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275:... I in my astonishment said: What do you mean Dionysodorus? I have often heard, and have been amazed to hear, this thesis of yours, which is maintained and employed by the disciples of Protagoras and others before them, and which to me appears to be quite wonderful, and suicidal as well as destructive, and I think that I am most likely to hear the truth about it from you. The dictum is that there is no such thing as a falsehood; a man must either say what is true or say nothing. Is not that your position?
43:
1740:—usually refers to an opposition inherently existing within one realm, one unified force or object. This contradiction, as opposed to metaphysical thinking, is not an objectively impossible thing, because these contradicting forces exist in objective reality, not cancelling each other out, but actually defining each other's existence. According to
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The prime requisite of a set of postulates is that it be consistent. Since the ordinary notion of consistency involves that of contradiction, which again involves negation, and since this function does not appear in general as a primitive in a new definition must be
1144:, but in the absence of EFQ it does not yield full classical logic. Minimal logic + LEM + EFQ yields classical logic. PR entails but is not entailed by LEM in minimal logic. If the formula B in Peirce's rule is restricted to absurdity, giving the axiom schema
279:
Indeed, Dionysodorus agrees that "there is no such thing as false opinion ... there is no such thing as ignorance", and demands of
Socrates to "Refute me." Socrates responds "But how can I refute you, if, as you say, to tell a falsehood is impossible?".
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In other words, the notion of "contradiction" can be dispensed when constructing a proof of consistency; what replaces it is the notion of "mutually exclusive and exhaustive" classes. An axiomatic system need not include the notion of "contradiction".
236:
can be derived from it, using the rules of the logic. It is a proposition that is unconditionally false (i.e., a self-contradictory proposition). This can be generalized to a collection of propositions, which is then said to "contain" a contradiction.
1723:
A pragmatic contradiction occurs when the very statement of the argument contradicts the claims it purports. An inconsistency arises, in this case, because the act of utterance, rather than the content of what is said, undermines its conclusion.
878:
and proof by contradiction, we can investigate the axiomatic strength and properties of various rules that treat contradiction by considering theorems of classical logic that are not theorems of minimal logic. Each of these extensions leads to an
1654:
Post observed that, if the system were inconsistent, a deduction in it (that is, the last formula in a sequence of formulas derived from the tautologies) could ultimately yield S itself. As an assignment to variable S can come from either class
1374:, or some of its variants, immediately after a contradiction symbol. In fact, this often occurs in a proof by contradiction to indicate that the original assumption was proved false—and hence that its negation must be true.
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don't hold, but where use of non-intuitionistic reasoning is marked by occurrences of double-negation in the conclusion. LEM entails but is not entailed by WLEM in minimal logic. WLEM is equivalent to the instance of
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and ~S (negation), one is forced to define the axioms in terms of these primitive notions. In a thorough manner, Post demonstrates in PM, and defines (as do Nagel and Newman, see below) that the property of
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set of postulates (i.e. axioms), he would no longer be able to automatically invoke the notion of "contradiction"—such a notion might not be contained in the postulates:
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In mathematics, the symbol used to represent a contradiction within a proof varies. Some symbols that may be used to represent a contradiction include ↯, Opq,
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1663:, the deduction violates the inheritance characteristic of tautology (i.e., the derivation must yield an evaluation of a formula that will fall into class K
1469:. They too observed a problem with respect to the notion of "contradiction" with its usual "truth values" of "truth" and "falsity". They observed that:
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1485:". ... proofs of consistency which are based on models, and which argue from the truth of axioms to their consistency, merely shift the problem.
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The property of being a tautology has been defined in notions of truth and falsity. Yet these notions obviously involve a reference to something
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states that "It is impossible that the same thing can at the same time both belong and not belong to the same object and in the same respect."
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Post's solution to the problem is described in the demonstration "An
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that captures proof by contradiction without explicitly referring to absurdity. Minimal logic + PR + EFQ yields classical logic.
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of the calculus, by supplying a model for the system. This being so, the authors have not done what they promised, namely, "
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are assigned from these classes). This also applies to the primitive formulas. For example: "A formula having the form S
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and yields a system where disjunction behaves more like in classical logic than intuitionistic logic, i.e. the
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as if this notion would simultaneously have to be "outside" the formal system in the definition of tautology.
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one can evaluate the formula and place its outcome in one or the other of the classes. The assignment of K
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2257:, translated from the French and German editions by Otto Bird, D. Reidel, Dordrecht, South Holland.
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Post 1921 "Introduction to a
General Theory of Elementary Propositions" in van Heijenoort 1967:272.
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A system will be said to be inconsistent if it yields the assertion of the unmodified variable p .
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to define a property of formulas in terms of purely structural features of the formulas themselves
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can label actions or statements as contradicting each other when due (or perceived as due) to
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the formula calculus. Therefore, the procedure mentioned in the text in effect offers an
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used to denote an arbitrary tautology. Contradiction is sometimes symbolized by "O
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The notion of contradiction in an axiomatic system and a proof of its consistency
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1667:). From this, Post was able to derive the following definition of inconsistency—
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of society, thus resolving the prior contradiction between (a) and (b).
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typically claim that as a necessary condition of the justification of a
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But by whatever method one goes about it, all consistency proofs would
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1948:(Winter 2018 ed.), Metaphysics Research Lab, Stanford University
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1108:. Peirce's rule entails but is not entailed by GD over minimal logic.
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1744:, such a contradiction can be found, for example, in the fact that:
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1858: – View that there are statements that are both true and false
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From Frege to Gödel: A Source Book in
Mathematical Logic 1879-1931
2161:(1921) Introduction to a General Theory of Elementary Propositions
1878:
1598:: "a formula is a tautology if and only if it falls in the class K
250:
138:
2622:
1919:
1754:(c) the existence of (a) being contrary to the existence of (b).
1703:, that belief must form a part of a logically non-contradictory
180:
172:
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921:, and when it is added to minimal logic yields classical logic.
523:", or "ex falso quodlibet" ("from falsity, anything follows").
36:
874:, a logic with similar axioms to classical logic but without
1748:(a) enormous wealth and productive powers coexist alongside:
267:
denies the existence of "contradiction", all the while that
143:
This diagram shows the contradictory relationships between
2279:, New York University Press, Card Catalog Number: 58-5610.
1715:, have argued that coherence may not require consistency.
1489:
Given some "primitive formulas" such as PM's primitives S
1296:{\displaystyle \neg (A\land B)\iff (\neg A)\vee (\neg B)}
780:{\displaystyle \Sigma \cup \{\neg \varphi \}\vdash \bot }
692:{\displaystyle \Sigma \cup \{\neg \varphi \}\vdash \bot }
2346:
1904: – Type of formal logic without explosion principle
1366:", as symbolized, for instance, by "0" (as is common in
27:
Logical incompatibility between two or more propositions
2782:
2169:(3):163—185 (1921) The Johns Hopkins University Press
1851:
Pages displaying short descriptions of redirect targets
1842:
Pages displaying short descriptions of redirect targets
530:
logic, a formula is contradictory if and only if it is
312:) represents an arbitrary contradiction, with the dual
2222:
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Weak law of the excluded middle (WLEM) is axiomatised
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Classifying
Material Implications over Minimal Logic
1910: – Statement that apparently contradicts itself
1650:. Thus by definition our formula is not a tautology.
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Hence Nagel and Newman can now define the notion of
183:. Illustrating a general tendency in applied logic,
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1577:", and "A formula having the form ~S is placed in K
1052:{\displaystyle ((A\implies B)\implies A)\implies A}
67:. Unsourced material may be challenged and removed.
1916: – In logic, a statement which is always true
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2218:"Capital and Labour: Can the Conflict Be Solved?"
2132:boldface italics added, Nagel and Newman:109-110.
1758:Hegelian and Marxist theories stipulate that the
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259:dialogue demonstrates the need for the notion of
2179:In Contradiction: A Study of the Transconsistent
1990:"Tautologies, contradictions, and contingencies"
1881: – Rhetorical device and literary technique
1355:{\displaystyle \leftrightarrow \ \!\!\!\!\!\!\!}
466:{\displaystyle \vdash \varphi \rightarrow \psi }
1673:
1604:
1471:
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1864: – Inconsistent application of principles
1801:which are contradictory in the logical sense.
1669:without the use of the notion of contradiction
1111:Law of the excluded middle (LEM), axiomatised
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2195:. Oxford University Press - U.S. p. 87.
1518:system will yield only tautologous formulas.
1140:, is the most often cited formulation of the
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1236:that distributes negation over conjunction:
1182:{\displaystyle (\neg A\implies A)\implies A}
827:. The use of this fact forms the basis of a
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171:conflicts either with itself or established
2264:, Harvard University Press, Cambridge, MA,
1840: – Word that has two opposing meanings
1097:{\displaystyle A\implies B\vee B\implies A}
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2078:Diener and Maarten McKubre-Jordens, 2020.
1822:uses contradiction to falsify bad theory.
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127:Learn how and when to remove this message
742:leads to a contradiction). Therefore, a
202:, the term is mainly used instead for a
2056:"Ex falso quodlibet - Oxford Reference"
1946:The Stanford Encyclopedia of Philosophy
1932:
1606:For example, given a formula such as ~S
1416:to necessitate the primitive notion of
1325:{\displaystyle \Rightarrow \Leftarrow }
990:{\displaystyle A\land \neg A\implies B}
175:. It is often used as a tool to detect
2275:Ernest Nagel and James R. Newman 1958
1220:{\displaystyle \neg A\vee \neg \neg A}
924:Ex falso quodlibet (EFQ), axiomatized
611:{\displaystyle \Sigma \vdash \varphi }
364:is often read as "yields" or "proves".
232:; a proposition is a contradiction if
2098:"The Comprehensive LATEX Symbol List"
914:{\displaystyle \neg \neg A\implies A}
7:
1229:disjunction and existence properties
417:{\displaystyle \varphi \vdash \bot }
367:In classical logic, particularly in
65:adding citations to reliable sources
2544:Analytic and synthetic propositions
2415:Formal semantics (natural language)
2339:Stanford Encyclopedia of Philosophy
2018:The Dialogs of Plato translated by
1762:nature of history will lead to the
1386:requires the following two things:
1874:Graham's hierarchy of disagreement
1521:On the topic of the definition of
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2096:Pakin, Scott (January 19, 2017).
1736:: Contradiction—as derived from
548:For a set of consistent premises
512:{\displaystyle \bot \vdash \psi }
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2034:Great Books of the Western World
1892: – 1937 essay by Mao Zedong
1435:(i.e. the logic) beyond that of
41:
2291:"Contradiction (inconsistency)"
2216:Sørensen, Michael Kuur (2006).
2164:American Journal of Mathematics
1751:(b) extreme poverty and misery;
1400:the case that both the formula
945:{\displaystyle \bot \implies A}
52:needs additional citations for
2084:Archive for Mathematical Logic
1525:, Nagel and Newman create two
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32:Contradiction (disambiguation)
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4496:History of mathematical logic
1944:, in Zalta, Edward N. (ed.),
1922: – Problem-solving tools
1849: – Type of logic diagram
1585:; otherwise it is placed in K
1573:; otherwise it is placed in K
1408:can be derived in the system.
1370:). It is not uncommon to see
1062:The Gödel-Dummett (GD) axiom
735:{\displaystyle \neg \varphi }
4421:Primitive recursive function
2255:Précis of Mathematical Logic
1133:{\displaystyle A\vee \neg A}
860:{\displaystyle A\vee \neg A}
2314:Encyclopedia of Mathematics
2296:Encyclopedia of Mathematics
1784:working and producing class
1396:A demonstration that it is
807:is true under the premises
263:. In the ensuing dialogue,
5034:
3485:Schröder–Bernstein theorem
3212:Monadic predicate calculus
2871:Foundations of mathematics
1940:Horn, Laurence R. (2018),
958:ex contradiction quodlibet
541:
424:. Since for contradictory
344:". The turnstile symbol,
287:
29:
4973:
4724:
4616:
4531:
4518:Philosophy of mathematics
4467:Automated theorem proving
3638:
3592:Von Neumann–Bernays–Gödel
3233:
2751:
2628:Necessity and sufficiency
2384:
2260:Jean van Heijenoort 1967
2193:Ignorance and Imagination
2031:, editor in chief, 1952,
1994:www.skillfulreasoning.com
867:is accepted as an axiom.
2191:Stoljar, Daniel (2006).
2141:Nagel and Newman:110-111
1898: – Figure of speech
1782:would equally serve the
1719:Pragmatic contradictions
800:{\displaystyle \varphi }
651:{\displaystyle \varphi }
581:{\displaystyle \varphi }
437:{\displaystyle \varphi }
388:{\displaystyle \varphi }
145:categorical propositions
4168:Self-verifying theories
3989:Tarski's axiomatization
2940:Tarski's undefinability
2935:incompleteness theorems
2309:"Contradiction, law of"
2060:www.oxfordreference.com
2039:Encyclopædia Britannica
2029:Robert Maynard Hutchins
1966:"Contradiction (logic)"
1885:Law of noncontradiction
1734:dialectical materialism
1728:Dialectical materialism
1308:Symbolic representation
956:. EFQ is equivalent to
820:{\displaystyle \Sigma }
712:{\displaystyle \Sigma }
631:{\displaystyle \Sigma }
561:{\displaystyle \Sigma }
357:{\displaystyle \vdash }
189:law of noncontradiction
4994:Mathematics portal
4542:Mathematics portal
4153:Proof of impossibility
3801:propositional variable
3111:Propositional calculus
1805:Proof by contradiction
1680:
1652:
1614:and an assignment of K
1557:is placed into class K
1487:
1452:
1433:propositional calculus
1356:
1326:
1297:
1221:
1183:
1142:principle of bivalence
1134:
1098:
1053:
991:
946:
915:
861:
837:law of excluded middle
833:proof by contradiction
821:
801:
781:
736:
713:
693:
652:
632:
612:
582:
562:
544:Proof by contradiction
538:Proof by contradiction
521:principle of explosion
513:
487:
467:
438:
418:
389:
358:
340:", and tautology by "V
330:
302:
277:
271:is contradicting him:
225:
156:
4983:Philosophy portal
4411:Kolmogorov complexity
4364:Computably enumerable
4264:Model complete theory
4056:Principia Mathematica
3116:Propositional formula
2945:Banach–Tarski paradox
2763:Philosophy portal
2251:Józef Maria Bocheński
1970:TheFreeDictionary.com
1438:Principia Mathematica
1357:
1327:
1298:
1222:
1184:
1135:
1099:
1054:
997:, over minimal logic.
992:
947:
916:
862:
822:
802:
782:
737:
714:
694:
653:
633:
613:
583:
563:
514:
488:
486:{\displaystyle \psi }
468:
439:
419:
390:
359:
331:
329:{\displaystyle \top }
303:
301:{\displaystyle \bot }
226:
224:{\displaystyle \bot }
142:
5018:Cognitive dissonance
4359:Church–Turing thesis
4346:Computability theory
3555:continuum hypothesis
3073:Square of opposition
2931:Gödel's completeness
2150:Nagel and Newman:111
2105:ctan.mirror.rafal.ca
1927:Notes and references
1902:Paraconsistent logic
1790:Outside formal logic
1336:
1316:
1240:
1196:
1148:
1115:
1066:
1007:
964:
954:intuitionistic logic
928:
891:
842:
811:
791:
750:
723:
703:
662:
642:
622:
596:
572:
552:
497:
477:
448:
428:
402:
379:
348:
320:
292:
215:
149:square of opposition
61:improve this article
30:For other uses, see
4513:Mathematical object
4404:P versus NP problem
4369:Computable function
4163:Reverse mathematics
4089:Logical consequence
3966:primitive recursive
3961:elementary function
3734:Free/bound variable
3587:Tarski–Grothendieck
3106:Logical connectives
3036:Logical equivalence
2886:Logical consequence
2425:Philosophy of logic
2023:appearing in: BK 7
1780:means of production
1106:Gödel-Dummett logic
395:is a contradiction
4311:Transfer principle
4274:Semantics of logic
4259:Categorical theory
4235:Non-standard model
3749:Logical connective
2876:Information theory
2825:Mathematical logic
2724:Rules of inference
2693:Mathematical logic
2435:Semantics of logic
1778:society where the
1707:of beliefs. Some
1527:mutually exclusive
1352:
1322:
1320:⇒ ⇐
1293:
1217:
1179:
1130:
1094:
1049:
987:
942:
911:
881:intermediate logic
876:ex falso quodlibet
857:
817:
797:
777:
732:
709:
689:
648:
628:
608:
578:
568:and a proposition
558:
509:
483:
463:
434:
414:
385:
354:
326:
298:
221:
157:
153:Aristotelian logic
5000:
4999:
4968:
4967:
4683:
4682:
4549:
4548:
4481:Abstract category
4284:Theories of truth
4094:Rule of inference
4084:Natural deduction
4065:
4064:
3610:
3609:
3315:Cartesian product
3220:
3219:
3126:Many-valued logic
3101:Boolean functions
2984:Russell's paradox
2959:diagonal argument
2856:First-order logic
2791:
2790:
2747:
2746:
2581:Deductive closure
2527:
2526:
2466:Critical thinking
2327:Horn, Laurence R.
2086:59 (7-8):905-924.
1832:"Argument Clinic"
1820:scientific method
1691:Adherents of the
1404:and its negation
1384:consistency proof
1344:
1003:(PR) is an axiom
787:also proves that
658:) if and only if
373:first-order logic
288:Note: The symbol
245:By creation of a
161:traditional logic
137:
136:
129:
111:
16:(Redirected from
5025:
4992:
4991:
4981:
4980:
4979:
4911:
4860:
4740:
4727:
4710:
4703:
4696:
4687:
4622:
4576:
4569:
4562:
4553:
4540:
4539:
4491:History of logic
4486:Category of sets
4379:Decision problem
4158:Ordinal analysis
4099:Sequent calculus
3997:Boolean algebras
3937:
3936:
3911:
3882:logical/constant
3636:
3622:
3545:Zermelo–Fraenkel
3296:Set operations:
3231:
3168:
2999:
2979:Löwenheim–Skolem
2866:Formal semantics
2818:
2811:
2804:
2795:
2761:
2760:
2759:
2681:
2446:
2410:Computer science
2371:
2364:
2357:
2348:
2343:
2334:Zalta, Edward N.
2322:
2304:
2238:
2237:
2235:
2233:
2213:
2207:
2206:
2188:
2182:
2181:By Graham Priest
2176:
2170:
2157:
2151:
2148:
2142:
2139:
2133:
2130:
2124:
2121:
2115:
2114:
2112:
2111:
2102:
2093:
2087:
2076:
2070:
2069:
2067:
2066:
2052:
2046:
2010:
2004:
2003:
2001:
2000:
1986:
1980:
1979:
1977:
1976:
1962:
1956:
1955:
1954:
1953:
1937:
1890:On Contradiction
1852:
1847:Contrary (logic)
1843:
1795:Colloquial usage
1504:deduction system
1392:axiomatic system
1361:
1359:
1358:
1353:
1342:
1331:
1329:
1328:
1323:
1302:
1300:
1299:
1294:
1226:
1224:
1223:
1218:
1188:
1186:
1185:
1180:
1139:
1137:
1136:
1131:
1103:
1101:
1100:
1095:
1058:
1056:
1055:
1050:
996:
994:
993:
988:
951:
949:
948:
943:
920:
918:
917:
912:
866:
864:
863:
858:
826:
824:
823:
818:
806:
804:
803:
798:
786:
784:
783:
778:
741:
739:
738:
733:
718:
716:
715:
710:
698:
696:
695:
690:
657:
655:
654:
649:
637:
635:
634:
629:
617:
615:
614:
609:
588:, it is true in
587:
585:
584:
579:
567:
565:
564:
559:
518:
516:
515:
510:
492:
490:
489:
484:
472:
470:
469:
464:
444:it is true that
443:
441:
440:
435:
423:
421:
420:
415:
394:
392:
391:
386:
375:, a proposition
363:
361:
360:
355:
335:
333:
332:
327:
307:
305:
304:
299:
230:
228:
227:
222:
132:
125:
121:
118:
112:
110:
69:
45:
37:
21:
5033:
5032:
5028:
5027:
5026:
5024:
5023:
5022:
5003:
5002:
5001:
4996:
4986:
4985:
4977:
4975:
4969:
4964:
4963:
4960:
4956:
4948:
4947:
4944:
4940:
4932:
4928:
4920:
4916:
4907:
4898:
4894:
4889:
4881:
4877:
4869:
4865:
4856:
4847:
4843:
4835:
4831:
4823:
4819:
4811:
4808:
4805:
4797:
4794:
4791:
4783:
4779:
4774:
4766:
4762:
4757:
4749:
4745:
4736:
4720:
4718:logical symbols
4714:
4684:
4679:
4649:
4623:
4614:
4587:
4580:
4550:
4545:
4534:
4527:
4472:Category theory
4462:Algebraic logic
4445:
4416:Lambda calculus
4354:Church encoding
4340:
4316:Truth predicate
4172:
4138:Complete theory
4061:
3930:
3926:
3922:
3917:
3909:
3629: and
3625:
3620:
3606:
3582:New Foundations
3550:axiom of choice
3533:
3495:Gödel numbering
3435: and
3427:
3331:
3216:
3166:
3147:
3096:Boolean algebra
3082:
3046:Equiconsistency
3011:Classical logic
2988:
2969:Halting problem
2957: and
2933: and
2921: and
2920:
2915:Theorems (
2910:
2827:
2822:
2792:
2787:
2757:
2755:
2743:
2707:
2698:Boolean algebra
2672:
2523:
2514:Metamathematics
2492:
2444:
2398:
2380:
2375:
2330:"Contradiction"
2325:
2307:
2289:
2286:
2247:
2242:
2241:
2231:
2229:
2215:
2214:
2210:
2203:
2190:
2189:
2185:
2177:
2173:
2158:
2154:
2149:
2145:
2140:
2136:
2131:
2127:
2122:
2118:
2109:
2107:
2100:
2095:
2094:
2090:
2077:
2073:
2064:
2062:
2054:
2053:
2049:
2020:Benjamin Jowett
2011:
2007:
1998:
1996:
1988:
1987:
1983:
1974:
1972:
1964:
1963:
1959:
1951:
1949:
1942:"Contradiction"
1939:
1938:
1934:
1929:
1862:Double standard
1850:
1841:
1828:
1799:presuppositions
1792:
1730:
1721:
1693:epistemological
1689:
1666:
1662:
1658:
1649:
1645:
1641:
1637:
1633:
1629:
1625:
1621:
1617:
1613:
1609:
1601:
1588:
1584:
1580:
1576:
1572:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1496:
1492:
1460:James R. Newman
1380:
1368:Boolean algebra
1334:
1333:
1314:
1313:
1310:
1238:
1237:
1234:De Morgan's law
1194:
1193:
1146:
1145:
1113:
1112:
1064:
1063:
1005:
1004:
962:
961:
926:
925:
889:
888:
840:
839:
829:proof technique
809:
808:
789:
788:
748:
747:
721:
720:
701:
700:
660:
659:
640:
639:
620:
619:
594:
593:
590:classical logic
570:
569:
550:
549:
546:
540:
495:
494:
475:
474:
446:
445:
426:
425:
400:
399:
377:
376:
365:
346:
345:
318:
317:
290:
289:
286:
284:In formal logic
243:
213:
212:
133:
122:
116:
113:
76:"Contradiction"
70:
68:
58:
46:
35:
28:
23:
22:
15:
12:
11:
5:
5031:
5029:
5021:
5020:
5015:
5005:
5004:
4998:
4997:
4974:
4971:
4970:
4966:
4965:
4961:quantification
4957:
4952:
4951:
4949:
4945:quantification
4941:
4936:
4935:
4933:
4924:
4923:
4921:
4902:
4901:
4899:
4885:
4884:
4882:
4873:
4872:
4870:
4851:
4850:
4848:
4839:
4838:
4836:
4827:
4826:
4824:
4815:
4814:
4812:
4801:
4800:
4798:
4787:
4786:
4784:
4770:
4769:
4767:
4753:
4752:
4750:
4731:
4730:
4725:
4722:
4721:
4715:
4713:
4712:
4705:
4698:
4690:
4681:
4680:
4678:
4677:
4672:
4667:
4657:
4655:
4654:Negation
4651:
4650:
4648:
4647:
4642:
4637:
4631:
4629:
4625:
4624:
4617:
4615:
4613:
4612:
4606:
4604:truth function
4601:
4595:
4593:
4589:
4588:
4581:
4579:
4578:
4571:
4564:
4556:
4547:
4546:
4532:
4529:
4528:
4526:
4525:
4520:
4515:
4510:
4505:
4504:
4503:
4493:
4488:
4483:
4474:
4469:
4464:
4459:
4457:Abstract logic
4453:
4451:
4447:
4446:
4444:
4443:
4438:
4436:Turing machine
4433:
4428:
4423:
4418:
4413:
4408:
4407:
4406:
4401:
4396:
4391:
4386:
4376:
4374:Computable set
4371:
4366:
4361:
4356:
4350:
4348:
4342:
4341:
4339:
4338:
4333:
4328:
4323:
4318:
4313:
4308:
4303:
4302:
4301:
4296:
4291:
4281:
4276:
4271:
4269:Satisfiability
4266:
4261:
4256:
4255:
4254:
4244:
4243:
4242:
4232:
4231:
4230:
4225:
4220:
4215:
4210:
4200:
4199:
4198:
4193:
4186:Interpretation
4182:
4180:
4174:
4173:
4171:
4170:
4165:
4160:
4155:
4150:
4140:
4135:
4134:
4133:
4132:
4131:
4121:
4116:
4106:
4101:
4096:
4091:
4086:
4081:
4075:
4073:
4067:
4066:
4063:
4062:
4060:
4059:
4051:
4050:
4049:
4048:
4043:
4042:
4041:
4036:
4031:
4011:
4010:
4009:
4007:minimal axioms
4004:
3993:
3992:
3991:
3980:
3979:
3978:
3973:
3968:
3963:
3958:
3953:
3940:
3938:
3919:
3918:
3916:
3915:
3914:
3913:
3901:
3896:
3895:
3894:
3889:
3884:
3879:
3869:
3864:
3859:
3854:
3853:
3852:
3847:
3837:
3836:
3835:
3830:
3825:
3820:
3810:
3805:
3804:
3803:
3798:
3793:
3783:
3782:
3781:
3776:
3771:
3766:
3761:
3756:
3746:
3741:
3736:
3731:
3730:
3729:
3724:
3719:
3714:
3704:
3699:
3697:Formation rule
3694:
3689:
3688:
3687:
3682:
3672:
3671:
3670:
3660:
3655:
3650:
3645:
3639:
3633:
3616:Formal systems
3612:
3611:
3608:
3607:
3605:
3604:
3599:
3594:
3589:
3584:
3579:
3574:
3569:
3564:
3559:
3558:
3557:
3552:
3541:
3539:
3535:
3534:
3532:
3531:
3530:
3529:
3519:
3514:
3513:
3512:
3505:Large cardinal
3502:
3497:
3492:
3487:
3482:
3468:
3467:
3466:
3461:
3456:
3441:
3439:
3429:
3428:
3426:
3425:
3424:
3423:
3418:
3413:
3403:
3398:
3393:
3388:
3383:
3378:
3373:
3368:
3363:
3358:
3353:
3348:
3342:
3340:
3333:
3332:
3330:
3329:
3328:
3327:
3322:
3317:
3312:
3307:
3302:
3294:
3293:
3292:
3287:
3277:
3272:
3270:Extensionality
3267:
3265:Ordinal number
3262:
3252:
3247:
3246:
3245:
3234:
3228:
3222:
3221:
3218:
3217:
3215:
3214:
3209:
3204:
3199:
3194:
3189:
3184:
3183:
3182:
3172:
3171:
3170:
3157:
3155:
3149:
3148:
3146:
3145:
3144:
3143:
3138:
3133:
3123:
3118:
3113:
3108:
3103:
3098:
3092:
3090:
3084:
3083:
3081:
3080:
3075:
3070:
3065:
3060:
3055:
3050:
3049:
3048:
3038:
3033:
3028:
3023:
3018:
3013:
3007:
3005:
2996:
2990:
2989:
2987:
2986:
2981:
2976:
2971:
2966:
2961:
2949:Cantor's
2947:
2942:
2937:
2927:
2925:
2912:
2911:
2909:
2908:
2903:
2898:
2893:
2888:
2883:
2878:
2873:
2868:
2863:
2858:
2853:
2848:
2847:
2846:
2835:
2833:
2829:
2828:
2823:
2821:
2820:
2813:
2806:
2798:
2789:
2788:
2786:
2785:
2780:
2770:
2765:
2752:
2749:
2748:
2745:
2744:
2742:
2741:
2736:
2731:
2726:
2721:
2715:
2713:
2709:
2708:
2706:
2705:
2700:
2695:
2689:
2687:
2678:
2674:
2673:
2671:
2670:
2665:
2660:
2655:
2650:
2645:
2640:
2635:
2630:
2625:
2620:
2615:
2610:
2605:
2604:
2603:
2593:
2588:
2583:
2578:
2573:
2572:
2571:
2566:
2556:
2551:
2546:
2541:
2535:
2533:
2529:
2528:
2525:
2524:
2522:
2521:
2516:
2511:
2506:
2500:
2498:
2494:
2493:
2491:
2490:
2485:
2480:
2475:
2474:
2473:
2468:
2458:
2452:
2450:
2443:
2442:
2437:
2432:
2427:
2422:
2417:
2412:
2406:
2404:
2400:
2399:
2397:
2396:
2391:
2385:
2382:
2381:
2376:
2374:
2373:
2366:
2359:
2351:
2345:
2344:
2323:
2305:
2285:
2284:External links
2282:
2281:
2280:
2273:
2258:
2246:
2243:
2240:
2239:
2208:
2201:
2183:
2171:
2152:
2143:
2134:
2125:
2116:
2088:
2071:
2047:
2005:
1981:
1957:
1931:
1930:
1928:
1925:
1924:
1923:
1917:
1911:
1905:
1899:
1893:
1887:
1882:
1876:
1871:
1865:
1859:
1853:
1844:
1835:
1827:
1824:
1791:
1788:
1774:evolve into a
1756:
1755:
1752:
1749:
1742:Marxist theory
1729:
1726:
1720:
1717:
1688:
1685:
1664:
1660:
1656:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1619:
1615:
1611:
1607:
1599:
1586:
1582:
1581:, if S is in K
1578:
1574:
1570:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1538:
1534:
1506:that contains
1494:
1490:
1479:interpretation
1462:in their 1958
1418:contradiction.
1410:
1409:
1394:
1382:In general, a
1379:
1376:
1341:
1321:
1309:
1306:
1305:
1304:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1264:
1260:
1257:
1254:
1251:
1248:
1245:
1216:
1213:
1210:
1207:
1204:
1201:
1190:
1178:
1174:
1170:
1167:
1163:
1159:
1156:
1153:
1129:
1126:
1123:
1120:
1109:
1093:
1089:
1085:
1082:
1079:
1075:
1071:
1060:
1048:
1044:
1040:
1037:
1033:
1029:
1026:
1022:
1018:
1015:
1012:
998:
986:
982:
978:
975:
972:
969:
960:, axiomatized
941:
937:
933:
922:
910:
906:
902:
899:
896:
856:
853:
850:
847:
816:
796:
776:
773:
770:
767:
764:
761:
758:
755:
731:
728:
708:
688:
685:
682:
679:
676:
673:
670:
667:
647:
627:
607:
604:
601:
577:
557:
542:Main article:
539:
536:
508:
505:
502:
482:
462:
459:
456:
453:
433:
413:
410:
407:
397:if and only if
384:
353:
325:
297:
285:
282:
242:
239:
220:
167:occurs when a
135:
134:
49:
47:
40:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5030:
5019:
5016:
5014:
5011:
5010:
5008:
4995:
4990:
4984:
4972:
4962:
4955:
4950:
4946:
4939:
4934:
4931:
4927:
4922:
4919:
4915:
4910:
4905:
4900:
4897:
4892:
4888:
4883:
4880:
4876:
4871:
4868:
4864:
4859:
4854:
4849:
4846:
4842:
4837:
4834:
4830:
4825:
4822:
4818:
4813:
4810:
4804:
4799:
4796:
4790:
4785:
4782:
4781:contradiction
4777:
4773:
4768:
4765:
4760:
4756:
4751:
4748:
4744:
4739:
4734:
4729:
4728:
4723:
4719:
4711:
4706:
4704:
4699:
4697:
4692:
4691:
4688:
4676:
4675:inconsistency
4673:
4671:
4670:contradiction
4668:
4666:
4662:
4659:
4658:
4656:
4652:
4646:
4643:
4641:
4638:
4636:
4633:
4632:
4630:
4626:
4621:
4611:
4608:⊨
4607:
4605:
4602:
4600:
4597:
4596:
4594:
4590:
4585:
4584:Logical truth
4577:
4572:
4570:
4565:
4563:
4558:
4557:
4554:
4544:
4543:
4538:
4530:
4524:
4521:
4519:
4516:
4514:
4511:
4509:
4506:
4502:
4499:
4498:
4497:
4494:
4492:
4489:
4487:
4484:
4482:
4478:
4475:
4473:
4470:
4468:
4465:
4463:
4460:
4458:
4455:
4454:
4452:
4448:
4442:
4439:
4437:
4434:
4432:
4431:Recursive set
4429:
4427:
4424:
4422:
4419:
4417:
4414:
4412:
4409:
4405:
4402:
4400:
4397:
4395:
4392:
4390:
4387:
4385:
4382:
4381:
4380:
4377:
4375:
4372:
4370:
4367:
4365:
4362:
4360:
4357:
4355:
4352:
4351:
4349:
4347:
4343:
4337:
4334:
4332:
4329:
4327:
4324:
4322:
4319:
4317:
4314:
4312:
4309:
4307:
4304:
4300:
4297:
4295:
4292:
4290:
4287:
4286:
4285:
4282:
4280:
4277:
4275:
4272:
4270:
4267:
4265:
4262:
4260:
4257:
4253:
4250:
4249:
4248:
4245:
4241:
4240:of arithmetic
4238:
4237:
4236:
4233:
4229:
4226:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4205:
4204:
4201:
4197:
4194:
4192:
4189:
4188:
4187:
4184:
4183:
4181:
4179:
4175:
4169:
4166:
4164:
4161:
4159:
4156:
4154:
4151:
4148:
4147:from ZFC
4144:
4141:
4139:
4136:
4130:
4127:
4126:
4125:
4122:
4120:
4117:
4115:
4112:
4111:
4110:
4107:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4085:
4082:
4080:
4077:
4076:
4074:
4072:
4068:
4058:
4057:
4053:
4052:
4047:
4046:non-Euclidean
4044:
4040:
4037:
4035:
4032:
4030:
4029:
4025:
4024:
4022:
4019:
4018:
4016:
4012:
4008:
4005:
4003:
4000:
3999:
3998:
3994:
3990:
3987:
3986:
3985:
3981:
3977:
3974:
3972:
3969:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3948:
3946:
3942:
3941:
3939:
3934:
3928:
3923:Example
3920:
3912:
3907:
3906:
3905:
3902:
3900:
3897:
3893:
3890:
3888:
3885:
3883:
3880:
3878:
3875:
3874:
3873:
3870:
3868:
3865:
3863:
3860:
3858:
3855:
3851:
3848:
3846:
3843:
3842:
3841:
3838:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3815:
3814:
3811:
3809:
3806:
3802:
3799:
3797:
3794:
3792:
3789:
3788:
3787:
3784:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3760:
3757:
3755:
3752:
3751:
3750:
3747:
3745:
3742:
3740:
3737:
3735:
3732:
3728:
3725:
3723:
3720:
3718:
3715:
3713:
3710:
3709:
3708:
3705:
3703:
3700:
3698:
3695:
3693:
3690:
3686:
3683:
3681:
3680:by definition
3678:
3677:
3676:
3673:
3669:
3666:
3665:
3664:
3661:
3659:
3656:
3654:
3651:
3649:
3646:
3644:
3641:
3640:
3637:
3634:
3632:
3628:
3623:
3617:
3613:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3578:
3575:
3573:
3570:
3568:
3567:Kripke–Platek
3565:
3563:
3560:
3556:
3553:
3551:
3548:
3547:
3546:
3543:
3542:
3540:
3536:
3528:
3525:
3524:
3523:
3520:
3518:
3515:
3511:
3508:
3507:
3506:
3503:
3501:
3498:
3496:
3493:
3491:
3488:
3486:
3483:
3480:
3476:
3472:
3469:
3465:
3462:
3460:
3457:
3455:
3452:
3451:
3450:
3446:
3443:
3442:
3440:
3438:
3434:
3430:
3422:
3419:
3417:
3414:
3412:
3411:constructible
3409:
3408:
3407:
3404:
3402:
3399:
3397:
3394:
3392:
3389:
3387:
3384:
3382:
3379:
3377:
3374:
3372:
3369:
3367:
3364:
3362:
3359:
3357:
3354:
3352:
3349:
3347:
3344:
3343:
3341:
3339:
3334:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3297:
3295:
3291:
3288:
3286:
3283:
3282:
3281:
3278:
3276:
3273:
3271:
3268:
3266:
3263:
3261:
3257:
3253:
3251:
3248:
3244:
3241:
3240:
3239:
3236:
3235:
3232:
3229:
3227:
3223:
3213:
3210:
3208:
3205:
3203:
3200:
3198:
3195:
3193:
3190:
3188:
3185:
3181:
3178:
3177:
3176:
3173:
3169:
3164:
3163:
3162:
3159:
3158:
3156:
3154:
3150:
3142:
3139:
3137:
3134:
3132:
3129:
3128:
3127:
3124:
3122:
3119:
3117:
3114:
3112:
3109:
3107:
3104:
3102:
3099:
3097:
3094:
3093:
3091:
3089:
3088:Propositional
3085:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3047:
3044:
3043:
3042:
3039:
3037:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3016:Logical truth
3014:
3012:
3009:
3008:
3006:
3004:
3000:
2997:
2995:
2991:
2985:
2982:
2980:
2977:
2975:
2972:
2970:
2967:
2965:
2962:
2960:
2956:
2952:
2948:
2946:
2943:
2941:
2938:
2936:
2932:
2929:
2928:
2926:
2924:
2918:
2913:
2907:
2904:
2902:
2899:
2897:
2894:
2892:
2889:
2887:
2884:
2882:
2879:
2877:
2874:
2872:
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2845:
2842:
2841:
2840:
2837:
2836:
2834:
2830:
2826:
2819:
2814:
2812:
2807:
2805:
2800:
2799:
2796:
2784:
2781:
2778:
2774:
2771:
2769:
2766:
2764:
2754:
2753:
2750:
2740:
2739:Logic symbols
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2716:
2714:
2710:
2704:
2701:
2699:
2696:
2694:
2691:
2690:
2688:
2686:
2682:
2679:
2675:
2669:
2666:
2664:
2661:
2659:
2656:
2654:
2651:
2649:
2646:
2644:
2641:
2639:
2636:
2634:
2631:
2629:
2626:
2624:
2621:
2619:
2618:Logical truth
2616:
2614:
2611:
2609:
2606:
2602:
2599:
2598:
2597:
2594:
2592:
2589:
2587:
2584:
2582:
2579:
2577:
2574:
2570:
2567:
2565:
2562:
2561:
2560:
2559:Contradiction
2557:
2555:
2552:
2550:
2547:
2545:
2542:
2540:
2537:
2536:
2534:
2530:
2520:
2517:
2515:
2512:
2510:
2507:
2505:
2504:Argumentation
2502:
2501:
2499:
2495:
2489:
2488:Philosophical
2486:
2484:
2483:Non-classical
2481:
2479:
2476:
2472:
2469:
2467:
2464:
2463:
2462:
2459:
2457:
2454:
2453:
2451:
2447:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2421:
2418:
2416:
2413:
2411:
2408:
2407:
2405:
2401:
2395:
2392:
2390:
2387:
2386:
2383:
2379:
2372:
2367:
2365:
2360:
2358:
2353:
2352:
2349:
2341:
2340:
2335:
2331:
2328:
2324:
2320:
2316:
2315:
2310:
2306:
2302:
2298:
2297:
2292:
2288:
2287:
2283:
2278:
2277:Gödel's Proof
2274:
2271:
2270:0-674-32449-8
2267:
2263:
2259:
2256:
2252:
2249:
2248:
2244:
2227:
2223:
2219:
2212:
2209:
2204:
2202:0-19-530658-9
2198:
2194:
2187:
2184:
2180:
2175:
2172:
2168:
2165:
2162:
2159:Emil L. Post
2156:
2153:
2147:
2144:
2138:
2135:
2129:
2126:
2120:
2117:
2106:
2099:
2092:
2089:
2085:
2081:
2075:
2072:
2061:
2057:
2051:
2048:
2044:
2040:
2036:
2035:
2030:
2026:
2022:
2021:
2015:
2009:
2006:
1995:
1991:
1985:
1982:
1971:
1967:
1961:
1958:
1947:
1943:
1936:
1933:
1926:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1886:
1883:
1880:
1877:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1848:
1845:
1839:
1836:
1833:
1830:
1829:
1825:
1823:
1821:
1816:
1814:
1811:to construct
1810:
1806:
1802:
1800:
1796:
1789:
1787:
1785:
1781:
1777:
1773:
1769:
1765:
1761:
1753:
1750:
1747:
1746:
1745:
1743:
1739:
1735:
1727:
1725:
1718:
1716:
1714:
1713:Graham Priest
1710:
1706:
1702:
1698:
1694:
1686:
1684:
1679:
1678:
1672:
1670:
1651:
1603:
1597:
1596:
1590:
1532:
1528:
1524:
1519:
1517:
1513:
1509:
1505:
1501:
1486:
1484:
1480:
1476:
1470:
1468:
1466:
1461:
1457:
1451:
1446:
1444:
1440:
1439:
1434:
1430:
1425:
1423:
1420:Moreover, it
1419:
1415:
1407:
1403:
1399:
1395:
1393:
1389:
1388:
1387:
1385:
1377:
1375:
1373:
1369:
1365:
1307:
1287:
1278:
1272:
1255:
1252:
1249:
1235:
1230:
1214:
1205:
1202:
1191:
1176:
1165:
1157:
1143:
1127:
1121:
1118:
1110:
1107:
1091:
1083:
1080:
1077:
1069:
1061:
1046:
1035:
1024:
1016:
1002:
1001:Peirce's rule
999:
984:
976:
970:
967:
959:
955:
939:
923:
908:
900:
886:
885:
884:
882:
877:
873:
872:minimal logic
868:
854:
848:
845:
838:
834:
830:
794:
771:
765:
756:
745:
729:
683:
677:
668:
645:
605:
602:
591:
575:
545:
537:
535:
533:
532:unsatisfiable
529:
524:
522:
506:
503:
480:
460:
454:
451:
431:
408:
405:
398:
382:
374:
370:
369:propositional
351:
343:
339:
315:
311:
283:
281:
276:
272:
270:
266:
262:
261:contradiction
258:
257:
252:
248:
240:
238:
235:
231:
209:
205:
201:
197:
192:
190:
186:
182:
178:
174:
170:
166:
165:contradiction
162:
154:
150:
146:
141:
131:
128:
120:
109:
106:
102:
99:
95:
92:
88:
85:
81:
78: –
77:
73:
72:Find sources:
66:
62:
56:
55:
50:This article
48:
44:
39:
38:
33:
19:
18:Contradictory
5013:Propositions
4908:
4857:
4780:
4737:
4669:
4660:
4640:formal proof
4533:
4331:Ultraproduct
4178:Model theory
4143:Independence
4079:Formal proof
4071:Proof theory
4054:
4027:
3984:real numbers
3956:second-order
3867:Substitution
3744:Metalanguage
3685:conservative
3658:Axiom schema
3602:Constructive
3572:Morse–Kelley
3538:Set theories
3517:Aleph number
3510:inaccessible
3416:Grothendieck
3300:intersection
3187:Higher-order
3175:Second-order
3121:Truth tables
3078:Venn diagram
2861:Formal proof
2658:Substitution
2558:
2478:Mathematical
2403:Major fields
2337:
2312:
2294:
2276:
2261:
2254:
2245:Bibliography
2230:. Retrieved
2225:
2221:
2211:
2192:
2186:
2174:
2166:
2163:
2155:
2146:
2137:
2128:
2119:
2108:. Retrieved
2104:
2091:
2074:
2063:. Retrieved
2059:
2050:
2032:
2024:
2017:
2013:
2008:
1997:. Retrieved
1993:
1984:
1973:. Retrieved
1969:
1960:
1950:, retrieved
1945:
1935:
1838:Auto-antonym
1817:
1803:
1793:
1757:
1731:
1722:
1711:, including
1709:dialetheists
1690:
1681:
1676:
1675:Definition.
1674:
1668:
1653:
1605:
1593:
1591:
1522:
1520:
1515:
1512:modus ponens
1508:substitution
1499:
1488:
1482:
1478:
1474:
1472:
1463:
1456:Ernest Nagel
1453:
1448:
1442:
1436:
1426:
1421:
1417:
1413:
1411:
1405:
1401:
1397:
1381:
1311:
957:
875:
869:
547:
525:
366:
341:
337:
278:
274:
265:Dionysodorus
260:
254:
244:
203:
196:formal logic
193:
179:beliefs and
177:disingenuous
164:
158:
123:
114:
104:
97:
90:
83:
71:
59:Please help
54:verification
51:
4959:existential
4599:truth value
4592:Functional:
4441:Type theory
4389:undecidable
4321:Truth value
4208:equivalence
3887:non-logical
3500:Enumeration
3490:Isomorphism
3437:cardinality
3421:Von Neumann
3386:Ultrafilter
3351:Uncountable
3285:equivalence
3202:Quantifiers
3192:Fixed-point
3161:First-order
3041:Consistency
3026:Proposition
3003:Traditional
2974:Lindström's
2964:Compactness
2906:Type theory
2851:Cardinality
2773:WikiProject
2643:Proposition
2638:Probability
2591:Description
2532:Foundations
1868:Doublethink
1856:Dialetheism
1809:mathematics
1807:is used in
1738:Hegelianism
1697:coherentism
1595:tautologous
1561:, if both S
1523:tautologous
1500:tautologous
1443:generalized
200:type theory
169:proposition
5007:Categories
4252:elementary
3945:arithmetic
3813:Quantifier
3791:functional
3663:Expression
3381:Transitive
3325:identities
3310:complement
3243:hereditary
3226:Set theory
2703:Set theory
2601:Linguistic
2596:Entailment
2586:Definition
2554:Consequent
2549:Antecedent
2228:(1): 29–48
2110:2019-12-10
2065:2019-12-10
2014:Euthydemus
1999:2020-08-14
1975:2020-08-14
1952:2019-12-10
1772:capitalism
1695:theory of
1687:Philosophy
1531:exhaustive
1516:consistent
256:Euthydemus
194:In modern
87:newspapers
4943:universal
4821:therefore
4809:therefore
4764:tautology
4610:tautology
4523:Supertask
4426:Recursion
4384:decidable
4218:saturated
4196:of models
4119:deductive
4114:axiomatic
4034:Hilbert's
4021:Euclidean
4002:canonical
3925:axiomatic
3857:Signature
3786:Predicate
3675:Extension
3597:Ackermann
3522:Operation
3401:Universal
3391:Recursive
3366:Singleton
3361:Inhabited
3346:Countable
3336:Types of
3320:power set
3290:partition
3207:Predicate
3153:Predicate
3068:Syllogism
3058:Soundness
3031:Inference
3021:Tautology
2923:paradoxes
2734:Fallacies
2729:Paradoxes
2719:Logicians
2653:Statement
2648:Reference
2613:Induction
2576:Deduction
2539:Abduction
2509:Metalogic
2456:Classical
2420:Inference
2319:EMS Press
2301:EMS Press
1914:Tautology
1776:socialist
1768:synthesis
1764:sublation
1760:dialectic
1638:places ~S
1533:classes K
1514:, then a
1429:Emil Post
1340:↔
1285:¬
1279:∨
1270:¬
1263:⟺
1253:∧
1244:¬
1212:¬
1209:¬
1206:∨
1200:¬
1173:⟹
1162:⟹
1155:¬
1125:¬
1122:∨
1088:⟹
1081:∨
1074:⟹
1043:⟹
1032:⟹
1021:⟹
981:⟹
974:¬
971:∧
936:⟹
932:⊥
905:⟹
898:¬
895:¬
852:¬
849:∨
815:Σ
795:φ
775:⊥
772:⊢
766:φ
763:¬
757:∪
754:Σ
730:φ
727:¬
707:Σ
687:⊥
684:⊢
678:φ
675:¬
669:∪
666:Σ
646:φ
626:Σ
606:φ
603:⊢
600:Σ
576:φ
556:Σ
507:ψ
504:⊢
501:⊥
493:(because
481:ψ
461:ψ
458:→
455:φ
452:⊢
432:φ
412:⊥
409:⊢
406:φ
383:φ
352:⊢
324:⊤
296:⊥
219:⊥
185:Aristotle
117:July 2008
4896:superset
4807:entails,
4793:entails,
4508:Logicism
4501:timeline
4477:Concrete
4336:Validity
4306:T-schema
4299:Kripke's
4294:Tarski's
4289:semantic
4279:Strength
4228:submodel
4223:spectrum
4191:function
4039:Tarski's
4028:Elements
4015:geometry
3971:Robinson
3892:variable
3877:function
3850:spectrum
3840:Sentence
3796:variable
3739:Language
3692:Relation
3653:Automata
3643:Alphabet
3627:language
3481:-jection
3459:codomain
3445:Function
3406:Universe
3376:Infinite
3280:Relation
3063:Validity
3053:Argument
2951:theorem,
2768:Category
2668:Validity
2569:Antinomy
2497:Theories
2461:Informal
2041:, Inc.,
1896:Oxymoron
1826:See also
1569:are in K
1467:'s Proof
528:complete
473:for all
269:Socrates
4912:
4891:implies
4879:implies
4861:
4833:because
4741:
4716:Common
4645:theorem
4628:Formal:
4586: ⊤
4450:Related
4247:Diagram
4145: (
4124:Hilbert
4109:Systems
4104:Theorem
3982:of the
3927:systems
3707:Formula
3702:Grammar
3618: (
3562:General
3275:Forcing
3260:Element
3180:Monadic
2955:paradox
2896:Theorem
2832:General
2783:changes
2775: (
2633:Premise
2564:Paradox
2394:History
2389:Outline
2336:(ed.).
2321:, 2001
2303:, 2001
2043:Chicago
2012:Dialog
1908:Paradox
1475:outside
831:called
699:(i.e.,
638:proves
618:(i.e.,
316:symbol
247:paradox
241:History
210:symbol
147:in the
101:scholar
4906:
4855:
4795:proves
4735:
4663:
4635:theory
4213:finite
3976:Skolem
3929:
3904:Theory
3872:Symbol
3862:String
3845:atomic
3722:ground
3717:closed
3712:atomic
3668:ground
3631:syntax
3527:binary
3454:domain
3371:Finite
3136:finite
2994:Logics
2953:
2901:Theory
2685:topics
2471:Reason
2449:Logics
2440:Syntax
2272:(pbk.)
2268:
2232:28 May
2199:
1813:proofs
1705:system
1701:belief
1450:given.
1372:Q.E.D.
1343:
870:Using
310:falsum
208:falsum
204:single
103:
96:
89:
82:
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4776:false
4743:&
4665:false
4203:Model
3951:Peano
3808:Proof
3648:Arity
3577:Naive
3464:image
3396:Fuzzy
3356:Empty
3305:union
3250:Class
2891:Model
2881:Lemma
2839:Axiom
2712:other
2677:Lists
2663:Truth
2430:Proof
2378:Logic
2332:. In
2253:1960
2101:(PDF)
2025:Plato
2016:from
1879:Irony
1766:, or
1622:and K
1565:and S
1545:and S
1537:and K
1465:Gödel
1427:When
1422:seems
1364:false
1332:, ⊥,
746:that
744:proof
592:that
526:In a
251:Plato
234:false
108:JSTOR
94:books
4930:nand
4759:true
4326:Type
4129:list
3933:list
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3899:Term
3833:rank
3727:open
3621:list
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3338:sets
3197:Free
3167:list
2917:list
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2777:talk
2623:Name
2608:Form
2266:ISBN
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2197:ISBN
1920:TRIZ
1818:The
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1634:to S
1626:to S
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1529:and
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719:and
371:and
198:and
181:bias
173:fact
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314:tee
253:'s
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4887:⊃
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