1440:
1834:
797:
In 1973, Padmanabhan and
Quackenbush demonstrated a method that, in principle, would yield a 1-basis for Boolean algebra. Applying this method in a straightforward manner yielded "axioms of enormous length", thereby prompting the question of how shorter axioms might be found. This search yielded the
921:
263:
635:
456:
140:
156:, by enumerating the Sheffer identities of length less or equal to 15 elements (excluding mirror images) that have no noncommutative models with four or fewer variables, and was first proven equivalent by
792:
370:
549:
489:. This proof established that the Robbins axiom, together with associativity and commutativity, form a 3-basis for Boolean algebra. The existence of a 2-basis was established in 1967 by
700:
1625:
172:, a site associated with Wolfram, has named the axiom the "Wolfram axiom". McCune et al. also found a longer single axiom for Boolean algebra based on disjunction and negation.
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1085:
Huntington, E. V. (1933). "New Sets of
Independent Postulates for the Algebra of Logic, with Special Reference to Whitehead and Russell's
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485:, who took considerable interest in it later. The conjecture was eventually proved in 1996 with the aid of
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916:{\displaystyle \neg (\neg (\neg (x\lor y)\lor z)\lor \neg (x\lor \neg (\neg z\lor \neg (z\lor u))))=z,}
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as being equivalent to
Boolean algebra, when combined with the commutativity of the
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where the vertical bar represents the NAND logical operation (also known as the
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258:{\displaystyle {\neg ({\neg x}\lor {y})}\lor {\neg ({\neg x}\lor {\neg y})}=x}
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The following year, Meredith found a 2-basis in terms of the
Sheffer stroke:
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1-basis in terms of the
Sheffer stroke given above, as well as the 1-basis
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1571:
481:. Neither Robbins nor Huntington could prove this conjecture; nor could
1538:
43:), chosen to be as short as possible. For example, an axiom with six
630:{\displaystyle \neg ({\neg x}\lor y)\lor (z\lor y)=y\lor (z\lor x).}
451:{\displaystyle \neg (\neg (x\lor y)\lor \neg (x\lor {\neg y}))=x,}
152:
It is one of 25 candidate axioms for this property identified by
135:{\displaystyle ((a\mid b)\mid c)\mid (a\mid ((a\mid c)\mid a))=c}
47:
operations and three variables is equivalent to
Boolean algebra:
1366:
1334:"Equational theories of algebras with distributive congruences"
461:
which requires one fewer use of the logical negation operator
787:{\displaystyle x|(y\mid (x\mid z))=((z\mid y)\mid y)\mid x.}
1157:
McCune, William (1997). "Solution of the
Robbins Problem".
376:
conjectured that
Huntington's axiom could be replaced by
953:"Logic, Explainability and the Future of Understanding"
35:
are assumptions which are equivalent to the axioms of
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1018:(2002), "Short single axioms for Boolean algebra",
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365:{\displaystyle (x\lor y)\lor z=x\lor (y\lor z)}
1291:"Equational postulates for the Sheffer stroke"
544:{\displaystyle \neg ({\neg x}\lor y)\lor x=x,}
1378:
8:
1332:Padmanabhan, R.; Quackenbush, R. W. (1973).
1199:"Computer Math Proof Shows Reasoning Power"
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695:{\displaystyle (x\mid x)\mid (y\mid x)=x,}
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1620:{\displaystyle \not \leftrightarrow }
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1014:; Harris, Kenneth; Feist, Andrew;
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33:minimal axioms for Boolean algebra
25:
1352:10.1090/S0002-9939-1973-0325498-2
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1649:{\displaystyle \leftrightarrow }
1438:
303:{\displaystyle x\lor y=y\lor x}
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1211:McCune, William (1997-01-23).
1160:Journal of Automated Reasoning
1021:Journal of Automated Reasoning
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1725:{\displaystyle \nrightarrow }
926:which is written in terms of
1750:{\displaystyle \nleftarrow }
1525:{\displaystyle \rightarrow }
1696:{\displaystyle \downarrow }
1496:{\displaystyle \leftarrow }
1218:Argonne National Laboratory
1213:"Comments on Robbins Story"
1889:
1296:Notre Dame J. Formal Logic
1250:Notre Dame J. Formal Logic
1120:Cylindric Algebras, Part I
177:Edward Vermilye Huntington
1829:
1792:
1672:
1567:
1471:{\displaystyle \uparrow }
1447:
1436:
1401:
1310:10.1305/ndjfl/1093893713
1264:10.1305/ndjfl/1093893457
957:Stephen Worfram Writings
487:theorem-proving software
1735:Converse nonimplication
1173:10.1023/A:1005843212881
1092:Trans. Amer. Math. Soc.
1034:10.1023/A:1020542009983
1873:Propositional calculus
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1779:{\displaystyle \land }
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41:propositional calculus
1839:Philosophy portal
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1816:{\displaystyle \bot }
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1591:{\displaystyle \neg }
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1554:{\displaystyle \lor }
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1425:{\displaystyle \top }
1087:Principia Mathematica
976:A New Kind of Science
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491:Carew Arthur Meredith
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474:{\displaystyle \neg }
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1395:logical connectives
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1452:Alternative denial
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1245:"Equational logic"
1204:The New York Times
1010:; Veroff, Robert;
951:Wolfram, Stephen.
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29:mathematical logic
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1844:
1135:978-0-7204-2043-2
1012:Fitelson, Branden
979:. Wolfram Media.
16:(Redirected from
1880:
1863:History of logic
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1065:"Wolfram Axiom"
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1008:McCune, William
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154:Stephen Wolfram
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37:Boolean algebra
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1706:Nonimplication
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1663:Digital buffer
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1345:(2): 373–377.
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1303:(3): 266–270.
1278:
1257:(3): 212–226.
1228:
1225:on 1997-06-05.
1197:(1996-12-10).
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1167:(3): 263–276.
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1114:Tarski, Alfred
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18:Wolfram axiom
1677:Joint denial
1601:Exclusive or
1342:
1337:
1327:
1300:
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1281:
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1241:Prior, A. N.
1231:
1223:the original
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1195:Kolata, Gina
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1110:Henkin, Leon
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1025:
1019:
974:
965:
956:
946:
925:
796:
639:
460:
267:
174:
151:
144:
32:
26:
1868:Logic gates
1760:Conjunction
1710:NIMPLY gate
1535:Disjunction
1506:Implication
1028:(1): 1–16,
272:operation,
1852:Categories
1510:IMPLY gate
1144:1024041028
1098:: 247–304.
1016:Wos, Larry
938:References
1811:⊥
1774:∧
1745:↚
1720:↛
1691:↓
1659:Statement
1644:↔
1634:XNOR gate
1586:¬
1549:∨
1520:→
1491:←
1466:↑
1456:NAND gate
1420:⊤
1406:Tautology
1070:MathWorld
1050:207582048
887:∨
878:¬
875:∨
869:¬
863:¬
860:∨
851:¬
848:∨
839:∨
830:∨
821:¬
815:¬
809:¬
776:∣
767:∣
758:∣
734:∣
725:∣
675:∣
666:∣
657:∣
616:∨
607:∨
592:∨
583:∨
574:∨
567:¬
560:¬
527:∨
518:∨
511:¬
504:¬
469:¬
427:¬
423:∨
414:¬
411:∨
402:∨
393:¬
387:¬
354:∨
345:∨
333:∨
324:∨
295:∨
283:∨
239:¬
235:∨
228:¬
221:¬
217:∨
205:∨
198:¬
191:¬
175:In 1933,
170:MathWorld
166:Larry Wos
115:∣
106:∣
94:∣
85:∣
76:∣
67:∣
1764:AND gate
1681:NOR gate
1615:↮
1605:XOR gate
1576:NOT gate
1572:Negation
1289:(1969).
1243:(1968).
1181:30847540
1116:(1971).
973:(2002).
1766:)
1762: (
1712:)
1708: (
1683:)
1679: (
1661: (
1636:)
1632: (
1607:)
1603: (
1578:)
1574: (
1541:)
1539:OR gate
1537: (
1512:)
1508: (
1458:)
1454: (
1393:Common
1319:0245423
1273:0246753
1042:1940227
1803:
1737:
1483:
1412:
1317:
1271:
1179:
1142:
1132:
1048:
1040:
983:
164:, and
1801:False
1177:S2CID
1046:S2CID
1410:True
1140:OCLC
1130:ISBN
981:ISBN
930:and
45:NAND
39:(or
1347:doi
1305:doi
1259:doi
1169:doi
1089:".
1030:doi
932:NOT
149:).
27:In
1854::
1343:41
1336:.
1315:MR
1313:.
1301:10
1299:.
1293:.
1269:MR
1267:.
1253:.
1247:.
1239:;
1215:.
1201:.
1175:.
1165:19
1163:.
1138:.
1128:.
1124:.
1096:25
1067:.
1044:,
1038:MR
1036:,
1026:29
1024:,
995:^
955:.
934:.
928:OR
493::
372:.
270:OR
168:.
160:,
31:,
1799:/
1665:)
1408:/
1386:e
1379:t
1372:v
1355:.
1349::
1321:.
1307::
1275:.
1261::
1255:9
1207:.
1183:.
1171::
1146:.
1073:.
1032::
989:.
959:.
911:,
908:z
905:=
902:)
899:)
896:)
893:)
890:u
884:z
881:(
872:z
866:(
857:x
854:(
845:)
842:z
836:)
833:y
827:x
824:(
818:(
812:(
782:.
779:x
773:)
770:y
764:)
761:y
755:z
752:(
749:(
746:=
743:)
740:)
737:z
731:x
728:(
722:y
719:(
715:|
711:x
690:,
687:x
684:=
681:)
678:x
672:y
669:(
663:)
660:x
654:x
651:(
625:.
622:)
619:x
613:z
610:(
604:y
601:=
598:)
595:y
589:z
586:(
580:)
577:y
570:x
563:(
539:,
536:x
533:=
530:x
524:)
521:y
514:x
507:(
446:,
443:x
440:=
437:)
434:)
430:y
420:x
417:(
408:)
405:y
399:x
396:(
390:(
360:)
357:z
351:y
348:(
342:x
339:=
336:z
330:)
327:y
321:x
318:(
298:x
292:y
289:=
286:y
280:x
253:x
250:=
246:)
242:y
231:x
224:(
213:)
209:y
201:x
194:(
130:c
127:=
124:)
121:)
118:a
112:)
109:c
103:a
100:(
97:(
91:a
88:(
82:)
79:c
73:)
70:b
64:a
61:(
58:(
20:)
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