2151:, 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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2076:(∃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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1558:{\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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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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1797:{\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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1367:{\displaystyle \lnot \ \exists {x}{\in }\mathbf {X} \,P(x)\equiv \ \forall {x}{\in }\mathbf {X} \,\lnot P(x)}
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1652:{\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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2259:{\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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234:∃, which, when used together with a predicate variable, is called an
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is greater than 0 and less than 1", then, for a domain of discourse
777:, which exhibits an object satisfying the "some" statement, or by a
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265:("for all"), which asserts that the property or relation holds for
41:"∄" redirects here. For the Ukrainian nightclub of that name, see
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2326:) 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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2814:
2138:{\displaystyle P(a)\to \ \exists {x}{\in }\mathbf {X} \,P(x)}
302:
1976:
1934:
1900:
1855:
1139:{\displaystyle \lnot \ \exists {x}{\in }\mathbf {X} \,P(x)}
1089:
that is greater than 0 and less than 1", or, symbolically:
972:
to respectively denote the intersection and union of sets.
847:{\displaystyle \exists {n}{\in }\mathbb {N} :n\times n=25}
31:
1926:
1277:
of that propositional function's negation; symbolically,
1259:{\displaystyle \forall {x}{\in }\mathbf {X} \,\lnot P(x)}
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2007:
1962:
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276:
Quantification in general is covered in the article on
2454:, the existential quantifier can be understood as the
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614:" is false, because there are no even solutions. The
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347:
327:
173:
146:
117:
82:
2373:{\displaystyle \exists {x}{\in }\varnothing \,P(x)}
1204:is logically equivalent to "For any natural number
1194:{\displaystyle \exists {x}{\in }\mathbf {X} \,P(x)}
1075:{\displaystyle \exists {x}{\in }\mathbf {X} \,P(x)}
2470:functor of a function between sets; likewise, the
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1920:
477:to be the natural numbers, not, for example, the
269:members of the domain. Some sources use the term
30:"∃" redirects here. For the letter turned E, see
4800:
2000:
1969:
1861:
261:). Existential quantification is distinct from
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2739:
2014:
1983:
1881:
1874:
918:The symbol's first usage is thought to be by
2636:
1212:is not greater than 0 and less than 1", or:
280:. The existential quantifier is encoded as
3248:
3063:
3049:
2753:
2746:
2732:
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2442:Universal quantification § As adjoint
618:, which specifies the values the variable
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1502:
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822:
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273:to refer to existential quantification.
2630:
14:
4801:
3070:
2314:; else, the logic does not follow: If
3044:
2727:
2428:) – exist in the empty set. See also
1806:
2610:Allen, Colin; Hand, Michael (2001).
2643:. Springer Cham. pp. 210–211.
1273:'s existential quantification is a
1269:Generally, then, the negation of a
1008:symbol is used to denote negation.
24:
2706:Fundamentals of Mathematical Logic
2416:of any description – let alone an
2341:
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1760:
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174:
140:is true for at least one value of
83:
27:Mathematical use of "there exists"
25:
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854:represents the (true) statement
488:, we produce the true statement
2380:is always false, regardless of
2306:must be true for all values of
34:. For the Japanese kana ヨ, see
2665:
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189:
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127:
121:
98:
92:
13:
1:
4743:History of mathematical logic
2696:
2676:Sheaves in Geometry and Logic
2435:
2420:fulfilling a given predicate
1377:(This is a generalization of
975:
630:For some positive odd number
305:and related formula editors.
195:{\displaystyle \exists xP(x)}
104:{\displaystyle \exists xP(x)}
4668:Primitive recursive function
2563:"Predicates and Quantifiers"
2401:{\displaystyle \varnothing }
907:{\displaystyle n\times n=25}
758:{\displaystyle n\times n=25}
675:{\displaystyle n\times n=25}
607:{\displaystyle n\times n=25}
545:{\displaystyle n\times n=25}
513:{\displaystyle 5\times 5=25}
466:{\displaystyle 2\times 2=25}
434:{\displaystyle 1\times 1=25}
402:{\displaystyle 0\times 0=25}
366:{\displaystyle n\times n=25}
7:
2481:
2299:{\displaystyle P(c)\to \ Q}
980:
784:
520:. It does not matter that "
10:
4830:
3732:Schröder–Bernstein theorem
3459:Monadic predicate calculus
3118:Foundations of mathematics
2510:– for the unicode symbol ∃
2439:
2051:Existential generalization
1856:Biconditional introduction
213:existential quantification
49:Existential quantification
40:
29:
4778:
4765:Philosophy of mathematics
4714:Automated theorem proving
4696:
4591:
4423:
4316:
4168:
3885:
3861:
3839:Von Neumann–Bernays–Gödel
3784:
3678:
3582:
3480:
3471:
3398:
3333:
3239:
3161:
3078:
3011:
2762:
2674:, Ieke Moerdijk, (1992):
2649:10.1007/978-3-319-71350-2
2537:Bergmann, Merrie (2014).
2519:Uniqueness quantification
2149:Existential instantiation
308:
164:
73:
63:
53:
2524:
2073:Existential introduction
2042:Universal generalization
1882:Disjunction introduction
1869:Conjunction introduction
1839:Implication introduction
1275:universal quantification
1001:{\displaystyle \lnot \ }
793:, "∃" (a turned letter "
693:For some natural number
321:For some natural number
263:universal quantification
4415:Self-verifying theories
4236:Tarski's axiomatization
3187:Tarski's undefinability
3182:incompleteness theorems
559:In contrast, "For some
4789:Mathematics portal
4400:Proof of impossibility
4048:propositional variable
3358:Propositional calculus
3032:Mathematics portal
2432:for more information.
2402:
2374:
2300:
2260:
2159:and for a proposition
2139:
1901:hypothetical syllogism
1822:Propositional calculus
1798:
1653:
1559:
1368:
1271:propositional function
1260:
1195:
1140:
1076:
1002:
966:
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925:Formulario mathematico
908:
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848:
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727:
707:
676:
644:
608:
576:
546:
514:
467:
435:
403:
367:
335:
278:quantification (logic)
236:existential quantifier
196:
154:
134:
105:
4658:Kolmogorov complexity
4611:Computably enumerable
4511:Model complete theory
4303:Principia Mathematica
3363:Propositional formula
3192:Banach–Tarski paradox
3021:Philosophy portal
2637:Stephen Webb (2018).
2508:List of logic symbols
2403:
2375:
2310:over the same domain
2301:
2261:
2140:
1943:Negation introduction
1936:modus ponendo tollens
1799:
1654:
1560:
1381:to predicate logic.)
1369:
1261:
1196:
1141:
1077:
1003:
967:
965:{\displaystyle \cup }
947:
945:{\displaystyle \cap }
909:
873:
849:
779:nonconstructive proof
760:
728:
708:
677:
645:
609:
577:
547:
515:
468:
436:
404:
368:
336:
197:
155:
135:
106:
4606:Church–Turing thesis
4593:Computability theory
3802:continuum hypothesis
3320:Square of opposition
3178:Gödel's completeness
2503:Lindström quantifier
2472:universal quantifier
2392:
2338:
2272:
2174:
2083:
2001:Material implication
1952:Rules of replacement
1815:Transformation rules
1668:
1575:
1392:
1284:
1219:
1157:
1096:
1038:
1019:) is the predicate "
989:
956:
936:
928:(1896). Afterwards,
886:
862:
805:
737:
717:
697:
687:logically equivalent
654:
634:
624:Logical conjunctions
586:
566:
524:
492:
445:
413:
381:
345:
325:
171:
144:
133:{\displaystyle P(x)}
115:
80:
4760:Mathematical object
4651:P versus NP problem
4616:Computable function
4410:Reverse mathematics
4336:Logical consequence
4213:primitive recursive
4208:elementary function
3981:Free/bound variable
3834:Tarski–Grothendieck
3353:Logical connectives
3283:Logical equivalence
3133:Logical consequence
2704:Hinman, P. (2005).
2514:Quantifier variance
2388:). This is because
1914:destructive dilemma
616:domain of discourse
475:domain of discourse
50:
4814:Quantifier (logic)
4558:Transfer principle
4521:Semantics of logic
4506:Categorical theory
4482:Non-standard model
3996:Logical connective
3123:Information theory
3072:Mathematical logic
2488:Existential clause
2450:and the theory of
2398:
2370:
2296:
2256:
2135:
2033:Rules of inference
1829:Rules of inference
1807:Rules of inference
1794:
1649:
1555:
1364:
1256:
1191:
1136:
1072:
998:
962:
942:
904:
868:
858:There exists some
844:
775:constructive proof
771:mathematical proof
755:
723:
703:
672:
640:
604:
572:
542:
510:
463:
431:
399:
363:
331:
271:existentialization
192:
165:Symbolic statement
150:
130:
101:
68:Mathematical logic
48:
4796:
4795:
4728:Abstract category
4531:Theories of truth
4341:Rule of inference
4331:Natural deduction
4312:
4311:
3857:
3856:
3562:Cartesian product
3467:
3466:
3373:Many-valued logic
3348:Boolean functions
3231:Russell's paradox
3206:diagonal argument
3103:First-order logic
3038:
3037:
3006:
3005:
2672:Saunders Mac Lane
2658:978-3-319-71349-6
2587:"1.2 Quantifiers"
2548:978-0-07-803841-9
2498:First-order logic
2493:Existence theorem
2292:
2249:
2237:
2213:
2167:does not appear:
2103:
2066:rule of inference
2062:
2061:
1722:
1617:
1520:
1483:
1477:
1437:
1400:
1329:
1292:
1104:
997:
871:{\displaystyle n}
726:{\displaystyle n}
706:{\displaystyle n}
643:{\displaystyle n}
575:{\displaystyle n}
334:{\displaystyle n}
205:
204:
153:{\displaystyle x}
16:(Redirected from
4821:
4787:
4786:
4738:History of logic
4733:Category of sets
4626:Decision problem
4405:Ordinal analysis
4346:Sequent calculus
4244:Boolean algebras
4184:
4183:
4158:
4129:logical/constant
3883:
3882:
3869:
3792:Zermelo–Fraenkel
3543:Set operations:
3478:
3477:
3415:
3246:
3245:
3226:Löwenheim–Skolem
3113:Formal semantics
3065:
3058:
3051:
3042:
3041:
3030:
3029:
3019:
3018:
3017:
2863:
2812:
2778:
2765:
2764:
2748:
2741:
2734:
2725:
2724:
2719:
2691:
2678:Springer-Verlag
2669:
2663:
2662:
2640:Clash of Symbols
2634:
2628:
2627:
2607:
2601:
2600:
2598:
2597:
2583:
2577:
2576:
2574:
2573:
2567:www.csm.ornl.gov
2559:
2553:
2552:
2534:
2452:elementary topoi
2407:
2405:
2404:
2399:
2379:
2377:
2376:
2371:
2353:
2348:
2305:
2303:
2302:
2297:
2290:
2265:
2263:
2262:
2257:
2247:
2235:
2211:
2194:
2189:
2184:
2153:necessarily true
2144:
2142:
2141:
2136:
2121:
2116:
2111:
2101:
2016:
2009:
2002:
1990:De Morgan's laws
1985:
1978:
1971:
1964:
1938:
1930:
1922:
1915:
1909:
1902:
1896:
1889:
1883:
1876:
1870:
1863:
1857:
1850:
1840:
1811:
1810:
1803:
1801:
1800:
1795:
1777:
1772:
1767:
1743:
1738:
1733:
1720:
1688:
1683:
1678:
1658:
1656:
1655:
1650:
1635:
1630:
1625:
1615:
1595:
1590:
1585:
1564:
1562:
1561:
1556:
1538:
1533:
1528:
1518:
1501:
1496:
1491:
1481:
1475:
1455:
1450:
1445:
1435:
1418:
1413:
1408:
1398:
1379:De Morgan's laws
1373:
1371:
1370:
1365:
1347:
1342:
1337:
1327:
1310:
1305:
1300:
1290:
1265:
1263:
1262:
1257:
1239:
1234:
1229:
1200:
1198:
1197:
1192:
1177:
1172:
1167:
1145:
1143:
1142:
1137:
1122:
1117:
1112:
1102:
1081:
1079:
1078:
1073:
1058:
1053:
1048:
1011:For example, if
1007:
1005:
1004:
999:
995:
971:
969:
968:
963:
951:
949:
948:
943:
930:Bertrand Russell
913:
911:
910:
905:
877:
875:
874:
869:
853:
851:
850:
845:
825:
820:
815:
764:
762:
761:
756:
732:
730:
729:
724:
712:
710:
709:
704:
689:to the sentence
681:
679:
678:
673:
649:
647:
646:
641:
613:
611:
610:
605:
581:
579:
578:
573:
551:
549:
548:
543:
519:
517:
516:
511:
472:
470:
469:
464:
440:
438:
437:
432:
408:
406:
405:
400:
372:
370:
369:
364:
340:
338:
337:
332:
300:
292:
289:
286:
284:
260:
252:
244:
229:logical operator
221:logical constant
201:
199:
198:
193:
159:
157:
156:
151:
139:
137:
136:
131:
110:
108:
107:
102:
51:
47:
21:
4829:
4828:
4824:
4823:
4822:
4820:
4819:
4818:
4799:
4798:
4797:
4792:
4781:
4774:
4719:Category theory
4709:Algebraic logic
4692:
4663:Lambda calculus
4601:Church encoding
4587:
4563:Truth predicate
4419:
4385:Complete theory
4308:
4177:
4173:
4169:
4164:
4156:
3876: and
3872:
3867:
3853:
3829:New Foundations
3797:axiom of choice
3780:
3742:Gödel numbering
3682: and
3674:
3578:
3463:
3413:
3394:
3343:Boolean algebra
3329:
3293:Equiconsistency
3258:Classical logic
3235:
3216:Halting problem
3204: and
3180: and
3168: and
3167:
3162:Theorems (
3157:
3074:
3069:
3039:
3034:
3024:
3023:
3015:
3013:
3007:
3002:
2998:
2990:
2986:
2978:
2975:
2972:
2964:
2961:
2958:
2950:
2946:
2941:
2933:
2929:
2924:
2916:
2915:
2912:
2908:
2900:
2899:
2896:
2892:
2884:
2880:
2872:
2868:
2859:
2850:
2846:
2841:
2833:
2829:
2821:
2817:
2808:
2799:
2795:
2787:
2783:
2774:
2758:
2756:logical symbols
2752:
2722:
2716:
2699:
2694:
2670:
2666:
2659:
2635:
2631:
2624:
2608:
2604:
2595:
2593:
2591:www.whitman.edu
2585:
2584:
2580:
2571:
2569:
2561:
2560:
2556:
2549:
2541:. McGraw Hill.
2535:
2531:
2527:
2484:
2448:category theory
2444:
2438:
2393:
2390:
2389:
2349:
2344:
2339:
2336:
2335:
2332:
2273:
2270:
2269:
2190:
2185:
2180:
2175:
2172:
2171:
2117:
2112:
2107:
2084:
2081:
2080:
2026:Predicate logic
2020:
1984:Double negation
1838:
1809:
1773:
1768:
1763:
1739:
1734:
1729:
1684:
1679:
1674:
1669:
1666:
1665:
1631:
1626:
1621:
1591:
1586:
1581:
1576:
1573:
1572:
1534:
1529:
1524:
1497:
1492:
1487:
1451:
1446:
1441:
1414:
1409:
1404:
1393:
1390:
1389:
1343:
1338:
1333:
1306:
1301:
1296:
1285:
1282:
1281:
1235:
1230:
1225:
1220:
1217:
1216:
1173:
1168:
1163:
1158:
1155:
1154:
1118:
1113:
1108:
1097:
1094:
1093:
1054:
1049:
1044:
1039:
1036:
1035:
990:
987:
986:
983:
978:
957:
954:
953:
937:
934:
933:
887:
884:
883:
880:natural numbers
863:
860:
859:
821:
816:
811:
806:
803:
802:
787:
738:
735:
734:
718:
715:
714:
698:
695:
694:
655:
652:
651:
635:
632:
631:
587:
584:
583:
567:
564:
563:
525:
522:
521:
493:
490:
489:
446:
443:
442:
414:
411:
410:
382:
379:
378:
346:
343:
342:
326:
323:
322:
311:
298:
290:
287:
282:
281:
254:
246:
239:
209:predicate logic
172:
169:
168:
145:
142:
141:
116:
113:
112:
81:
78:
77:
46:
43:K41 (nightclub)
39:
28:
23:
22:
15:
12:
11:
5:
4827:
4817:
4816:
4811:
4794:
4793:
4779:
4776:
4775:
4773:
4772:
4767:
4762:
4757:
4752:
4751:
4750:
4740:
4735:
4730:
4721:
4716:
4711:
4706:
4704:Abstract logic
4700:
4698:
4694:
4693:
4691:
4690:
4685:
4683:Turing machine
4680:
4675:
4670:
4665:
4660:
4655:
4654:
4653:
4648:
4643:
4638:
4633:
4623:
4621:Computable set
4618:
4613:
4608:
4603:
4597:
4595:
4589:
4588:
4586:
4585:
4580:
4575:
4570:
4565:
4560:
4555:
4550:
4549:
4548:
4543:
4538:
4528:
4523:
4518:
4516:Satisfiability
4513:
4508:
4503:
4502:
4501:
4491:
4490:
4489:
4479:
4478:
4477:
4472:
4467:
4462:
4457:
4447:
4446:
4445:
4440:
4433:Interpretation
4429:
4427:
4421:
4420:
4418:
4417:
4412:
4407:
4402:
4397:
4387:
4382:
4381:
4380:
4379:
4378:
4368:
4363:
4353:
4348:
4343:
4338:
4333:
4328:
4322:
4320:
4314:
4313:
4310:
4309:
4307:
4306:
4298:
4297:
4296:
4295:
4290:
4289:
4288:
4283:
4278:
4258:
4257:
4256:
4254:minimal axioms
4251:
4240:
4239:
4238:
4227:
4226:
4225:
4220:
4215:
4210:
4205:
4200:
4187:
4185:
4166:
4165:
4163:
4162:
4161:
4160:
4148:
4143:
4142:
4141:
4136:
4131:
4126:
4116:
4111:
4106:
4101:
4100:
4099:
4094:
4084:
4083:
4082:
4077:
4072:
4067:
4057:
4052:
4051:
4050:
4045:
4040:
4030:
4029:
4028:
4023:
4018:
4013:
4008:
4003:
3993:
3988:
3983:
3978:
3977:
3976:
3971:
3966:
3961:
3951:
3946:
3944:Formation rule
3941:
3936:
3935:
3934:
3929:
3919:
3918:
3917:
3907:
3902:
3897:
3892:
3886:
3880:
3863:Formal systems
3859:
3858:
3855:
3854:
3852:
3851:
3846:
3841:
3836:
3831:
3826:
3821:
3816:
3811:
3806:
3805:
3804:
3799:
3788:
3786:
3782:
3781:
3779:
3778:
3777:
3776:
3766:
3761:
3760:
3759:
3752:Large cardinal
3749:
3744:
3739:
3734:
3729:
3715:
3714:
3713:
3708:
3703:
3688:
3686:
3676:
3675:
3673:
3672:
3671:
3670:
3665:
3660:
3650:
3645:
3640:
3635:
3630:
3625:
3620:
3615:
3610:
3605:
3600:
3595:
3589:
3587:
3580:
3579:
3577:
3576:
3575:
3574:
3569:
3564:
3559:
3554:
3549:
3541:
3540:
3539:
3534:
3524:
3519:
3517:Extensionality
3514:
3512:Ordinal number
3509:
3499:
3494:
3493:
3492:
3481:
3475:
3469:
3468:
3465:
3464:
3462:
3461:
3456:
3451:
3446:
3441:
3436:
3431:
3430:
3429:
3419:
3418:
3417:
3404:
3402:
3396:
3395:
3393:
3392:
3391:
3390:
3385:
3380:
3370:
3365:
3360:
3355:
3350:
3345:
3339:
3337:
3331:
3330:
3328:
3327:
3322:
3317:
3312:
3307:
3302:
3297:
3296:
3295:
3285:
3280:
3275:
3270:
3265:
3260:
3254:
3252:
3243:
3237:
3236:
3234:
3233:
3228:
3223:
3218:
3213:
3208:
3196:Cantor's
3194:
3189:
3184:
3174:
3172:
3159:
3158:
3156:
3155:
3150:
3145:
3140:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3095:
3094:
3093:
3082:
3080:
3076:
3075:
3068:
3067:
3060:
3053:
3045:
3036:
3035:
3012:
3009:
3008:
3004:
3003:
2994:
2993:
2991:
2982:
2981:
2979:
2968:
2967:
2965:
2954:
2953:
2951:
2937:
2936:
2934:
2920:
2919:
2917:
2913:quantification
2909:
2904:
2903:
2901:
2897:quantification
2893:
2888:
2887:
2885:
2876:
2875:
2873:
2854:
2853:
2851:
2837:
2836:
2834:
2825:
2824:
2822:
2803:
2802:
2800:
2791:
2790:
2788:
2769:
2768:
2763:
2760:
2759:
2751:
2750:
2743:
2736:
2728:
2721:
2720:
2714:
2708:. A K Peters.
2700:
2698:
2695:
2693:
2692:
2664:
2657:
2629:
2622:
2602:
2578:
2554:
2547:
2539:The Logic Book
2528:
2526:
2523:
2522:
2521:
2516:
2511:
2505:
2500:
2495:
2490:
2483:
2480:
2440:Main article:
2437:
2434:
2397:
2369:
2366:
2363:
2360:
2356:
2352:
2347:
2343:
2331:
2328:
2295:
2289:
2286:
2283:
2280:
2277:
2267:
2266:
2255:
2252:
2246:
2243:
2240:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2210:
2207:
2204:
2201:
2198:
2193:
2188:
2183:
2179:
2146:
2145:
2134:
2131:
2128:
2125:
2120:
2115:
2110:
2106:
2100:
2097:
2094:
2091:
2088:
2060:
2059:
2058:
2057:
2048:
2036:
2035:
2029:
2028:
2022:
2021:
2019:
2018:
2011:
2004:
1997:
1992:
1987:
1980:
1977:Distributivity
1973:
1966:
1958:
1955:
1954:
1948:
1947:
1946:
1945:
1940:
1917:
1904:
1891:
1878:
1865:
1852:
1832:
1831:
1825:
1824:
1818:
1817:
1808:
1805:
1793:
1790:
1787:
1784:
1781:
1776:
1771:
1766:
1762:
1759:
1756:
1753:
1750:
1747:
1742:
1737:
1732:
1728:
1725:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1687:
1682:
1677:
1673:
1660:
1659:
1648:
1645:
1642:
1639:
1634:
1629:
1624:
1620:
1614:
1611:
1608:
1605:
1602:
1599:
1594:
1589:
1584:
1580:
1566:
1565:
1554:
1551:
1548:
1545:
1542:
1537:
1532:
1527:
1523:
1517:
1514:
1511:
1508:
1505:
1500:
1495:
1490:
1486:
1480:
1474:
1471:
1468:
1465:
1462:
1459:
1454:
1449:
1444:
1440:
1434:
1431:
1428:
1425:
1422:
1417:
1412:
1407:
1403:
1397:
1375:
1374:
1363:
1360:
1357:
1354:
1351:
1346:
1341:
1336:
1332:
1326:
1323:
1320:
1317:
1314:
1309:
1304:
1299:
1295:
1289:
1267:
1266:
1255:
1252:
1249:
1246:
1243:
1238:
1233:
1228:
1224:
1202:
1201:
1190:
1187:
1184:
1181:
1176:
1171:
1166:
1162:
1148:
1147:
1135:
1132:
1129:
1126:
1121:
1116:
1111:
1107:
1101:
1083:
1082:
1071:
1068:
1065:
1062:
1057:
1052:
1047:
1043:
994:
982:
979:
977:
974:
961:
941:
920:Giuseppe Peano
916:
915:
903:
900:
897:
894:
891:
878:in the set of
867:
843:
840:
837:
834:
831:
828:
824:
819:
814:
810:
791:symbolic logic
786:
783:
767:
766:
754:
751:
748:
745:
742:
722:
702:
683:
682:
671:
668:
665:
662:
659:
639:
603:
600:
597:
594:
591:
571:
541:
538:
535:
532:
529:
509:
506:
503:
500:
497:
462:
459:
456:
453:
450:
430:
427:
424:
421:
418:
398:
395:
392:
389:
386:
375:
374:
362:
359:
356:
353:
350:
330:
310:
307:
203:
202:
191:
188:
185:
182:
179:
176:
166:
162:
161:
149:
129:
126:
123:
120:
100:
97:
94:
91:
88:
85:
75:
71:
70:
65:
61:
60:
55:
26:
9:
6:
4:
3:
2:
4826:
4815:
4812:
4810:
4809:Logic symbols
4807:
4806:
4804:
4791:
4790:
4785:
4777:
4771:
4768:
4766:
4763:
4761:
4758:
4756:
4753:
4749:
4746:
4745:
4744:
4741:
4739:
4736:
4734:
4731:
4729:
4725:
4722:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4701:
4699:
4695:
4689:
4686:
4684:
4681:
4679:
4678:Recursive set
4676:
4674:
4671:
4669:
4666:
4664:
4661:
4659:
4656:
4652:
4649:
4647:
4644:
4642:
4639:
4637:
4634:
4632:
4629:
4628:
4627:
4624:
4622:
4619:
4617:
4614:
4612:
4609:
4607:
4604:
4602:
4599:
4598:
4596:
4594:
4590:
4584:
4581:
4579:
4576:
4574:
4571:
4569:
4566:
4564:
4561:
4559:
4556:
4554:
4551:
4547:
4544:
4542:
4539:
4537:
4534:
4533:
4532:
4529:
4527:
4524:
4522:
4519:
4517:
4514:
4512:
4509:
4507:
4504:
4500:
4497:
4496:
4495:
4492:
4488:
4487:of arithmetic
4485:
4484:
4483:
4480:
4476:
4473:
4471:
4468:
4466:
4463:
4461:
4458:
4456:
4453:
4452:
4451:
4448:
4444:
4441:
4439:
4436:
4435:
4434:
4431:
4430:
4428:
4426:
4422:
4416:
4413:
4411:
4408:
4406:
4403:
4401:
4398:
4395:
4394:from ZFC
4391:
4388:
4386:
4383:
4377:
4374:
4373:
4372:
4369:
4367:
4364:
4362:
4359:
4358:
4357:
4354:
4352:
4349:
4347:
4344:
4342:
4339:
4337:
4334:
4332:
4329:
4327:
4324:
4323:
4321:
4319:
4315:
4305:
4304:
4300:
4299:
4294:
4293:non-Euclidean
4291:
4287:
4284:
4282:
4279:
4277:
4276:
4272:
4271:
4269:
4266:
4265:
4263:
4259:
4255:
4252:
4250:
4247:
4246:
4245:
4241:
4237:
4234:
4233:
4232:
4228:
4224:
4221:
4219:
4216:
4214:
4211:
4209:
4206:
4204:
4201:
4199:
4196:
4195:
4193:
4189:
4188:
4186:
4181:
4175:
4170:Example
4167:
4159:
4154:
4153:
4152:
4149:
4147:
4144:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4121:
4120:
4117:
4115:
4112:
4110:
4107:
4105:
4102:
4098:
4095:
4093:
4090:
4089:
4088:
4085:
4081:
4078:
4076:
4073:
4071:
4068:
4066:
4063:
4062:
4061:
4058:
4056:
4053:
4049:
4046:
4044:
4041:
4039:
4036:
4035:
4034:
4031:
4027:
4024:
4022:
4019:
4017:
4014:
4012:
4009:
4007:
4004:
4002:
3999:
3998:
3997:
3994:
3992:
3989:
3987:
3984:
3982:
3979:
3975:
3972:
3970:
3967:
3965:
3962:
3960:
3957:
3956:
3955:
3952:
3950:
3947:
3945:
3942:
3940:
3937:
3933:
3930:
3928:
3927:by definition
3925:
3924:
3923:
3920:
3916:
3913:
3912:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3887:
3884:
3881:
3879:
3875:
3870:
3864:
3860:
3850:
3847:
3845:
3842:
3840:
3837:
3835:
3832:
3830:
3827:
3825:
3822:
3820:
3817:
3815:
3814:Kripke–Platek
3812:
3810:
3807:
3803:
3800:
3798:
3795:
3794:
3793:
3790:
3789:
3787:
3783:
3775:
3772:
3771:
3770:
3767:
3765:
3762:
3758:
3755:
3754:
3753:
3750:
3748:
3745:
3743:
3740:
3738:
3735:
3733:
3730:
3727:
3723:
3719:
3716:
3712:
3709:
3707:
3704:
3702:
3699:
3698:
3697:
3693:
3690:
3689:
3687:
3685:
3681:
3677:
3669:
3666:
3664:
3661:
3659:
3658:constructible
3656:
3655:
3654:
3651:
3649:
3646:
3644:
3641:
3639:
3636:
3634:
3631:
3629:
3626:
3624:
3621:
3619:
3616:
3614:
3611:
3609:
3606:
3604:
3601:
3599:
3596:
3594:
3591:
3590:
3588:
3586:
3581:
3573:
3570:
3568:
3565:
3563:
3560:
3558:
3555:
3553:
3550:
3548:
3545:
3544:
3542:
3538:
3535:
3533:
3530:
3529:
3528:
3525:
3523:
3520:
3518:
3515:
3513:
3510:
3508:
3504:
3500:
3498:
3495:
3491:
3488:
3487:
3486:
3483:
3482:
3479:
3476:
3474:
3470:
3460:
3457:
3455:
3452:
3450:
3447:
3445:
3442:
3440:
3437:
3435:
3432:
3428:
3425:
3424:
3423:
3420:
3416:
3411:
3410:
3409:
3406:
3405:
3403:
3401:
3397:
3389:
3386:
3384:
3381:
3379:
3376:
3375:
3374:
3371:
3369:
3366:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3340:
3338:
3336:
3335:Propositional
3332:
3326:
3323:
3321:
3318:
3316:
3313:
3311:
3308:
3306:
3303:
3301:
3298:
3294:
3291:
3290:
3289:
3286:
3284:
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3264:
3263:Logical truth
3261:
3259:
3256:
3255:
3253:
3251:
3247:
3244:
3242:
3238:
3232:
3229:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3203:
3199:
3195:
3193:
3190:
3188:
3185:
3183:
3179:
3176:
3175:
3173:
3171:
3165:
3160:
3154:
3151:
3149:
3146:
3144:
3141:
3139:
3136:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3092:
3089:
3088:
3087:
3084:
3083:
3081:
3077:
3073:
3066:
3061:
3059:
3054:
3052:
3047:
3046:
3043:
3033:
3028:
3022:
3010:
3001:
2997:
2992:
2989:
2985:
2980:
2977:
2971:
2966:
2963:
2957:
2952:
2949:
2948:contradiction
2944:
2940:
2935:
2932:
2927:
2923:
2918:
2914:
2907:
2902:
2898:
2891:
2886:
2883:
2879:
2874:
2871:
2867:
2862:
2857:
2852:
2849:
2844:
2840:
2835:
2832:
2828:
2823:
2820:
2816:
2811:
2806:
2801:
2798:
2794:
2789:
2786:
2782:
2777:
2772:
2767:
2766:
2761:
2757:
2749:
2744:
2742:
2737:
2735:
2730:
2729:
2726:
2717:
2715:1-56881-262-0
2711:
2707:
2702:
2701:
2689:
2685:
2684:0-387-97710-4
2681:
2677:
2673:
2668:
2660:
2654:
2650:
2646:
2642:
2641:
2633:
2625:
2619:
2616:. MIT Press.
2615:
2614:
2606:
2592:
2588:
2582:
2568:
2564:
2558:
2550:
2544:
2540:
2533:
2529:
2520:
2517:
2515:
2512:
2509:
2506:
2504:
2501:
2499:
2496:
2494:
2491:
2489:
2486:
2485:
2479:
2477:
2476:right adjoint
2473:
2469:
2468:inverse image
2465:
2461:
2457:
2453:
2449:
2443:
2433:
2431:
2430:Vacuous truth
2427:
2423:
2419:
2415:
2411:
2395:
2387:
2383:
2364:
2358:
2350:
2345:
2330:The empty set
2327:
2325:
2321:
2317:
2313:
2309:
2293:
2281:
2275:
2250:
2238:
2226:
2220:
2202:
2196:
2186:
2181:
2170:
2169:
2168:
2166:
2162:
2158:
2154:
2150:
2129:
2123:
2113:
2108:
2092:
2086:
2079:
2078:
2077:
2075:
2074:
2069:
2067:
2056:
2055:instantiation
2052:
2049:
2047:
2046:instantiation
2043:
2040:
2039:
2038:
2037:
2034:
2031:
2030:
2027:
2024:
2023:
2017:
2012:
2010:
2005:
2003:
1998:
1996:
1995:Transposition
1993:
1991:
1988:
1986:
1981:
1979:
1974:
1972:
1970:Commutativity
1967:
1965:
1963:Associativity
1960:
1959:
1957:
1956:
1953:
1950:
1949:
1944:
1941:
1939:
1937:
1931:
1929:
1928:modus tollens
1923:
1918:
1916:
1910:
1905:
1903:
1897:
1892:
1890:
1884:
1879:
1877:
1871:
1866:
1864:
1858:
1853:
1851:
1848:
1845:elimination (
1841:
1836:
1835:
1834:
1833:
1830:
1827:
1826:
1823:
1820:
1819:
1816:
1813:
1812:
1804:
1785:
1779:
1769:
1764:
1757:
1751:
1745:
1735:
1730:
1711:
1705:
1702:
1696:
1690:
1680:
1675:
1663:
1643:
1637:
1627:
1622:
1609:
1603:
1597:
1587:
1582:
1578:
1571:
1570:
1569:
1549:
1543:
1530:
1525:
1515:
1509:
1503:
1493:
1488:
1472:
1466:
1460:
1447:
1442:
1432:
1426:
1420:
1410:
1405:
1388:
1387:
1386:
1382:
1380:
1358:
1352:
1339:
1334:
1324:
1318:
1312:
1302:
1297:
1280:
1279:
1278:
1276:
1272:
1250:
1244:
1231:
1226:
1215:
1214:
1213:
1211:
1207:
1185:
1179:
1169:
1164:
1153:
1152:
1151:
1130:
1124:
1114:
1109:
1092:
1091:
1090:
1088:
1066:
1060:
1050:
1045:
1034:
1033:
1032:
1030:
1026:
1022:
1018:
1014:
1009:
973:
959:
939:
931:
927:
926:
921:
901:
898:
895:
892:
889:
881:
865:
857:
856:
855:
841:
838:
835:
832:
829:
826:
817:
812:
800:
796:
792:
782:
780:
776:
772:
752:
749:
746:
743:
740:
720:
700:
692:
691:
690:
688:
669:
666:
663:
660:
657:
637:
629:
628:
627:
625:
621:
617:
601:
598:
595:
592:
589:
569:
562:
557:
555:
539:
536:
533:
530:
527:
507:
504:
501:
498:
495:
487:
482:
480:
476:
460:
457:
454:
451:
448:
428:
425:
422:
419:
416:
396:
393:
390:
387:
384:
360:
357:
354:
351:
348:
328:
320:
319:
318:
316:
313:Consider the
306:
304:
296:
279:
274:
272:
268:
264:
258:
250:
243:
237:
233:
230:
226:
222:
218:
215:is a type of
214:
210:
186:
180:
177:
167:
163:
147:
124:
118:
111:is true when
95:
89:
86:
76:
72:
69:
66:
62:
59:
56:
52:
44:
37:
33:
19:
4780:
4578:Ultraproduct
4425:Model theory
4390:Independence
4326:Formal proof
4318:Proof theory
4301:
4274:
4231:real numbers
4203:second-order
4114:Substitution
4064:
3991:Metalanguage
3932:conservative
3905:Axiom schema
3849:Constructive
3819:Morse–Kelley
3785:Set theories
3764:Aleph number
3757:inaccessible
3663:Grothendieck
3547:intersection
3434:Higher-order
3422:Second-order
3368:Truth tables
3325:Venn diagram
3108:Formal proof
2910:
2905:
2860:
2809:
2775:
2705:
2687:
2675:
2667:
2639:
2632:
2613:Logic Primer
2612:
2605:
2594:. Retrieved
2590:
2581:
2570:. Retrieved
2566:
2557:
2538:
2532:
2456:left adjoint
2445:
2425:
2421:
2417:
2413:
2408:denotes the
2385:
2381:
2334:The formula
2333:
2323:
2319:
2315:
2311:
2307:
2268:
2164:
2160:
2156:
2147:
2071:
2070:
2063:
2053: /
2044: /
1935:
1932: /
1927:
1924: /
1911: /
1908:Constructive
1898: /
1885: /
1872: /
1859: /
1847:modus ponens
1846:
1842: /
1664:
1661:
1567:
1383:
1376:
1268:
1209:
1205:
1203:
1149:
1086:
1084:
1028:
1024:
1020:
1016:
1012:
1010:
984:
923:
917:
788:
768:
684:
619:
558:
485:
483:
479:real numbers
376:
312:
291:THERE EXISTS
275:
270:
266:
256:
248:
241:
235:
212:
206:
18:There exists
4688:Type theory
4636:undecidable
4568:Truth value
4455:equivalence
4134:non-logical
3747:Enumeration
3737:Isomorphism
3684:cardinality
3668:Von Neumann
3633:Ultrafilter
3598:Uncountable
3532:equivalence
3449:Quantifiers
3439:Fixed-point
3408:First-order
3288:Consistency
3273:Proposition
3250:Traditional
3221:Lindström's
3211:Compactness
3153:Type theory
3098:Cardinality
2911:existential
2008:Exportation
1895:Disjunctive
1888:elimination
1875:elimination
1862:elimination
733:is odd and
561:even number
225:interpreted
4803:Categories
4499:elementary
4192:arithmetic
4060:Quantifier
4038:functional
3910:Expression
3628:Transitive
3572:identities
3557:complement
3490:hereditary
3473:Set theory
2697:References
2623:0262303965
2596:2020-09-04
2572:2020-09-04
2464:power sets
2436:As adjoint
1921:Absorption
976:Properties
882:such that
799:sans-serif
217:quantifier
58:Quantifier
4770:Supertask
4673:Recursion
4631:decidable
4465:saturated
4443:of models
4366:deductive
4361:axiomatic
4281:Hilbert's
4268:Euclidean
4249:canonical
4172:axiomatic
4104:Signature
4033:Predicate
3922:Extension
3844:Ackermann
3769:Operation
3648:Universal
3638:Recursive
3613:Singleton
3608:Inhabited
3593:Countable
3583:Types of
3567:power set
3537:partition
3454:Predicate
3400:Predicate
3315:Syllogism
3305:Soundness
3278:Inference
3268:Tautology
3170:paradoxes
2988:therefore
2976:therefore
2931:tautology
2895:universal
2688:See p. 58
2412:, and no
2410:empty set
2396:∅
2355:∅
2351:∈
2342:∃
2288:→
2245:→
2233:→
2209:→
2187:∈
2178:∃
2163:in which
2114:∈
2105:∃
2099:→
2015:Tautology
1770:∈
1761:∃
1758:∨
1736:∈
1727:∃
1718:→
1703:∨
1681:∈
1672:∃
1628:∈
1619:∃
1613:¬
1610:≡
1588:∈
1579:∄
1541:¬
1531:∈
1522:∃
1516:≡
1494:∈
1485:∀
1479:¬
1458:¬
1448:∈
1439:∀
1433:≡
1411:∈
1402:∃
1396:¬
1350:¬
1340:∈
1331:∀
1325:≡
1303:∈
1294:∃
1288:¬
1242:¬
1232:∈
1223:∀
1170:∈
1161:∃
1115:∈
1106:∃
1100:¬
1051:∈
1042:∃
993:¬
960:∪
940:∩
893:×
833:×
818:∈
809:∃
744:×
661:×
593:×
531:×
499:×
452:×
420:×
388:×
352:×
317:sentence
297:, and as
223:which is
175:∃
84:∃
74:Statement
36:Yo (kana)
4755:Logicism
4748:timeline
4724:Concrete
4583:Validity
4553:T-schema
4546:Kripke's
4541:Tarski's
4536:semantic
4526:Strength
4475:submodel
4470:spectrum
4438:function
4286:Tarski's
4275:Elements
4262:geometry
4218:Robinson
4139:variable
4124:function
4097:spectrum
4087:Sentence
4043:variable
3986:Language
3939:Relation
3900:Automata
3890:Alphabet
3874:language
3728:-jection
3706:codomain
3692:Function
3653:Universe
3623:Infinite
3527:Relation
3310:Validity
3300:Argument
3198:theorem,
2974:entails,
2960:entails,
2848:superset
2482:See also
2462:between
1473:≢
981:Negation
785:Notation
554:solution
288:∃
4697:Related
4494:Diagram
4392: (
4371:Hilbert
4356:Systems
4351:Theorem
4229:of the
4174:systems
3954:Formula
3949:Grammar
3865: (
3809:General
3522:Forcing
3507:Element
3427:Monadic
3202:paradox
3143:Theorem
3079:General
3000:because
2864:
2843:implies
2831:implies
2813:
2779:
2754:Common
2474:is the
2460:functor
797:" in a
299:\exists
295:Unicode
4460:finite
4223:Skolem
4176:
4151:Theory
4119:Symbol
4109:String
4092:atomic
3969:ground
3964:closed
3959:atomic
3915:ground
3878:syntax
3774:binary
3701:domain
3618:Finite
3383:finite
3241:Logics
3200:
3148:Theory
2962:proves
2858:
2807:
2773:
2712:
2682:
2655:
2620:
2545:
2466:, the
2291:
2248:
2236:
2212:
2102:
1721:
1616:
1519:
1482:
1476:
1436:
1399:
1328:
1291:
1103:
996:
315:formal
309:Basics
285:
283:U+2203
253:" or "
245:" or "
232:symbol
4450:Model
4198:Peano
4055:Proof
3895:Arity
3824:Naive
3711:image
3643:Fuzzy
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