6894:
5137:
4426:
2743:
1491:
50:
4319:
130:
5126:
4820:
3622:, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris is in France". These classic problems have been called the
3704:
study, where less than 10% of participants reasoned according to the material conditional. Some researchers have interpreted this result as a failure of the participants to conform to normative laws of reasoning, while others interpret the participants as reasoning normatively according to
3192:
3085:
2401:
3439:
1511:
3360:
2898:
3600:
3661:
has generally eschewed material implication as an analysis for natural-language conditionals. In particular, such work has often rejected the assumption that natural-language conditionals are
2950:
2724:
3539:
3253:
2831:
2993:
854:
1281:
796:
709:
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921:
582:
3293:
1919:
1240:
880:
556:
261:
982:
218:
177:
4640:
1063:
3481:
2512:
2439:
2305:
1595:
828:
668:
616:
496:
83:
4716:
1393:
1341:
947:
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1761:
1565:
1198:
4687:
4487:
735:
2241:
2203:
2165:
2043:
1367:
642:
4462:
1307:
530:
1741:
467:
418:
392:
1172:
1120:
761:
4770:
2267:
2121:
1817:
1086:
1013:
117:
4807:
4582:
4545:
4416:
1721:
4164:
Stenning, K.; van
Lambalgen, M. (2004). "A little logic goes a long way: basing experiment on semantic theory in the cognitive science of conditional reasoning".
1791:
1146:
1036:
441:
2485:
2464:
2083:
2063:
2017:
1997:
1977:
1957:
1871:
of the conditional. Conditional statements may be nested such that the antecedent or the consequent may themselves be conditional statements, as in the formula
1863:
1837:
1635:
1615:
3095:
1518:
337:
5273:
3000:
2671:
Unlike the semantic definition, this approach to logical connectives permits the examination of structurally identical propositional forms in various
4855:
5948:
3645:
principles could explain the discrepancies between natural language conditionals and the material conditional. On their accounts, conditionals
2334:
operator which returns "true" unless its first argument is true and its second argument is false. This semantics can be shown graphically in a
4370:
1931:
6031:
5172:
2341:
3364:
3626:. In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis. For instance,
3938:"elementary set theory – Is there any connection between the symbol ⊃ when it means implication and its meaning as superset?"
3757:
330:
31:
3370:
6345:
3912:
20:
6503:
4323:
4101:
3870:
5291:
1671:. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many
6358:
5681:
4848:
4028:
3735:
1504:
3303:
4340:
3700:
Similar discrepancies have been observed by psychologists studying conditional reasoning, for instance, by the notorious
3654:
2841:
323:
3549:
6363:
6353:
6090:
5943:
5296:
2905:
5287:
3649:
material implication but end up conveying additional information when they interact with conversational norms such as
2682:
6499:
4363:
4051:
3895:
3623:
1684:
5841:
3681:. Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as
6596:
6340:
5165:
3491:
3208:
2792:
5901:
5594:
4841:
5335:
3650:
6857:
6559:
6322:
6317:
6142:
5563:
5247:
3197:
Similarly, on classical interpretations of the other connectives, material implication validates the following
2957:
2727:
6852:
6635:
6552:
6265:
6196:
6073:
5315:
833:
1253:
766:
681:
6928:
6918:
6777:
6603:
6289:
5923:
5522:
4596:
4356:
2754:
1494:
893:
561:
3263:
1874:
6655:
6650:
6260:
5999:
5928:
5257:
5158:
3865:(6th ed.). Boca Raton: CRC Press/Taylor & Francis Group (A Chapman & Hall Book). p. 2.
1680:
1211:
859:
4202:
4117:
Oaksford, M.; Chater, N. (1994). "A rational analysis of the selection task as optimal data selection".
535:
231:
49:
6584:
6174:
5568:
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5227:
5020:
5015:
4004:
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3627:
952:
190:
149:
4625:
6874:
6823:
6720:
6218:
6179:
5656:
5301:
3785:
1041:
5330:
4133:
3457:
2491:
2418:
2284:
1703:
In logic and related fields, the material conditional is customarily notated with an infix operator
1574:
807:
647:
595:
472:
62:
6715:
6645:
6184:
6036:
6019:
5742:
5222:
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4284:
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1372:
1320:
926:
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4501:
1746:
1550:
1177:
6547:
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5958:
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5722:
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4720:
4691:
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1441:
714:
183:
142:
2220:
2182:
2144:
2022:
1346:
621:
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6509:
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5801:
5767:
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4248:
4173:
4128:
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3730:
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1568:
1411:
1286:
509:
1726:
446:
397:
371:
6923:
6767:
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6130:
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5772:
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5109:
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2129:
1151:
1099:
740:
502:
302:
4755:
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2100:
1796:
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and related problems, material implication is not generally considered a viable analysis of
1068:
995:
96:
6740:
6702:
6579:
6383:
6223:
6147:
5953:
5911:
5810:
5777:
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4792:
4567:
4530:
4401:
4204:
Towards a
Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules
4119:
3701:
3611:
2676:
2662:
1688:
1672:
1654:
1469:
224:
1706:
8:
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6682:
6569:
6519:
6445:
6390:
6327:
6120:
6115:
6063:
5831:
5820:
5492:
5392:
5320:
5311:
5307:
5242:
5237:
5097:
4906:
4894:
4745:
4520:
4068:
3752:
3297:
3089:
2778:
1770:
1479:
1125:
1092:
364:
3187:{\displaystyle {\big (}R\to (P\to Q){\big )}\equiv {\big (}(R\to P)\to (R\to Q){\big )}}
1018:
423:
6933:
6898:
6667:
6630:
6615:
6608:
6591:
6395:
6377:
6243:
6169:
6152:
6105:
5918:
5827:
5661:
5646:
5606:
5558:
5543:
5531:
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5462:
5232:
5181:
5141:
5069:
5065:
4948:
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4252:
4218:
4146:
3800:
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2470:
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2324:
2068:
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2002:
1982:
1962:
1942:
1848:
1841:
1822:
1676:
1668:
1620:
1600:
1544:
1474:
353:
292:
27:
5851:
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6640:
6450:
6440:
6332:
6213:
6048:
6024:
5805:
5789:
5694:
5671:
5548:
5517:
5482:
5377:
5212:
5136:
5130:
5040:
4824:
4644:
4391:
4262:
4256:
4222:
4097:
4047:
3891:
3866:
3858:
3444:
2652:
1675:. However, many logics replace material implication with other operators such as the
1650:
1416:
267:
4425:
6847:
6842:
6735:
6692:
6514:
6475:
6470:
6455:
6281:
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5933:
5883:
5457:
5419:
5083:
5079:
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4331:
4294:
4208:
4183:
4150:
4138:
4087:
4079:
4039:
3720:
3615:
2270:
2124:
1692:
1436:
1313:
3937:
6828:
6818:
6772:
6755:
6710:
6672:
6574:
6494:
6301:
6228:
6201:
6189:
6095:
6009:
5983:
5938:
5906:
5707:
5509:
5452:
5402:
5367:
5325:
4335:
4270:
4187:
3762:
3686:
3485:
3080:{\displaystyle {\big (}P\to (Q\to R){\big )}\equiv {\big (}Q\to (P\to R){\big )}}
2619:
2598:
2328:
2321:
2168:
2019:
with the symbol Ɔ, which is the opposite of C. He also expressed the proposition
1764:
1664:
1540:
1421:
4142:
6813:
6792:
6750:
6730:
6625:
6480:
6078:
6068:
6058:
6053:
5987:
5861:
5737:
5626:
5621:
5599:
5200:
5093:
4991:
4987:
4979:
4965:
4936:
4880:
4648:
4437:
4083:
4043:
3715:
3662:
2657:
2331:
1936:
1431:
674:
3819:
Note that the horseshoe symbol Ɔ has been flipped to become a subset symbol ⊂.
6912:
6787:
6465:
5972:
5757:
5747:
5717:
5702:
5372:
5057:
5052:
4786:
4782:
4395:
4348:
3747:
3725:
3619:
3257:
2672:
2578:
2206:
2086:
1645:
1446:
3618:. For example, even though material conditionals with false antecedents are
6687:
6534:
6435:
6427:
6307:
6255:
6164:
6100:
6083:
6014:
5873:
5732:
5434:
5217:
5035:
5031:
4902:
4586:
3741:
3694:
2675:, where somewhat different properties may be demonstrated. For example, in
2646:
2634:
1639:
1246:
2785:
are classical, material implication validates the following equivalences:
1637:
is false. Material implication can also be characterized inferentially by
6797:
6677:
5856:
5846:
5793:
5477:
5397:
5382:
5262:
5207:
4833:
4695:
4662:
4276:
4213:
3682:
3634:
2774:
2615:
2594:
2412:
2335:
1426:
886:
89:
4092:
3913:"English Translation of 'Arithmetices Principia, Nova Methodo Exposita'"
2742:
5727:
5582:
5553:
5359:
4975:
4495:
3646:
3642:
3198:
1867:
1464:
123:
2679:, which rejects proofs by contraposition as valid rules of inference,
19:"Logical conditional" redirects here. For other related meanings, see
6879:
6782:
5835:
5752:
5712:
5676:
5612:
5424:
5414:
5387:
5150:
4890:
4619:
4441:
4074:. In von Heusinger, Klaus; Maienborn, Claudia; Portner, Paul (eds.).
1204:
2396:{\displaystyle A\to B\equiv \neg (A\land \neg B)\equiv \neg A\lor B}
6864:
6662:
6110:
5815:
5409:
4999:
4928:
4914:
4749:
4666:
4590:
4557:
3888:
From Frege to Gödel: A Source Book in
Mathematical Logic, 1879–1931
2782:
988:
6460:
5252:
5048:
4524:
4957:
4318:
4283:, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition,
4244:
3845:
Prinzipien der
Mathematik (Lecture Notes edited by Bernays, P.)
6004:
5350:
5195:
4924:
2338:
such as the one below. One can also consider the equivalence
1723:. The material conditional is also notated using the infixes
129:
3434:{\displaystyle (P\lor Q)\to R\models (P\to R)\land (Q\to R)}
3633:
In the mid-20th century, a number of researchers including
4207:(doctoralThesis). Göttingen: Göttingen University Press.
3610:
Material implication does not closely match the usage of
1663:
Material implication is used in all the basic systems of
2133:(1910–1913), in which he expressed the proposition "If
3605:
4795:
4758:
4729:
4704:
4675:
4628:
4599:
4570:
4533:
4504:
4475:
4450:
4404:
4163:
3552:
3494:
3460:
3373:
3306:
3266:
3211:
3098:
3003:
2960:
2908:
2844:
2795:
2685:
2494:
2473:
2452:
2421:
2344:
2287:
2249:
2223:
2185:
2147:
2103:
2071:
2051:
2025:
2005:
1985:
1965:
1945:
1877:
1851:
1825:
1799:
1773:
1749:
1729:
1709:
1623:
1603:
1577:
1553:
1375:
1349:
1323:
1289:
1256:
1214:
1180:
1154:
1128:
1102:
1071:
1044:
1021:
998:
955:
929:
896:
862:
836:
810:
769:
743:
717:
684:
650:
624:
598:
564:
538:
512:
475:
449:
426:
400:
374:
234:
193:
152:
99:
65:
3355:{\displaystyle (P\to Q)\land (Q\to R)\models P\to R}
3885:
2893:{\displaystyle P\to (Q\to R)\equiv (P\land Q)\to R}
2728:
the material conditional is used to define negation
2601:, then Galileo Galilei is a brother of Marie Curie"
4801:
4764:
4735:
4710:
4681:
4634:
4605:
4576:
4539:
4510:
4481:
4456:
4410:
4241:Boolean Reasoning: The Logic of Boolean Equations
3979:
3977:
3975:
3973:
3595:{\displaystyle \models (P\to Q)\lor (P\to \neg Q)}
3594:
3533:
3475:
3433:
3354:
3287:
3247:
3186:
3079:
2987:
2944:
2892:
2825:
2718:
2506:
2479:
2458:
2433:
2395:
2299:
2261:
2235:
2197:
2159:
2115:
2077:
2057:
2037:
2011:
1991:
1971:
1951:
1913:
1857:
1831:
1811:
1785:
1755:
1735:
1715:
1629:
1609:
1589:
1559:
1387:
1361:
1335:
1301:
1275:
1234:
1192:
1166:
1140:
1114:
1080:
1057:
1030:
1007:
976:
941:
915:
874:
848:
822:
790:
755:
729:
703:
662:
636:
610:
576:
550:
524:
490:
461:
435:
412:
386:
255:
212:
171:
111:
77:
3966:. Paris: Hermann & Cie, Éditeurs. p. 14.
2945:{\displaystyle \neg (P\to Q)\equiv P\land \neg Q}
6910:
3935:
3630:would all be vacuously true on such an account.
2719:{\displaystyle (A\to B)\Rightarrow \neg A\lor B}
4076:Semantics: An international handbook of meaning
3970:
2633:Material implication can also be characterized
4378:
4116:
4034:. In Hale, B.; Wright, C.; Miller, A. (eds.).
3673:" is determined solely by the truth values of
5166:
4849:
4364:
3910:
3838:
3836:
3534:{\displaystyle \models (P\to Q)\lor (Q\to P)}
3248:{\displaystyle P\to Q\models (P\land R)\to Q}
3179:
3139:
3129:
3101:
3072:
3044:
3034:
3006:
2826:{\displaystyle P\to Q\equiv \neg Q\to \neg P}
1932:Arithmetices Principia: Nova Methodo Exposita
1512:
331:
4265:(2001), "Conditionals", in Lou Goble (ed.),
4194:
3955:
3890:. Harvard University Press. pp. 84–87.
5358:
5173:
5159:
4863:
4856:
4842:
4371:
4357:
4267:The Blackwell Guide to Philosophical Logic
4066:
4022:
4020:
4018:
3833:
1519:
1505:
338:
324:
4212:
4200:
4177:
4132:
4091:
4078:. de Gruyter Mouton. pp. 1515–1538.
4036:A Companion to the Philosophy of Language
3983:
3857:
3665:in the sense that the truth value of "If
3365:Simplification of disjunctive antecedents
2988:{\displaystyle P\to Q\equiv \neg P\lor Q}
2243:at first but later came to express it as
3961:
3758:Material implication (rule of inference)
3447:involving material implication include:
32:Material implication (rule of inference)
4026:
4015:
4009:The Stanford Encyclopedia of Philosophy
3990:The Stanford Encyclopedia of Philosophy
3842:
2628:
2563:The logical cases where the antecedent
849:{\displaystyle A\not \Leftrightarrow B}
6911:
5180:
1276:{\displaystyle A{\underline {\lor }}B}
791:{\displaystyle {\overline {A\cdot B}}}
704:{\displaystyle A{\overline {\land }}B}
5154:
4837:
4606:{\displaystyle \not \leftrightarrow }
4352:
4329:
4038:. Wiley Blackwell. pp. 401–436.
4002:
916:{\displaystyle A{\overline {\lor }}B}
577:{\displaystyle A\leftrightharpoons B}
3736:Implicational propositional calculus
3288:{\displaystyle \neg P\models P\to Q}
2737:
2733:
2726:is not a propositional theorem, but
1914:{\displaystyle (p\to q)\to (r\to s)}
4341:Stanford Encyclopedia of Philosophy
3606:Discrepancies with natural language
1571:as material implication, a formula
1235:{\displaystyle A\ {\text{XNOR}}\ B}
875:{\displaystyle A\nleftrightarrow B}
13:
4796:
4571:
4405:
4233:
3863:Introduction to Mathematical Logic
3583:
3267:
2973:
2936:
2909:
2817:
2808:
2704:
2622:, then Marie Curie has a sibling."
2381:
2369:
2357:
999:
551:{\displaystyle A\Leftrightarrow B}
482:
479:
453:
256:{\displaystyle 1\oplus x\oplus xy}
14:
6945:
4311:
3886:Jean van Heijenoort, ed. (1967).
3624:paradoxes of material implication
1685:paradoxes of material implication
977:{\displaystyle {\overline {A+B}}}
213:{\displaystyle {\overline {x}}+y}
172:{\displaystyle {\overline {x}}+y}
6892:
5135:
5124:
4818:
4635:{\displaystyle \leftrightarrow }
4424:
4317:
2741:
1490:
1489:
128:
48:
4157:
4110:
4060:
3936:Mauro ALLEGRANZA (2015-02-13).
3779:
1058:{\displaystyle {\overline {A}}}
4676:
4629:
4505:
4476:
4451:
3996:
3929:
3904:
3879:
3851:
3813:
3589:
3580:
3574:
3568:
3562:
3556:
3528:
3522:
3516:
3510:
3504:
3498:
3476:{\displaystyle \models P\to P}
3467:
3428:
3422:
3416:
3410:
3404:
3398:
3389:
3386:
3374:
3346:
3337:
3331:
3325:
3319:
3313:
3307:
3279:
3239:
3236:
3224:
3215:
3174:
3168:
3162:
3159:
3156:
3150:
3144:
3124:
3118:
3112:
3109:
3067:
3061:
3055:
3052:
3029:
3023:
3017:
3014:
2997:Commutativity of antecedents:
2964:
2924:
2918:
2912:
2884:
2881:
2869:
2863:
2857:
2851:
2848:
2814:
2799:
2701:
2698:
2692:
2686:
2507:{\displaystyle A\rightarrow B}
2498:
2434:{\displaystyle A\rightarrow B}
2425:
2406:
2375:
2360:
2348:
2327:, material implication is the
2310:
2300:{\displaystyle A\Rightarrow B}
2291:
2273:expressed the proposition "If
2253:
2209:expressed the proposition "If
2171:expressed the proposition "If
2107:
2089:expressed the proposition "If
1939:expressed the proposition "If
1908:
1902:
1896:
1893:
1890:
1884:
1878:
1803:
1767:, conditionals are notated as
1750:
1710:
1590:{\displaystyle P\rightarrow Q}
1581:
1554:
1547:. When the conditional symbol
1379:
1327:
933:
823:{\displaystyle A\not \equiv B}
721:
663:{\displaystyle A\rightarrow B}
654:
611:{\displaystyle A\Rightarrow B}
602:
568:
542:
491:{\displaystyle A\&\&B}
106:
100:
78:{\displaystyle x\rightarrow y}
69:
1:
6853:History of mathematical logic
4711:{\displaystyle \nrightarrow }
4297:, "Indicative Conditionals",
4239:Brown, Frank Markham (2003),
4007:. In Zalta, Edward N. (ed.).
3911:Michael Nahas (25 Apr 2022).
3826:
2269:with a right-pointing arrow.
1388:{\displaystyle A\leftarrow B}
1336:{\displaystyle A\Leftarrow B}
942:{\displaystyle A\downarrow B}
6778:Primitive recursive function
4736:{\displaystyle \nleftarrow }
4511:{\displaystyle \rightarrow }
4188:10.1016/j.cogsci.2004.02.002
3988:. In Edward N. Zalta (ed.).
3944:. Stack Exchange Inc. Answer
2315:
1756:{\displaystyle \Rightarrow }
1560:{\displaystyle \rightarrow }
1193:{\displaystyle A\parallel B}
1050:
969:
905:
783:
693:
199:
158:
7:
4682:{\displaystyle \downarrow }
4482:{\displaystyle \leftarrow }
4143:10.1037/0033-295X.101.4.608
3984:Edgington, Dorothy (2008).
3708:
3628:counterfactual conditionals
3544:Conditional excluded middle
1793:. In a conditional formula
1698:
1681:variably strict conditional
730:{\displaystyle A\uparrow B}
10:
6950:
5842:Schröder–Bernstein theorem
5569:Monadic predicate calculus
5228:Foundations of mathematics
4084:10.1515/9783110255072.1515
4044:10.1002/9781118972090.ch17
3942:Mathematics Stack Exchange
3791:Counterfactual conditional
3205:Antecedent strengthening:
2637:in terms of the following
2236:{\displaystyle A\supset B}
2198:{\displaystyle A\supset B}
2160:{\displaystyle A\supset B}
2038:{\displaystyle A\supset B}
1924:
1362:{\displaystyle A\subset B}
637:{\displaystyle A\supset B}
25:
18:
6888:
6875:Philosophy of mathematics
6824:Automated theorem proving
6806:
6701:
6533:
6426:
6278:
5995:
5971:
5949:Von Neumann–Bernays–Gödel
5894:
5788:
5692:
5590:
5581:
5508:
5443:
5349:
5271:
5188:
5121:
4872:
4815:
4778:
4658:
4553:
4457:{\displaystyle \uparrow }
4433:
4422:
4387:
3786:Corresponding conditional
1302:{\displaystyle A\oplus B}
525:{\displaystyle A\equiv B}
319:
311:
301:
291:
283:
275:
266:
223:
182:
141:
136:
122:
88:
56:
47:
42:
4285:Harvard University Press
4067:von Fintel, Kai (2011).
3806:
3773:Sole sufficient operator
2658:Classical contraposition
1736:{\displaystyle \supset }
462:{\displaystyle A\&B}
413:{\displaystyle A\cdot B}
387:{\displaystyle A\land B}
26:Not to be confused with
6525:Self-verifying theories
6346:Tarski's axiomatization
5297:Tarski's undefinability
5292:incompleteness theorems
4721:Converse nonimplication
4027:Gillies, Thony (2017).
3992:(Winter 2008 ed.).
1442:Functional completeness
1167:{\displaystyle A\mid B}
1115:{\displaystyle A\lor B}
756:{\displaystyle A\mid B}
6899:Mathematics portal
6510:Proof of impossibility
6158:propositional variable
5468:Propositional calculus
5142:Mathematics portal
4803:
4766:
4765:{\displaystyle \land }
4737:
4712:
4683:
4636:
4607:
4578:
4541:
4512:
4483:
4458:
4412:
4201:von Sydow, M. (2006).
3796:Indicative conditional
3768:Propositional calculus
3731:Conditional quantifier
3659:philosophy of language
3596:
3535:
3477:
3435:
3356:
3289:
3249:
3188:
3081:
2989:
2946:
2902:Negated conditionals:
2894:
2827:
2720:
2508:
2481:
2460:
2435:
2397:
2301:
2263:
2262:{\displaystyle A\to B}
2237:
2199:
2161:
2127:followed Peano in his
2117:
2116:{\displaystyle A\to B}
2079:
2059:
2039:
2013:
1993:
1973:
1953:
1915:
1859:
1839:is referred to as the
1833:
1813:
1812:{\displaystyle p\to q}
1787:
1757:
1737:
1717:
1631:
1611:
1591:
1561:
1412:Propositional calculus
1389:
1363:
1337:
1303:
1277:
1236:
1194:
1168:
1142:
1116:
1082:
1081:{\displaystyle \sim A}
1059:
1032:
1009:
1008:{\displaystyle \neg A}
978:
943:
917:
876:
850:
824:
792:
757:
731:
705:
664:
638:
612:
578:
552:
526:
492:
463:
437:
414:
388:
257:
214:
173:
113:
112:{\displaystyle (1011)}
79:
6768:Kolmogorov complexity
6721:Computably enumerable
6621:Model complete theory
6413:Principia Mathematica
5473:Propositional formula
5302:Banach–Tarski paradox
5131:Philosophy portal
4825:Philosophy portal
4804:
4802:{\displaystyle \bot }
4767:
4738:
4713:
4684:
4637:
4608:
4579:
4577:{\displaystyle \neg }
4542:
4540:{\displaystyle \lor }
4513:
4484:
4459:
4413:
4411:{\displaystyle \top }
4247:Academic Publishers,
3964:Théorie des ensembles
3962:Bourbaki, N. (1954).
3612:conditional sentences
3597:
3536:
3478:
3436:
3357:
3290:
3250:
3189:
3082:
2990:
2947:
2895:
2828:
2721:
2577:is true, are called "
2509:
2482:
2461:
2436:
2398:
2302:
2264:
2238:
2200:
2167:. Following Russell,
2162:
2130:Principia Mathematica
2118:
2080:
2060:
2040:
2014:
1994:
1974:
1954:
1916:
1860:
1834:
1814:
1788:
1758:
1738:
1718:
1689:conditional sentences
1673:programming languages
1632:
1612:
1592:
1562:
1470:Programming languages
1390:
1364:
1338:
1304:
1278:
1237:
1195:
1169:
1143:
1117:
1083:
1060:
1033:
1010:
979:
944:
918:
877:
851:
825:
793:
758:
732:
706:
665:
639:
613:
579:
553:
527:
493:
464:
438:
415:
389:
258:
215:
174:
114:
80:
21:Conditional statement
6716:Church–Turing thesis
6703:Computability theory
5912:continuum hypothesis
5430:Square of opposition
5288:Gödel's completeness
4793:
4756:
4727:
4702:
4673:
4626:
4597:
4568:
4531:
4502:
4473:
4467:Converse implication
4448:
4402:
4330:Edgington, Dorothy.
4326:at Wikimedia Commons
4324:Material conditional
4251:, MA. 2nd edition,
4214:10.53846/goediss-161
4120:Psychological Review
4003:Starr, Will (2019).
3918:. GitHub. p. VI
3843:Hilbert, D. (1918).
3702:Wason selection task
3550:
3492:
3458:
3371:
3304:
3264:
3209:
3096:
3001:
2958:
2906:
2842:
2793:
2683:
2677:intuitionistic logic
2665:reductio ad absurdum
2629:Deductive definition
2581:". Examples are ...
2492:
2471:
2450:
2419:
2342:
2325:semantic perspective
2285:
2247:
2221:
2183:
2145:
2101:
2069:
2049:
2023:
2003:
1983:
1963:
1943:
1875:
1849:
1823:
1797:
1771:
1747:
1727:
1716:{\displaystyle \to }
1707:
1657:reductio ad absurdum
1621:
1601:
1575:
1551:
1537:material implication
1533:material conditional
1373:
1347:
1321:
1287:
1254:
1212:
1178:
1152:
1126:
1100:
1069:
1042:
1019:
996:
953:
927:
894:
860:
834:
808:
767:
741:
715:
682:
648:
622:
596:
562:
536:
510:
473:
447:
424:
398:
372:
232:
225:Zhegalkin polynomial
191:
150:
97:
63:
38:Material conditional
6929:Logical consequence
6919:Logical connectives
6870:Mathematical object
6761:P versus NP problem
6726:Computable function
6520:Reverse mathematics
6446:Logical consequence
6323:primitive recursive
6318:elementary function
6091:Free/bound variable
5944:Tarski–Grothendieck
5463:Logical connectives
5393:Logical equivalence
5243:Logical consequence
4381:logical connectives
3753:Logical equivalence
3705:nonclassical laws.
3258:Vacuous conditional
3090:Left distributivity
1786:{\displaystyle Cpq}
1669:nonclassical logics
1480:Philosophy of logic
1141:{\displaystyle A+B}
354:Logical connectives
39:
6668:Transfer principle
6631:Semantics of logic
6616:Categorical theory
6592:Non-standard model
6106:Logical connective
5233:Information theory
5182:Mathematical logic
4799:
4762:
4733:
4708:
4679:
4632:
4603:
4574:
4537:
4508:
4479:
4454:
4438:Alternative denial
4408:
4263:Edgington, Dorothy
4253:Dover Publications
3859:Mendelson, Elliott
3801:Strict conditional
3691:probability theory
3592:
3531:
3473:
3431:
3352:
3285:
3245:
3184:
3077:
2985:
2942:
2890:
2823:
2753:. You can help by
2716:
2639:rules of inference
2504:
2477:
2456:
2431:
2393:
2297:
2259:
2233:
2195:
2157:
2113:
2075:
2055:
2035:
2009:
1989:
1969:
1949:
1911:
1855:
1829:
1809:
1783:
1763:. In the prefixed
1753:
1733:
1713:
1677:strict conditional
1627:
1607:
1587:
1557:
1475:Mathematical logic
1385:
1359:
1333:
1299:
1273:
1268:
1232:
1190:
1164:
1138:
1112:
1078:
1055:
1031:{\displaystyle -A}
1028:
1005:
974:
939:
913:
872:
846:
820:
788:
753:
727:
701:
660:
634:
608:
574:
548:
522:
488:
459:
436:{\displaystyle AB}
433:
410:
384:
253:
210:
169:
109:
75:
37:
28:Material inference
16:Logical connective
6906:
6905:
6838:Abstract category
6641:Theories of truth
6451:Rule of inference
6441:Natural deduction
6422:
6421:
5967:
5966:
5672:Cartesian product
5577:
5576:
5483:Many-valued logic
5458:Boolean functions
5341:Russell's paradox
5316:diagonal argument
5213:First-order logic
5148:
5147:
5116:
5115:
4831:
4830:
4322:Media related to
4295:Stalnaker, Robert
4166:Cognitive Science
4103:978-3-11-018523-2
4005:"Counterfactuals"
3872:978-1-4822-3778-8
3653:. Recent work in
2771:
2770:
2734:Formal properties
2653:Conditional proof
2561:
2560:
2480:{\displaystyle B}
2459:{\displaystyle A}
2078:{\displaystyle B}
2058:{\displaystyle A}
2012:{\displaystyle B}
1992:{\displaystyle A}
1972:{\displaystyle B}
1952:{\displaystyle A}
1858:{\displaystyle q}
1832:{\displaystyle p}
1819:, the subformula
1651:conditional proof
1630:{\displaystyle Q}
1610:{\displaystyle P}
1543:commonly used in
1529:
1528:
1398:
1397:
1261:
1228:
1224:
1220:
1053:
972:
908:
786:
696:
348:
347:
202:
161:
6941:
6897:
6896:
6848:History of logic
6843:Category of sets
6736:Decision problem
6515:Ordinal analysis
6456:Sequent calculus
6354:Boolean algebras
6294:
6293:
6268:
6239:logical/constant
5993:
5992:
5979:
5902:Zermelo–Fraenkel
5653:Set operations:
5588:
5587:
5525:
5356:
5355:
5336:Löwenheim–Skolem
5223:Formal semantics
5175:
5168:
5161:
5152:
5151:
5140:
5139:
5129:
5128:
5127:
4973:
4922:
4888:
4875:
4874:
4858:
4851:
4844:
4835:
4834:
4823:
4822:
4821:
4808:
4806:
4805:
4800:
4771:
4769:
4768:
4763:
4742:
4740:
4739:
4734:
4717:
4715:
4714:
4709:
4688:
4686:
4685:
4680:
4641:
4639:
4638:
4633:
4612:
4610:
4609:
4604:
4583:
4581:
4580:
4575:
4546:
4544:
4543:
4538:
4517:
4515:
4514:
4509:
4488:
4486:
4485:
4480:
4463:
4461:
4460:
4455:
4428:
4417:
4415:
4414:
4409:
4373:
4366:
4359:
4350:
4349:
4345:
4336:Zalta, Edward N.
4321:
4307:(1975): 269–286.
4281:Methods of Logic
4227:
4226:
4216:
4198:
4192:
4191:
4181:
4161:
4155:
4154:
4136:
4114:
4108:
4107:
4095:
4073:
4064:
4058:
4057:
4033:
4024:
4013:
4012:
4000:
3994:
3993:
3981:
3968:
3967:
3959:
3953:
3952:
3950:
3949:
3933:
3927:
3926:
3924:
3923:
3917:
3908:
3902:
3901:
3883:
3877:
3876:
3855:
3849:
3848:
3840:
3820:
3817:
3721:Boolean function
3663:truth functional
3655:formal semantics
3635:H. P. Grice
3616:natural language
3601:
3599:
3598:
3593:
3540:
3538:
3537:
3532:
3482:
3480:
3479:
3474:
3440:
3438:
3437:
3432:
3361:
3359:
3358:
3353:
3294:
3292:
3291:
3286:
3254:
3252:
3251:
3246:
3193:
3191:
3190:
3185:
3183:
3182:
3143:
3142:
3133:
3132:
3105:
3104:
3086:
3084:
3083:
3078:
3076:
3075:
3048:
3047:
3038:
3037:
3010:
3009:
2994:
2992:
2991:
2986:
2951:
2949:
2948:
2943:
2899:
2897:
2896:
2891:
2832:
2830:
2829:
2824:
2789:Contraposition:
2766:
2763:
2745:
2738:
2725:
2723:
2722:
2717:
2611:
2590:
2576:
2566:
2513:
2511:
2510:
2505:
2486:
2484:
2483:
2478:
2465:
2463:
2462:
2457:
2444:
2443:
2440:
2438:
2437:
2432:
2402:
2400:
2399:
2394:
2332:truth functional
2306:
2304:
2303:
2298:
2268:
2266:
2265:
2260:
2242:
2240:
2239:
2234:
2204:
2202:
2201:
2196:
2166:
2164:
2163:
2158:
2122:
2120:
2119:
2114:
2084:
2082:
2081:
2076:
2064:
2062:
2061:
2056:
2044:
2042:
2041:
2036:
2018:
2016:
2015:
2010:
1998:
1996:
1995:
1990:
1978:
1976:
1975:
1970:
1958:
1956:
1955:
1950:
1920:
1918:
1917:
1912:
1864:
1862:
1861:
1856:
1838:
1836:
1835:
1830:
1818:
1816:
1815:
1810:
1792:
1790:
1789:
1784:
1762:
1760:
1759:
1754:
1742:
1740:
1739:
1734:
1722:
1720:
1719:
1714:
1693:natural language
1667:as well as some
1636:
1634:
1633:
1628:
1616:
1614:
1613:
1608:
1596:
1594:
1593:
1588:
1566:
1564:
1563:
1558:
1521:
1514:
1507:
1493:
1492:
1437:Boolean function
1403:Related concepts
1394:
1392:
1391:
1386:
1368:
1366:
1365:
1360:
1342:
1340:
1339:
1334:
1308:
1306:
1305:
1300:
1282:
1280:
1279:
1274:
1269:
1241:
1239:
1238:
1233:
1226:
1225:
1222:
1218:
1199:
1197:
1196:
1191:
1173:
1171:
1170:
1165:
1147:
1145:
1144:
1139:
1121:
1119:
1118:
1113:
1087:
1085:
1084:
1079:
1064:
1062:
1061:
1056:
1054:
1046:
1037:
1035:
1034:
1029:
1014:
1012:
1011:
1006:
983:
981:
980:
975:
973:
968:
957:
948:
946:
945:
940:
922:
920:
919:
914:
909:
901:
881:
879:
878:
873:
855:
853:
852:
847:
829:
827:
826:
821:
797:
795:
794:
789:
787:
782:
771:
762:
760:
759:
754:
736:
734:
733:
728:
710:
708:
707:
702:
697:
689:
669:
667:
666:
661:
643:
641:
640:
635:
617:
615:
614:
609:
583:
581:
580:
575:
557:
555:
554:
549:
531:
529:
528:
523:
497:
495:
494:
489:
468:
466:
465:
460:
442:
440:
439:
434:
419:
417:
416:
411:
393:
391:
390:
385:
361:
360:
350:
349:
340:
333:
326:
270:
262:
260:
259:
254:
219:
217:
216:
211:
203:
195:
178:
176:
175:
170:
162:
154:
132:
118:
116:
115:
110:
84:
82:
81:
76:
52:
40:
36:
6949:
6948:
6944:
6943:
6942:
6940:
6939:
6938:
6909:
6908:
6907:
6902:
6891:
6884:
6829:Category theory
6819:Algebraic logic
6802:
6773:Lambda calculus
6711:Church encoding
6697:
6673:Truth predicate
6529:
6495:Complete theory
6418:
6287:
6283:
6279:
6274:
6266:
5986: and
5982:
5977:
5963:
5939:New Foundations
5907:axiom of choice
5890:
5852:Gödel numbering
5792: and
5784:
5688:
5573:
5523:
5504:
5453:Boolean algebra
5439:
5403:Equiconsistency
5368:Classical logic
5345:
5326:Halting problem
5314: and
5290: and
5278: and
5277:
5272:Theorems (
5267:
5184:
5179:
5149:
5144:
5134:
5133:
5125:
5123:
5117:
5112:
5108:
5100:
5096:
5088:
5085:
5082:
5074:
5071:
5068:
5060:
5056:
5051:
5043:
5039:
5034:
5026:
5025:
5022:
5018:
5010:
5009:
5006:
5002:
4994:
4990:
4982:
4978:
4969:
4960:
4956:
4951:
4943:
4939:
4931:
4927:
4918:
4909:
4905:
4897:
4893:
4884:
4868:
4866:logical symbols
4862:
4832:
4827:
4819:
4817:
4811:
4794:
4791:
4790:
4774:
4757:
4754:
4753:
4728:
4725:
4724:
4703:
4700:
4699:
4674:
4671:
4670:
4654:
4627:
4624:
4623:
4598:
4595:
4594:
4569:
4566:
4565:
4549:
4532:
4529:
4528:
4503:
4500:
4499:
4474:
4471:
4470:
4449:
4446:
4445:
4429:
4420:
4403:
4400:
4399:
4383:
4377:
4314:
4243:, 1st edition,
4236:
4234:Further reading
4231:
4230:
4199:
4195:
4162:
4158:
4134:10.1.1.174.4085
4115:
4111:
4104:
4071:
4065:
4061:
4054:
4031:
4025:
4016:
4001:
3997:
3982:
3971:
3960:
3956:
3947:
3945:
3934:
3930:
3921:
3919:
3915:
3909:
3905:
3898:
3884:
3880:
3873:
3856:
3852:
3841:
3834:
3829:
3824:
3823:
3818:
3814:
3809:
3782:
3777:
3711:
3687:relevance logic
3608:
3551:
3548:
3547:
3493:
3490:
3489:
3459:
3456:
3455:
3372:
3369:
3368:
3305:
3302:
3301:
3265:
3262:
3261:
3210:
3207:
3206:
3178:
3177:
3138:
3137:
3128:
3127:
3100:
3099:
3097:
3094:
3093:
3071:
3070:
3043:
3042:
3033:
3032:
3005:
3004:
3002:
2999:
2998:
2959:
2956:
2955:
2907:
2904:
2903:
2843:
2840:
2839:
2794:
2791:
2790:
2767:
2761:
2758:
2751:needs expansion
2736:
2684:
2681:
2680:
2673:logical systems
2631:
2620:Galileo Galilei
2618:is a sister of
2607:
2599:Galileo Galilei
2597:is a sister of
2586:
2568:
2564:
2493:
2490:
2489:
2472:
2469:
2468:
2451:
2448:
2447:
2420:
2417:
2416:
2409:
2343:
2340:
2339:
2318:
2313:
2286:
2283:
2282:
2248:
2245:
2244:
2222:
2219:
2218:
2184:
2181:
2180:
2146:
2143:
2142:
2102:
2099:
2098:
2070:
2067:
2066:
2050:
2047:
2046:
2024:
2021:
2020:
2004:
2001:
2000:
1984:
1981:
1980:
1964:
1961:
1960:
1944:
1941:
1940:
1927:
1876:
1873:
1872:
1850:
1847:
1846:
1824:
1821:
1820:
1798:
1795:
1794:
1772:
1769:
1768:
1765:Polish notation
1748:
1745:
1744:
1728:
1725:
1724:
1708:
1705:
1704:
1701:
1665:classical logic
1622:
1619:
1618:
1602:
1599:
1598:
1597:is true unless
1576:
1573:
1572:
1552:
1549:
1548:
1535:(also known as
1525:
1484:
1451:
1422:Boolean algebra
1417:Predicate logic
1374:
1371:
1370:
1348:
1345:
1344:
1322:
1319:
1318:
1288:
1285:
1284:
1260:
1255:
1252:
1251:
1221:
1213:
1210:
1209:
1179:
1176:
1175:
1153:
1150:
1149:
1127:
1124:
1123:
1101:
1098:
1097:
1070:
1067:
1066:
1045:
1043:
1040:
1039:
1020:
1017:
1016:
997:
994:
993:
958:
956:
954:
951:
950:
928:
925:
924:
900:
895:
892:
891:
861:
858:
857:
835:
832:
831:
809:
806:
805:
772:
770:
768:
765:
764:
742:
739:
738:
716:
713:
712:
688:
683:
680:
679:
649:
646:
645:
623:
620:
619:
597:
594:
593:
563:
560:
559:
537:
534:
533:
511:
508:
507:
474:
471:
470:
448:
445:
444:
425:
422:
421:
399:
396:
395:
373:
370:
369:
344:
269:Post's lattices
268:
233:
230:
229:
194:
192:
189:
188:
153:
151:
148:
147:
98:
95:
94:
64:
61:
60:
35:
24:
17:
12:
11:
5:
6947:
6937:
6936:
6931:
6926:
6921:
6904:
6903:
6889:
6886:
6885:
6883:
6882:
6877:
6872:
6867:
6862:
6861:
6860:
6850:
6845:
6840:
6831:
6826:
6821:
6816:
6814:Abstract logic
6810:
6808:
6804:
6803:
6801:
6800:
6795:
6793:Turing machine
6790:
6785:
6780:
6775:
6770:
6765:
6764:
6763:
6758:
6753:
6748:
6743:
6733:
6731:Computable set
6728:
6723:
6718:
6713:
6707:
6705:
6699:
6698:
6696:
6695:
6690:
6685:
6680:
6675:
6670:
6665:
6660:
6659:
6658:
6653:
6648:
6638:
6633:
6628:
6626:Satisfiability
6623:
6618:
6613:
6612:
6611:
6601:
6600:
6599:
6589:
6588:
6587:
6582:
6577:
6572:
6567:
6557:
6556:
6555:
6550:
6543:Interpretation
6539:
6537:
6531:
6530:
6528:
6527:
6522:
6517:
6512:
6507:
6497:
6492:
6491:
6490:
6489:
6488:
6478:
6473:
6463:
6458:
6453:
6448:
6443:
6438:
6432:
6430:
6424:
6423:
6420:
6419:
6417:
6416:
6408:
6407:
6406:
6405:
6400:
6399:
6398:
6393:
6388:
6368:
6367:
6366:
6364:minimal axioms
6361:
6350:
6349:
6348:
6337:
6336:
6335:
6330:
6325:
6320:
6315:
6310:
6297:
6295:
6276:
6275:
6273:
6272:
6271:
6270:
6258:
6253:
6252:
6251:
6246:
6241:
6236:
6226:
6221:
6216:
6211:
6210:
6209:
6204:
6194:
6193:
6192:
6187:
6182:
6177:
6167:
6162:
6161:
6160:
6155:
6150:
6140:
6139:
6138:
6133:
6128:
6123:
6118:
6113:
6103:
6098:
6093:
6088:
6087:
6086:
6081:
6076:
6071:
6061:
6056:
6054:Formation rule
6051:
6046:
6045:
6044:
6039:
6029:
6028:
6027:
6017:
6012:
6007:
6002:
5996:
5990:
5973:Formal systems
5969:
5968:
5965:
5964:
5962:
5961:
5956:
5951:
5946:
5941:
5936:
5931:
5926:
5921:
5916:
5915:
5914:
5909:
5898:
5896:
5892:
5891:
5889:
5888:
5887:
5886:
5876:
5871:
5870:
5869:
5862:Large cardinal
5859:
5854:
5849:
5844:
5839:
5825:
5824:
5823:
5818:
5813:
5798:
5796:
5786:
5785:
5783:
5782:
5781:
5780:
5775:
5770:
5760:
5755:
5750:
5745:
5740:
5735:
5730:
5725:
5720:
5715:
5710:
5705:
5699:
5697:
5690:
5689:
5687:
5686:
5685:
5684:
5679:
5674:
5669:
5664:
5659:
5651:
5650:
5649:
5644:
5634:
5629:
5627:Extensionality
5624:
5622:Ordinal number
5619:
5609:
5604:
5603:
5602:
5591:
5585:
5579:
5578:
5575:
5574:
5572:
5571:
5566:
5561:
5556:
5551:
5546:
5541:
5540:
5539:
5529:
5528:
5527:
5514:
5512:
5506:
5505:
5503:
5502:
5501:
5500:
5495:
5490:
5480:
5475:
5470:
5465:
5460:
5455:
5449:
5447:
5441:
5440:
5438:
5437:
5432:
5427:
5422:
5417:
5412:
5407:
5406:
5405:
5395:
5390:
5385:
5380:
5375:
5370:
5364:
5362:
5353:
5347:
5346:
5344:
5343:
5338:
5333:
5328:
5323:
5318:
5306:Cantor's
5304:
5299:
5294:
5284:
5282:
5269:
5268:
5266:
5265:
5260:
5255:
5250:
5245:
5240:
5235:
5230:
5225:
5220:
5215:
5210:
5205:
5204:
5203:
5192:
5190:
5186:
5185:
5178:
5177:
5170:
5163:
5155:
5146:
5145:
5122:
5119:
5118:
5114:
5113:
5104:
5103:
5101:
5092:
5091:
5089:
5078:
5077:
5075:
5064:
5063:
5061:
5047:
5046:
5044:
5030:
5029:
5027:
5023:quantification
5019:
5014:
5013:
5011:
5007:quantification
5003:
4998:
4997:
4995:
4986:
4985:
4983:
4964:
4963:
4961:
4947:
4946:
4944:
4935:
4934:
4932:
4913:
4912:
4910:
4901:
4900:
4898:
4879:
4878:
4873:
4870:
4869:
4861:
4860:
4853:
4846:
4838:
4829:
4828:
4816:
4813:
4812:
4810:
4809:
4798:
4779:
4776:
4775:
4773:
4772:
4761:
4743:
4732:
4718:
4707:
4692:Nonimplication
4689:
4678:
4659:
4656:
4655:
4653:
4652:
4649:Digital buffer
4642:
4631:
4613:
4602:
4584:
4573:
4554:
4551:
4550:
4548:
4547:
4536:
4518:
4507:
4489:
4478:
4464:
4453:
4434:
4431:
4430:
4423:
4421:
4419:
4418:
4407:
4388:
4385:
4384:
4376:
4375:
4368:
4361:
4353:
4347:
4346:
4332:"Conditionals"
4327:
4313:
4312:External links
4310:
4309:
4308:
4292:
4274:
4260:
4235:
4232:
4229:
4228:
4193:
4179:10.1.1.13.1854
4172:(4): 481–530.
4156:
4127:(4): 608–631.
4109:
4102:
4069:"Conditionals"
4059:
4052:
4029:"Conditionals"
4014:
3995:
3986:"Conditionals"
3969:
3954:
3928:
3903:
3896:
3878:
3871:
3850:
3831:
3830:
3828:
3825:
3822:
3821:
3811:
3810:
3808:
3805:
3804:
3803:
3798:
3793:
3788:
3781:
3778:
3776:
3775:
3770:
3765:
3760:
3755:
3750:
3745:
3738:
3733:
3728:
3723:
3718:
3716:Boolean domain
3712:
3710:
3707:
3651:Grice's maxims
3641:proposed that
3620:vacuously true
3607:
3604:
3603:
3602:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3541:
3530:
3527:
3524:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3483:
3472:
3469:
3466:
3463:
3442:
3441:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3362:
3351:
3348:
3345:
3342:
3339:
3336:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3295:
3284:
3281:
3278:
3275:
3272:
3269:
3255:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3195:
3194:
3181:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3141:
3136:
3131:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3103:
3087:
3074:
3069:
3066:
3063:
3060:
3057:
3054:
3051:
3046:
3041:
3036:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3008:
2995:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2952:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2900:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2833:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2769:
2768:
2748:
2746:
2735:
2732:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2669:
2668:
2660:
2655:
2650:
2630:
2627:
2626:
2625:
2604:
2579:vacuous truths
2559:
2558:
2555:
2552:
2548:
2547:
2544:
2541:
2537:
2536:
2533:
2530:
2526:
2525:
2522:
2519:
2515:
2514:
2503:
2500:
2497:
2487:
2476:
2466:
2455:
2430:
2427:
2424:
2408:
2405:
2392:
2389:
2386:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2359:
2356:
2353:
2350:
2347:
2317:
2314:
2312:
2309:
2296:
2293:
2290:
2258:
2255:
2252:
2232:
2229:
2226:
2194:
2191:
2188:
2156:
2153:
2150:
2112:
2109:
2106:
2074:
2054:
2034:
2031:
2028:
2008:
1988:
1968:
1948:
1926:
1923:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1865:is termed the
1854:
1828:
1808:
1805:
1802:
1782:
1779:
1776:
1752:
1732:
1712:
1700:
1697:
1626:
1606:
1586:
1583:
1580:
1556:
1527:
1526:
1524:
1523:
1516:
1509:
1501:
1498:
1497:
1486:
1485:
1483:
1482:
1477:
1472:
1467:
1461:
1458:
1457:
1453:
1452:
1450:
1449:
1444:
1439:
1434:
1432:Truth function
1429:
1424:
1419:
1414:
1408:
1405:
1404:
1400:
1399:
1396:
1395:
1384:
1381:
1378:
1358:
1355:
1352:
1332:
1329:
1326:
1316:
1310:
1309:
1298:
1295:
1292:
1272:
1267:
1264:
1259:
1249:
1243:
1242:
1231:
1217:
1207:
1201:
1200:
1189:
1186:
1183:
1163:
1160:
1157:
1137:
1134:
1131:
1111:
1108:
1105:
1095:
1089:
1088:
1077:
1074:
1052:
1049:
1027:
1024:
1004:
1001:
991:
985:
984:
971:
967:
964:
961:
938:
935:
932:
912:
907:
904:
899:
889:
883:
882:
871:
868:
865:
845:
842:
839:
819:
816:
813:
803:
799:
798:
785:
781:
778:
775:
752:
749:
746:
726:
723:
720:
700:
695:
692:
687:
677:
671:
670:
659:
656:
653:
633:
630:
627:
607:
604:
601:
591:
585:
584:
573:
570:
567:
547:
544:
541:
521:
518:
515:
505:
499:
498:
487:
484:
481:
478:
458:
455:
452:
432:
429:
409:
406:
403:
383:
380:
377:
367:
357:
356:
346:
345:
343:
342:
335:
328:
320:
317:
316:
313:
309:
308:
305:
299:
298:
295:
289:
288:
285:
281:
280:
277:
273:
272:
264:
263:
252:
249:
246:
243:
240:
237:
227:
221:
220:
209:
206:
201:
198:
186:
180:
179:
168:
165:
160:
157:
145:
139:
138:
134:
133:
126:
120:
119:
108:
105:
102:
92:
86:
85:
74:
71:
68:
58:
54:
53:
45:
44:
15:
9:
6:
4:
3:
2:
6946:
6935:
6932:
6930:
6927:
6925:
6922:
6920:
6917:
6916:
6914:
6901:
6900:
6895:
6887:
6881:
6878:
6876:
6873:
6871:
6868:
6866:
6863:
6859:
6856:
6855:
6854:
6851:
6849:
6846:
6844:
6841:
6839:
6835:
6832:
6830:
6827:
6825:
6822:
6820:
6817:
6815:
6812:
6811:
6809:
6805:
6799:
6796:
6794:
6791:
6789:
6788:Recursive set
6786:
6784:
6781:
6779:
6776:
6774:
6771:
6769:
6766:
6762:
6759:
6757:
6754:
6752:
6749:
6747:
6744:
6742:
6739:
6738:
6737:
6734:
6732:
6729:
6727:
6724:
6722:
6719:
6717:
6714:
6712:
6709:
6708:
6706:
6704:
6700:
6694:
6691:
6689:
6686:
6684:
6681:
6679:
6676:
6674:
6671:
6669:
6666:
6664:
6661:
6657:
6654:
6652:
6649:
6647:
6644:
6643:
6642:
6639:
6637:
6634:
6632:
6629:
6627:
6624:
6622:
6619:
6617:
6614:
6610:
6607:
6606:
6605:
6602:
6598:
6597:of arithmetic
6595:
6594:
6593:
6590:
6586:
6583:
6581:
6578:
6576:
6573:
6571:
6568:
6566:
6563:
6562:
6561:
6558:
6554:
6551:
6549:
6546:
6545:
6544:
6541:
6540:
6538:
6536:
6532:
6526:
6523:
6521:
6518:
6516:
6513:
6511:
6508:
6505:
6504:from ZFC
6501:
6498:
6496:
6493:
6487:
6484:
6483:
6482:
6479:
6477:
6474:
6472:
6469:
6468:
6467:
6464:
6462:
6459:
6457:
6454:
6452:
6449:
6447:
6444:
6442:
6439:
6437:
6434:
6433:
6431:
6429:
6425:
6415:
6414:
6410:
6409:
6404:
6403:non-Euclidean
6401:
6397:
6394:
6392:
6389:
6387:
6386:
6382:
6381:
6379:
6376:
6375:
6373:
6369:
6365:
6362:
6360:
6357:
6356:
6355:
6351:
6347:
6344:
6343:
6342:
6338:
6334:
6331:
6329:
6326:
6324:
6321:
6319:
6316:
6314:
6311:
6309:
6306:
6305:
6303:
6299:
6298:
6296:
6291:
6285:
6280:Example
6277:
6269:
6264:
6263:
6262:
6259:
6257:
6254:
6250:
6247:
6245:
6242:
6240:
6237:
6235:
6232:
6231:
6230:
6227:
6225:
6222:
6220:
6217:
6215:
6212:
6208:
6205:
6203:
6200:
6199:
6198:
6195:
6191:
6188:
6186:
6183:
6181:
6178:
6176:
6173:
6172:
6171:
6168:
6166:
6163:
6159:
6156:
6154:
6151:
6149:
6146:
6145:
6144:
6141:
6137:
6134:
6132:
6129:
6127:
6124:
6122:
6119:
6117:
6114:
6112:
6109:
6108:
6107:
6104:
6102:
6099:
6097:
6094:
6092:
6089:
6085:
6082:
6080:
6077:
6075:
6072:
6070:
6067:
6066:
6065:
6062:
6060:
6057:
6055:
6052:
6050:
6047:
6043:
6040:
6038:
6037:by definition
6035:
6034:
6033:
6030:
6026:
6023:
6022:
6021:
6018:
6016:
6013:
6011:
6008:
6006:
6003:
6001:
5998:
5997:
5994:
5991:
5989:
5985:
5980:
5974:
5970:
5960:
5957:
5955:
5952:
5950:
5947:
5945:
5942:
5940:
5937:
5935:
5932:
5930:
5927:
5925:
5924:Kripke–Platek
5922:
5920:
5917:
5913:
5910:
5908:
5905:
5904:
5903:
5900:
5899:
5897:
5893:
5885:
5882:
5881:
5880:
5877:
5875:
5872:
5868:
5865:
5864:
5863:
5860:
5858:
5855:
5853:
5850:
5848:
5845:
5843:
5840:
5837:
5833:
5829:
5826:
5822:
5819:
5817:
5814:
5812:
5809:
5808:
5807:
5803:
5800:
5799:
5797:
5795:
5791:
5787:
5779:
5776:
5774:
5771:
5769:
5768:constructible
5766:
5765:
5764:
5761:
5759:
5756:
5754:
5751:
5749:
5746:
5744:
5741:
5739:
5736:
5734:
5731:
5729:
5726:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5706:
5704:
5701:
5700:
5698:
5696:
5691:
5683:
5680:
5678:
5675:
5673:
5670:
5668:
5665:
5663:
5660:
5658:
5655:
5654:
5652:
5648:
5645:
5643:
5640:
5639:
5638:
5635:
5633:
5630:
5628:
5625:
5623:
5620:
5618:
5614:
5610:
5608:
5605:
5601:
5598:
5597:
5596:
5593:
5592:
5589:
5586:
5584:
5580:
5570:
5567:
5565:
5562:
5560:
5557:
5555:
5552:
5550:
5547:
5545:
5542:
5538:
5535:
5534:
5533:
5530:
5526:
5521:
5520:
5519:
5516:
5515:
5513:
5511:
5507:
5499:
5496:
5494:
5491:
5489:
5486:
5485:
5484:
5481:
5479:
5476:
5474:
5471:
5469:
5466:
5464:
5461:
5459:
5456:
5454:
5451:
5450:
5448:
5446:
5445:Propositional
5442:
5436:
5433:
5431:
5428:
5426:
5423:
5421:
5418:
5416:
5413:
5411:
5408:
5404:
5401:
5400:
5399:
5396:
5394:
5391:
5389:
5386:
5384:
5381:
5379:
5376:
5374:
5373:Logical truth
5371:
5369:
5366:
5365:
5363:
5361:
5357:
5354:
5352:
5348:
5342:
5339:
5337:
5334:
5332:
5329:
5327:
5324:
5322:
5319:
5317:
5313:
5309:
5305:
5303:
5300:
5298:
5295:
5293:
5289:
5286:
5285:
5283:
5281:
5275:
5270:
5264:
5261:
5259:
5256:
5254:
5251:
5249:
5246:
5244:
5241:
5239:
5236:
5234:
5231:
5229:
5226:
5224:
5221:
5219:
5216:
5214:
5211:
5209:
5206:
5202:
5199:
5198:
5197:
5194:
5193:
5191:
5187:
5183:
5176:
5171:
5169:
5164:
5162:
5157:
5156:
5153:
5143:
5138:
5132:
5120:
5111:
5107:
5102:
5099:
5095:
5090:
5087:
5081:
5076:
5073:
5067:
5062:
5059:
5058:contradiction
5054:
5050:
5045:
5042:
5037:
5033:
5028:
5024:
5017:
5012:
5008:
5001:
4996:
4993:
4989:
4984:
4981:
4977:
4972:
4967:
4962:
4959:
4954:
4950:
4945:
4942:
4938:
4933:
4930:
4926:
4921:
4916:
4911:
4908:
4904:
4899:
4896:
4892:
4887:
4882:
4877:
4876:
4871:
4867:
4859:
4854:
4852:
4847:
4845:
4840:
4839:
4836:
4826:
4814:
4788:
4784:
4783:Contradiction
4781:
4780:
4777:
4759:
4751:
4747:
4744:
4730:
4722:
4719:
4705:
4697:
4693:
4690:
4668:
4664:
4661:
4660:
4657:
4650:
4646:
4643:
4621:
4617:
4616:Biconditional
4614:
4600:
4592:
4588:
4585:
4563:
4559:
4556:
4555:
4552:
4534:
4526:
4522:
4519:
4497:
4493:
4490:
4468:
4465:
4443:
4439:
4436:
4435:
4432:
4427:
4397:
4393:
4390:
4389:
4386:
4382:
4374:
4369:
4367:
4362:
4360:
4355:
4354:
4351:
4343:
4342:
4337:
4333:
4328:
4325:
4320:
4316:
4315:
4306:
4302:
4301:
4296:
4293:
4290:
4286:
4282:
4278:
4275:
4272:
4268:
4264:
4261:
4258:
4254:
4250:
4246:
4242:
4238:
4237:
4224:
4220:
4215:
4210:
4206:
4205:
4197:
4189:
4185:
4180:
4175:
4171:
4167:
4160:
4152:
4148:
4144:
4140:
4135:
4130:
4126:
4122:
4121:
4113:
4105:
4099:
4094:
4089:
4085:
4081:
4077:
4070:
4063:
4055:
4053:9781118972090
4049:
4045:
4041:
4037:
4030:
4023:
4021:
4019:
4010:
4006:
3999:
3991:
3987:
3980:
3978:
3976:
3974:
3965:
3958:
3943:
3939:
3932:
3914:
3907:
3899:
3897:0-674-32449-8
3893:
3889:
3882:
3874:
3868:
3864:
3860:
3854:
3846:
3839:
3837:
3832:
3816:
3812:
3802:
3799:
3797:
3794:
3792:
3789:
3787:
3784:
3783:
3774:
3771:
3769:
3766:
3764:
3761:
3759:
3756:
3754:
3751:
3749:
3748:Logical graph
3746:
3744:
3743:
3739:
3737:
3734:
3732:
3729:
3727:
3726:Boolean logic
3724:
3722:
3719:
3717:
3714:
3713:
3706:
3703:
3698:
3696:
3695:causal models
3692:
3688:
3684:
3680:
3676:
3672:
3668:
3664:
3660:
3656:
3652:
3648:
3644:
3640:
3639:Frank Jackson
3636:
3631:
3629:
3625:
3621:
3617:
3613:
3586:
3577:
3571:
3565:
3559:
3553:
3545:
3542:
3525:
3519:
3513:
3507:
3501:
3495:
3487:
3484:
3470:
3464:
3461:
3453:
3450:
3449:
3448:
3446:
3425:
3419:
3413:
3407:
3401:
3395:
3392:
3383:
3380:
3377:
3366:
3363:
3349:
3343:
3340:
3334:
3328:
3322:
3316:
3310:
3299:
3296:
3282:
3276:
3273:
3270:
3259:
3256:
3242:
3233:
3230:
3227:
3221:
3218:
3212:
3204:
3203:
3202:
3200:
3171:
3165:
3153:
3147:
3134:
3121:
3115:
3106:
3091:
3088:
3064:
3058:
3049:
3039:
3026:
3020:
3011:
2996:
2982:
2979:
2976:
2970:
2967:
2961:
2953:
2939:
2933:
2930:
2927:
2921:
2915:
2901:
2887:
2878:
2875:
2872:
2866:
2860:
2854:
2845:
2837:
2836:Import-export
2834:
2820:
2811:
2805:
2802:
2796:
2788:
2787:
2786:
2784:
2780:
2776:
2765:
2762:February 2021
2756:
2752:
2749:This section
2747:
2744:
2740:
2739:
2731:
2729:
2713:
2710:
2707:
2695:
2689:
2678:
2674:
2667:
2666:
2661:
2659:
2656:
2654:
2651:
2649:
2648:
2644:
2643:
2642:
2640:
2636:
2623:
2621:
2617:
2610:
2605:
2602:
2600:
2596:
2589:
2584:
2583:
2582:
2580:
2575:
2571:
2567:is false and
2556:
2553:
2550:
2549:
2545:
2542:
2539:
2538:
2534:
2531:
2528:
2527:
2523:
2520:
2517:
2516:
2501:
2495:
2488:
2474:
2467:
2453:
2446:
2445:
2442:
2428:
2422:
2414:
2404:
2390:
2387:
2384:
2378:
2372:
2366:
2363:
2354:
2351:
2345:
2337:
2333:
2330:
2326:
2323:
2308:
2294:
2288:
2280:
2276:
2272:
2256:
2250:
2230:
2227:
2224:
2216:
2212:
2208:
2192:
2189:
2186:
2178:
2174:
2170:
2154:
2151:
2148:
2140:
2136:
2132:
2131:
2126:
2110:
2104:
2096:
2092:
2088:
2072:
2052:
2032:
2029:
2026:
2006:
1986:
1966:
1946:
1938:
1934:
1933:
1922:
1905:
1899:
1887:
1881:
1870:
1869:
1852:
1844:
1843:
1826:
1806:
1800:
1780:
1777:
1774:
1766:
1730:
1696:
1694:
1690:
1686:
1683:. Due to the
1682:
1678:
1674:
1670:
1666:
1661:
1659:
1658:
1652:
1648:
1647:
1646:modus tollens
1642:
1641:
1624:
1604:
1584:
1578:
1570:
1546:
1542:
1538:
1534:
1522:
1517:
1515:
1510:
1508:
1503:
1502:
1500:
1499:
1496:
1488:
1487:
1481:
1478:
1476:
1473:
1471:
1468:
1466:
1465:Digital logic
1463:
1462:
1460:
1459:
1455:
1454:
1448:
1447:Scope (logic)
1445:
1443:
1440:
1438:
1435:
1433:
1430:
1428:
1425:
1423:
1420:
1418:
1415:
1413:
1410:
1409:
1407:
1406:
1402:
1401:
1382:
1376:
1356:
1353:
1350:
1330:
1324:
1317:
1315:
1312:
1311:
1296:
1293:
1290:
1270:
1265:
1262:
1257:
1250:
1248:
1245:
1244:
1229:
1215:
1208:
1206:
1203:
1202:
1187:
1184:
1181:
1161:
1158:
1155:
1135:
1132:
1129:
1109:
1106:
1103:
1096:
1094:
1091:
1090:
1075:
1072:
1047:
1025:
1022:
1002:
992:
990:
987:
986:
965:
962:
959:
936:
930:
910:
902:
897:
890:
888:
885:
884:
869:
866:
863:
843:
840:
837:
817:
814:
811:
804:
802:nonequivalent
801:
800:
779:
776:
773:
750:
747:
744:
724:
718:
698:
690:
685:
678:
676:
673:
672:
657:
651:
631:
628:
625:
605:
599:
592:
590:
587:
586:
571:
565:
545:
539:
519:
516:
513:
506:
504:
501:
500:
485:
476:
456:
450:
430:
427:
407:
404:
401:
381:
378:
375:
368:
366:
363:
362:
359:
358:
355:
352:
351:
341:
336:
334:
329:
327:
322:
321:
318:
314:
310:
306:
304:
300:
296:
294:
290:
286:
282:
278:
274:
271:
265:
250:
247:
244:
241:
238:
235:
228:
226:
222:
207:
204:
196:
187:
185:
181:
166:
163:
155:
146:
144:
140:
135:
131:
127:
125:
121:
103:
93:
91:
87:
72:
66:
59:
55:
51:
46:
41:
33:
29:
22:
6924:Conditionals
6890:
6688:Ultraproduct
6535:Model theory
6500:Independence
6436:Formal proof
6428:Proof theory
6411:
6384:
6341:real numbers
6313:second-order
6224:Substitution
6125:
6101:Metalanguage
6042:conservative
6015:Axiom schema
5959:Constructive
5929:Morse–Kelley
5895:Set theories
5874:Aleph number
5867:inaccessible
5773:Grothendieck
5657:intersection
5544:Higher-order
5532:Second-order
5478:Truth tables
5435:Venn diagram
5218:Formal proof
4970:
4952:
4940:
4919:
4885:
4663:Joint denial
4587:Exclusive or
4491:
4339:
4304:
4298:
4280:
4266:
4240:
4203:
4196:
4169:
4165:
4159:
4124:
4118:
4112:
4093:1721.1/95781
4075:
4062:
4035:
4008:
3998:
3989:
3963:
3957:
3946:. Retrieved
3941:
3931:
3920:. Retrieved
3906:
3887:
3881:
3862:
3853:
3844:
3815:
3780:Conditionals
3763:Peirce's law
3742:Laws of Form
3740:
3699:
3678:
3674:
3670:
3666:
3632:
3609:
3443:
3298:Transitivity
3196:
2772:
2759:
2755:adding to it
2750:
2670:
2664:
2647:Modus ponens
2645:
2632:
2613:
2608:
2592:
2587:
2573:
2569:
2562:
2410:
2319:
2278:
2274:
2214:
2210:
2176:
2172:
2138:
2134:
2128:
2094:
2090:
1930:
1928:
1866:
1840:
1702:
1662:
1656:
1644:
1640:modus ponens
1638:
1617:is true and
1536:
1532:
1530:
1456:Applications
588:
284:1-preserving
276:0-preserving
137:Normal forms
6798:Type theory
6746:undecidable
6678:Truth value
6565:equivalence
6244:non-logical
5857:Enumeration
5847:Isomorphism
5794:cardinality
5778:Von Neumann
5743:Ultrafilter
5708:Uncountable
5642:equivalence
5559:Quantifiers
5549:Fixed-point
5518:First-order
5398:Consistency
5383:Proposition
5360:Traditional
5331:Lindström's
5321:Compactness
5263:Type theory
5208:Cardinality
5021:existential
4746:Conjunction
4696:NIMPLY gate
4521:Disjunction
4492:Implication
4300:Philosophia
4277:Quine, W.V.
4259:, NY, 2003.
3683:modal logic
3452:Reflexivity
3445:Tautologies
3199:entailments
2954:Or-and-if:
2779:conjunction
2775:disjunction
2635:deductively
2616:Marie Curie
2595:Marie Curie
2413:truth table
2407:Truth table
2336:truth table
2311:Definitions
1569:interpreted
1427:Truth table
184:Conjunctive
143:Disjunctive
90:Truth table
6913:Categories
6609:elementary
6302:arithmetic
6170:Quantifier
6148:functional
6020:Expression
5738:Transitive
5682:identities
5667:complement
5600:hereditary
5583:Set theory
4496:IMPLY gate
3948:2022-08-10
3922:2022-08-10
3827:References
2663:Classical
1868:consequent
1842:antecedent
1655:classical
503:equivalent
124:Logic gate
57:Definition
6934:Semantics
6880:Supertask
6783:Recursion
6741:decidable
6575:saturated
6553:of models
6476:deductive
6471:axiomatic
6391:Hilbert's
6378:Euclidean
6359:canonical
6282:axiomatic
6214:Signature
6143:Predicate
6032:Extension
5954:Ackermann
5879:Operation
5758:Universal
5748:Recursive
5723:Singleton
5718:Inhabited
5703:Countable
5693:Types of
5677:power set
5647:partition
5564:Predicate
5510:Predicate
5425:Syllogism
5415:Soundness
5388:Inference
5378:Tautology
5280:paradoxes
5098:therefore
5086:therefore
5041:tautology
5005:universal
4797:⊥
4760:∧
4731:↚
4706:↛
4677:↓
4645:Statement
4630:↔
4620:XNOR gate
4572:¬
4535:∨
4506:→
4477:←
4452:↑
4442:NAND gate
4406:⊤
4392:Tautology
4289:Cambridge
4271:Blackwell
4223:246924881
4174:CiteSeerX
4129:CiteSeerX
3643:pragmatic
3584:¬
3581:→
3572:∨
3563:→
3554:⊨
3523:→
3514:∨
3505:→
3496:⊨
3468:→
3462:⊨
3423:→
3414:∧
3405:→
3396:⊨
3390:→
3381:∨
3347:→
3341:⊨
3332:→
3323:∧
3314:→
3280:→
3274:⊨
3268:¬
3240:→
3231:∧
3222:⊨
3216:→
3169:→
3160:→
3151:→
3135:≡
3119:→
3110:→
3062:→
3053:→
3040:≡
3024:→
3015:→
2980:∨
2974:¬
2971:≡
2965:→
2937:¬
2934:∧
2928:≡
2919:→
2910:¬
2885:→
2876:∧
2867:≡
2858:→
2849:→
2818:¬
2815:→
2809:¬
2806:≡
2800:→
2711:∨
2705:¬
2702:⇒
2693:→
2606:... with
2585:... with
2499:→
2426:→
2388:∨
2382:¬
2379:≡
2370:¬
2367:∧
2358:¬
2355:≡
2349:→
2322:classical
2316:Semantics
2307:in 1954.
2292:⇒
2254:→
2228:⊃
2190:⊃
2152:⊃
2123:in 1918.
2108:→
2030:⊃
1903:→
1894:→
1885:→
1804:→
1751:⇒
1731:⊃
1711:→
1582:→
1555:→
1541:operation
1380:←
1354:⊂
1328:⇐
1294:⊕
1266:_
1263:∨
1185:∥
1159:∣
1107:∨
1073:∼
1051:¯
1023:−
1000:¬
970:¯
934:↓
906:¯
903:∨
867:↮
784:¯
777:⋅
748:∣
722:↑
694:¯
691:∧
655:→
629:⊃
603:⇒
569:⇋
543:⇔
517:≡
483:&
480:&
454:&
405:⋅
379:∧
312:Self-dual
245:⊕
239:⊕
200:¯
159:¯
70:→
6865:Logicism
6858:timeline
6834:Concrete
6693:Validity
6663:T-schema
6656:Kripke's
6651:Tarski's
6646:semantic
6636:Strength
6585:submodel
6580:spectrum
6548:function
6396:Tarski's
6385:Elements
6372:geometry
6328:Robinson
6249:variable
6234:function
6207:spectrum
6197:Sentence
6153:variable
6096:Language
6049:Relation
6010:Automata
6000:Alphabet
5984:language
5838:-jection
5816:codomain
5802:Function
5763:Universe
5733:Infinite
5637:Relation
5420:Validity
5410:Argument
5308:theorem,
5084:entails,
5070:entails,
4958:superset
4750:AND gate
4667:NOR gate
4601:↮
4591:XOR gate
4562:NOT gate
4558:Negation
4279:(1982),
3861:(2015).
3709:See also
3486:Totality
2783:negation
2271:Bourbaki
1935:(1889),
1699:Notation
1679:and the
1539:) is an
1495:Category
1314:converse
841:⇎
815:≢
293:Monotone
6807:Related
6604:Diagram
6502: (
6481:Hilbert
6466:Systems
6461:Theorem
6339:of the
6284:systems
6064:Formula
6059:Grammar
5975: (
5919:General
5632:Forcing
5617:Element
5537:Monadic
5312:paradox
5253:Theorem
5189:General
5110:because
4974:
4953:implies
4941:implies
4923:
4889:
4864:Common
4752:)
4748: (
4698:)
4694: (
4669:)
4665: (
4647: (
4622:)
4618: (
4593:)
4589: (
4564:)
4560: (
4527:)
4525:OR gate
4523: (
4498:)
4494: (
4444:)
4440: (
4379:Common
4338:(ed.).
4257:Mineola
4249:Norwell
4151:2912209
3669:, then
2591:false:
2320:From a
2277:, then
2213:, then
2207:Heyting
2175:, then
2169:Gentzen
2137:, then
2125:Russell
2093:, then
2087:Hilbert
1959:, then
1925:History
589:implies
6570:finite
6333:Skolem
6286:
6261:Theory
6229:Symbol
6219:String
6202:atomic
6079:ground
6074:closed
6069:atomic
6025:ground
5988:syntax
5884:binary
5811:domain
5728:Finite
5493:finite
5351:Logics
5310:
5258:Theory
5072:proves
4968:
4917:
4883:
4789:
4723:
4469:
4398:
4245:Kluwer
4221:
4176:
4149:
4131:
4100:
4050:
3894:
3869:
3693:, and
3647:denote
2612:true:
2329:binary
1653:, and
1227:
1219:
303:Affine
6560:Model
6308:Peano
6165:Proof
6005:Arity
5934:Naive
5821:image
5753:Fuzzy
5713:Empty
5662:union
5607:Class
5248:Model
5238:Lemma
5196:Axiom
5053:false
4891:&
4787:False
4334:. In
4291:, MA.
4219:S2CID
4147:S2CID
4072:(PDF)
4032:(PDF)
3916:(PDF)
3807:Notes
2773:When
2281:" as
2217:" as
2179:" as
2141:" as
2097:" as
1979:" as
1937:Peano
1545:logic
43:IMPLY
6683:Type
6486:list
6290:list
6267:list
6256:Term
6190:rank
6084:open
5978:list
5790:Maps
5695:sets
5554:Free
5524:list
5274:list
5201:list
5036:true
4992:nand
4396:True
4098:ISBN
4048:ISBN
3892:ISBN
3867:ISBN
3677:and
3657:and
3637:and
2781:and
2614:"If
2593:"If
2411:The
1845:and
1743:and
1531:The
1223:XNOR
1205:XNOR
675:NAND
104:1011
6370:of
6352:of
6300:of
5832:Sur
5806:Map
5613:Ur-
5595:Set
4980:iff
4929:not
4895:and
4209:doi
4184:doi
4139:doi
4125:101
4088:hdl
4080:doi
4040:doi
3614:in
2757:.
2415:of
2045:as
1929:In
1691:in
1567:is
1247:XOR
989:NOT
887:NOR
365:AND
287:yes
30:or
6915::
6756:NP
6380::
6374::
6304::
5981:),
5836:Bi
5828:In
4971:or
4920:or
4907:or
4886:or
4303:,
4287:,
4269:,
4255:,
4217:.
4182:.
4170:28
4168:.
4145:.
4137:.
4123:.
4096:.
4086:.
4046:.
4017:^
3972:^
3940:.
3835:^
3697:.
3689:,
3685:,
3546::
3488::
3454::
3367::
3300::
3260::
3201::
3092::
2838::
2777:,
2730:.
2641:.
2572:→
2441::
2403:.
2205:.
2085:.
2065:Ɔ
1999:Ɔ
1921:.
1695:.
1660:.
1649:,
1643:,
1369:,
1343:,
1283:,
1174:,
1148:,
1122:,
1093:OR
1065:,
1038:,
1015:,
949:,
923:,
856:,
830:,
763:,
737:,
711:,
644:,
618:,
558:,
532:,
469:,
443:,
420:,
394:,
315:no
307:no
297:no
279:no
6836:/
6751:P
6506:)
6292:)
6288:(
6185:∀
6180:!
6175:∃
6136:=
6131:↔
6126:→
6121:∧
6116:∨
6111:¬
5834:/
5830:/
5804:/
5615:)
5611:(
5498:∞
5488:3
5276:)
5174:e
5167:t
5160:v
5106:∵
5094:∴
5080:⊨
5066:⊢
5055:,
5049:⊥
5038:,
5032:⊤
5016:∃
5000:∀
4988:|
4976:≡
4966:↔
4955:,
4949:⊃
4937:→
4925:~
4915:¬
4903:∨
4881:∧
4857:e
4850:t
4843:v
4785:/
4651:)
4394:/
4372:e
4365:t
4358:v
4344:.
4305:5
4273:.
4225:.
4211::
4190:.
4186::
4153:.
4141::
4106:.
4090::
4082::
4056:.
4042::
4011:.
3951:.
3925:.
3900:.
3875:.
3847:.
3679:Q
3675:P
3671:Q
3667:P
3590:)
3587:Q
3578:P
3575:(
3569:)
3566:Q
3560:P
3557:(
3529:)
3526:P
3520:Q
3517:(
3511:)
3508:Q
3502:P
3499:(
3471:P
3465:P
3429:)
3426:R
3420:Q
3417:(
3411:)
3408:R
3402:P
3399:(
3393:R
3387:)
3384:Q
3378:P
3375:(
3350:R
3344:P
3338:)
3335:R
3329:Q
3326:(
3320:)
3317:Q
3311:P
3308:(
3283:Q
3277:P
3271:P
3243:Q
3237:)
3234:R
3228:P
3225:(
3219:Q
3213:P
3180:)
3175:)
3172:Q
3166:R
3163:(
3157:)
3154:P
3148:R
3145:(
3140:(
3130:)
3125:)
3122:Q
3116:P
3113:(
3107:R
3102:(
3073:)
3068:)
3065:R
3059:P
3056:(
3050:Q
3045:(
3035:)
3030:)
3027:R
3021:Q
3018:(
3012:P
3007:(
2983:Q
2977:P
2968:Q
2962:P
2940:Q
2931:P
2925:)
2922:Q
2916:P
2913:(
2888:R
2882:)
2879:Q
2873:P
2870:(
2864:)
2861:R
2855:Q
2852:(
2846:P
2821:P
2812:Q
2803:Q
2797:P
2764:)
2760:(
2714:B
2708:A
2699:)
2696:B
2690:A
2687:(
2624:.
2609:B
2603:,
2588:B
2574:B
2570:A
2565:A
2557:T
2554:T
2551:T
2546:F
2543:F
2540:T
2535:T
2532:T
2529:F
2524:T
2521:F
2518:F
2502:B
2496:A
2475:B
2454:A
2429:B
2423:A
2391:B
2385:A
2376:)
2373:B
2364:A
2361:(
2352:B
2346:A
2295:B
2289:A
2279:B
2275:A
2257:B
2251:A
2231:B
2225:A
2215:B
2211:A
2193:B
2187:A
2177:B
2173:A
2155:B
2149:A
2139:B
2135:A
2111:B
2105:A
2095:B
2091:A
2073:B
2053:A
2033:B
2027:A
2007:B
1987:A
1967:B
1947:A
1909:)
1906:s
1900:r
1897:(
1891:)
1888:q
1882:p
1879:(
1853:q
1827:p
1807:q
1801:p
1781:q
1778:p
1775:C
1625:Q
1605:P
1585:Q
1579:P
1520:e
1513:t
1506:v
1383:B
1377:A
1357:B
1351:A
1331:B
1325:A
1297:B
1291:A
1271:B
1258:A
1230:B
1216:A
1188:B
1182:A
1162:B
1156:A
1136:B
1133:+
1130:A
1110:B
1104:A
1076:A
1048:A
1026:A
1003:A
966:B
963:+
960:A
937:B
931:A
911:B
898:A
870:B
864:A
844:B
838:A
818:B
812:A
780:B
774:A
751:B
745:A
725:B
719:A
699:B
686:A
658:B
652:A
632:B
626:A
606:B
600:A
572:B
566:A
546:B
540:A
520:B
514:A
486:B
477:A
457:B
451:A
431:B
428:A
408:B
402:A
382:B
376:A
339:e
332:t
325:v
251:y
248:x
242:x
236:1
208:y
205:+
197:x
167:y
164:+
156:x
107:)
101:(
73:y
67:x
34:.
23:.
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