "this sentence is false" paradox to kill another artificial intelligence,
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in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the
138:
102:
2322:
2287:
2262:
2214:
2139:
2114:
2109:
2021:
2011:
1986:
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1332:
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1203:
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956:
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2006:
1996:
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1679:
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653:) â arises, in case of the liar and other paradoxes such as the unsignifiability paradox (
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2169:
2144:
2134:
2129:
2059:
1859:
1748:
1031:
880:
692:
613:
394:
253:
249:
1109:
HĂĄjek, P.; Paris, J.; Shepherdson, J. (Mar 2000). "The Liar
Paradox and Fuzzy Logic".
947:
Jan E.M. Houben (1995). "Bhartrhari's solution to the Liar and some other paradoxes".
756:
has since sketched an alternative proof of the first incompleteness theorem that uses
2405:
2149:
2016:
1864:
1592:
1529:
1344:
1154:"Computation of fuzzy truth values for the liar and related self-referential systems"
1095:
976:
753:
747:
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596:
476:
324:
223:
158:
91:
50:
1486:
Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972).
1472:
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three-sentence version, with each statement asserting the falsity of its successor:
1946:
1919:
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1138:
841:
779:
696:
511:
503:
149:
The problem of the liar paradox is that it seems to show that common beliefs about
1077:
1269:
1260:
Glanzberg, Michael (2015-06-17). "Complexity and
Hierarchy in Truth Predicates".
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600:
572:
535:
492:
398:
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46:
786:
to confuse and ultimately disable an android holding them captive. In the 1973
685:". Now, the liar paradox can be expressed as the statement A, that A is false:
1336:
788:
783:
638:
123:
72:
69:
20:
661:) and its extension in time, by accepting a simultaneous, opposite function (
330:
Let the truth value of the statement "This statement is false" be denoted by
1417:
ed. by Eli Franco and Karin
Preisendanz, Amsterdam â Atlanta: Rodopi, 1997;
154:
110:
828:
The next thing I say to you will be true / The last thing I said was false.
803:
760:
rather than the liar paradox to construct a true but unprovable formula.
166:
94:. One version of the liar paradox is attributed to the Greek philosopher
968:
1724:
1608:
1588:. MIT Press. Especially chapter 9. ISBN 0262611082, ISBN 978-0262611084
1215:
1130:
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775:
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41:
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1227:
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808:
76:
1692:
1207:
1122:
1559:
Greenough, P. M. (2001). " Free
Assumptions and the Liar Paradox,"
1433:
The Logic of Reflection. German Philosophy in the Twentieth Century
994:
137:
could have been the first logician to identify the liar paradox as
1660:
256:), the dialetheic approach can overcome this version of the Liar.
248:, it is a paradox. However, it has been argued that by adopting a
211:. Since this is what (B) itself states, it means that (B) must be
150:
114:
24:
1788:
1389:
819:
1696:
699:
claims that the equation should be "A = 'A = false and A = true
564:
statements are ungrounded, and therefore have no truth value.
1236:"Liar Paradox: Section 4.3.1 Tarski's hierarchy of languages"
734:
To prove the first incompleteness theorem, Gödel represented
1578:. Buridan's detailed solution to a number of such paradoxes.
1234:
Beall, Jc; Glanzberg, Michael; Ripley, David (2016-12-12) .
1323:
Mills, Eugene (1998). "A simple solution to the Liar".
236:
and false, then (C) is only false. But then, it is not
222:
Another reaction to the paradox of (A) is to posit, as
1413:
Jan E. M. Houben, "Bhartrhari's Perspectivism (1)" in
1374:. New York: Oxford University Press. pp. 6, 188.
1647:. A collection of logic puzzles exploring this theme.
1447:"Computational verb systems: The paradox of the liar"
1435:. New Haven and London: Yale University Press. p. 43.
678:
it is possible to give the condition by an equation.
448:
410:
355:
172:
The simplest version of the paradox is the sentence:
1046:
1044:
544:
and Jones says only these three things about Smith:
2223:
2052:
1852:
1731:
1561:
American Philosophical Quarterly 38/2, pp. 115-135.
1302:. MIT Press. section 9.6 "A. N. Prior's Solution".
1161:
Journal of Multiple-Valued Logic and Soft Computing
523:
This statement is true and this statement is false.
393:by generalizing the NOT operator to the equivalent
517:Thus the following two statements are equivalent:
460:
428:
382:
338:
1525:"Fightin' Words: Robert Earl Keen v. Toby Keith"
1152:Kehagias, Athanasios; Vezerides, K. (Aug 2006).
898:Epimenides paradox has "All Cretans are liars."
657:), when abstraction is made from this function (
541:A majority of what Jones says about me is false.
487:of the hierarchy, there is a statement at level
107:
1356:
1354:
1190:
1188:
1708:
8:
1451:International Journal of Intelligent Systems
711:
1371:The Liar: An Essay on Truth and Circularity
1050:Andrew Irvine, "Gaps, Gluts, and Paradox",
2376:
1715:
1701:
1693:
1595:(1984). "The Logic of Paradox Revisited".
1299:Theories of Truth: A Critical Introduction
988:
986:
1462:
1085:
447:
409:
354:
219:and is now true, another paradox arises.
1054:, supplementary vol. 18 (1992), 273â299
1023:Ahmed Alwishah and David Sanson (2009).
556:If Smith really is a big spender but is
111:I said in my alarm, Every man is a liar!
891:
553:Everything Smith says about me is true.
131:liar paradox in early Islamic tradition
1422:Paris â La Haye: Mouton, 1964, p. 263)
306:The preceding statement is false. (G2)
303:The following statement is false. (G1)
265:The preceding statement is false. (D2)
262:The following statement is true. (D1)
7:
712:Gödel's first incompleteness theorem
252:relational semantics (as opposed to
1685:Stanford Encyclopedia of Philosophy
1666:Internet Encyclopedia of Philosophy
2033:What the Tortoise Said to Achilles
1066:Australasian Journal of Philosophy
637:The Indian grammarian-philosopher
122:The Indian grammarian-philosopher
101:The paradox was once discussed by
14:
1523:Cohen, Jason (January 25, 2012).
229:This statement is only false. (C)
86:The paradox's name translates as
2386:
2385:
2375:
1674:Beall, J C; Glanzberg, Michael.
1262:Unifying the Philosophy of Truth
719:are two fundamental theorems of
1396:from the original on 2020-04-22
1242:from the original on 2021-01-12
1174:from the original on 2021-07-08
1005:from the original on 2022-05-15
919:from the original on 2012-11-11
778:", the liar paradox is used by
744:Tarski's undefinability theorem
717:Gödel's incompleteness theorems
568:Jon Barwise and John Etchemendy
207:nor false, then it must be not
200:This statement is not true. (B)
175:A: This statement (A) is false.
1638:What Is the Name of This Book?
1597:Journal of Philosophical Logic
1052:Canadian Journal of Philosophy
915:. About.com (New York Times).
771:Star Trek: The Original Series
377:
371:
215:. Since initially (B) was not
1:
1111:The Journal of Symbolic Logic
1078:10.1080/00048402.2015.1093010
632:
1270:10.1007/978-94-017-9673-6_10
1200:Outline of a theory of truth
949:Journal of Indian Philosophy
729:first incompleteness theorem
665:) undoing the previous one.
129:There was discussion of the
2412:Communication of falsehoods
1488:What is mathematical logic?
1030:. p. 1. Archived from
1001:(in French) (19): 173â199.
999:Bulletin d'Ătudes Indiennes
586:non-well-founded set theory
439:from which it follows that
296:This statement is true. (F)
190:, a concept related to the
2440:
2422:Self-referential paradoxes
1628:Papers in Logic and Ethics
1490:. London-Oxford-New York:
876:Performative contradiction
832:In the seventh episode of
807:, artificial intelligence
633:Bhartrhari's perspectivism
240:. Since initially (C) was
192:law of the excluded middle
2371:
1570:, Cambridge Univ. Press,
993:Jan E.M. Houben (2001).
913:"Paradoxes of Eubulides"
520:This statement is false.
401:, the statement becomes
383:{\displaystyle x=NOT(x)}
346:. The statement becomes
145:Explanation and variants
1952:Paradoxes of set theory
1492:Oxford University Press
1431:Roberts, Julian. 1992.
1337:10.1023/a:1004232928938
861:HilbertâBernays paradox
801:In the 2011 video game
649:) or infinite regress (
550:Smith is soft on crime.
547:Smith is a big spender.
1566:Hughes, G. E. (1992).
826:, includes the lyrics
605:principle of explosion
508:Charles Sanders Peirce
462:
430:
384:
340:
188:principle of bivalence
120:
90:(ÏΔÏ
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1445:Yang, T. (Sep 2001).
1325:Philosophical Studies
835:Minecraft: Story Mode
736:statements by numbers
643:sarvam mithyÄ bravÄ«mi
463:
461:{\displaystyle x=0.5}
431:
429:{\displaystyle x=1-x}
385:
341:
16:Paradoxical assertion
2318:Kavka's toxin puzzle
2090:Income and fertility
655:Bhartrhari's paradox
446:
408:
353:
334:
310:Possible resolutions
254:functional semantics
135:NaáčŁÄ«r al-DÄ«n al-áčŹĆ«sÄ«
96:Eubulides of Miletus
37:antinomy of the liar
1977:Temperature paradox
1900:Free choice paradox
1764:Fitch's knowability
1037:on August 16, 2011.
582:situation semantics
157:actually lead to a
2353:Prisoner's dilemma
2039:Heat death paradox
2027:Unexpected hanging
1992:Chicken or the egg
1609:10.1007/bf00453020
1494:. pp. 52â53.
1419:Madeleine Biardeau
1415:Beyond Orientalism
961:10.1007/bf01880219
871:Knights and Knaves
764:In popular culture
721:mathematical logic
458:
426:
380:
203:If (B) is neither
66:Epimenides paradox
2399:
2398:
2070:Arrow information
1659:Dowden, Bradley.
1634:Smullyan, Raymond
1586:Theories of Truth
1501:978-0-19-888087-5
1279:978-94-017-9672-9
911:Andrea Borghini.
669:Logical structure
284:E1 is false. (E3)
281:E3 is false. (E2)
278:E2 is false. (E1)
103:Jerome of Stridon
88:pseudĂłmenos lĂłgos
2429:
2389:
2388:
2379:
2378:
2190:Service recovery
2044:Olbers's paradox
1744:Buridan's bridge
1717:
1710:
1703:
1694:
1689:
1680:Zalta, Edward N.
1670:
1620:
1582:Kirkham, Richard
1547:
1546:
1544:
1542:
1533:. Archived from
1520:
1514:
1513:
1483:
1477:
1476:
1466:
1464:10.1002/int.1049
1457:(9): 1053â1067.
1442:
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1402:
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1358:
1349:
1348:
1331:(2/3): 197â212.
1320:
1314:
1313:
1294:Kirkham, Richard
1290:
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1167:(5â6): 539â559.
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684:
676:self-referential
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139:self-referential
27:, the classical
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2367:
2278:Decision-making
2224:Decision theory
2219:
2048:
1972:Hilbert's Hotel
1905:GrellingâNelson
1848:
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1366:John Etchemendy
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1123:10.2307/2586541
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857:
795:The Green Death
766:
758:Berry's paradox
714:
709:
700:
689:
688:A = "A = false"
682:
671:
635:
623:
621:Non-cognitivism
594:
577:John Etchemendy
570:
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244:and is now not
232:If (C) is both
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2195:St. Petersburg
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1824:Rule-following
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1801:
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1776:
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1749:Dream argument
1746:
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1735:
1733:
1729:
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1722:
1720:
1719:
1712:
1705:
1697:
1691:
1690:
1676:"Liar Paradox"
1671:
1661:"Liar Paradox"
1654:
1653:External links
1651:
1649:
1648:
1631:
1621:
1603:(2): 153â179.
1593:Priest, Graham
1589:
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1221:
1198:(1975-11-06).
1184:
1144:
1117:(1): 339â346.
1101:
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1040:
1015:
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955:(4): 381â401.
939:
929:
903:
890:
888:
885:
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881:Self-reference
878:
873:
868:
863:
856:
853:
844:'s 1994 song "
765:
762:
713:
710:
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693:Boolean domain
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614:paraconsistent
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33:liar's paradox
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2236:Apportionment
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2095:DownsâThomson
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2018:
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2013:
2012:Plato's beard
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1910:KleeneâRosser
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1834:Theseus' ship
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1794:Mere addition
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1732:Philosophical
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1645:0-671-62832-1
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1577:
1576:0-521-28864-9
1573:
1569:
1565:
1562:
1558:
1557:
1552:
1537:on 2015-10-03
1536:
1532:
1531:
1530:Texas Monthly
1526:
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1381:9780195059441
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1309:0-262-61108-2
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761:
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755:
754:George Boolos
751:
749:
748:Alfred Tarski
745:
739:
737:
732:
730:
726:
722:
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704:
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664:
663:apara vyÄpÄra
660:
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652:
648:
644:
640:
630:
627:
626:Andrew Irvine
620:
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616:
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597:Graham Priest
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583:
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477:Alfred Tarski
472:Alfred Tarski
471:
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325:Boolean logic
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159:contradiction
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132:
127:
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106:
105:in a sermon:
104:
99:
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92:Ancient Greek
89:
84:
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78:
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71:
67:
59:
57:
54:
52:
51:contradiction
48:
43:
38:
34:
30:
26:
22:
2343:Preparedness
2175:Productivity
2155:Mandeville's
1947:Opposite Day
1914:
1875:Burali-Forti
1870:Bhartrhari's
1683:
1664:
1637:
1630:. Duckworth.
1627:
1600:
1596:
1585:
1567:
1560:
1539:. Retrieved
1535:the original
1528:
1518:
1487:
1481:
1454:
1450:
1440:
1432:
1427:
1414:
1409:
1398:. Retrieved
1370:
1328:
1324:
1318:
1297:
1288:
1261:
1255:
1244:. Retrieved
1199:
1196:Kripke, Saul
1176:. Retrieved
1164:
1160:
1147:
1114:
1110:
1104:
1069:
1065:
1059:
1051:
1032:the original
1018:
1007:. Retrieved
998:
952:
948:
942:
932:
921:. Retrieved
906:
894:
850:
842:Rollins Band
840:
833:
831:
827:
823:
817:
802:
800:
793:
787:
780:Captain Kirk
769:
767:
752:
740:
733:
715:
707:Applications
697:Arthur Prior
690:
680:
672:
662:
658:
650:
646:
642:
636:
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612:
608:
595:
571:
562:
557:
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512:John Buridan
504:Arthur Prior
502:
499:Arthur Prior
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128:
121:
108:
100:
87:
85:
80:
63:
55:
36:
32:
29:liar paradox
28:
18:
2273:Condorcet's
2125:Giffen good
2085:Competition
1839:White horse
1814:Omnipotence
1624:A. N. Prior
1362:Jon Barwise
866:Insolubilia
824:Enough Said
601:dialetheism
592:Dialetheism
573:Jon Barwise
536:Saul Kripke
531:Saul Kripke
493:Saul Kripke
399:fuzzy logic
321:fuzzy logic
315:Fuzzy logic
49:leads to a
47:truth value
2406:Categories
2348:Prevention
2338:Parrondo's
2328:Navigation
2313:Inventor's
2308:Hedgehog's
2268:Chainstore
2251:Population
2246:New states
2180:Prosperity
2160:Mayfield's
2002:Entailment
1982:Barbershop
1895:Epimenides
1553:References
1510:0251.02001
1400:2016-02-22
1246:2021-01-16
1178:2021-02-17
1087:2292/30988
1009:2018-08-04
923:2012-09-04
900:Titus 1:12
789:Doctor Who
784:Harry Mudd
725:Kurt Gödel
639:Bhartrhari
250:two-valued
124:Bhartrhari
73:Epimenides
21:philosophy
2363:Willpower
2358:Tolerance
2333:Newcomb's
2298:Fredkin's
2185:Scitovsky
2105:Edgeworth
2100:Easterlin
2065:Antitrust
1962:Russell's
1957:Richard's
1930:Pinocchio
1885:Crocodile
1804:Newcomb's
1774:Goodman's
1769:Free will
1754:Epicurean
1725:paradoxes
1345:169981769
1096:170137819
977:170337976
774:episode "
651:anavasthÄ
421:−
2391:Category
2288:Ellsberg
2140:Leontief
2120:Gibson's
2115:European
2110:Ellsberg
2080:Braess's
2075:Bertrand
2053:Economic
1987:Catch-22
1967:Socratic
1809:Nihilism
1779:Hedonism
1739:Analysis
1723:Notable
1636:(1986).
1626:(1976).
1584:(1992).
1541:July 12,
1473:41448750
1394:Archived
1390:86031260
1368:(1989).
1296:(1992).
1240:Archived
1169:Archived
1072:(3): 7.
1003:Archived
969:23447805
917:Archived
855:See also
813:Wheatley
804:Portal 2
167:semantic
2293:Fenno's
2258:Arrow's
2241:Alabama
2231:Abilene
2210:Tullock
2165:Metzler
2007:Lottery
1997:Drinker
1940:Yablo's
1935:Quine's
1890:Curry's
1853:Logical
1829:Sorites
1819:Preface
1799:Moore's
1784:Liberal
1759:Fiction
1682:(ed.).
1617:2442524
1216:2024634
1139:6865763
1131:2586541
792:serial
776:I, Mudd
659:vyÄpÄra
647:virodha
169:rules.
163:grammar
155:falsity
83:lies).
60:History
42:paradox
2200:Thrift
2170:Plenty
2145:Lerner
2135:Jevons
2130:Icarus
2060:Allais
2022:Ross's
1860:Barber
1844:Zeno's
1789:Meno's
1643:
1615:
1574:
1508:
1498:
1471:
1388:
1378:
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1276:
1214:
1137:
1129:
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967:
822:song,
809:GLaDOS
77:Cretan
2417:Lying
2303:Green
2283:Downs
2215:Value
2150:Lucas
2017:Raven
1925:No-no
1880:Court
1865:Berry
1678:. In
1613:S2CID
1469:S2CID
1341:S2CID
1212:JSTOR
1172:(PDF)
1157:(PDF)
1135:S2CID
1127:JSTOR
1092:S2CID
1035:(PDF)
1028:(PDF)
973:S2CID
965:JSTOR
887:Notes
397:from
151:truth
115:David
113:" Is
25:logic
2381:List
2205:Toil
1920:Card
1915:Liar
1641:ISBN
1572:ISBN
1543:2021
1496:ISBN
1386:LCCN
1376:ISBN
1304:ISBN
1274:ISBN
846:Liar
820:Devo
818:The
782:and
575:and
510:and
246:true
242:true
238:true
234:true
217:true
213:true
209:true
205:true
165:and
153:and
81:only
75:, a
70:seer
64:The
23:and
1605:doi
1506:Zbl
1459:doi
1333:doi
1266:doi
1204:doi
1119:doi
1082:hdl
1074:doi
957:doi
937:294
609:all
558:not
456:0.5
319:In
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