5607:
4896:
1492:
34:
123:
4008:
3980:
3950:
3793:
3765:
3735:
3574:
3546:
3516:
3335:
3307:
3277:
3249:
3219:
2340:
2312:
2282:
2102:
2053:
5596:
5290:
2970:
2942:
2912:
2753:. (An operator is truth-preserving if its value is truth whenever all of its arguments are truth, or falsity-preserving if its value is falsity whenever all of its arguments are falsity.) Therefore {NAND} is a functionally complete set.
2695:
who first used the stroke as a sign for non-conjunction (NAND) in a paper of 1917 and which has since become current practice. Russell and
Whitehead used the Sheffer stroke in the 1927 second edition of
4809:
1512:
3508:
225:
177:
2737:
NAND does not possess any of the following five properties, each of which is required to be absent from, and the absence of all of which is sufficient for, at least one member of a set of
2408:
2381:
4208:
2687:
of
Boolean algebras, Sheffer's axioms are equally valid for either of the NAND or NOR operations in place of the stroke. Sheffer interpreted the stroke as a sign for nondisjunction (
855:
4246:
1282:
797:
710:
76:
5081:
1645:
922:
583:
2488:
for non-disjunction. Many people, beginning with Nicod in 1917, and followed by
Whitehead, Russell and many others, mistakenly thought Sheffer has described non-conjunction using
1241:
881:
3374:
557:
983:
5110:
3973:
3854:
3758:
3639:
3539:
3396:
3300:
3242:
3132:
3031:
2935:
2840:
2305:
2204:
2076:
2008:
4141:
2045:
1064:
3009:
829:
669:
617:
497:
5186:
3942:
3888:
3727:
3673:
3430:
3211:
3110:
1394:
1342:
948:
5211:
4986:
1199:
5157:
4957:
4106:
4080:
2639:
2182:
1986:
1861:
1788:
736:
1368:
643:
262:
4932:
4002:
3909:
3787:
3694:
3568:
3451:
3329:
3271:
3178:
3077:
2964:
2883:
2786:
2619:
1618:
1308:
531:
3617:
468:
419:
393:
4167:
3832:
2486:
1173:
1121:
762:
5240:
4054:
2818:
2274:
2229:
1087:
1014:
110:
5277:
5052:
5015:
4886:
2506:
2466:
2435:
2334:
2250:
2098:
1598:
2592:
1671:
1147:
3157:
3056:
1037:
442:
2904:
2862:
2566:
2539:
2152:
2132:
1955:
1935:
1834:
1813:
1519:
338:
4609:
4567:
4525:
2655:
5325:
2719:
4840:
331:
4421:
4711:
4343:
4768:
5318:
3457:
1505:
4505:(Revised ed.). Cambridge, London, New York, New Rochelle, Melbourne and Sydney: Harvard University Press. p. 45.
4793:
4749:
324:
4283:
4833:
4627:
190:
142:
5311:
4249:
2729:
also described the NAND and NOR operators and showed that the other
Boolean operations could be expressed by it.
2756:
This can also be realized as follows: All three elements of the functionally complete set {AND, OR, NOT} can be
2386:
2359:
4475:. Translated by Hammond, L. M.; Leckie, G. G.; Steinhardt, F. New York: Chelsea Publishing Company. p. 11.
4172:
4393:
834:
4669:
4213:
1254:
767:
682:
46:
5635:
5066:
4826:
1623:
1495:
894:
562:
4416:
2724:
1708:). Like its dual, NAND can be used by itself, without any other logical operator, to constitute a logical
2660:
1212:
860:
3353:
2691:) in his paper, mentioning non-conjunction only in a footnote and without a special sign for it. It was
536:
5490:
5485:
4731:
4594:
2664:
953:
5095:
3958:
3839:
3743:
3624:
3524:
3381:
3285:
3227:
3117:
3016:
2920:
2825:
2290:
2189:
2061:
1993:
4111:
2015:
1042:
2988:
808:
648:
596:
473:
5474:
5171:
4804:
4774:
4764:
4735:
4562:
4359:
Peirce, C. S. (1933) . "A Boolian
Algebra with One Constant". In Hartshorne, C.; Weiss, P. (eds.).
4310:
3915:
3861:
3700:
3646:
3403:
3184:
3083:
1373:
1321:
927:
5196:
4971:
4799:
4521:"A set of five independent postulates for Boolean algebras, with application to logical constants"
1178:
33:
5190:
5161:
5142:
4942:
4085:
4059:
4026:
2709:
2624:
2161:
1965:
1840:
1767:
1442:
715:
183:
135:
2417:
was the first to publish a proof of the completeness of non-conjunction, representing this with
1347:
622:
238:
5640:
4917:
4756:
3987:
3894:
3772:
3679:
3553:
3436:
3314:
3256:
3163:
3062:
2949:
2868:
2771:
2705:
2684:
2604:
2353:
1603:
1537:
1412:
1287:
510:
3596:
447:
398:
372:
5579:
5575:
5335:
5085:
5031:
4516:
4146:
3811:
2742:
2738:
2698:
2650:
2471:
2445:
1713:
1689:
1577:
1152:
1100:
741:
503:
303:
5225:
4036:
2800:
2256:
2211:
1069:
996:
89:
5630:
5422:
5410:
5262:
5037:
5000:
4961:
4871:
2491:
2451:
2441:) and non-disjunction in print at the first time and showed their functional completeness.
2420:
2356:
was the first to show the functional completeness of non-conjunction (representing this as
2319:
2235:
2083:
1583:
1470:
589:
231:
2702:
and suggested it as a replacement for the "OR" and "NOT" operations of the first edition.
8:
5567:
5376:
5364:
5215:
4990:
2718:(for 'cutting both ways'), but he never published his finding. Two years before Sheffer,
2676:
2672:
2668:
2571:
2111:
1721:
1682:
1650:
1569:
1553:
1480:
1126:
1093:
365:
4377:
Peirce, C. S. (1933) . "The
Simplest Mathematics". In Hartshorne, C.; Weiss, P. (eds.).
3139:
3038:
1019:
424:
5611:
5539:
5535:
5418:
4850:
4760:
4544:
4438:
2889:
2847:
2551:
2522:
2137:
2117:
1940:
1920:
1819:
1798:
1725:
1475:
354:
293:
4742:. Translated by Bird, Otto (revised ed.). Dordrecht, South Holland, Netherlands:
2545:
1678:
5606:
5600:
5510:
5294:
5114:
4861:
4707:
4623:
4614:. Annals of Mathematics studies. Vol. 5. Princeton: Princeton University Press.
4442:
4339:
2516:
1741:
1545:
1417:
268:
4895:
5553:
5549:
4936:
4615:
4534:
4430:
1549:
1533:
1437:
1314:
2663:
using the stroke, and proved its equivalence to a standard formulation thereof by
4299:
4272:
2621:. It is not clear who first introduced this notation, although the corresponding
2595:
1674:
1422:
5563:
5449:
5435:
5406:
5350:
5118:
4262:
1432:
4379:
Collected Papers of
Charles Sanders Peirce, Volume IV The Simplest Mathematics
4361:
Collected Papers of
Charles Sanders Peirce, Volume IV The Simplest Mathematics
5624:
5527:
5522:
5256:
5252:
4865:
4818:
4586:
4278:
2512:
1745:
1709:
1705:
1447:
5505:
5501:
5372:
5056:
4304:
4289:
2750:
1701:
1697:
1556:
operation, expressed in ordinary language as "not both". It is also called
1247:
4619:
2763:
5303:
5165:
5132:
4788:
4030:
2688:
1761:
1749:
1693:
1564:(since it says in effect that at least one of its operands is false), or
1427:
887:
82:
4565:(1917). "A Reduction in the Number of Primitive Propositions of Logic".
4460:(in German) (1 ed.). Berlin: Verlag von Julius Springer. p. 9.
5445:
4965:
4548:
4434:
4294:
2692:
1748:. It produces a value of true, if — and only if — at least one of the
1465:
116:
5360:
5089:
4911:
4743:
2746:
1717:
1573:
1205:
4539:
4520:
5469:
5398:
5384:
5219:
5136:
5060:
5027:
4676:(Fall 2023 ed.), Metaphysics Research Lab, Stanford University
2680:
989:
4490:(in Polish) (2 ed.). Warszawa: Państwowe Wydawnictwo Naukowe.
5518:
4994:
4644:
2757:
2714:
2114:, this is also equivalent to the disjunction of the negations of
4748:(NB. Edited and translated from the French and German editions:
4029:
set of connectives. This can be proved by first showing, with a
5427:
4800:
http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nand.html
4210:, the Sheffer stroke suffices to define the set of connectives
5394:
2760:. Thus the set {NAND} must be functionally complete as well.
2414:
4381:. Massachusetts: Harvard University Press. pp. 189–262.
122:
4813:
4267:
4007:
3979:
3949:
3792:
3764:
3734:
3573:
3545:
3515:
3334:
3306:
3276:
3248:
3218:
2339:
2311:
2281:
2101:
2052:
4363:. Massachusetts: Harvard University Press. pp. 13–18.
4248:, which is shown to be truth-functionally complete by the
2712:
of NAND or NOR more than 30 years earlier, using the term
2468:
and showed its functional completeness. Sheffer also used
2969:
2941:
2911:
4759:(1931–1935) . "A Boolian Algebra with One Constant". In
2764:
Other
Boolean operations in terms of the Sheffer stroke
2383:) but didn't publish his result. Peirce's editor added
4810:
Proofs of some axioms by Stroke function by Yasuo Setô
4611:
The Two-Valued
Iterative Systems of Mathematical Logic
4471:
Hilbert, D.; Ackermann, W. (1950). Luce, R. E. (ed.).
5265:
5228:
5199:
5174:
5145:
5098:
5069:
5040:
5003:
4974:
4945:
4920:
4874:
4216:
4175:
4149:
4114:
4088:
4062:
4039:
3990:
3961:
3918:
3897:
3864:
3842:
3814:
3775:
3746:
3703:
3682:
3649:
3627:
3599:
3556:
3527:
3503:{\displaystyle ((P\uparrow P)\uparrow (Q\uparrow Q))}
3460:
3439:
3406:
3384:
3356:
3317:
3288:
3259:
3230:
3187:
3166:
3142:
3120:
3086:
3065:
3041:
3019:
2991:
2952:
2923:
2892:
2871:
2850:
2828:
2803:
2774:
2741:
operators: truth-preservation, falsity-preservation,
2627:
2607:
2574:
2554:
2525:
2494:
2474:
2454:
2423:
2389:
2362:
2322:
2293:
2259:
2238:
2214:
2192:
2164:
2140:
2120:
2086:
2064:
2018:
1996:
1968:
1943:
1923:
1843:
1822:
1801:
1770:
1653:
1626:
1606:
1586:
1376:
1350:
1324:
1290:
1257:
1215:
1181:
1155:
1129:
1103:
1072:
1045:
1022:
999:
956:
930:
897:
863:
837:
811:
770:
744:
718:
685:
651:
625:
599:
565:
539:
513:
476:
450:
427:
401:
375:
241:
193:
145:
92:
49:
4464:
4449:
4704:
Logic with trees: an introduction to symbolic logic
4336:
Logic with trees: an introduction to symbolic logic
5271:
5234:
5205:
5180:
5151:
5104:
5075:
5046:
5009:
4980:
4951:
4926:
4880:
4568:Proceedings of the Cambridge Philosophical Society
4240:
4202:
4161:
4135:
4100:
4074:
4048:
3996:
3967:
3936:
3903:
3882:
3848:
3826:
3781:
3752:
3721:
3688:
3667:
3633:
3611:
3562:
3533:
3502:
3445:
3424:
3390:
3368:
3323:
3294:
3265:
3236:
3205:
3172:
3151:
3126:
3104:
3071:
3050:
3025:
3003:
2958:
2929:
2898:
2877:
2856:
2834:
2812:
2788:, the usual operators of propositional logic are:
2780:
2633:
2613:
2586:
2560:
2533:
2500:
2480:
2460:
2429:
2402:
2375:
2348:
2328:
2299:
2268:
2244:
2223:
2198:
2176:
2146:
2126:
2092:
2070:
2039:
2002:
1980:
1949:
1929:
1855:
1828:
1807:
1782:
1665:
1639:
1612:
1592:
1388:
1362:
1336:
1302:
1276:
1235:
1193:
1167:
1141:
1115:
1081:
1058:
1031:
1008:
977:
942:
916:
875:
849:
823:
791:
756:
730:
704:
663:
637:
611:
577:
551:
525:
491:
462:
436:
413:
387:
256:
219:
171:
104:
70:
4526:Transactions of the American Mathematical Society
2656:Transactions of the American Mathematical Society
5622:
4470:
4455:
2641:for non-disjunction was used by Quine in 1940,.
4706:. London; New York: Routledge. pp. 41–43.
2601:An alternative notation for non-conjunction is
220:{\displaystyle {\overline {x}}+{\overline {y}}}
172:{\displaystyle {\overline {x}}+{\overline {y}}}
4848:
4730:
4607:
4479:
5319:
4834:
4672:, in Zalta, Edward N.; Nodelman, Uri (eds.),
4394:"Sheffer stroke before Sheffer: Edward Stamm"
1513:
332:
4805:Implementations of 2- and 4-input NAND gates
4509:
4411:
4409:
4372:
4370:
4235:
4217:
2519:described non-conjunction with the operator
4755:
4579:
4485:
4338:. London; New York: Routledge. p. 43.
5333:
5326:
5312:
4841:
4827:
4770:Collected Papers of Charles Sanders Peirce
4555:
4025:The Sheffer stroke, taken by itself, is a
4020:
2403:{\displaystyle {\overline {\curlywedge }}}
2376:{\displaystyle {\overline {\curlywedge }}}
1520:
1506:
339:
325:
4538:
4419:(1911). "Beitrag zur Algebra der Logik".
4406:
4367:
4352:
4203:{\displaystyle \neg (\neg A\land \neg B)}
4494:
1681:(but not as ||, often used to represent
4697:
4695:
4693:
4691:
4689:
4674:The Stanford Encyclopedia of Philosophy
4515:
4385:
2653:, who in 1913 published a paper in the
1912:
850:{\displaystyle A\not \Leftrightarrow B}
5623:
4701:
4667:
4585:
4376:
4358:
4333:
4241:{\displaystyle \{\land ,\lor ,\neg \}}
1277:{\displaystyle A{\underline {\lor }}B}
792:{\displaystyle {\overline {A\cdot B}}}
705:{\displaystyle A{\overline {\land }}B}
71:{\displaystyle {\overline {x\cdot y}}}
5307:
5076:{\displaystyle \not \leftrightarrow }
4822:
4642:
4561:
4500:
4422:Monatshefte für Mathematik und Physik
4415:
4329:
4327:
1957:is the negation of their conjunction
1640:{\displaystyle {\overline {\wedge }}}
917:{\displaystyle A{\overline {\lor }}B}
578:{\displaystyle A\leftrightharpoons B}
4686:
4391:
4108:is truth-functionally equivalent to
4056:is truth-functionally equivalent to
2667:employing the familiar operators of
4794:Internet Encyclopedia of Philosophy
4636:
4456:Hilbert, D.; Ackermann, W. (1928).
1236:{\displaystyle A\ {\text{XNOR}}\ B}
876:{\displaystyle A\nleftrightarrow B}
13:
5266:
5041:
4875:
4724:
4661:
4591:Introduction to mathematical logic
4458:Grundzügen der theoretischen Logik
4324:
4284:Minimal axioms for Boolean algebra
4232:
4191:
4182:
4176:
4115:
4040:
3369:{\displaystyle P\leftrightarrow Q}
2804:
2508:, naming this the Sheffer Stroke.
2260:
2215:
2087:
2019:
1000:
552:{\displaystyle A\Leftrightarrow B}
483:
480:
454:
14:
5652:
4782:
978:{\displaystyle {\overline {A+B}}}
5605:
5594:
5288:
5105:{\displaystyle \leftrightarrow }
4894:
4473:Principles of Mathematical Logic
4006:
3978:
3968:{\displaystyle \Leftrightarrow }
3948:
3849:{\displaystyle \Leftrightarrow }
3791:
3763:
3753:{\displaystyle \Leftrightarrow }
3733:
3634:{\displaystyle \Leftrightarrow }
3572:
3544:
3534:{\displaystyle \Leftrightarrow }
3514:
3391:{\displaystyle \Leftrightarrow }
3333:
3305:
3295:{\displaystyle \Leftrightarrow }
3275:
3247:
3237:{\displaystyle \Leftrightarrow }
3217:
3127:{\displaystyle \Leftrightarrow }
3026:{\displaystyle \Leftrightarrow }
2968:
2940:
2930:{\displaystyle \Leftrightarrow }
2910:
2835:{\displaystyle \Leftrightarrow }
2448:described non-disjunction using
2338:
2310:
2300:{\displaystyle \Leftrightarrow }
2280:
2199:{\displaystyle \Leftrightarrow }
2100:
2071:{\displaystyle \Leftrightarrow }
2051:
2003:{\displaystyle \Leftrightarrow }
1491:
1490:
121:
32:
4601:
4250:Disjunctive Normal Form Theorem
4136:{\displaystyle \neg (A\land B)}
2659:providing an axiomatization of
2349:Alternative notations and names
2040:{\displaystyle \neg (P\land Q)}
1059:{\displaystyle {\overline {A}}}
5146:
5099:
4975:
4946:
4921:
4750:Précis de logique mathématique
4197:
4179:
4130:
4118:
4092:
4066:
3991:
3962:
3931:
3925:
3919:
3898:
3877:
3871:
3865:
3843:
3776:
3747:
3716:
3710:
3704:
3683:
3662:
3656:
3650:
3628:
3557:
3528:
3497:
3494:
3488:
3482:
3479:
3476:
3470:
3464:
3461:
3440:
3419:
3413:
3407:
3385:
3360:
3318:
3289:
3260:
3231:
3200:
3194:
3188:
3167:
3121:
3099:
3093:
3087:
3066:
3020:
3004:{\displaystyle P\rightarrow Q}
2995:
2953:
2924:
2872:
2829:
2775:
2628:
2608:
2294:
2193:
2168:
2065:
2034:
2022:
1997:
1972:
1847:
1774:
1755:
1607:
1380:
1328:
934:
824:{\displaystyle A\not \equiv B}
722:
664:{\displaystyle A\rightarrow B}
655:
612:{\displaystyle A\Rightarrow B}
603:
569:
543:
492:{\displaystyle A\&\&B}
99:
93:
1:
5181:{\displaystyle \nrightarrow }
4488:Elementy logiki matematycznej
4317:
3937:{\displaystyle (Q\uparrow Q)}
3883:{\displaystyle (P\uparrow P)}
3722:{\displaystyle (P\uparrow Q)}
3668:{\displaystyle (P\uparrow Q)}
3425:{\displaystyle (P\uparrow Q)}
3206:{\displaystyle (P\uparrow Q)}
3105:{\displaystyle (Q\uparrow Q)}
2732:
1731:
1389:{\displaystyle A\leftarrow B}
1337:{\displaystyle A\Leftarrow B}
943:{\displaystyle A\downarrow B}
5206:{\displaystyle \nleftarrow }
4981:{\displaystyle \rightarrow }
4740:Precis of Mathematical Logic
2395:
2368:
1632:
1194:{\displaystyle A\parallel B}
1051:
970:
906:
784:
694:
212:
199:
164:
151:
63:
7:
5152:{\displaystyle \downarrow }
4952:{\displaystyle \leftarrow }
4255:
4101:{\displaystyle A\uparrow B}
4075:{\displaystyle A\uparrow A}
2768:Expressed in terms of NAND
2758:constructed using only NAND
2634:{\displaystyle \downarrow }
2594:for non-conjunction in his
2177:{\displaystyle P\uparrow Q}
1981:{\displaystyle P\uparrow Q}
1856:{\displaystyle A\uparrow B}
1783:{\displaystyle A\uparrow B}
1716:). This property makes the
731:{\displaystyle A\uparrow B}
10:
5657:
4773:. Vol. 4. Cambridge:
4595:Princeton University Press
2708:(1880) had discovered the
2649:The stroke is named after
2644:
1548:that is equivalent to the
1363:{\displaystyle A\subset B}
638:{\displaystyle A\supset B}
257:{\displaystyle 1\oplus xy}
5591:
5342:
5285:
5248:
5128:
5023:
4927:{\displaystyle \uparrow }
4903:
4892:
4857:
4563:Nicod, Jean George Pierre
4486:Łukasiewicz, J. (1958) .
3997:{\displaystyle \uparrow }
3975:
3904:{\displaystyle \uparrow }
3856:
3782:{\displaystyle \uparrow }
3760:
3689:{\displaystyle \uparrow }
3641:
3563:{\displaystyle \uparrow }
3541:
3446:{\displaystyle \uparrow }
3398:
3324:{\displaystyle \uparrow }
3302:
3266:{\displaystyle \uparrow }
3244:
3173:{\displaystyle \uparrow }
3134:
3072:{\displaystyle \uparrow }
3033:
2959:{\displaystyle \uparrow }
2937:
2878:{\displaystyle \uparrow }
2842:
2781:{\displaystyle \uparrow }
2614:{\displaystyle \uparrow }
1613:{\displaystyle \uparrow }
1303:{\displaystyle A\oplus B}
526:{\displaystyle A\equiv B}
320:
312:
302:
292:
284:
276:
267:
230:
182:
134:
129:
115:
81:
40:
31:
26:
4775:Harvard University Press
4645:"Propositional Calculus"
4311:Sole sufficient operator
3955:
3836:
3740:
3621:
3612:{\displaystyle P\land Q}
3521:
3378:
3282:
3224:
3114:
3013:
2917:
2822:
1572:, it corresponds to the
463:{\displaystyle A\&B}
414:{\displaystyle A\cdot B}
388:{\displaystyle A\land B}
5191:Converse nonimplication
4757:Peirce, Charles Sanders
4668:Franks, Curtis (2023),
4608:Emil Leon Post (1941).
4417:Stamm, Edward Bronisław
4392:Zach, R. (2023-02-18).
4162:{\displaystyle A\lor B}
4021:Functional completeness
3827:{\displaystyle P\lor Q}
2710:functional completeness
2481:{\displaystyle \wedge }
2410:) for non-disjunction.
1724:, including its use in
1443:Functional completeness
1168:{\displaystyle A\mid B}
1116:{\displaystyle A\lor B}
757:{\displaystyle A\mid B}
5612:Mathematics portal
5273:
5236:
5235:{\displaystyle \land }
5207:
5182:
5153:
5106:
5077:
5048:
5011:
4982:
4953:
4928:
4882:
4736:Menne, Albert Heinrich
4732:Bocheński, Józef Maria
4702:Howson, Colin (1997).
4517:Sheffer, Henry Maurice
4334:Howson, Colin (1997).
4242:
4204:
4163:
4137:
4102:
4076:
4050:
4049:{\displaystyle \neg A}
3998:
3969:
3938:
3905:
3884:
3850:
3828:
3783:
3754:
3723:
3690:
3669:
3635:
3613:
3564:
3535:
3504:
3447:
3426:
3392:
3370:
3325:
3296:
3267:
3238:
3207:
3174:
3153:
3128:
3106:
3073:
3052:
3027:
3005:
2960:
2931:
2900:
2879:
2858:
2836:
2814:
2813:{\displaystyle \neg P}
2782:
2706:Charles Sanders Peirce
2635:
2615:
2588:
2562:
2535:
2502:
2482:
2462:
2431:
2404:
2377:
2330:
2301:
2270:
2269:{\displaystyle \neg Q}
2246:
2225:
2224:{\displaystyle \neg P}
2200:
2178:
2148:
2128:
2094:
2072:
2041:
2004:
1982:
1951:
1931:
1917:The Sheffer stroke of
1857:
1830:
1809:
1784:
1667:
1641:
1614:
1594:
1538:propositional calculus
1413:Propositional calculus
1390:
1364:
1338:
1304:
1278:
1237:
1195:
1169:
1143:
1117:
1083:
1082:{\displaystyle \sim A}
1060:
1033:
1010:
1009:{\displaystyle \neg A}
979:
944:
918:
877:
851:
825:
793:
758:
732:
706:
665:
639:
613:
579:
553:
527:
493:
464:
438:
415:
389:
258:
221:
173:
106:
105:{\displaystyle (1110)}
72:
5601:Philosophy portal
5295:Philosophy portal
5274:
5272:{\displaystyle \bot }
5237:
5208:
5183:
5154:
5107:
5078:
5049:
5047:{\displaystyle \neg }
5012:
5010:{\displaystyle \lor }
4983:
4954:
4929:
4883:
4881:{\displaystyle \top }
4670:"Propositional Logic"
4649:mathworld.wolfram.com
4620:10.1515/9781400882366
4501:Quine, W. V (1981) .
4243:
4205:
4164:
4138:
4103:
4077:
4051:
4027:functionally complete
3999:
3970:
3939:
3906:
3885:
3851:
3829:
3784:
3755:
3724:
3691:
3670:
3636:
3614:
3565:
3536:
3505:
3448:
3427:
3393:
3371:
3326:
3297:
3268:
3239:
3208:
3175:
3154:
3129:
3107:
3074:
3053:
3028:
3006:
2961:
2932:
2901:
2880:
2859:
2837:
2815:
2783:
2739:functionally complete
2699:Principia Mathematica
2651:Henry Maurice Sheffer
2636:
2616:
2589:
2563:
2536:
2503:
2501:{\displaystyle \mid }
2483:
2463:
2461:{\displaystyle \mid }
2432:
2430:{\displaystyle \sim }
2405:
2378:
2331:
2329:{\displaystyle \lor }
2302:
2271:
2247:
2245:{\displaystyle \lor }
2226:
2201:
2179:
2149:
2129:
2095:
2093:{\displaystyle \neg }
2073:
2042:
2005:
1983:
1952:
1932:
1858:
1831:
1810:
1785:
1714:functionally complete
1668:
1642:
1615:
1595:
1593:{\displaystyle \mid }
1578:Henry Maurice Sheffer
1471:Programming languages
1391:
1365:
1339:
1305:
1279:
1238:
1196:
1170:
1144:
1118:
1084:
1061:
1034:
1011:
980:
945:
919:
878:
852:
826:
794:
759:
733:
707:
666:
640:
614:
580:
554:
528:
494:
465:
439:
416:
390:
259:
222:
174:
107:
73:
5263:
5226:
5197:
5172:
5143:
5096:
5067:
5038:
5001:
4972:
4943:
4937:Converse implication
4918:
4872:
4214:
4173:
4147:
4112:
4086:
4060:
4037:
3988:
3959:
3916:
3895:
3862:
3840:
3812:
3773:
3744:
3701:
3680:
3647:
3625:
3597:
3554:
3525:
3458:
3437:
3404:
3382:
3354:
3315:
3286:
3257:
3228:
3185:
3164:
3140:
3118:
3084:
3063:
3039:
3017:
2989:
2950:
2921:
2890:
2869:
2848:
2826:
2801:
2772:
2625:
2605:
2572:
2552:
2523:
2492:
2472:
2452:
2421:
2387:
2360:
2320:
2291:
2257:
2236:
2212:
2190:
2162:
2138:
2118:
2084:
2062:
2016:
1994:
1966:
1941:
1921:
1913:Logical equivalences
1841:
1820:
1799:
1768:
1651:
1624:
1604:
1584:
1576:. It is named after
1374:
1348:
1322:
1288:
1255:
1213:
1179:
1153:
1127:
1101:
1070:
1043:
1020:
997:
954:
928:
895:
861:
835:
809:
768:
742:
716:
683:
649:
623:
597:
563:
537:
511:
474:
448:
425:
399:
373:
239:
232:Zhegalkin polynomial
191:
143:
90:
47:
5636:Logical connectives
4851:logical connectives
4761:Hartshorne, Charles
4643:Weisstein, Eric W.
3802:
3344:
2979:
2683:). Because of self-
2669:propositional logic
2587:{\displaystyle Dpq}
1722:digital electronics
1696:(also known as the
1666:{\displaystyle Dpq}
1570:digital electronics
1481:Philosophy of logic
1142:{\displaystyle A+B}
355:Logical connectives
23:
5269:
5232:
5203:
5178:
5149:
5102:
5073:
5044:
5007:
4978:
4949:
4924:
4908:Alternative denial
4878:
4503:Mathematical Logic
4435:10.1007/BF01742795
4238:
4200:
4159:
4133:
4098:
4072:
4046:
3994:
3965:
3934:
3901:
3880:
3846:
3824:
3779:
3750:
3719:
3686:
3665:
3631:
3609:
3560:
3531:
3500:
3443:
3422:
3388:
3366:
3321:
3292:
3263:
3234:
3203:
3170:
3152:{\displaystyle ~P}
3149:
3124:
3102:
3069:
3051:{\displaystyle ~P}
3048:
3023:
3001:
2956:
2927:
2896:
2875:
2854:
2832:
2810:
2778:
2631:
2611:
2584:
2558:
2531:
2498:
2478:
2458:
2427:
2400:
2373:
2326:
2297:
2266:
2242:
2221:
2196:
2174:
2144:
2124:
2090:
2068:
2037:
2000:
1978:
1947:
1927:
1853:
1826:
1805:
1780:
1726:computer processor
1720:crucial to modern
1663:
1637:
1610:
1590:
1562:alternative denial
1476:Mathematical logic
1386:
1360:
1334:
1300:
1274:
1269:
1233:
1191:
1165:
1139:
1113:
1079:
1056:
1032:{\displaystyle -A}
1029:
1006:
975:
940:
914:
873:
847:
821:
789:
754:
728:
702:
661:
635:
609:
575:
549:
523:
489:
460:
437:{\displaystyle AB}
434:
411:
385:
254:
217:
169:
102:
68:
21:
5618:
5617:
5586:
5585:
5301:
5300:
4777:. pp. 12–20.
4713:978-0-415-13342-5
4345:978-0-415-13342-5
4169:is equivalent to
4018:
4017:
4014:
4013:
3799:
3798:
3580:
3579:
3341:
3340:
3145:
3044:
2976:
2975:
2899:{\displaystyle P}
2857:{\displaystyle P}
2561:{\displaystyle D}
2534:{\displaystyle /}
2398:
2371:
2346:
2345:
2147:{\displaystyle Q}
2127:{\displaystyle P}
2108:
2107:
1950:{\displaystyle Q}
1930:{\displaystyle P}
1910:
1909:
1829:{\displaystyle B}
1808:{\displaystyle A}
1742:logical operation
1635:
1546:logical operation
1534:Boolean functions
1530:
1529:
1399:
1398:
1262:
1229:
1225:
1221:
1054:
973:
909:
787:
697:
349:
348:
215:
202:
167:
154:
66:
16:Logical operation
5648:
5610:
5609:
5599:
5598:
5597:
5443:
5392:
5358:
5345:
5344:
5328:
5321:
5314:
5305:
5304:
5293:
5292:
5291:
5278:
5276:
5275:
5270:
5241:
5239:
5238:
5233:
5212:
5210:
5209:
5204:
5187:
5185:
5184:
5179:
5158:
5156:
5155:
5150:
5111:
5109:
5108:
5103:
5082:
5080:
5079:
5074:
5053:
5051:
5050:
5045:
5016:
5014:
5013:
5008:
4987:
4985:
4984:
4979:
4958:
4956:
4955:
4950:
4933:
4931:
4930:
4925:
4898:
4887:
4885:
4884:
4879:
4843:
4836:
4829:
4820:
4819:
4778:
4747:
4718:
4717:
4699:
4684:
4683:
4682:
4681:
4665:
4659:
4658:
4656:
4655:
4640:
4634:
4633:
4605:
4599:
4598:
4583:
4577:
4576:
4559:
4553:
4552:
4542:
4513:
4507:
4506:
4498:
4492:
4491:
4483:
4477:
4476:
4468:
4462:
4461:
4453:
4447:
4446:
4413:
4404:
4403:
4401:
4400:
4389:
4383:
4382:
4374:
4365:
4364:
4356:
4350:
4349:
4331:
4247:
4245:
4244:
4239:
4209:
4207:
4206:
4201:
4168:
4166:
4165:
4160:
4142:
4140:
4139:
4134:
4107:
4105:
4104:
4099:
4081:
4079:
4078:
4073:
4055:
4053:
4052:
4047:
4010:
4003:
4001:
4000:
3995:
3982:
3974:
3972:
3971:
3966:
3952:
3943:
3941:
3940:
3935:
3910:
3908:
3907:
3902:
3889:
3887:
3886:
3881:
3855:
3853:
3852:
3847:
3833:
3831:
3830:
3825:
3806:
3805:
3795:
3788:
3786:
3785:
3780:
3767:
3759:
3757:
3756:
3751:
3737:
3728:
3726:
3725:
3720:
3695:
3693:
3692:
3687:
3674:
3672:
3671:
3666:
3640:
3638:
3637:
3632:
3618:
3616:
3615:
3610:
3591:
3590:
3576:
3569:
3567:
3566:
3561:
3548:
3540:
3538:
3537:
3532:
3518:
3509:
3507:
3506:
3501:
3452:
3450:
3449:
3444:
3431:
3429:
3428:
3423:
3397:
3395:
3394:
3389:
3375:
3373:
3372:
3367:
3348:
3347:
3337:
3330:
3328:
3327:
3322:
3309:
3301:
3299:
3298:
3293:
3279:
3272:
3270:
3269:
3264:
3251:
3243:
3241:
3240:
3235:
3221:
3212:
3210:
3209:
3204:
3179:
3177:
3176:
3171:
3158:
3156:
3155:
3150:
3143:
3133:
3131:
3130:
3125:
3111:
3109:
3108:
3103:
3078:
3076:
3075:
3070:
3057:
3055:
3054:
3049:
3042:
3032:
3030:
3029:
3024:
3010:
3008:
3007:
3002:
2983:
2982:
2972:
2965:
2963:
2962:
2957:
2944:
2936:
2934:
2933:
2928:
2914:
2905:
2903:
2902:
2897:
2884:
2882:
2881:
2876:
2863:
2861:
2860:
2855:
2841:
2839:
2838:
2833:
2819:
2817:
2816:
2811:
2795:
2794:
2791:
2790:
2787:
2785:
2784:
2779:
2728:
2661:Boolean algebras
2640:
2638:
2637:
2632:
2620:
2618:
2617:
2612:
2593:
2591:
2590:
2585:
2567:
2565:
2564:
2559:
2540:
2538:
2537:
2532:
2530:
2507:
2505:
2504:
2499:
2487:
2485:
2484:
2479:
2467:
2465:
2464:
2459:
2436:
2434:
2433:
2428:
2409:
2407:
2406:
2401:
2399:
2391:
2382:
2380:
2379:
2374:
2372:
2364:
2342:
2335:
2333:
2332:
2327:
2314:
2306:
2304:
2303:
2298:
2284:
2275:
2273:
2272:
2267:
2251:
2249:
2248:
2243:
2230:
2228:
2227:
2222:
2205:
2203:
2202:
2197:
2183:
2181:
2180:
2175:
2156:
2155:
2153:
2151:
2150:
2145:
2133:
2131:
2130:
2125:
2112:De Morgan's laws
2104:
2099:
2097:
2096:
2091:
2077:
2075:
2074:
2069:
2055:
2046:
2044:
2043:
2038:
2009:
2007:
2006:
2001:
1987:
1985:
1984:
1979:
1960:
1959:
1956:
1954:
1953:
1948:
1936:
1934:
1933:
1928:
1862:
1860:
1859:
1854:
1835:
1833:
1832:
1827:
1814:
1812:
1811:
1806:
1793:
1792:
1789:
1787:
1786:
1781:
1672:
1670:
1669:
1664:
1646:
1644:
1643:
1638:
1636:
1628:
1619:
1617:
1616:
1611:
1599:
1597:
1596:
1591:
1568:("not and"). In
1522:
1515:
1508:
1494:
1493:
1438:Boolean function
1404:Related concepts
1395:
1393:
1392:
1387:
1369:
1367:
1366:
1361:
1343:
1341:
1340:
1335:
1309:
1307:
1306:
1301:
1283:
1281:
1280:
1275:
1270:
1242:
1240:
1239:
1234:
1227:
1226:
1223:
1219:
1200:
1198:
1197:
1192:
1174:
1172:
1171:
1166:
1148:
1146:
1145:
1140:
1122:
1120:
1119:
1114:
1088:
1086:
1085:
1080:
1065:
1063:
1062:
1057:
1055:
1047:
1038:
1036:
1035:
1030:
1015:
1013:
1012:
1007:
984:
982:
981:
976:
974:
969:
958:
949:
947:
946:
941:
923:
921:
920:
915:
910:
902:
882:
880:
879:
874:
856:
854:
853:
848:
830:
828:
827:
822:
798:
796:
795:
790:
788:
783:
772:
763:
761:
760:
755:
737:
735:
734:
729:
711:
709:
708:
703:
698:
690:
670:
668:
667:
662:
644:
642:
641:
636:
618:
616:
615:
610:
584:
582:
581:
576:
558:
556:
555:
550:
532:
530:
529:
524:
498:
496:
495:
490:
469:
467:
466:
461:
443:
441:
440:
435:
420:
418:
417:
412:
394:
392:
391:
386:
362:
361:
351:
350:
341:
334:
327:
271:
263:
261:
260:
255:
226:
224:
223:
218:
216:
208:
203:
195:
178:
176:
175:
170:
168:
160:
155:
147:
125:
111:
109:
108:
103:
77:
75:
74:
69:
67:
62:
51:
36:
24:
20:
5656:
5655:
5651:
5650:
5649:
5647:
5646:
5645:
5621:
5620:
5619:
5614:
5604:
5603:
5595:
5593:
5587:
5582:
5578:
5570:
5566:
5558:
5555:
5552:
5544:
5541:
5538:
5530:
5526:
5521:
5513:
5509:
5504:
5496:
5495:
5492:
5488:
5480:
5479:
5476:
5472:
5464:
5460:
5452:
5448:
5439:
5430:
5426:
5421:
5413:
5409:
5401:
5397:
5388:
5379:
5375:
5367:
5363:
5354:
5338:
5336:logical symbols
5332:
5302:
5297:
5289:
5287:
5281:
5264:
5261:
5260:
5244:
5227:
5224:
5223:
5198:
5195:
5194:
5173:
5170:
5169:
5144:
5141:
5140:
5124:
5097:
5094:
5093:
5068:
5065:
5064:
5039:
5036:
5035:
5019:
5002:
4999:
4998:
4973:
4970:
4969:
4944:
4941:
4940:
4919:
4916:
4915:
4899:
4890:
4873:
4870:
4869:
4853:
4847:
4791:article in the
4785:
4727:
4725:Further reading
4722:
4721:
4714:
4700:
4687:
4679:
4677:
4666:
4662:
4653:
4651:
4641:
4637:
4630:
4606:
4602:
4593:. Vol. 1.
4584:
4580:
4560:
4556:
4540:10.2307/1988701
4514:
4510:
4499:
4495:
4484:
4480:
4469:
4465:
4454:
4450:
4414:
4407:
4398:
4396:
4390:
4386:
4375:
4368:
4357:
4353:
4346:
4332:
4325:
4320:
4315:
4273:Gate equivalent
4258:
4215:
4212:
4211:
4174:
4171:
4170:
4148:
4145:
4144:
4113:
4110:
4109:
4087:
4084:
4083:
4061:
4058:
4057:
4038:
4035:
4034:
4023:
3989:
3986:
3985:
3960:
3957:
3956:
3917:
3914:
3913:
3896:
3893:
3892:
3863:
3860:
3859:
3841:
3838:
3837:
3813:
3810:
3809:
3774:
3771:
3770:
3745:
3742:
3741:
3702:
3699:
3698:
3681:
3678:
3677:
3648:
3645:
3644:
3626:
3623:
3622:
3598:
3595:
3594:
3555:
3552:
3551:
3526:
3523:
3522:
3459:
3456:
3455:
3438:
3435:
3434:
3405:
3402:
3401:
3383:
3380:
3379:
3355:
3352:
3351:
3316:
3313:
3312:
3287:
3284:
3283:
3258:
3255:
3254:
3229:
3226:
3225:
3186:
3183:
3182:
3165:
3162:
3161:
3141:
3138:
3137:
3119:
3116:
3115:
3085:
3082:
3081:
3064:
3061:
3060:
3040:
3037:
3036:
3018:
3015:
3014:
2990:
2987:
2986:
2951:
2948:
2947:
2922:
2919:
2918:
2891:
2888:
2887:
2870:
2867:
2866:
2849:
2846:
2845:
2827:
2824:
2823:
2802:
2799:
2798:
2773:
2770:
2769:
2766:
2735:
2722:
2647:
2626:
2623:
2622:
2606:
2603:
2602:
2596:Polish notation
2573:
2570:
2569:
2553:
2550:
2549:
2526:
2524:
2521:
2520:
2493:
2490:
2489:
2473:
2470:
2469:
2453:
2450:
2449:
2422:
2419:
2418:
2390:
2388:
2385:
2384:
2363:
2361:
2358:
2357:
2351:
2321:
2318:
2317:
2307:
2292:
2289:
2288:
2258:
2255:
2254:
2237:
2234:
2233:
2213:
2210:
2209:
2206:
2191:
2188:
2187:
2163:
2160:
2159:
2139:
2136:
2135:
2119:
2116:
2115:
2085:
2082:
2081:
2078:
2063:
2060:
2059:
2017:
2014:
2013:
2010:
1995:
1992:
1991:
1967:
1964:
1963:
1942:
1939:
1938:
1922:
1919:
1918:
1915:
1842:
1839:
1838:
1821:
1818:
1817:
1800:
1797:
1796:
1790:is as follows.
1769:
1766:
1765:
1758:
1738:non-conjunction
1734:
1675:Polish notation
1652:
1649:
1648:
1627:
1625:
1622:
1621:
1605:
1602:
1601:
1585:
1582:
1581:
1580:and written as
1558:non-conjunction
1526:
1485:
1452:
1423:Boolean algebra
1418:Predicate logic
1375:
1372:
1371:
1349:
1346:
1345:
1323:
1320:
1319:
1289:
1286:
1285:
1261:
1256:
1253:
1252:
1222:
1214:
1211:
1210:
1180:
1177:
1176:
1154:
1151:
1150:
1128:
1125:
1124:
1102:
1099:
1098:
1071:
1068:
1067:
1046:
1044:
1041:
1040:
1021:
1018:
1017:
998:
995:
994:
959:
957:
955:
952:
951:
929:
926:
925:
901:
896:
893:
892:
862:
859:
858:
836:
833:
832:
810:
807:
806:
773:
771:
769:
766:
765:
743:
740:
739:
717:
714:
713:
689:
684:
681:
680:
650:
647:
646:
624:
621:
620:
598:
595:
594:
564:
561:
560:
538:
535:
534:
512:
509:
508:
475:
472:
471:
449:
446:
445:
426:
423:
422:
400:
397:
396:
374:
371:
370:
345:
270:Post's lattices
269:
240:
237:
236:
207:
194:
192:
189:
188:
159:
146:
144:
141:
140:
91:
88:
87:
52:
50:
48:
45:
44:
17:
12:
11:
5:
5654:
5644:
5643:
5638:
5633:
5616:
5615:
5592:
5589:
5588:
5584:
5583:
5574:
5573:
5571:
5562:
5561:
5559:
5548:
5547:
5545:
5534:
5533:
5531:
5517:
5516:
5514:
5500:
5499:
5497:
5493:quantification
5489:
5484:
5483:
5481:
5477:quantification
5473:
5468:
5467:
5465:
5456:
5455:
5453:
5434:
5433:
5431:
5417:
5416:
5414:
5405:
5404:
5402:
5383:
5382:
5380:
5371:
5370:
5368:
5349:
5348:
5343:
5340:
5339:
5331:
5330:
5323:
5316:
5308:
5299:
5298:
5286:
5283:
5282:
5280:
5279:
5268:
5249:
5246:
5245:
5243:
5242:
5231:
5213:
5202:
5188:
5177:
5162:Nonimplication
5159:
5148:
5129:
5126:
5125:
5123:
5122:
5119:Digital buffer
5112:
5101:
5083:
5072:
5054:
5043:
5024:
5021:
5020:
5018:
5017:
5006:
4988:
4977:
4959:
4948:
4934:
4923:
4904:
4901:
4900:
4893:
4891:
4889:
4888:
4877:
4858:
4855:
4854:
4846:
4845:
4838:
4831:
4823:
4817:
4816:
4814:Project Euclid
4807:
4802:
4797:
4789:Sheffer Stroke
4784:
4783:External links
4781:
4780:
4779:
4753:
4726:
4723:
4720:
4719:
4712:
4685:
4660:
4635:
4628:
4600:
4597:. p. 134.
4587:Church, Alonzo
4578:
4554:
4533:(4): 481–488.
4508:
4493:
4478:
4463:
4448:
4429:(1): 137–149.
4405:
4384:
4366:
4351:
4344:
4322:
4321:
4319:
4316:
4314:
4313:
4308:
4302:
4297:
4292:
4286:
4281:
4276:
4270:
4265:
4263:Boolean domain
4259:
4257:
4254:
4237:
4234:
4231:
4228:
4225:
4222:
4219:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4158:
4155:
4152:
4132:
4129:
4126:
4123:
4120:
4117:
4097:
4094:
4091:
4082:. Then, since
4071:
4068:
4065:
4045:
4042:
4022:
4019:
4016:
4015:
4012:
4011:
4004:
3993:
3983:
3976:
3964:
3953:
3945:
3944:
3933:
3930:
3927:
3924:
3921:
3911:
3900:
3890:
3879:
3876:
3873:
3870:
3867:
3857:
3845:
3834:
3823:
3820:
3817:
3803:
3800:
3797:
3796:
3789:
3778:
3768:
3761:
3749:
3738:
3730:
3729:
3718:
3715:
3712:
3709:
3706:
3696:
3685:
3675:
3664:
3661:
3658:
3655:
3652:
3642:
3630:
3619:
3608:
3605:
3602:
3587:
3586:
3582:
3581:
3578:
3577:
3570:
3559:
3549:
3542:
3530:
3519:
3511:
3510:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3463:
3453:
3442:
3432:
3421:
3418:
3415:
3412:
3409:
3399:
3387:
3376:
3365:
3362:
3359:
3345:
3342:
3339:
3338:
3331:
3320:
3310:
3303:
3291:
3280:
3273:
3262:
3252:
3245:
3233:
3222:
3214:
3213:
3202:
3199:
3196:
3193:
3190:
3180:
3169:
3159:
3148:
3135:
3123:
3112:
3101:
3098:
3095:
3092:
3089:
3079:
3068:
3058:
3047:
3034:
3022:
3011:
3000:
2997:
2994:
2980:
2977:
2974:
2973:
2966:
2955:
2945:
2938:
2926:
2915:
2907:
2906:
2895:
2885:
2874:
2864:
2853:
2843:
2831:
2820:
2809:
2806:
2777:
2765:
2762:
2734:
2731:
2646:
2643:
2630:
2610:
2583:
2580:
2577:
2557:
2529:
2497:
2477:
2457:
2426:
2397:
2394:
2370:
2367:
2350:
2347:
2344:
2343:
2336:
2325:
2315:
2308:
2296:
2287:
2285:
2277:
2276:
2265:
2262:
2252:
2241:
2231:
2220:
2217:
2207:
2195:
2186:
2184:
2173:
2170:
2167:
2143:
2123:
2106:
2105:
2089:
2079:
2067:
2058:
2056:
2048:
2047:
2036:
2033:
2030:
2027:
2024:
2021:
2011:
1999:
1990:
1988:
1977:
1974:
1971:
1946:
1926:
1914:
1911:
1908:
1907:
1904:
1901:
1897:
1896:
1893:
1890:
1886:
1885:
1882:
1879:
1875:
1874:
1871:
1868:
1864:
1863:
1852:
1849:
1846:
1836:
1825:
1815:
1804:
1779:
1776:
1773:
1757:
1754:
1746:logical values
1733:
1730:
1662:
1659:
1656:
1634:
1631:
1609:
1589:
1542:Sheffer stroke
1528:
1527:
1525:
1524:
1517:
1510:
1502:
1499:
1498:
1487:
1486:
1484:
1483:
1478:
1473:
1468:
1462:
1459:
1458:
1454:
1453:
1451:
1450:
1445:
1440:
1435:
1433:Truth function
1430:
1425:
1420:
1415:
1409:
1406:
1405:
1401:
1400:
1397:
1396:
1385:
1382:
1379:
1359:
1356:
1353:
1333:
1330:
1327:
1317:
1311:
1310:
1299:
1296:
1293:
1273:
1268:
1265:
1260:
1250:
1244:
1243:
1232:
1218:
1208:
1202:
1201:
1190:
1187:
1184:
1164:
1161:
1158:
1138:
1135:
1132:
1112:
1109:
1106:
1096:
1090:
1089:
1078:
1075:
1053:
1050:
1028:
1025:
1005:
1002:
992:
986:
985:
972:
968:
965:
962:
939:
936:
933:
913:
908:
905:
900:
890:
884:
883:
872:
869:
866:
846:
843:
840:
820:
817:
814:
804:
800:
799:
786:
782:
779:
776:
753:
750:
747:
727:
724:
721:
701:
696:
693:
688:
678:
672:
671:
660:
657:
654:
634:
631:
628:
608:
605:
602:
592:
586:
585:
574:
571:
568:
548:
545:
542:
522:
519:
516:
506:
500:
499:
488:
485:
482:
479:
459:
456:
453:
433:
430:
410:
407:
404:
384:
381:
378:
368:
358:
357:
347:
346:
344:
343:
336:
329:
321:
318:
317:
314:
310:
309:
306:
300:
299:
296:
290:
289:
286:
282:
281:
278:
274:
273:
265:
264:
253:
250:
247:
244:
234:
228:
227:
214:
211:
206:
201:
198:
186:
180:
179:
166:
163:
158:
153:
150:
138:
132:
131:
127:
126:
119:
113:
112:
101:
98:
95:
85:
79:
78:
65:
61:
58:
55:
42:
38:
37:
29:
28:
22:Sheffer stroke
15:
9:
6:
4:
3:
2:
5653:
5642:
5641:Logic symbols
5639:
5637:
5634:
5632:
5629:
5628:
5626:
5613:
5608:
5602:
5590:
5581:
5577:
5572:
5569:
5565:
5560:
5557:
5551:
5546:
5543:
5537:
5532:
5529:
5528:contradiction
5524:
5520:
5515:
5512:
5507:
5503:
5498:
5494:
5487:
5482:
5478:
5471:
5466:
5463:
5459:
5454:
5451:
5447:
5442:
5437:
5432:
5429:
5424:
5420:
5415:
5412:
5408:
5403:
5400:
5396:
5391:
5386:
5381:
5378:
5374:
5369:
5366:
5362:
5357:
5352:
5347:
5346:
5341:
5337:
5329:
5324:
5322:
5317:
5315:
5310:
5309:
5306:
5296:
5284:
5258:
5254:
5253:Contradiction
5251:
5250:
5247:
5229:
5221:
5217:
5214:
5200:
5192:
5189:
5175:
5167:
5163:
5160:
5138:
5134:
5131:
5130:
5127:
5120:
5116:
5113:
5091:
5087:
5086:Biconditional
5084:
5070:
5062:
5058:
5055:
5033:
5029:
5026:
5025:
5022:
5004:
4996:
4992:
4989:
4967:
4963:
4960:
4938:
4935:
4913:
4909:
4906:
4905:
4902:
4897:
4867:
4863:
4860:
4859:
4856:
4852:
4844:
4839:
4837:
4832:
4830:
4825:
4824:
4821:
4815:
4811:
4808:
4806:
4803:
4801:
4798:
4796:
4795:
4790:
4787:
4786:
4776:
4772:
4771:
4766:
4762:
4758:
4754:
4751:
4745:
4741:
4737:
4733:
4729:
4728:
4715:
4709:
4705:
4698:
4696:
4694:
4692:
4690:
4675:
4671:
4664:
4650:
4646:
4639:
4631:
4629:9781400882366
4625:
4621:
4617:
4613:
4612:
4604:
4596:
4592:
4588:
4582:
4574:
4570:
4569:
4564:
4558:
4550:
4546:
4541:
4536:
4532:
4528:
4527:
4522:
4518:
4512:
4504:
4497:
4489:
4482:
4474:
4467:
4459:
4452:
4444:
4440:
4436:
4432:
4428:
4425:(in German).
4424:
4423:
4418:
4412:
4410:
4395:
4388:
4380:
4373:
4371:
4362:
4355:
4347:
4341:
4337:
4330:
4328:
4323:
4312:
4309:
4306:
4303:
4301:
4298:
4296:
4293:
4291:
4287:
4285:
4282:
4280:
4279:Logical graph
4277:
4274:
4271:
4269:
4266:
4264:
4261:
4260:
4253:
4251:
4229:
4226:
4223:
4220:
4194:
4188:
4185:
4156:
4153:
4150:
4127:
4124:
4121:
4095:
4089:
4069:
4063:
4043:
4032:
4028:
4009:
4005:
3984:
3981:
3977:
3954:
3951:
3947:
3946:
3928:
3922:
3912:
3891:
3874:
3868:
3858:
3835:
3821:
3818:
3815:
3808:
3807:
3804:
3801:
3794:
3790:
3769:
3766:
3762:
3739:
3736:
3732:
3731:
3713:
3707:
3697:
3676:
3659:
3653:
3643:
3620:
3606:
3603:
3600:
3593:
3592:
3589:
3588:
3584:
3583:
3575:
3571:
3550:
3547:
3543:
3520:
3517:
3513:
3512:
3491:
3485:
3473:
3467:
3454:
3433:
3416:
3410:
3400:
3377:
3363:
3357:
3350:
3349:
3346:
3343:
3336:
3332:
3311:
3308:
3304:
3281:
3278:
3274:
3253:
3250:
3246:
3223:
3220:
3216:
3215:
3197:
3191:
3181:
3160:
3146:
3136:
3113:
3096:
3090:
3080:
3059:
3045:
3035:
3012:
2998:
2992:
2985:
2984:
2981:
2978:
2971:
2967:
2946:
2943:
2939:
2916:
2913:
2909:
2908:
2893:
2886:
2865:
2851:
2844:
2821:
2807:
2797:
2796:
2793:
2792:
2789:
2761:
2759:
2754:
2752:
2748:
2744:
2740:
2730:
2726:
2721:
2717:
2716:
2711:
2707:
2703:
2701:
2700:
2694:
2690:
2686:
2682:
2678:
2674:
2670:
2666:
2662:
2658:
2657:
2652:
2642:
2599:
2597:
2581:
2578:
2575:
2555:
2547:
2542:
2527:
2518:
2514:
2509:
2495:
2475:
2455:
2447:
2442:
2440:
2424:
2416:
2411:
2392:
2365:
2355:
2341:
2337:
2323:
2316:
2313:
2309:
2286:
2283:
2279:
2278:
2263:
2253:
2239:
2232:
2218:
2208:
2185:
2171:
2165:
2158:
2157:
2154:
2141:
2121:
2113:
2103:
2080:
2057:
2054:
2050:
2049:
2031:
2028:
2025:
2012:
1989:
1975:
1969:
1962:
1961:
1958:
1944:
1924:
1905:
1902:
1899:
1898:
1894:
1891:
1888:
1887:
1883:
1880:
1877:
1876:
1872:
1869:
1866:
1865:
1850:
1844:
1837:
1823:
1816:
1802:
1795:
1794:
1791:
1777:
1771:
1763:
1753:
1751:
1747:
1743:
1739:
1729:
1727:
1723:
1719:
1715:
1712:(making NAND
1711:
1710:formal system
1707:
1706:Webb operator
1703:
1699:
1695:
1691:
1686:
1684:
1680:
1676:
1660:
1657:
1654:
1629:
1587:
1579:
1575:
1571:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1523:
1518:
1516:
1511:
1509:
1504:
1503:
1501:
1500:
1497:
1489:
1488:
1482:
1479:
1477:
1474:
1472:
1469:
1467:
1466:Digital logic
1464:
1463:
1461:
1460:
1456:
1455:
1449:
1448:Scope (logic)
1446:
1444:
1441:
1439:
1436:
1434:
1431:
1429:
1426:
1424:
1421:
1419:
1416:
1414:
1411:
1410:
1408:
1407:
1403:
1402:
1383:
1377:
1357:
1354:
1351:
1331:
1325:
1318:
1316:
1313:
1312:
1297:
1294:
1291:
1271:
1266:
1263:
1258:
1251:
1249:
1246:
1245:
1230:
1216:
1209:
1207:
1204:
1203:
1188:
1185:
1182:
1162:
1159:
1156:
1136:
1133:
1130:
1110:
1107:
1104:
1097:
1095:
1092:
1091:
1076:
1073:
1048:
1026:
1023:
1003:
993:
991:
988:
987:
966:
963:
960:
937:
931:
911:
903:
898:
891:
889:
886:
885:
870:
867:
864:
844:
841:
838:
818:
815:
812:
805:
803:nonequivalent
802:
801:
780:
777:
774:
751:
748:
745:
725:
719:
699:
691:
686:
679:
677:
674:
673:
658:
652:
632:
629:
626:
606:
600:
593:
591:
588:
587:
572:
566:
546:
540:
520:
517:
514:
507:
505:
502:
501:
486:
477:
457:
451:
431:
428:
408:
405:
402:
382:
379:
376:
369:
367:
364:
363:
360:
359:
356:
353:
352:
342:
337:
335:
330:
328:
323:
322:
319:
315:
311:
307:
305:
301:
297:
295:
291:
287:
283:
279:
275:
272:
266:
251:
248:
245:
242:
235:
233:
229:
209:
204:
196:
187:
185:
181:
161:
156:
148:
139:
137:
133:
128:
124:
120:
118:
114:
96:
86:
84:
80:
59:
56:
53:
43:
39:
35:
30:
25:
19:
5461:
5457:
5440:
5389:
5355:
5133:Joint denial
5057:Exclusive or
4907:
4792:
4769:
4739:
4703:
4678:, retrieved
4673:
4663:
4652:. Retrieved
4648:
4638:
4610:
4603:
4590:
4581:
4572:
4566:
4557:
4530:
4524:
4511:
4502:
4496:
4487:
4481:
4472:
4466:
4457:
4451:
4426:
4420:
4397:. Retrieved
4387:
4378:
4360:
4354:
4335:
4305:Peirce arrow
4300:Peirce's law
4290:flash memory
4024:
2767:
2755:
2751:self-duality
2747:monotonicity
2736:
2720:Edward Stamm
2713:
2704:
2697:
2654:
2648:
2600:
2543:
2510:
2443:
2438:
2412:
2352:
2109:
1916:
1759:
1750:propositions
1737:
1735:
1702:Quine dagger
1698:Peirce arrow
1694:NOR operator
1687:
1565:
1561:
1557:
1541:
1531:
1457:Applications
675:
285:1-preserving
277:0-preserving
130:Normal forms
18:
5631:Logic gates
5491:existential
5216:Conjunction
5166:NIMPLY gate
4991:Disjunction
4962:Implication
4765:Weiss, Paul
4031:truth table
2723: [
2546:Łukasiewicz
1762:truth table
1756:Truth table
1683:disjunction
1679:Łukasiewicz
1554:conjunction
1428:Truth table
184:Conjunctive
136:Disjunctive
83:Truth table
5625:Categories
4966:IMPLY gate
4680:2024-03-22
4654:2024-03-22
4399:2023-07-02
4318:References
4295:NAND logic
2733:Properties
2693:Jean Nicod
2665:Huntington
2439:Stamm hook
1752:is false.
1732:Definition
1544:denotes a
504:equivalent
117:Logic gate
41:Definition
5568:therefore
5556:therefore
5511:tautology
5475:universal
5267:⊥
5230:∧
5201:↚
5176:↛
5147:↓
5115:Statement
5100:↔
5090:XNOR gate
5042:¬
5005:∨
4976:→
4947:←
4922:↑
4912:NAND gate
4876:⊤
4862:Tautology
4744:D. Reidel
4443:119816758
4233:¬
4227:∨
4221:∧
4192:¬
4189:∧
4183:¬
4177:¬
4154:∨
4125:∧
4116:¬
4093:↑
4067:↑
4041:¬
3992:↑
3963:⇔
3926:↑
3899:↑
3872:↑
3844:⇔
3819:∨
3777:↑
3748:⇔
3711:↑
3684:↑
3657:↑
3629:⇔
3604:∧
3558:↑
3529:⇔
3489:↑
3480:↑
3471:↑
3441:↑
3414:↑
3386:⇔
3361:↔
3319:↑
3290:⇔
3261:↑
3232:⇔
3195:↑
3168:↑
3122:⇔
3094:↑
3067:↑
3021:⇔
2996:→
2954:↑
2925:⇔
2873:↑
2830:⇔
2805:¬
2776:↑
2743:linearity
2629:↓
2609:↑
2544:In 1929,
2517:Ackermann
2511:In 1928,
2496:∣
2476:∧
2456:∣
2444:In 1913,
2425:∼
2413:In 1911,
2396:¯
2393:⋏
2369:¯
2366:⋏
2324:∨
2295:⇔
2261:¬
2240:∨
2216:¬
2194:⇔
2169:↑
2088:¬
2066:⇔
2029:∧
2020:¬
1998:⇔
1973:↑
1848:↑
1775:↑
1718:NAND gate
1633:¯
1630:∧
1608:↑
1588:∣
1574:NAND gate
1381:←
1355:⊂
1329:⇐
1295:⊕
1267:_
1264:∨
1186:∥
1160:∣
1108:∨
1074:∼
1052:¯
1024:−
1001:¬
971:¯
935:↓
907:¯
904:∨
868:↮
785:¯
778:⋅
749:∣
723:↑
695:¯
692:∧
656:→
630:⊃
604:⇒
570:⇋
544:⇔
518:≡
484:&
481:&
455:&
406:⋅
380:∧
313:Self-dual
246:⊕
213:¯
200:¯
165:¯
152:¯
64:¯
57:⋅
5554:entails,
5540:entails,
5428:superset
5220:AND gate
5137:NOR gate
5071:↮
5061:XOR gate
5032:NOT gate
5028:Negation
4767:(eds.).
4738:(1960).
4589:(1956).
4575:: 32–41.
4519:(1913).
4256:See also
1728:design.
1550:negation
1496:Category
1315:converse
842:⇎
816:≢
294:Monotone
5580:because
5444:
5423:implies
5411:implies
5393:
5359:
5334:Common
5222:)
5218: (
5168:)
5164: (
5139:)
5135: (
5117: (
5092:)
5088: (
5063:)
5059: (
5034:)
5030: (
4997:)
4995:OR gate
4993: (
4968:)
4964: (
4914:)
4910: (
4849:Common
4549:1988701
4033:, that
3585:
2715:ampheck
2685:duality
2645:History
2513:Hilbert
2446:Sheffer
1744:on two
1692:is the
1552:of the
590:implies
5542:proves
5438:
5387:
5353:
5259:
5193:
4939:
4868:
4710:
4626:
4547:
4441:
4342:
4143:, and
3144:
3043:
2354:Peirce
1647:or as
1620:or as
1600:or as
1540:, the
1228:
1220:
304:Affine
5523:false
5361:&
5257:False
4545:JSTOR
4439:S2CID
4307:= NOR
4288:NAND
2727:]
2548:used
2437:(the
2415:Stamm
1740:is a
1560:, or
5506:true
5462:nand
4866:True
4708:ISBN
4624:ISBN
4340:ISBN
4275:(GE)
4268:CMOS
2515:and
2134:and
1937:and
1760:The
1736:The
1690:dual
1688:Its
1566:NAND
1536:and
1224:XNOR
1206:XNOR
676:NAND
97:1110
27:NAND
5450:iff
5399:not
5365:and
4616:doi
4535:doi
4431:doi
2689:NOR
2681:NOT
2673:AND
2568:in
2110:By
1764:of
1704:or
1685:).
1677:by
1673:in
1532:In
1248:XOR
990:NOT
888:NOR
366:AND
5627::
5441:or
5390:or
5377:or
5356:or
4812:@
4763:;
4734:;
4688:^
4647:.
4622:.
4573:19
4571:.
4543:.
4531:14
4529:.
4523:.
4437:.
4427:22
4408:^
4369:^
4326:^
4252:.
2749:,
2745:,
2725:pl
2679:,
2677:OR
2675:,
2598:.
2541:.
1700:,
1370:,
1344:,
1284:,
1175:,
1149:,
1123:,
1094:OR
1066:,
1039:,
1016:,
950:,
924:,
857:,
831:,
764:,
738:,
712:,
645:,
619:,
559:,
533:,
470:,
444:,
421:,
395:,
316:no
308:no
298:no
288:no
280:no
5576:∵
5564:∴
5550:⊨
5536:⊢
5525:,
5519:⊥
5508:,
5502:⊤
5486:∃
5470:∀
5458:|
5446:≡
5436:↔
5425:,
5419:⊃
5407:→
5395:~
5385:¬
5373:∨
5351:∧
5327:e
5320:t
5313:v
5255:/
5121:)
4864:/
4842:e
4835:t
4828:v
4752:)
4746:.
4716:.
4657:.
4632:.
4618::
4551:.
4537::
4445:.
4433::
4402:.
4348:.
4236:}
4230:,
4224:,
4218:{
4198:)
4195:B
4186:A
4180:(
4157:B
4151:A
4131:)
4128:B
4122:A
4119:(
4096:B
4090:A
4070:A
4064:A
4044:A
3932:)
3929:Q
3923:Q
3920:(
3878:)
3875:P
3869:P
3866:(
3822:Q
3816:P
3717:)
3714:Q
3708:P
3705:(
3663:)
3660:Q
3654:P
3651:(
3607:Q
3601:P
3498:)
3495:)
3492:Q
3486:Q
3483:(
3477:)
3474:P
3468:P
3465:(
3462:(
3420:)
3417:Q
3411:P
3408:(
3364:Q
3358:P
3201:)
3198:Q
3192:P
3189:(
3147:P
3100:)
3097:Q
3091:Q
3088:(
3046:P
2999:Q
2993:P
2894:P
2852:P
2808:P
2671:(
2582:q
2579:p
2576:D
2556:D
2528:/
2264:Q
2219:P
2172:Q
2166:P
2142:Q
2122:P
2035:)
2032:Q
2026:P
2023:(
1976:Q
1970:P
1945:Q
1925:P
1906:F
1903:T
1900:T
1895:T
1892:F
1889:T
1884:T
1881:T
1878:F
1873:T
1870:F
1867:F
1851:B
1845:A
1824:B
1803:A
1778:B
1772:A
1661:q
1658:p
1655:D
1521:e
1514:t
1507:v
1384:B
1378:A
1358:B
1352:A
1332:B
1326:A
1298:B
1292:A
1272:B
1259:A
1231:B
1217:A
1189:B
1183:A
1163:B
1157:A
1137:B
1134:+
1131:A
1111:B
1105:A
1077:A
1049:A
1027:A
1004:A
967:B
964:+
961:A
938:B
932:A
912:B
899:A
871:B
865:A
845:B
839:A
819:B
813:A
781:B
775:A
752:B
746:A
726:B
720:A
700:B
687:A
659:B
653:A
633:B
627:A
607:B
601:A
573:B
567:A
547:B
541:A
521:B
515:A
487:B
478:A
458:B
452:A
432:B
429:A
409:B
403:A
383:B
377:A
340:e
333:t
326:v
252:y
249:x
243:1
210:y
205:+
197:x
162:y
157:+
149:x
100:)
94:(
60:y
54:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
