2140:, when conducted in a Fitch style deduction, proceeds by entering a new sub-derivation while substituting an existentially quantified variable for a subject—which does not appear within any active sub-derivation. If a conclusion can be reached within this sub-derivation in which the substituted subject does not appear, then one can exit that sub-derivation with that conclusion. The reasoning behind existential elimination (∃E) is as follows: If it is given that there exists an element for which the proposition function is true, and if a conclusion can be reached by giving that element an arbitrary name, that conclusion is
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2065:(∃I) concludes that, if the propositional function is known to be true for a particular element of the domain of discourse, then it must be true that there exists an element for which the proposition function is true. Symbolically,
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1547:{\displaystyle \lnot \ \exists {x}{\in }\mathbf {X} \,P(x)\equiv \ \forall {x}{\in }\mathbf {X} \,\lnot P(x)\not \equiv \ \lnot \ \forall {x}{\in }\mathbf {X} \,P(x)\equiv \ \exists {x}{\in }\mathbf {X} \,\lnot P(x)}
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A common error is stating "all persons are not married" (i.e., "there exists no person who is married"), when "not all persons are married" (i.e., "there exists a person who is not married") is intended:
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462:, or... and so on," but more precise, because it doesn't need us to infer the meaning of the phrase "and so on." (In particular, the sentence explicitly specifies its
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If there is no element of the domain of discourse for which the statement is true, then it must be false for all of those elements. That is, the negation of
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is a rule justifying a logical step from hypothesis to conclusion. There are several rules of inference which utilize the existential quantifier.
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1786:{\displaystyle \exists {x}{\in }\mathbf {X} \,P(x)\lor Q(x)\to \ (\exists {x}{\in }\mathbf {X} \,P(x)\lor \exists {x}{\in }\mathbf {X} \,Q(x))}
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popularised its use as the existential quantifier. Through his research in set theory, Peano also introduced the symbols
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This can be demonstrated to be false. Truthfully, it must be said, "It is not the case that there is a natural number
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This is a single statement using existential quantification. It is roughly analogous to the informal sentence "Either
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A quantified propositional function is a statement; thus, like statements, quantified functions can be negated. The
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1356:{\displaystyle \lnot \ \exists {x}{\in }\mathbf {X} \,P(x)\equiv \ \forall {x}{\in }\mathbf {X} \,\lnot P(x)}
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1641:{\displaystyle \nexists {x}{\in }\mathbf {X} \,P(x)\equiv \lnot \ \exists {x}{\in }\mathbf {X} \,P(x)}
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are used to restrict the domain of discourse to fulfill a given predicate. For example, the sentence
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Unlike the universal quantifier, the existential quantifier distributes over logical disjunctions:
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font, Unicode U+2203) is used to indicate existential quantification. For example, the notation
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is not arbitrary, and is instead a specific element of the domain of discourse, then stating
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This particular example is true, because 5 is a natural number, and when we substitute 5 for
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Negation is also expressible through a statement of "for no", as opposed to "for some":
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as "there exists", "there is at least one", or "for some". It is usually denoted by the
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of all natural numbers, the existential quantification "There exists a natural number
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is allowed to take, is therefore critical to a statement's trueness or falseness.
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2248:{\displaystyle \exists {x}{\in }\mathbf {X} \,P(x)\to \ ((P(c)\to \ Q)\to \ Q)}
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of an existential statement about "some" object may be achieved either by a
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which is greater than 0 and less than 1" can be symbolically stated as:
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223:∃, which, when used together with a predicate variable, is called an
24:
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is greater than 0 and less than 1", then, for a domain of discourse
766:, which exhibits an object satisfying the "some" statement, or by a
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2964:
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254:("for all"), which asserts that the property or relation holds for
30:"∄" redirects here. For the Ukrainian nightclub of that name, see
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2798:
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283:
2922:
2315:) might unjustifiably give more information about that object.
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is enough to prove this existential quantification to be true.
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3229:
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2127:{\displaystyle P(a)\to \ \exists {x}{\in }\mathbf {X} \,P(x)}
291:
1965:
1923:
1889:
1844:
1128:{\displaystyle \lnot \ \exists {x}{\in }\mathbf {X} \,P(x)}
1078:
that is greater than 0 and less than 1", or, symbolically:
961:
to respectively denote the intersection and union of sets.
836:{\displaystyle \exists {n}{\in }\mathbb {N} :n\times n=25}
20:
1915:
1266:
of that propositional function's negation; symbolically,
1248:{\displaystyle \forall {x}{\in }\mathbf {X} \,\lnot P(x)}
1883:
1996:
1951:
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265:
Quantification in general is covered in the article on
2443:, the existential quantifier can be understood as the
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603:" is false, because there are no even solutions. The
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372:
336:
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71:
2362:{\displaystyle \exists {x}{\in }\varnothing \,P(x)}
1193:is logically equivalent to "For any natural number
1183:{\displaystyle \exists {x}{\in }\mathbf {X} \,P(x)}
1064:{\displaystyle \exists {x}{\in }\mathbf {X} \,P(x)}
2459:functor of a function between sets; likewise, the
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1909:
466:to be the natural numbers, not, for example, the
258:members of the domain. Some sources use the term
19:"∃" redirects here. For the letter turned E, see
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1989:
1958:
1850:
250:). Existential quantification is distinct from
3045:
2728:
2003:
1972:
1870:
1863:
907:The symbol's first usage is thought to be by
2625:
1201:is not greater than 0 and less than 1", or:
269:. The existential quantifier is encoded as
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3052:
3038:
2742:
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2721:
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2431:Universal quantification § As adjoint
607:, which specifies the values the variable
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1167:
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811:
2525:
262:to refer to existential quantification.
2619:
4790:
3059:
2303:; else, the logic does not follow: If
3033:
2716:
2417:) – exist in the empty set. See also
1795:
2599:Allen, Colin; Hand, Michael (2001).
2632:. Springer Cham. pp. 210–211.
1262:'s existential quantification is a
1258:Generally, then, the negation of a
997:symbol is used to denote negation.
13:
2695:Fundamentals of Mathematical Logic
2405:of any description – let alone an
2330:
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2093:
1749:
1715:
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163:
129:is true for at least one value of
72:
16:Mathematical use of "there exists"
14:
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2343:
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1729:
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843:represents the (true) statement
477:, we produce the true statement
2369:is always false, regardless of
2295:must be true for all values of
23:. For the Japanese kana ヨ, see
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178:
172:
116:
110:
87:
81:
1:
4732:History of mathematical logic
2685:
2665:Sheaves in Geometry and Logic
2424:
2409:fulfilling a given predicate
1366:(This is a generalization of
964:
619:For some positive odd number
294:and related formula editors.
184:{\displaystyle \exists xP(x)}
93:{\displaystyle \exists xP(x)}
4657:Primitive recursive function
2552:"Predicates and Quantifiers"
2390:{\displaystyle \varnothing }
896:{\displaystyle n\times n=25}
747:{\displaystyle n\times n=25}
664:{\displaystyle n\times n=25}
596:{\displaystyle n\times n=25}
534:{\displaystyle n\times n=25}
502:{\displaystyle 5\times 5=25}
455:{\displaystyle 2\times 2=25}
423:{\displaystyle 1\times 1=25}
391:{\displaystyle 0\times 0=25}
355:{\displaystyle n\times n=25}
7:
2470:
2288:{\displaystyle P(c)\to \ Q}
969:
773:
509:. It does not matter that "
10:
4819:
3721:Schröder–Bernstein theorem
3448:Monadic predicate calculus
3107:Foundations of mathematics
2499:– for the unicode symbol ∃
2428:
2040:Existential generalization
1845:Biconditional introduction
202:existential quantification
38:Existential quantification
29:
18:
4767:
4754:Philosophy of mathematics
4703:Automated theorem proving
4685:
4580:
4412:
4305:
4157:
3874:
3850:
3828:Von Neumann–Bernays–Gödel
3773:
3667:
3571:
3469:
3460:
3387:
3322:
3228:
3150:
3067:
3000:
2751:
2663:, Ieke Moerdijk, (1992):
2638:10.1007/978-3-319-71350-2
2526:Bergmann, Merrie (2014).
2508:Uniqueness quantification
2138:Existential instantiation
297:
153:
62:
52:
42:
2513:
2062:Existential introduction
2031:Universal generalization
1871:Disjunction introduction
1858:Conjunction introduction
1828:Implication introduction
1264:universal quantification
990:{\displaystyle \lnot \ }
782:, "∃" (a turned letter "
682:For some natural number
310:For some natural number
252:universal quantification
4404:Self-verifying theories
4225:Tarski's axiomatization
3176:Tarski's undefinability
3171:incompleteness theorems
548:In contrast, "For some
4778:Mathematics portal
4389:Proof of impossibility
4037:propositional variable
3347:Propositional calculus
3021:Mathematics portal
2421:for more information.
2391:
2363:
2289:
2249:
2148:and for a proposition
2128:
1890:hypothetical syllogism
1811:Propositional calculus
1787:
1642:
1548:
1357:
1260:propositional function
1249:
1184:
1129:
1065:
991:
955:
935:
914:Formulario mathematico
897:
861:
837:
748:
716:
696:
665:
633:
597:
565:
535:
503:
456:
424:
392:
356:
324:
267:quantification (logic)
225:existential quantifier
185:
143:
123:
94:
4647:Kolmogorov complexity
4600:Computably enumerable
4500:Model complete theory
4292:Principia Mathematica
3352:Propositional formula
3181:Banach–Tarski paradox
3010:Philosophy portal
2626:Stephen Webb (2018).
2497:List of logic symbols
2392:
2364:
2299:over the same domain
2290:
2250:
2129:
1932:Negation introduction
1925:modus ponendo tollens
1788:
1643:
1549:
1370:to predicate logic.)
1358:
1250:
1185:
1130:
1066:
992:
956:
954:{\displaystyle \cup }
936:
934:{\displaystyle \cap }
898:
862:
838:
768:nonconstructive proof
749:
717:
697:
666:
634:
598:
566:
536:
504:
457:
425:
393:
357:
325:
186:
144:
124:
95:
4595:Church–Turing thesis
4582:Computability theory
3791:continuum hypothesis
3309:Square of opposition
3167:Gödel's completeness
2492:Lindström quantifier
2461:universal quantifier
2381:
2327:
2261:
2163:
2072:
1990:Material implication
1941:Rules of replacement
1804:Transformation rules
1657:
1564:
1381:
1273:
1208:
1146:
1085:
1027:
1008:) is the predicate "
978:
945:
925:
917:(1896). Afterwards,
875:
851:
794:
726:
706:
686:
676:logically equivalent
643:
623:
613:Logical conjunctions
575:
555:
513:
481:
434:
402:
370:
334:
314:
160:
133:
122:{\displaystyle P(x)}
104:
69:
4749:Mathematical object
4640:P versus NP problem
4605:Computable function
4399:Reverse mathematics
4325:Logical consequence
4202:primitive recursive
4197:elementary function
3970:Free/bound variable
3823:Tarski–Grothendieck
3342:Logical connectives
3272:Logical equivalence
3122:Logical consequence
2693:Hinman, P. (2005).
2503:Quantifier variance
2377:). This is because
1903:destructive dilemma
605:domain of discourse
464:domain of discourse
39:
4803:Quantifier (logic)
4547:Transfer principle
4510:Semantics of logic
4495:Categorical theory
4471:Non-standard model
3985:Logical connective
3112:Information theory
3061:Mathematical logic
2477:Existential clause
2439:and the theory of
2387:
2359:
2285:
2245:
2124:
2022:Rules of inference
1818:Rules of inference
1796:Rules of inference
1783:
1638:
1544:
1353:
1245:
1180:
1125:
1061:
987:
951:
931:
893:
857:
847:There exists some
833:
764:constructive proof
760:mathematical proof
744:
712:
692:
661:
629:
593:
561:
531:
499:
452:
420:
388:
352:
320:
260:existentialization
181:
154:Symbolic statement
139:
119:
90:
57:Mathematical logic
37:
4785:
4784:
4717:Abstract category
4520:Theories of truth
4330:Rule of inference
4320:Natural deduction
4301:
4300:
3846:
3845:
3551:Cartesian product
3456:
3455:
3362:Many-valued logic
3337:Boolean functions
3220:Russell's paradox
3195:diagonal argument
3092:First-order logic
3027:
3026:
2995:
2994:
2661:Saunders Mac Lane
2647:978-3-319-71349-6
2576:"1.2 Quantifiers"
2537:978-0-07-803841-9
2487:First-order logic
2482:Existence theorem
2281:
2238:
2226:
2202:
2156:does not appear:
2092:
2055:rule of inference
2051:
2050:
1711:
1606:
1509:
1472:
1466:
1426:
1389:
1318:
1281:
1093:
986:
860:{\displaystyle n}
715:{\displaystyle n}
695:{\displaystyle n}
632:{\displaystyle n}
564:{\displaystyle n}
323:{\displaystyle n}
194:
193:
142:{\displaystyle x}
4810:
4776:
4775:
4727:History of logic
4722:Category of sets
4615:Decision problem
4394:Ordinal analysis
4335:Sequent calculus
4233:Boolean algebras
4173:
4172:
4147:
4118:logical/constant
3872:
3871:
3858:
3781:Zermelo–Fraenkel
3532:Set operations:
3467:
3466:
3404:
3235:
3234:
3215:Löwenheim–Skolem
3102:Formal semantics
3054:
3047:
3040:
3031:
3030:
3019:
3018:
3008:
3007:
3006:
2938:
2887:
2767:
2754:
2753:
2737:
2730:
2723:
2714:
2713:
2708:
2680:
2667:Springer-Verlag
2658:
2652:
2651:
2629:Clash of Symbols
2623:
2617:
2616:
2596:
2590:
2589:
2587:
2586:
2572:
2566:
2565:
2563:
2562:
2556:www.csm.ornl.gov
2548:
2542:
2541:
2523:
2441:elementary topoi
2396:
2394:
2393:
2388:
2368:
2366:
2365:
2360:
2342:
2337:
2294:
2292:
2291:
2286:
2279:
2254:
2252:
2251:
2246:
2236:
2224:
2200:
2183:
2178:
2173:
2142:necessarily true
2133:
2131:
2130:
2125:
2110:
2105:
2100:
2090:
2005:
1998:
1991:
1979:De Morgan's laws
1974:
1967:
1960:
1953:
1927:
1919:
1911:
1904:
1898:
1891:
1885:
1878:
1872:
1865:
1859:
1852:
1846:
1839:
1829:
1800:
1799:
1792:
1790:
1789:
1784:
1766:
1761:
1756:
1732:
1727:
1722:
1709:
1677:
1672:
1667:
1647:
1645:
1644:
1639:
1624:
1619:
1614:
1604:
1584:
1579:
1574:
1553:
1551:
1550:
1545:
1527:
1522:
1517:
1507:
1490:
1485:
1480:
1470:
1464:
1444:
1439:
1434:
1424:
1407:
1402:
1397:
1387:
1368:De Morgan's laws
1362:
1360:
1359:
1354:
1336:
1331:
1326:
1316:
1299:
1294:
1289:
1279:
1254:
1252:
1251:
1246:
1228:
1223:
1218:
1189:
1187:
1186:
1181:
1166:
1161:
1156:
1134:
1132:
1131:
1126:
1111:
1106:
1101:
1091:
1070:
1068:
1067:
1062:
1047:
1042:
1037:
1000:For example, if
996:
994:
993:
988:
984:
960:
958:
957:
952:
940:
938:
937:
932:
919:Bertrand Russell
902:
900:
899:
894:
866:
864:
863:
858:
842:
840:
839:
834:
814:
809:
804:
753:
751:
750:
745:
721:
719:
718:
713:
701:
699:
698:
693:
678:to the sentence
670:
668:
667:
662:
638:
636:
635:
630:
602:
600:
599:
594:
570:
568:
567:
562:
540:
538:
537:
532:
508:
506:
505:
500:
461:
459:
458:
453:
429:
427:
426:
421:
397:
395:
394:
389:
361:
359:
358:
353:
329:
327:
326:
321:
289:
281:
278:
275:
273:
249:
241:
233:
218:logical operator
210:logical constant
190:
188:
187:
182:
148:
146:
145:
140:
128:
126:
125:
120:
99:
97:
96:
91:
40:
36:
4818:
4817:
4813:
4812:
4811:
4809:
4808:
4807:
4788:
4787:
4786:
4781:
4770:
4763:
4708:Category theory
4698:Algebraic logic
4681:
4652:Lambda calculus
4590:Church encoding
4576:
4552:Truth predicate
4408:
4374:Complete theory
4297:
4166:
4162:
4158:
4153:
4145:
3865: and
3861:
3856:
3842:
3818:New Foundations
3786:axiom of choice
3769:
3731:Gödel numbering
3671: and
3663:
3567:
3452:
3402:
3383:
3332:Boolean algebra
3318:
3282:Equiconsistency
3247:Classical logic
3224:
3205:Halting problem
3193: and
3169: and
3157: and
3156:
3151:Theorems (
3146:
3063:
3058:
3028:
3023:
3013:
3012:
3004:
3002:
2996:
2991:
2990:
2987:
2983:
2975:
2974:
2971:
2967:
2959:
2955:
2947:
2943:
2934:
2925:
2921:
2916:
2908:
2904:
2896:
2892:
2883:
2874:
2870:
2862:
2858:
2850:
2846:
2838:
2835:
2832:
2824:
2821:
2818:
2810:
2806:
2801:
2793:
2789:
2784:
2776:
2772:
2763:
2747:
2745:logical symbols
2741:
2711:
2705:
2688:
2683:
2659:
2655:
2648:
2624:
2620:
2613:
2597:
2593:
2584:
2582:
2580:www.whitman.edu
2574:
2573:
2569:
2560:
2558:
2550:
2549:
2545:
2538:
2530:. McGraw Hill.
2524:
2520:
2516:
2473:
2437:category theory
2433:
2427:
2382:
2379:
2378:
2338:
2333:
2328:
2325:
2324:
2321:
2262:
2259:
2258:
2179:
2174:
2169:
2164:
2161:
2160:
2106:
2101:
2096:
2073:
2070:
2069:
2015:Predicate logic
2009:
1973:Double negation
1827:
1798:
1762:
1757:
1752:
1728:
1723:
1718:
1673:
1668:
1663:
1658:
1655:
1654:
1620:
1615:
1610:
1580:
1575:
1570:
1565:
1562:
1561:
1523:
1518:
1513:
1486:
1481:
1476:
1440:
1435:
1430:
1403:
1398:
1393:
1382:
1379:
1378:
1332:
1327:
1322:
1295:
1290:
1285:
1274:
1271:
1270:
1224:
1219:
1214:
1209:
1206:
1205:
1162:
1157:
1152:
1147:
1144:
1143:
1107:
1102:
1097:
1086:
1083:
1082:
1043:
1038:
1033:
1028:
1025:
1024:
979:
976:
975:
972:
967:
946:
943:
942:
926:
923:
922:
876:
873:
872:
869:natural numbers
852:
849:
848:
810:
805:
800:
795:
792:
791:
776:
727:
724:
723:
707:
704:
703:
687:
684:
683:
644:
641:
640:
624:
621:
620:
576:
573:
572:
556:
553:
552:
514:
511:
510:
482:
479:
478:
435:
432:
431:
403:
400:
399:
371:
368:
367:
335:
332:
331:
315:
312:
311:
300:
287:
279:
276:
271:
270:
243:
235:
228:
198:predicate logic
161:
158:
157:
134:
131:
130:
105:
102:
101:
70:
67:
66:
35:
32:K41 (nightclub)
28:
17:
12:
11:
5:
4816:
4806:
4805:
4800:
4783:
4782:
4768:
4765:
4764:
4762:
4761:
4756:
4751:
4746:
4741:
4740:
4739:
4729:
4724:
4719:
4710:
4705:
4700:
4695:
4693:Abstract logic
4689:
4687:
4683:
4682:
4680:
4679:
4674:
4672:Turing machine
4669:
4664:
4659:
4654:
4649:
4644:
4643:
4642:
4637:
4632:
4627:
4622:
4612:
4610:Computable set
4607:
4602:
4597:
4592:
4586:
4584:
4578:
4577:
4575:
4574:
4569:
4564:
4559:
4554:
4549:
4544:
4539:
4538:
4537:
4532:
4527:
4517:
4512:
4507:
4505:Satisfiability
4502:
4497:
4492:
4491:
4490:
4480:
4479:
4478:
4468:
4467:
4466:
4461:
4456:
4451:
4446:
4436:
4435:
4434:
4429:
4422:Interpretation
4418:
4416:
4410:
4409:
4407:
4406:
4401:
4396:
4391:
4386:
4376:
4371:
4370:
4369:
4368:
4367:
4357:
4352:
4342:
4337:
4332:
4327:
4322:
4317:
4311:
4309:
4303:
4302:
4299:
4298:
4296:
4295:
4287:
4286:
4285:
4284:
4279:
4278:
4277:
4272:
4267:
4247:
4246:
4245:
4243:minimal axioms
4240:
4229:
4228:
4227:
4216:
4215:
4214:
4209:
4204:
4199:
4194:
4189:
4176:
4174:
4155:
4154:
4152:
4151:
4150:
4149:
4137:
4132:
4131:
4130:
4125:
4120:
4115:
4105:
4100:
4095:
4090:
4089:
4088:
4083:
4073:
4072:
4071:
4066:
4061:
4056:
4046:
4041:
4040:
4039:
4034:
4029:
4019:
4018:
4017:
4012:
4007:
4002:
3997:
3992:
3982:
3977:
3972:
3967:
3966:
3965:
3960:
3955:
3950:
3940:
3935:
3933:Formation rule
3930:
3925:
3924:
3923:
3918:
3908:
3907:
3906:
3896:
3891:
3886:
3881:
3875:
3869:
3852:Formal systems
3848:
3847:
3844:
3843:
3841:
3840:
3835:
3830:
3825:
3820:
3815:
3810:
3805:
3800:
3795:
3794:
3793:
3788:
3777:
3775:
3771:
3770:
3768:
3767:
3766:
3765:
3755:
3750:
3749:
3748:
3741:Large cardinal
3738:
3733:
3728:
3723:
3718:
3704:
3703:
3702:
3697:
3692:
3677:
3675:
3665:
3664:
3662:
3661:
3660:
3659:
3654:
3649:
3639:
3634:
3629:
3624:
3619:
3614:
3609:
3604:
3599:
3594:
3589:
3584:
3578:
3576:
3569:
3568:
3566:
3565:
3564:
3563:
3558:
3553:
3548:
3543:
3538:
3530:
3529:
3528:
3523:
3513:
3508:
3506:Extensionality
3503:
3501:Ordinal number
3498:
3488:
3483:
3482:
3481:
3470:
3464:
3458:
3457:
3454:
3453:
3451:
3450:
3445:
3440:
3435:
3430:
3425:
3420:
3419:
3418:
3408:
3407:
3406:
3393:
3391:
3385:
3384:
3382:
3381:
3380:
3379:
3374:
3369:
3359:
3354:
3349:
3344:
3339:
3334:
3328:
3326:
3320:
3319:
3317:
3316:
3311:
3306:
3301:
3296:
3291:
3286:
3285:
3284:
3274:
3269:
3264:
3259:
3254:
3249:
3243:
3241:
3232:
3226:
3225:
3223:
3222:
3217:
3212:
3207:
3202:
3197:
3185:Cantor's
3183:
3178:
3173:
3163:
3161:
3148:
3147:
3145:
3144:
3139:
3134:
3129:
3124:
3119:
3114:
3109:
3104:
3099:
3094:
3089:
3084:
3083:
3082:
3071:
3069:
3065:
3064:
3057:
3056:
3049:
3042:
3034:
3025:
3024:
3001:
2998:
2997:
2993:
2992:
2988:quantification
2984:
2979:
2978:
2976:
2972:quantification
2968:
2963:
2962:
2960:
2951:
2950:
2948:
2929:
2928:
2926:
2912:
2911:
2909:
2900:
2899:
2897:
2878:
2877:
2875:
2866:
2865:
2863:
2854:
2853:
2851:
2842:
2841:
2839:
2828:
2827:
2825:
2814:
2813:
2811:
2797:
2796:
2794:
2780:
2779:
2777:
2758:
2757:
2752:
2749:
2748:
2740:
2739:
2732:
2725:
2717:
2710:
2709:
2703:
2697:. A K Peters.
2689:
2687:
2684:
2682:
2681:
2653:
2646:
2618:
2611:
2591:
2567:
2543:
2536:
2528:The Logic Book
2517:
2515:
2512:
2511:
2510:
2505:
2500:
2494:
2489:
2484:
2479:
2472:
2469:
2429:Main article:
2426:
2423:
2386:
2358:
2355:
2352:
2349:
2345:
2341:
2336:
2332:
2320:
2317:
2284:
2278:
2275:
2272:
2269:
2266:
2256:
2255:
2244:
2241:
2235:
2232:
2229:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2199:
2196:
2193:
2190:
2187:
2182:
2177:
2172:
2168:
2135:
2134:
2123:
2120:
2117:
2114:
2109:
2104:
2099:
2095:
2089:
2086:
2083:
2080:
2077:
2049:
2048:
2047:
2046:
2037:
2025:
2024:
2018:
2017:
2011:
2010:
2008:
2007:
2000:
1993:
1986:
1981:
1976:
1969:
1966:Distributivity
1962:
1955:
1947:
1944:
1943:
1937:
1936:
1935:
1934:
1929:
1906:
1893:
1880:
1867:
1854:
1841:
1821:
1820:
1814:
1813:
1807:
1806:
1797:
1794:
1782:
1779:
1776:
1773:
1770:
1765:
1760:
1755:
1751:
1748:
1745:
1742:
1739:
1736:
1731:
1726:
1721:
1717:
1714:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1676:
1671:
1666:
1662:
1649:
1648:
1637:
1634:
1631:
1628:
1623:
1618:
1613:
1609:
1603:
1600:
1597:
1594:
1591:
1588:
1583:
1578:
1573:
1569:
1555:
1554:
1543:
1540:
1537:
1534:
1531:
1526:
1521:
1516:
1512:
1506:
1503:
1500:
1497:
1494:
1489:
1484:
1479:
1475:
1469:
1463:
1460:
1457:
1454:
1451:
1448:
1443:
1438:
1433:
1429:
1423:
1420:
1417:
1414:
1411:
1406:
1401:
1396:
1392:
1386:
1364:
1363:
1352:
1349:
1346:
1343:
1340:
1335:
1330:
1325:
1321:
1315:
1312:
1309:
1306:
1303:
1298:
1293:
1288:
1284:
1278:
1256:
1255:
1244:
1241:
1238:
1235:
1232:
1227:
1222:
1217:
1213:
1191:
1190:
1179:
1176:
1173:
1170:
1165:
1160:
1155:
1151:
1137:
1136:
1124:
1121:
1118:
1115:
1110:
1105:
1100:
1096:
1090:
1072:
1071:
1060:
1057:
1054:
1051:
1046:
1041:
1036:
1032:
983:
971:
968:
966:
963:
950:
930:
909:Giuseppe Peano
905:
904:
892:
889:
886:
883:
880:
867:in the set of
856:
832:
829:
826:
823:
820:
817:
813:
808:
803:
799:
780:symbolic logic
775:
772:
756:
755:
743:
740:
737:
734:
731:
711:
691:
672:
671:
660:
657:
654:
651:
648:
628:
592:
589:
586:
583:
580:
560:
530:
527:
524:
521:
518:
498:
495:
492:
489:
486:
451:
448:
445:
442:
439:
419:
416:
413:
410:
407:
387:
384:
381:
378:
375:
364:
363:
351:
348:
345:
342:
339:
319:
299:
296:
192:
191:
180:
177:
174:
171:
168:
165:
155:
151:
150:
138:
118:
115:
112:
109:
89:
86:
83:
80:
77:
74:
64:
60:
59:
54:
50:
49:
44:
15:
9:
6:
4:
3:
2:
4815:
4804:
4801:
4799:
4798:Logic symbols
4796:
4795:
4793:
4780:
4779:
4774:
4766:
4760:
4757:
4755:
4752:
4750:
4747:
4745:
4742:
4738:
4735:
4734:
4733:
4730:
4728:
4725:
4723:
4720:
4718:
4714:
4711:
4709:
4706:
4704:
4701:
4699:
4696:
4694:
4691:
4690:
4688:
4684:
4678:
4675:
4673:
4670:
4668:
4667:Recursive set
4665:
4663:
4660:
4658:
4655:
4653:
4650:
4648:
4645:
4641:
4638:
4636:
4633:
4631:
4628:
4626:
4623:
4621:
4618:
4617:
4616:
4613:
4611:
4608:
4606:
4603:
4601:
4598:
4596:
4593:
4591:
4588:
4587:
4585:
4583:
4579:
4573:
4570:
4568:
4565:
4563:
4560:
4558:
4555:
4553:
4550:
4548:
4545:
4543:
4540:
4536:
4533:
4531:
4528:
4526:
4523:
4522:
4521:
4518:
4516:
4513:
4511:
4508:
4506:
4503:
4501:
4498:
4496:
4493:
4489:
4486:
4485:
4484:
4481:
4477:
4476:of arithmetic
4474:
4473:
4472:
4469:
4465:
4462:
4460:
4457:
4455:
4452:
4450:
4447:
4445:
4442:
4441:
4440:
4437:
4433:
4430:
4428:
4425:
4424:
4423:
4420:
4419:
4417:
4415:
4411:
4405:
4402:
4400:
4397:
4395:
4392:
4390:
4387:
4384:
4383:from ZFC
4380:
4377:
4375:
4372:
4366:
4363:
4362:
4361:
4358:
4356:
4353:
4351:
4348:
4347:
4346:
4343:
4341:
4338:
4336:
4333:
4331:
4328:
4326:
4323:
4321:
4318:
4316:
4313:
4312:
4310:
4308:
4304:
4294:
4293:
4289:
4288:
4283:
4282:non-Euclidean
4280:
4276:
4273:
4271:
4268:
4266:
4265:
4261:
4260:
4258:
4255:
4254:
4252:
4248:
4244:
4241:
4239:
4236:
4235:
4234:
4230:
4226:
4223:
4222:
4221:
4217:
4213:
4210:
4208:
4205:
4203:
4200:
4198:
4195:
4193:
4190:
4188:
4185:
4184:
4182:
4178:
4177:
4175:
4170:
4164:
4159:Example
4156:
4148:
4143:
4142:
4141:
4138:
4136:
4133:
4129:
4126:
4124:
4121:
4119:
4116:
4114:
4111:
4110:
4109:
4106:
4104:
4101:
4099:
4096:
4094:
4091:
4087:
4084:
4082:
4079:
4078:
4077:
4074:
4070:
4067:
4065:
4062:
4060:
4057:
4055:
4052:
4051:
4050:
4047:
4045:
4042:
4038:
4035:
4033:
4030:
4028:
4025:
4024:
4023:
4020:
4016:
4013:
4011:
4008:
4006:
4003:
4001:
3998:
3996:
3993:
3991:
3988:
3987:
3986:
3983:
3981:
3978:
3976:
3973:
3971:
3968:
3964:
3961:
3959:
3956:
3954:
3951:
3949:
3946:
3945:
3944:
3941:
3939:
3936:
3934:
3931:
3929:
3926:
3922:
3919:
3917:
3916:by definition
3914:
3913:
3912:
3909:
3905:
3902:
3901:
3900:
3897:
3895:
3892:
3890:
3887:
3885:
3882:
3880:
3877:
3876:
3873:
3870:
3868:
3864:
3859:
3853:
3849:
3839:
3836:
3834:
3831:
3829:
3826:
3824:
3821:
3819:
3816:
3814:
3811:
3809:
3806:
3804:
3803:Kripke–Platek
3801:
3799:
3796:
3792:
3789:
3787:
3784:
3783:
3782:
3779:
3778:
3776:
3772:
3764:
3761:
3760:
3759:
3756:
3754:
3751:
3747:
3744:
3743:
3742:
3739:
3737:
3734:
3732:
3729:
3727:
3724:
3722:
3719:
3716:
3712:
3708:
3705:
3701:
3698:
3696:
3693:
3691:
3688:
3687:
3686:
3682:
3679:
3678:
3676:
3674:
3670:
3666:
3658:
3655:
3653:
3650:
3648:
3647:constructible
3645:
3644:
3643:
3640:
3638:
3635:
3633:
3630:
3628:
3625:
3623:
3620:
3618:
3615:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3580:
3579:
3577:
3575:
3570:
3562:
3559:
3557:
3554:
3552:
3549:
3547:
3544:
3542:
3539:
3537:
3534:
3533:
3531:
3527:
3524:
3522:
3519:
3518:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3493:
3489:
3487:
3484:
3480:
3477:
3476:
3475:
3472:
3471:
3468:
3465:
3463:
3459:
3449:
3446:
3444:
3441:
3439:
3436:
3434:
3431:
3429:
3426:
3424:
3421:
3417:
3414:
3413:
3412:
3409:
3405:
3400:
3399:
3398:
3395:
3394:
3392:
3390:
3386:
3378:
3375:
3373:
3370:
3368:
3365:
3364:
3363:
3360:
3358:
3355:
3353:
3350:
3348:
3345:
3343:
3340:
3338:
3335:
3333:
3330:
3329:
3327:
3325:
3324:Propositional
3321:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3297:
3295:
3292:
3290:
3287:
3283:
3280:
3279:
3278:
3275:
3273:
3270:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3252:Logical truth
3250:
3248:
3245:
3244:
3242:
3240:
3236:
3233:
3231:
3227:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3203:
3201:
3198:
3196:
3192:
3188:
3184:
3182:
3179:
3177:
3174:
3172:
3168:
3165:
3164:
3162:
3160:
3154:
3149:
3143:
3140:
3138:
3135:
3133:
3130:
3128:
3125:
3123:
3120:
3118:
3115:
3113:
3110:
3108:
3105:
3103:
3100:
3098:
3095:
3093:
3090:
3088:
3085:
3081:
3078:
3077:
3076:
3073:
3072:
3070:
3066:
3062:
3055:
3050:
3048:
3043:
3041:
3036:
3035:
3032:
3022:
3017:
3011:
2999:
2989:
2982:
2977:
2973:
2966:
2961:
2958:
2954:
2949:
2946:
2942:
2937:
2932:
2927:
2924:
2919:
2915:
2910:
2907:
2903:
2898:
2895:
2891:
2886:
2881:
2876:
2873:
2869:
2864:
2861:
2857:
2852:
2849:
2845:
2840:
2837:
2831:
2826:
2823:
2817:
2812:
2809:
2808:contradiction
2804:
2800:
2795:
2792:
2787:
2783:
2778:
2775:
2771:
2766:
2761:
2756:
2755:
2750:
2746:
2738:
2733:
2731:
2726:
2724:
2719:
2718:
2715:
2706:
2704:1-56881-262-0
2700:
2696:
2691:
2690:
2678:
2674:
2673:0-387-97710-4
2670:
2666:
2662:
2657:
2649:
2643:
2639:
2635:
2631:
2630:
2622:
2614:
2608:
2605:. MIT Press.
2604:
2603:
2595:
2581:
2577:
2571:
2557:
2553:
2547:
2539:
2533:
2529:
2522:
2518:
2509:
2506:
2504:
2501:
2498:
2495:
2493:
2490:
2488:
2485:
2483:
2480:
2478:
2475:
2474:
2468:
2466:
2465:right adjoint
2462:
2458:
2457:inverse image
2454:
2450:
2446:
2442:
2438:
2432:
2422:
2420:
2419:Vacuous truth
2416:
2412:
2408:
2404:
2400:
2384:
2376:
2372:
2353:
2347:
2339:
2334:
2319:The empty set
2316:
2314:
2310:
2306:
2302:
2298:
2282:
2270:
2264:
2239:
2227:
2215:
2209:
2191:
2185:
2175:
2170:
2159:
2158:
2157:
2155:
2151:
2147:
2143:
2139:
2118:
2112:
2102:
2097:
2081:
2075:
2068:
2067:
2066:
2064:
2063:
2058:
2056:
2045:
2044:instantiation
2041:
2038:
2036:
2035:instantiation
2032:
2029:
2028:
2027:
2026:
2023:
2020:
2019:
2016:
2013:
2012:
2006:
2001:
1999:
1994:
1992:
1987:
1985:
1984:Transposition
1982:
1980:
1977:
1975:
1970:
1968:
1963:
1961:
1959:Commutativity
1956:
1954:
1952:Associativity
1949:
1948:
1946:
1945:
1942:
1939:
1938:
1933:
1930:
1928:
1926:
1920:
1918:
1917:modus tollens
1912:
1907:
1905:
1899:
1894:
1892:
1886:
1881:
1879:
1873:
1868:
1866:
1860:
1855:
1853:
1847:
1842:
1840:
1837:
1834:elimination (
1830:
1825:
1824:
1823:
1822:
1819:
1816:
1815:
1812:
1809:
1808:
1805:
1802:
1801:
1793:
1774:
1768:
1758:
1753:
1746:
1740:
1734:
1724:
1719:
1700:
1694:
1691:
1685:
1679:
1669:
1664:
1652:
1632:
1626:
1616:
1611:
1598:
1592:
1586:
1576:
1571:
1567:
1560:
1559:
1558:
1538:
1532:
1519:
1514:
1504:
1498:
1492:
1482:
1477:
1461:
1455:
1449:
1436:
1431:
1421:
1415:
1409:
1399:
1394:
1377:
1376:
1375:
1371:
1369:
1347:
1341:
1328:
1323:
1313:
1307:
1301:
1291:
1286:
1269:
1268:
1267:
1265:
1261:
1239:
1233:
1220:
1215:
1204:
1203:
1202:
1200:
1196:
1174:
1168:
1158:
1153:
1142:
1141:
1140:
1119:
1113:
1103:
1098:
1081:
1080:
1079:
1077:
1055:
1049:
1039:
1034:
1023:
1022:
1021:
1019:
1015:
1011:
1007:
1003:
998:
962:
948:
928:
920:
916:
915:
910:
890:
887:
884:
881:
878:
870:
854:
846:
845:
844:
830:
827:
824:
821:
818:
815:
806:
801:
789:
785:
781:
771:
769:
765:
761:
741:
738:
735:
732:
729:
709:
689:
681:
680:
679:
677:
658:
655:
652:
649:
646:
626:
618:
617:
616:
614:
610:
606:
590:
587:
584:
581:
578:
558:
551:
546:
544:
528:
525:
522:
519:
516:
496:
493:
490:
487:
484:
476:
471:
469:
465:
449:
446:
443:
440:
437:
417:
414:
411:
408:
405:
385:
382:
379:
376:
373:
349:
346:
343:
340:
337:
317:
309:
308:
307:
305:
302:Consider the
295:
293:
285:
268:
263:
261:
257:
253:
247:
239:
232:
226:
222:
219:
215:
211:
207:
204:is a type of
203:
199:
175:
169:
166:
156:
152:
136:
113:
107:
100:is true when
84:
78:
75:
65:
61:
58:
55:
51:
48:
45:
41:
33:
26:
22:
4769:
4567:Ultraproduct
4414:Model theory
4379:Independence
4315:Formal proof
4307:Proof theory
4290:
4263:
4220:real numbers
4192:second-order
4103:Substitution
4053:
3980:Metalanguage
3921:conservative
3894:Axiom schema
3838:Constructive
3808:Morse–Kelley
3774:Set theories
3753:Aleph number
3746:inaccessible
3652:Grothendieck
3536:intersection
3423:Higher-order
3411:Second-order
3357:Truth tables
3314:Venn diagram
3097:Formal proof
2985:
2980:
2935:
2884:
2764:
2694:
2676:
2664:
2656:
2628:
2621:
2602:Logic Primer
2601:
2594:
2583:. Retrieved
2579:
2570:
2559:. Retrieved
2555:
2546:
2527:
2521:
2445:left adjoint
2434:
2414:
2410:
2406:
2402:
2397:denotes the
2374:
2370:
2323:The formula
2322:
2312:
2308:
2304:
2300:
2296:
2257:
2153:
2149:
2145:
2136:
2060:
2059:
2052:
2042: /
2033: /
1924:
1921: /
1916:
1913: /
1900: /
1897:Constructive
1887: /
1874: /
1861: /
1848: /
1836:modus ponens
1835:
1831: /
1653:
1650:
1556:
1372:
1365:
1257:
1198:
1194:
1192:
1138:
1075:
1073:
1017:
1013:
1009:
1005:
1001:
999:
973:
912:
906:
777:
757:
673:
608:
547:
474:
472:
468:real numbers
365:
301:
280:THERE EXISTS
264:
259:
255:
245:
237:
230:
224:
201:
195:
4677:Type theory
4625:undecidable
4557:Truth value
4444:equivalence
4123:non-logical
3736:Enumeration
3726:Isomorphism
3673:cardinality
3657:Von Neumann
3622:Ultrafilter
3587:Uncountable
3521:equivalence
3438:Quantifiers
3428:Fixed-point
3397:First-order
3277:Consistency
3262:Proposition
3239:Traditional
3210:Lindström's
3200:Compactness
3142:Type theory
3087:Cardinality
2986:existential
1997:Exportation
1884:Disjunctive
1877:elimination
1864:elimination
1851:elimination
722:is odd and
550:even number
214:interpreted
4792:Categories
4488:elementary
4181:arithmetic
4049:Quantifier
4027:functional
3899:Expression
3617:Transitive
3561:identities
3546:complement
3479:hereditary
3462:Set theory
2686:References
2612:0262303965
2585:2020-09-04
2561:2020-09-04
2453:power sets
2425:As adjoint
1910:Absorption
965:Properties
871:such that
788:sans-serif
206:quantifier
47:Quantifier
4759:Supertask
4662:Recursion
4620:decidable
4454:saturated
4432:of models
4355:deductive
4350:axiomatic
4270:Hilbert's
4257:Euclidean
4238:canonical
4161:axiomatic
4093:Signature
4022:Predicate
3911:Extension
3833:Ackermann
3758:Operation
3637:Universal
3627:Recursive
3602:Singleton
3597:Inhabited
3582:Countable
3572:Types of
3556:power set
3526:partition
3443:Predicate
3389:Predicate
3304:Syllogism
3294:Soundness
3267:Inference
3257:Tautology
3159:paradoxes
2970:universal
2848:therefore
2836:therefore
2791:tautology
2677:See p. 58
2401:, and no
2399:empty set
2385:∅
2344:∅
2340:∈
2331:∃
2277:→
2234:→
2222:→
2198:→
2176:∈
2167:∃
2152:in which
2103:∈
2094:∃
2088:→
2004:Tautology
1759:∈
1750:∃
1747:∨
1725:∈
1716:∃
1707:→
1692:∨
1670:∈
1661:∃
1617:∈
1608:∃
1602:¬
1599:≡
1577:∈
1568:∄
1530:¬
1520:∈
1511:∃
1505:≡
1483:∈
1474:∀
1468:¬
1447:¬
1437:∈
1428:∀
1422:≡
1400:∈
1391:∃
1385:¬
1339:¬
1329:∈
1320:∀
1314:≡
1292:∈
1283:∃
1277:¬
1231:¬
1221:∈
1212:∀
1159:∈
1150:∃
1104:∈
1095:∃
1089:¬
1040:∈
1031:∃
982:¬
949:∪
929:∩
882:×
822:×
807:∈
798:∃
733:×
650:×
582:×
520:×
488:×
441:×
409:×
377:×
341:×
306:sentence
286:, and as
212:which is
164:∃
73:∃
63:Statement
25:Yo (kana)
4744:Logicism
4737:timeline
4713:Concrete
4572:Validity
4542:T-schema
4535:Kripke's
4530:Tarski's
4525:semantic
4515:Strength
4464:submodel
4459:spectrum
4427:function
4275:Tarski's
4264:Elements
4251:geometry
4207:Robinson
4128:variable
4113:function
4086:spectrum
4076:Sentence
4032:variable
3975:Language
3928:Relation
3889:Automata
3879:Alphabet
3863:language
3717:-jection
3695:codomain
3681:Function
3642:Universe
3612:Infinite
3516:Relation
3299:Validity
3289:Argument
3187:theorem,
2923:superset
2834:entails,
2820:entails,
2471:See also
2451:between
1462:≢
970:Negation
774:Notation
543:solution
277:∃
4686:Related
4483:Diagram
4381: (
4360:Hilbert
4345:Systems
4340:Theorem
4218:of the
4163:systems
3943:Formula
3938:Grammar
3854: (
3798:General
3511:Forcing
3496:Element
3416:Monadic
3191:paradox
3132:Theorem
3068:General
2939:
2918:implies
2906:implies
2888:
2860:because
2768:
2743:Common
2463:is the
2449:functor
786:" in a
288:\exists
284:Unicode
4449:finite
4212:Skolem
4165:
4140:Theory
4108:Symbol
4098:String
4081:atomic
3958:ground
3953:closed
3948:atomic
3904:ground
3867:syntax
3763:binary
3690:domain
3607:Finite
3372:finite
3230:Logics
3189:
3137:Theory
2933:
2882:
2822:proves
2762:
2701:
2671:
2644:
2609:
2534:
2455:, the
2280:
2237:
2225:
2201:
2091:
1710:
1605:
1508:
1471:
1465:
1425:
1388:
1317:
1280:
1092:
985:
304:formal
298:Basics
274:
272:U+2203
242:" or "
234:" or "
221:symbol
4439:Model
4187:Peano
4044:Proof
3884:Arity
3813:Naive
3700:image
3632:Fuzzy
3592:Empty
3541:union
3486:Class
3127:Model
3117:Lemma
3075:Axiom
2803:false
2770:&
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