6038:
1171:
3503:
There are no minimal functionally complete sets of more than three at most binary logical connectives. In order to keep the lists above readable, operators that ignore one or more inputs have been omitted. For example, an operator that ignores the first input and outputs the negation of the second
3807:. Then, if we map boolean operators into set operators, the "translated" above text are valid also for sets: there are many "minimal complete set of set-theory operators" that can generate any other set relations. The more popular "Minimal complete operator sets" are
1698:
2414:
2585:
2526:
2223:
together with the two nullary constant functions be functionally complete or, equivalently, functionally complete in the strong sense of the previous paragraph. The example of the
Boolean function given by
3457:
3343:
2070:
1582:
3498:
1822:
1599:
3263:
3228:
3419:
3381:
1509:
3305:
1594:
1955:
2016:
2940:
2908:
2363:
3036:
3164:
3100:
3068:
534:
3196:
3132:
2972:
1739:
961:
476:
389:
3004:
601:
262:
1198:
920:
560:
236:
2748:
662:
2876:
2812:
1442:
4417:
2716:
743:
2844:
2780:
508:
348:
296:
176:
4263:
1073:
1021:
627:
4243:
1905:
878:
1462:
1444:). Further connectives can be defined, if so desired, by defining them in terms of these primitives. For example, NOR (the negation of the disjunction, sometimes denoted
415:
1047:
322:
1532:
987:
210:
5092:
147:
98:
72:
852:
800:
441:
1845:
1346:
766:
693:
2445:
1978:
1885:
1865:
1763:
1394:
1370:
1418:
826:
716:
121:
5175:
4316:
3760:
is functionally complete reversible gate by itself – a sole sufficient operator. There are many other three-input universal logic gates, such as the
2159:. In other words, the set is functionally complete if every Boolean function that takes at least one variable can be expressed in terms of the functions
4101:
2216:
itself does not contain at least one nullary function. With this stronger definition, the smallest functionally complete sets would have 2 elements.
1308:
can be realized as a network of gates of the types prescribed by the set. In particular, all logic gates can be assembled from either only binary
5489:
2295:
proved that a set of logical connectives is functionally complete if and only if it is not a subset of any of the following sets of connectives:
5647:
4170:
3988:
3963:
3934:
3905:
4435:
5502:
4825:
1184:
2615:, which implies the above result as a simple corollary: the five mentioned sets of connectives are exactly the maximal nontrivial clones.
2375:
2546:
5087:
2487:
6067:
5507:
5497:
5234:
4440:
3749:
Apart from logical connectives (Boolean operators), functional completeness can be introduced in other domains. For example, a set of
4985:
4431:
3696:
Note that an electronic circuit or a software function can be optimized by reuse, to reduce the number of gates. For instance, the "
5643:
4119:
3822:, set operators are restricted to being falsity (Ø) preserving, and cannot be equivalent to functionally complete Boolean algebra.
3424:
3310:
2025:
1537:
5740:
5484:
4309:
5045:
4738:
4479:
6077:
6072:
3849:
2204:. However, the examples given above are not functionally complete in this stronger sense because it is not possible to write a
1742:
1191:
6001:
5703:
5466:
5461:
5286:
4707:
4391:
3855:
2372:
connectives, such that each connected variable either always or never affects the truth value these connectives return, e.g.
5996:
5779:
5696:
5409:
5340:
5217:
4459:
1282:
A gate (or set of gates) that is functionally complete can also be called a universal gate (or a universal set of gates).
5067:
3462:
1771:
1693:{\displaystyle {\begin{aligned}A\to B&:=\neg A\lor B\\A\leftrightarrow B&:=(A\to B)\land (B\to A).\end{aligned}}}
1588:
functionally complete set, because the conditional and biconditional can be defined in terms of the other connectives as
5921:
5747:
5433:
4666:
3872:
3233:
3201:
1174:
5072:
3386:
3348:
1470:
5404:
5143:
4401:
4302:
4225:: 117–32. In his list on the last page of the article, Wernick does not distinguish between ← and →, or between
3804:
3272:
5799:
5794:
1913:
2172:
is functionally complete if and only if every binary
Boolean function can be expressed in terms of the functions in
5728:
5318:
4712:
4680:
4371:
3955:
1983:
4445:
2913:
6018:
5967:
5864:
5362:
5323:
4800:
5859:
4474:
2881:
6062:
5789:
5328:
5180:
5163:
4886:
4366:
2330:
2168:. Since every Boolean function of at least one variable can be expressed in terms of binary Boolean functions,
3009:
3137:
3073:
3041:
513:
5691:
5668:
5629:
5515:
5456:
5102:
5022:
4866:
4810:
4423:
3169:
3105:
2945:
1706:
1252:
933:
446:
361:
2977:
1534:), can be defined in terms of disjunction and negation. Every binary connective can be defined in terms of
573:
241:
5981:
5708:
5686:
5653:
5546:
5392:
5377:
5350:
5301:
5185:
5120:
4945:
4911:
4906:
4780:
4611:
4588:
3750:
2648:
2427:; if the truth values of all variables are reversed, so is the truth value these connectives return, e.g.
1091:
891:
539:
5911:
5764:
5556:
5274:
5010:
4916:
4775:
4760:
4641:
4616:
2623:
When a single logical connective or
Boolean operator is functionally complete by itself, it is called a
2369:
1421:
215:
182:
6037:
2721:
632:
2849:
2785:
1427:
5884:
5846:
5723:
5527:
5367:
5291:
5269:
5097:
5055:
4954:
4921:
4785:
4573:
4484:
3969:. (Defines "expressively adequate", shortened to "adequate set of connectives" in a section heading.)
3843:
3819:
2689:
2664:
1397:
1149:
721:
268:
2817:
2753:
2599:(sets of operations closed under composition and containing all projections) on the two-element set
1584:, which means that set is functionally complete. However, it contains redundancy: this set is not a
487:
327:
275:
152:
6013:
5904:
5889:
5869:
5826:
5713:
5663:
5589:
5534:
5471:
5264:
5259:
5207:
4975:
4964:
4636:
4536:
4464:
4455:
4451:
4386:
4381:
4248:
1349:
1325:
1301:
1286:
1274:
1242:
1159:
1052:
1000:
772:
606:
44:
4275:
4228:
1890:
857:
6042:
5811:
5774:
5759:
5752:
5735:
5521:
5387:
5313:
5296:
5249:
5062:
4971:
4805:
4790:
4750:
4702:
4687:
4675:
4631:
4606:
4376:
4325:
4287:
3780:
3776:
1447:
1236:
1220:
1212:
1154:
394:
33:
5539:
4995:
1960:
No further simplifications are possible. Hence, every two-element set of connectives containing
1026:
301:
1517:
966:
189:
5977:
5784:
5594:
5584:
5476:
5357:
5192:
5168:
4949:
4933:
4838:
4815:
4692:
4661:
4626:
4521:
4356:
4166:
4115:
3984:
3959:
3930:
3901:
3768:
2612:
2449:
2287:
1232:
1096:
126:
77:
51:
831:
779:
420:
5991:
5986:
5879:
5836:
5658:
5619:
5614:
5599:
5425:
5382:
5279:
5077:
5027:
4601:
4563:
4199:
4107:
4081:
4049:
4017:
2652:
1830:
1331:
1224:
1116:
993:
748:
675:
3926:
2430:
1963:
1870:
1850:
1748:
1379:
1355:
5972:
5962:
5916:
5899:
5854:
5816:
5718:
5638:
5445:
5372:
5345:
5333:
5239:
5153:
5127:
5082:
5050:
4851:
4653:
4596:
4546:
4511:
4469:
3837:
3831:
3800:
3796:
2647:, are the only two binary Sheffer functions. These were discovered, but not published, by
2644:
2632:
2596:
2592:
2118:
1403:
1101:
805:
698:
103:
17:
5957:
5936:
5894:
5874:
5769:
5624:
5222:
5212:
5202:
5197:
5131:
5005:
4881:
4770:
4765:
4743:
4344:
3897:
2424:
2292:
2081:
1258:
1111:
354:
6056:
5931:
5609:
5116:
4901:
4891:
4861:
4846:
4516:
4106:. Annals of Mathematics studies. Vol. 5. Princeton: Princeton University Press.
3816:
3772:
2670:
The following are the minimal functionally complete sets of logical connectives with
2278:
otherwise shows that this condition is strictly weaker than functional completeness.
1126:
3753:
gates is called functionally complete, if it can express every reversible operator.
5831:
5678:
5579:
5571:
5451:
5399:
5308:
5244:
5227:
5158:
5017:
4876:
4578:
4361:
4187:
4126:
See p.105 for the theorem, pp.53, 59, 69, 70, 131 for a definition of the classes A
4069:
4037:
4005:
3761:
3757:
2636:
926:
1741:
is also functionally complete. (Its functional completeness is also proved by the
1324:
Modern texts on logic typically take as primitive some subset of the connectives:
5941:
5821:
5000:
4990:
4937:
4621:
4541:
4526:
4406:
4351:
3792:
2640:
2474:
1266:
1228:
1106:
566:
4204:
4086:
4054:
4022:
4871:
4726:
4697:
4503:
3860:
1305:
1144:
4159:
operations, but since the end of the 20th century it is used more generally.
6023:
5926:
4979:
4896:
4856:
4820:
4756:
4568:
4558:
4531:
3866:
2656:
2300:
1309:
884:
4111:
3706:" operation, when expressed by ↑ gates, is implemented with the reuse of "
6008:
5806:
5254:
4959:
4553:
2660:
1373:
1313:
1246:
668:
5604:
4396:
1514:
Similarly, the negation of the conjunction, NAND (sometimes denoted as
4146:, and pp.35, 43 for the definition of condition and α, β, γ function.
2303:
connectives; changing the truth value of any connected variables from
4294:
2019:
2409:{\displaystyle \neg ,\top ,\bot ,\leftrightarrow ,\nleftrightarrow }
2580:{\displaystyle \vee ,\wedge ,\bot ,\nrightarrow ,\nleftrightarrow }
5148:
4494:
4339:
2671:
2205:
2521:{\displaystyle \vee ,\wedge ,\top ,\rightarrow ,\leftrightarrow }
4188:"Axiomatization of propositional calculus with Sheffer functors"
4298:
4276:
http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nand.html
2219:
Another natural condition would be that the clone generated by
4288:
http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/nor.html
4221:
Wernick, William (1942) "Complete Sets of
Logical Functions,"
3852: – Properties linking logical conjunction and disjunction
2179:
A more natural condition would be that the clone generated by
3452:{\displaystyle \{\land ,\leftrightarrow ,\nleftrightarrow \}}
2651:
around 1880, and rediscovered independently and published by
2319:
never makes these connectives change their return value from
1289:, functionally complete sets of connectives are also called (
3338:{\displaystyle \{\lor ,\leftrightarrow ,\nleftrightarrow \}}
2065:{\displaystyle \{\neg ,\land ,\lor ,\to ,\leftrightarrow \}}
1577:{\displaystyle \{\neg ,\land ,\lor ,\to ,\leftrightarrow \}}
3940:. ("unctional completeness of set of logical operators").
2655:
in 1913. In digital electronics terminology, the binary
4070:"A Correction To My Paper" A. Sole Sufficient Operator"
3875: – Abstract machine that uses only one instruction
4103:
The Two-Valued
Iterative Systems of Mathematical Logic
4251:
4231:
3465:
3427:
3389:
3351:
3313:
3275:
3236:
3204:
3172:
3140:
3108:
3076:
3044:
3012:
2980:
2948:
2916:
2884:
2852:
2820:
2788:
2756:
2724:
2692:
2549:
2490:
2433:
2378:
2333:
2028:
1986:
1966:
1916:
1893:
1873:
1853:
1833:
1774:
1751:
1709:
1597:
1540:
1520:
1473:
1450:
1430:
1406:
1382:
1358:
1334:
1055:
1029:
1003:
969:
936:
894:
860:
834:
808:
782:
751:
724:
701:
678:
635:
609:
576:
542:
516:
490:
449:
423:
397:
364:
330:
304:
278:
244:
218:
192:
155:
129:
106:
80:
54:
4223:
Transactions of the
American Mathematical Society 51
3921:
Nolt, John; Rohatyn, Dennis; Varzi, Achille (1998),
3840: – Algebraic manipulation of "true" and "false"
1464:) can be expressed as conjunction of two negations:
1304:, functional completeness means that every possible
1279:
is incomplete, due to its inability to express NOT.
5950:
5845:
5677:
5570:
5422:
5115:
5038:
4932:
4836:
4725:
4652:
4587:
4502:
4493:
4415:
4332:
3981:
Logic with trees: an introduction to symbolic logic
3834: – Identities and relationships involving sets
3493:{\displaystyle \{\land ,\nleftrightarrow ,\top \}.}
1817:{\displaystyle A\lor B:=\neg (\neg A\land \neg B).}
4257:
4237:
3492:
3451:
3413:
3375:
3337:
3299:
3258:{\displaystyle \{\nleftarrow ,\leftrightarrow \}.}
3257:
3223:{\displaystyle \{\nrightarrow ,\leftrightarrow \}}
3222:
3190:
3158:
3126:
3094:
3062:
3030:
2998:
2966:
2934:
2902:
2870:
2838:
2806:
2774:
2742:
2710:
2579:
2520:
2439:
2408:
2357:
2208:function, i.e. a constant expression, in terms of
2064:
2010:
1972:
1949:
1899:
1879:
1859:
1839:
1816:
1757:
1733:
1692:
1576:
1526:
1503:
1456:
1436:
1412:
1388:
1364:
1340:
1067:
1041:
1015:
981:
955:
914:
872:
846:
820:
794:
760:
737:
710:
687:
656:
621:
595:
554:
528:
502:
470:
435:
409:
383:
342:
316:
290:
256:
230:
204:
170:
141:
115:
92:
66:
3414:{\displaystyle \{\land ,\leftrightarrow ,\bot \}}
3376:{\displaystyle \{\lor ,\nleftrightarrow ,\top \}}
1504:{\displaystyle A\downarrow B:=\neg A\land \neg B}
3923:Schaum's outline of theory and problems of logic
3300:{\displaystyle \{\lor ,\leftrightarrow ,\bot \}}
1227:is one that can be used to express all possible
3869: – Making other gates using just NOR gates
1950:{\displaystyle \ A\vee B:=\neg A\rightarrow B.}
3863: – Logic constructed only from NAND gates
3846: – Characteristic of some logical systems
1239:. A well-known complete set of connectives is
4310:
2011:{\displaystyle \{\land ,\lor ,\rightarrow \}}
1192:
8:
3484:
3466:
3446:
3428:
3408:
3390:
3370:
3352:
3332:
3314:
3294:
3276:
3249:
3237:
3217:
3205:
3185:
3173:
3153:
3141:
3121:
3109:
3089:
3077:
3057:
3045:
3025:
3013:
2993:
2981:
2961:
2949:
2935:{\displaystyle \{\gets ,\nleftrightarrow \}}
2929:
2917:
2897:
2885:
2865:
2853:
2833:
2821:
2801:
2789:
2769:
2757:
2737:
2725:
2705:
2693:
2059:
2029:
2005:
1987:
1728:
1710:
1571:
1541:
3983:. London; New York: Routledge. p. 41.
2619:Minimal functionally complete operator sets
2282:Characterization of functional completeness
1271:is functionally complete. However, the set
5136:
4731:
4499:
4317:
4303:
4295:
4217:
4215:
4165:, Cambridge University Press, p. 54,
3911:. ("Complete set of logical connectives").
2903:{\displaystyle \{\to ,\nleftrightarrow \}}
2535:connectives; they return the truth value
2423:connectives, which are equal to their own
1199:
1185:
29:
4250:
4230:
4203:
4085:
4053:
4021:
3464:
3426:
3388:
3350:
3312:
3274:
3235:
3203:
3171:
3139:
3107:
3075:
3043:
3011:
2979:
2947:
2915:
2883:
2851:
2819:
2787:
2755:
2723:
2691:
2548:
2489:
2432:
2377:
2358:{\displaystyle \vee ,\wedge ,\top ,\bot }
2332:
2027:
1985:
1965:
1915:
1892:
1872:
1852:
1832:
1773:
1750:
1708:
1598:
1596:
1539:
1519:
1472:
1449:
1429:
1405:
1381:
1357:
1333:
1054:
1028:
1002:
968:
940:
935:
901:
893:
859:
833:
807:
781:
750:
725:
723:
700:
677:
636:
634:
608:
580:
575:
541:
515:
489:
450:
448:
422:
396:
368:
363:
329:
303:
277:
243:
217:
191:
154:
128:
105:
79:
53:
4038:"Concerning an alleged Sheffer function"
3031:{\displaystyle \{\gets ,\nrightarrow \}}
2591:Post gave a complete description of the
3884:
3159:{\displaystyle \{\nrightarrow ,\top \}}
3095:{\displaystyle \{\nrightarrow ,\neg \}}
3063:{\displaystyle \{\gets ,\nleftarrow \}}
1135:
1082:
529:{\displaystyle A\not \Leftrightarrow B}
32:
4155:The term was originally restricted to
3783:than that of functional completeness.
3191:{\displaystyle \{\nleftarrow ,\top \}}
3127:{\displaystyle \{\nleftarrow ,\neg \}}
2967:{\displaystyle \{\to ,\nrightarrow \}}
2539:under any interpretation that assigns
2480:under any interpretation that assigns
1734:{\displaystyle \{\neg ,\land ,\lor \}}
956:{\displaystyle A{\underline {\lor }}B}
471:{\displaystyle {\overline {A\cdot B}}}
384:{\displaystyle A{\overline {\land }}B}
3608:(↓) completeness. As illustrated by,
3517:(↑) completeness. As illustrated by,
3504:can be replaced by a unary negation.
2999:{\displaystyle \{\to ,\nleftarrow \}}
1745:.) But this is still not minimal, as
596:{\displaystyle A{\overline {\lor }}B}
257:{\displaystyle A\leftrightharpoons B}
7:
3894:A mathematical introduction to logic
3781:slightly more restrictive definition
2018:is a minimal functionally complete
915:{\displaystyle A\ {\text{XNOR}}\ B}
555:{\displaystyle A\nleftrightarrow B}
4074:Notre Dame Journal of Formal Logic
4042:Notre Dame Journal of Formal Logic
4010:Notre Dame Journal of Formal Logic
3481:
3405:
3367:
3291:
3182:
3150:
3118:
3086:
2862:
2830:
2798:
2766:
2734:
2702:
2562:
2503:
2434:
2391:
2385:
2379:
2352:
2346:
2032:
1967:
1932:
1802:
1793:
1787:
1713:
1618:
1544:
1495:
1486:
1383:
679:
231:{\displaystyle A\Leftrightarrow B}
162:
159:
133:
25:
2743:{\displaystyle \{\wedge ,\neg \}}
2125:generated by the basic functions
657:{\displaystyle {\overline {A+B}}}
6036:
2871:{\displaystyle \{\gets ,\bot \}}
2807:{\displaystyle \{\gets ,\neg \}}
1703:It follows that the smaller set
1437:{\displaystyle \leftrightarrow }
1170:
1169:
3952:An introduction to formal logic
3850:Conjunction/disjunction duality
2711:{\displaystyle \{\vee ,\neg \}}
1743:Disjunctive Normal Form Theorem
738:{\displaystyle {\overline {A}}}
3856:List of Boolean algebra topics
3803:, that is, they have the same
3437:
3399:
3323:
3285:
3246:
3214:
3048:
3016:
2984:
2952:
2920:
2888:
2856:
2839:{\displaystyle \{\to ,\bot \}}
2824:
2792:
2775:{\displaystyle \{\to ,\neg \}}
2760:
2635:operators with this property.
2515:
2509:
2397:
2056:
2050:
2002:
1938:
1894:
1808:
1790:
1680:
1674:
1668:
1662:
1656:
1650:
1637:
1605:
1568:
1562:
1521:
1477:
1451:
1431:
1407:
1059:
1007:
613:
503:{\displaystyle A\not \equiv B}
401:
343:{\displaystyle A\rightarrow B}
334:
291:{\displaystyle A\Rightarrow B}
282:
248:
222:
171:{\displaystyle A\&\&B}
1:
5997:History of mathematical logic
4258:{\displaystyle \nrightarrow }
3779:are universal, albeit with a
2473:connectives; they return the
1068:{\displaystyle A\leftarrow B}
1016:{\displaystyle A\Leftarrow B}
622:{\displaystyle A\downarrow B}
27:Concept in mathematical logic
5922:Primitive recursive function
4238:{\displaystyle \nleftarrow }
4006:"A sole sufficient operator"
3896:(2nd ed.), Boston, MA:
3873:One-instruction set computer
1900:{\displaystyle \rightarrow }
1231:by combining members of the
873:{\displaystyle A\parallel B}
730:
649:
585:
463:
373:
4068:Wesselkamper, T.C. (1975),
4004:Wesselkamper, T.C. (1975),
1887:may be defined in terms of
1847:may be defined in terms of
1457:{\displaystyle \downarrow }
410:{\displaystyle A\uparrow B}
6094:
4986:Schröder–Bernstein theorem
4713:Monadic predicate calculus
4372:Foundations of mathematics
4274:"NAND Gate Operations" at
4192:Notre Dame J. Formal Logic
3956:Cambridge University Press
3925:(2nd ed.), New York:
3892:Enderton, Herbert (2001),
2311:without changing any from
2285:
1300:From the point of view of
1042:{\displaystyle A\subset B}
317:{\displaystyle A\supset B}
6068:Logic in computer science
6032:
6019:Philosophy of mathematics
5968:Automated theorem proving
5139:
5093:Von Neumann–Bernays–Gödel
4734:
4286:"NOR Gate Operations" at
2183:consist of all functions
1527:{\displaystyle \uparrow }
982:{\displaystyle A\oplus B}
205:{\displaystyle A\equiv B}
4205:10.1305/ndjfl/1093958259
4087:10.1305/ndjfl/1093891899
4055:10.1305/ndjfl/1093891898
4023:10.1305/ndjfl/1093891614
2663:(↓) are the only binary
2629:sole sufficient operator
1867:in a similar manner, or
142:{\displaystyle A\&B}
93:{\displaystyle A\cdot B}
67:{\displaystyle A\land B}
18:Sole sufficient operator
5669:Self-verifying theories
5490:Tarski's axiomatization
4441:Tarski's undefinability
4436:incompleteness theorems
4100:Emil Leon Post (1941).
2543:to all variables, e.g.
2484:to all variables, e.g.
2134:contains all functions
1122:Functional completeness
847:{\displaystyle A\mid B}
795:{\displaystyle A\lor B}
436:{\displaystyle A\mid B}
6078:Charles Sanders Peirce
6073:Propositional calculus
6043:Mathematics portal
5654:Proof of impossibility
5302:propositional variable
4612:Propositional calculus
4259:
4239:
4186:Scharle, T.W. (1965),
3979:Howson, Colin (1997).
3604:Examples of using the
3513:Examples of using the
3494:
3453:
3415:
3377:
3339:
3301:
3259:
3224:
3192:
3160:
3128:
3096:
3064:
3032:
3000:
2968:
2936:
2904:
2872:
2840:
2808:
2776:
2744:
2712:
2649:Charles Sanders Peirce
2581:
2522:
2441:
2410:
2359:
2066:
2012:
1974:
1951:
1901:
1881:
1861:
1841:
1840:{\displaystyle \land }
1818:
1759:
1735:
1694:
1578:
1528:
1505:
1458:
1438:
1414:
1390:
1366:
1342:
1341:{\displaystyle \land }
1092:Propositional calculus
1069:
1043:
1017:
983:
957:
916:
874:
848:
822:
796:
762:
761:{\displaystyle \sim A}
739:
712:
689:
688:{\displaystyle \neg A}
658:
623:
597:
556:
530:
504:
472:
437:
411:
385:
344:
318:
292:
258:
232:
206:
172:
143:
117:
94:
68:
5912:Kolmogorov complexity
5865:Computably enumerable
5765:Model complete theory
5557:Principia Mathematica
4617:Propositional formula
4446:Banach–Tarski paradox
4260:
4240:
4161:Martin, N.M. (1989),
4112:10.1515/9781400882366
4036:Massey, G.J. (1975),
3950:Smith, Peter (2003),
3495:
3454:
3416:
3378:
3340:
3302:
3260:
3225:
3193:
3161:
3129:
3097:
3065:
3033:
3001:
2969:
2937:
2905:
2873:
2841:
2809:
2777:
2745:
2713:
2665:universal logic gates
2582:
2523:
2442:
2440:{\displaystyle \neg }
2411:
2360:
2286:Further information:
2115:functionally complete
2094:of Boolean functions
2067:
2013:
1975:
1973:{\displaystyle \neg }
1952:
1902:
1882:
1880:{\displaystyle \lor }
1862:
1860:{\displaystyle \lor }
1842:
1819:
1760:
1758:{\displaystyle \lor }
1736:
1695:
1579:
1529:
1506:
1459:
1439:
1415:
1391:
1389:{\displaystyle \neg }
1367:
1365:{\displaystyle \lor }
1343:
1217:functionally complete
1150:Programming languages
1070:
1044:
1018:
984:
958:
917:
875:
849:
823:
797:
763:
740:
713:
690:
659:
624:
598:
557:
531:
505:
473:
438:
412:
386:
345:
319:
293:
259:
233:
207:
173:
144:
118:
95:
69:
5860:Church–Turing thesis
5847:Computability theory
5056:continuum hypothesis
4574:Square of opposition
4432:Gödel's completeness
4249:
4229:
3844:Completeness (logic)
3463:
3425:
3387:
3349:
3311:
3273:
3234:
3202:
3170:
3138:
3106:
3074:
3042:
3010:
2978:
2946:
2914:
2882:
2850:
2818:
2786:
2754:
2722:
2690:
2547:
2488:
2431:
2376:
2331:
2026:
1984:
1964:
1914:
1891:
1871:
1851:
1831:
1772:
1749:
1707:
1595:
1538:
1518:
1471:
1448:
1428:
1420:); and possibly the
1413:{\displaystyle \to }
1404:
1398:material conditional
1380:
1356:
1332:
1053:
1027:
1001:
967:
934:
892:
858:
832:
806:
780:
749:
722:
699:
676:
633:
607:
574:
540:
514:
488:
447:
421:
395:
362:
328:
302:
276:
242:
216:
190:
153:
127:
104:
78:
52:
6014:Mathematical object
5905:P versus NP problem
5870:Computable function
5664:Reverse mathematics
5590:Logical consequence
5467:primitive recursive
5462:elementary function
5235:Free/bound variable
5088:Tarski–Grothendieck
4607:Logical connectives
4537:Logical equivalence
4387:Logical consequence
2659:(↑) and the binary
2197:, for all integers
1302:digital electronics
1287:propositional logic
1221:logical connectives
1160:Philosophy of logic
821:{\displaystyle A+B}
34:Logical connectives
5812:Transfer principle
5775:Semantics of logic
5760:Categorical theory
5736:Non-standard model
5250:Logical connective
4377:Information theory
4326:Mathematical logic
4255:
4235:
3490:
3449:
3411:
3373:
3335:
3297:
3255:
3220:
3188:
3156:
3124:
3092:
3060:
3028:
2996:
2964:
2932:
2900:
2868:
2836:
2804:
2772:
2740:
2708:
2645:dual to each other
2611:, nowadays called
2577:
2533:falsity-preserving
2518:
2437:
2406:
2355:
2062:
2008:
1970:
1947:
1897:
1877:
1857:
1837:
1814:
1765:can be defined as
1755:
1731:
1690:
1688:
1574:
1524:
1501:
1454:
1434:
1410:
1386:
1362:
1338:
1237:Boolean expression
1155:Mathematical logic
1065:
1039:
1013:
979:
953:
948:
912:
870:
844:
818:
792:
758:
735:
711:{\displaystyle -A}
708:
685:
654:
619:
593:
552:
526:
500:
468:
433:
407:
381:
340:
314:
288:
254:
228:
202:
168:
139:
116:{\displaystyle AB}
113:
90:
64:
6050:
6049:
5982:Abstract category
5785:Theories of truth
5595:Rule of inference
5585:Natural deduction
5566:
5565:
5111:
5110:
4816:Cartesian product
4721:
4720:
4627:Many-valued logic
4602:Boolean functions
4485:Russell's paradox
4460:diagonal argument
4357:First-order logic
4172:978-0-521-36770-7
3990:978-0-415-13342-5
3965:978-0-521-00804-4
3936:978-0-07-046649-4
3907:978-0-12-238452-3
3769:quantum computing
2150:strictly positive
2076:Formal definition
1919:
1312:, or only binary
1225:Boolean operators
1209:
1208:
1078:
1077:
941:
908:
904:
900:
733:
652:
588:
466:
376:
16:(Redirected from
6085:
6041:
6040:
5992:History of logic
5987:Category of sets
5880:Decision problem
5659:Ordinal analysis
5600:Sequent calculus
5498:Boolean algebras
5438:
5437:
5412:
5383:logical/constant
5137:
5123:
5046:Zermelo–Fraenkel
4797:Set operations:
4732:
4669:
4500:
4480:Löwenheim–Skolem
4367:Formal semantics
4319:
4312:
4305:
4296:
4290:
4284:
4278:
4272:
4266:
4264:
4262:
4261:
4256:
4244:
4242:
4241:
4236:
4219:
4210:
4208:
4207:
4183:
4177:
4175:
4163:Systems of logic
4153:
4147:
4125:
4097:
4091:
4090:
4089:
4065:
4059:
4058:
4057:
4033:
4027:
4026:
4025:
4001:
3995:
3994:
3976:
3970:
3968:
3947:
3941:
3939:
3918:
3912:
3910:
3889:
3814:
3810:
3745:In other domains
3709:
3705:
3607:
3516:
3499:
3497:
3496:
3491:
3458:
3456:
3455:
3450:
3420:
3418:
3417:
3412:
3382:
3380:
3379:
3374:
3344:
3342:
3341:
3336:
3306:
3304:
3303:
3298:
3264:
3262:
3261:
3256:
3229:
3227:
3226:
3221:
3197:
3195:
3194:
3189:
3165:
3163:
3162:
3157:
3133:
3131:
3130:
3125:
3101:
3099:
3098:
3093:
3069:
3067:
3066:
3061:
3037:
3035:
3034:
3029:
3005:
3003:
3002:
2997:
2973:
2971:
2970:
2965:
2941:
2939:
2938:
2933:
2909:
2907:
2906:
2901:
2877:
2875:
2874:
2869:
2845:
2843:
2842:
2837:
2813:
2811:
2810:
2805:
2781:
2779:
2778:
2773:
2749:
2747:
2746:
2741:
2717:
2715:
2714:
2709:
2653:Henry M. Sheffer
2625:Sheffer function
2610:
2586:
2584:
2583:
2578:
2527:
2525:
2524:
2519:
2471:truth-preserving
2465:
2446:
2444:
2443:
2438:
2415:
2413:
2412:
2407:
2364:
2362:
2361:
2356:
2277:
2255:
2245:
2203:
2196:
2158:
2147:
2112:
2089:
2071:
2069:
2068:
2063:
2017:
2015:
2014:
2009:
1979:
1977:
1976:
1971:
1956:
1954:
1953:
1948:
1917:
1906:
1904:
1903:
1898:
1886:
1884:
1883:
1878:
1866:
1864:
1863:
1858:
1846:
1844:
1843:
1838:
1823:
1821:
1820:
1815:
1764:
1762:
1761:
1756:
1740:
1738:
1737:
1732:
1699:
1697:
1696:
1691:
1689:
1583:
1581:
1580:
1575:
1533:
1531:
1530:
1525:
1510:
1508:
1507:
1502:
1463:
1461:
1460:
1455:
1443:
1441:
1440:
1435:
1419:
1417:
1416:
1411:
1395:
1393:
1392:
1387:
1371:
1369:
1368:
1363:
1347:
1345:
1344:
1339:
1285:In a context of
1278:
1270:
1262:
1250:
1201:
1194:
1187:
1173:
1172:
1117:Boolean function
1083:Related concepts
1074:
1072:
1071:
1066:
1048:
1046:
1045:
1040:
1022:
1020:
1019:
1014:
988:
986:
985:
980:
962:
960:
959:
954:
949:
921:
919:
918:
913:
906:
905:
902:
898:
879:
877:
876:
871:
853:
851:
850:
845:
827:
825:
824:
819:
801:
799:
798:
793:
767:
765:
764:
759:
744:
742:
741:
736:
734:
726:
717:
715:
714:
709:
694:
692:
691:
686:
663:
661:
660:
655:
653:
648:
637:
628:
626:
625:
620:
602:
600:
599:
594:
589:
581:
561:
559:
558:
553:
535:
533:
532:
527:
509:
507:
506:
501:
477:
475:
474:
469:
467:
462:
451:
442:
440:
439:
434:
416:
414:
413:
408:
390:
388:
387:
382:
377:
369:
349:
347:
346:
341:
323:
321:
320:
315:
297:
295:
294:
289:
263:
261:
260:
255:
237:
235:
234:
229:
211:
209:
208:
203:
177:
175:
174:
169:
148:
146:
145:
140:
122:
120:
119:
114:
99:
97:
96:
91:
73:
71:
70:
65:
41:
40:
30:
21:
6093:
6092:
6088:
6087:
6086:
6084:
6083:
6082:
6063:Boolean algebra
6053:
6052:
6051:
6046:
6035:
6028:
5973:Category theory
5963:Algebraic logic
5946:
5917:Lambda calculus
5855:Church encoding
5841:
5817:Truth predicate
5673:
5639:Complete theory
5562:
5431:
5427:
5423:
5418:
5410:
5130: and
5126:
5121:
5107:
5083:New Foundations
5051:axiom of choice
5034:
4996:Gödel numbering
4936: and
4928:
4832:
4717:
4667:
4648:
4597:Boolean algebra
4583:
4547:Equiconsistency
4512:Classical logic
4489:
4470:Halting problem
4458: and
4434: and
4422: and
4421:
4416:Theorems (
4411:
4328:
4323:
4293:
4285:
4281:
4273:
4269:
4247:
4246:
4227:
4226:
4220:
4213:
4185:
4184:
4180:
4173:
4160:
4154:
4150:
4145:
4141:
4137:
4133:
4129:
4122:
4099:
4098:
4094:
4067:
4066:
4062:
4035:
4034:
4030:
4003:
4002:
3998:
3991:
3978:
3977:
3973:
3966:
3949:
3948:
3944:
3937:
3920:
3919:
3915:
3908:
3891:
3890:
3886:
3882:
3838:Boolean algebra
3832:Algebra of sets
3828:
3812:
3808:
3801:Boolean algebra
3797:algebra of sets
3789:
3747:
3707:
3697:
3605:
3514:
3510:
3461:
3460:
3423:
3422:
3385:
3384:
3347:
3346:
3309:
3308:
3271:
3270:
3232:
3231:
3200:
3199:
3168:
3167:
3136:
3135:
3104:
3103:
3072:
3071:
3040:
3039:
3008:
3007:
2976:
2975:
2944:
2943:
2912:
2911:
2880:
2879:
2848:
2847:
2816:
2815:
2784:
2783:
2752:
2751:
2720:
2719:
2688:
2687:
2631:. There are no
2627:or sometimes a
2621:
2600:
2545:
2544:
2486:
2485:
2448:
2429:
2428:
2374:
2373:
2329:
2328:
2290:
2284:
2257:
2247:
2225:
2198:
2184:
2167:
2153:
2135:
2133:
2103:
2095:
2084:
2078:
2024:
2023:
1982:
1981:
1962:
1961:
1912:
1911:
1889:
1888:
1869:
1868:
1849:
1848:
1829:
1828:
1827:Alternatively,
1770:
1769:
1747:
1746:
1705:
1704:
1687:
1686:
1643:
1631:
1630:
1611:
1593:
1592:
1536:
1535:
1516:
1515:
1469:
1468:
1446:
1445:
1426:
1425:
1402:
1401:
1378:
1377:
1354:
1353:
1330:
1329:
1322:
1272:
1264:
1256:
1240:
1205:
1164:
1131:
1102:Boolean algebra
1097:Predicate logic
1051:
1050:
1025:
1024:
999:
998:
965:
964:
932:
931:
890:
889:
856:
855:
830:
829:
804:
803:
778:
777:
747:
746:
720:
719:
697:
696:
674:
673:
638:
631:
630:
605:
604:
572:
571:
538:
537:
512:
511:
486:
485:
452:
445:
444:
419:
418:
393:
392:
360:
359:
326:
325:
300:
299:
274:
273:
240:
239:
214:
213:
188:
187:
151:
150:
125:
124:
102:
101:
76:
75:
50:
49:
28:
23:
22:
15:
12:
11:
5:
6091:
6089:
6081:
6080:
6075:
6070:
6065:
6055:
6054:
6048:
6047:
6033:
6030:
6029:
6027:
6026:
6021:
6016:
6011:
6006:
6005:
6004:
5994:
5989:
5984:
5975:
5970:
5965:
5960:
5958:Abstract logic
5954:
5952:
5948:
5947:
5945:
5944:
5939:
5937:Turing machine
5934:
5929:
5924:
5919:
5914:
5909:
5908:
5907:
5902:
5897:
5892:
5887:
5877:
5875:Computable set
5872:
5867:
5862:
5857:
5851:
5849:
5843:
5842:
5840:
5839:
5834:
5829:
5824:
5819:
5814:
5809:
5804:
5803:
5802:
5797:
5792:
5782:
5777:
5772:
5770:Satisfiability
5767:
5762:
5757:
5756:
5755:
5745:
5744:
5743:
5733:
5732:
5731:
5726:
5721:
5716:
5711:
5701:
5700:
5699:
5694:
5687:Interpretation
5683:
5681:
5675:
5674:
5672:
5671:
5666:
5661:
5656:
5651:
5641:
5636:
5635:
5634:
5633:
5632:
5622:
5617:
5607:
5602:
5597:
5592:
5587:
5582:
5576:
5574:
5568:
5567:
5564:
5563:
5561:
5560:
5552:
5551:
5550:
5549:
5544:
5543:
5542:
5537:
5532:
5512:
5511:
5510:
5508:minimal axioms
5505:
5494:
5493:
5492:
5481:
5480:
5479:
5474:
5469:
5464:
5459:
5454:
5441:
5439:
5420:
5419:
5417:
5416:
5415:
5414:
5402:
5397:
5396:
5395:
5390:
5385:
5380:
5370:
5365:
5360:
5355:
5354:
5353:
5348:
5338:
5337:
5336:
5331:
5326:
5321:
5311:
5306:
5305:
5304:
5299:
5294:
5284:
5283:
5282:
5277:
5272:
5267:
5262:
5257:
5247:
5242:
5237:
5232:
5231:
5230:
5225:
5220:
5215:
5205:
5200:
5198:Formation rule
5195:
5190:
5189:
5188:
5183:
5173:
5172:
5171:
5161:
5156:
5151:
5146:
5140:
5134:
5117:Formal systems
5113:
5112:
5109:
5108:
5106:
5105:
5100:
5095:
5090:
5085:
5080:
5075:
5070:
5065:
5060:
5059:
5058:
5053:
5042:
5040:
5036:
5035:
5033:
5032:
5031:
5030:
5020:
5015:
5014:
5013:
5006:Large cardinal
5003:
4998:
4993:
4988:
4983:
4969:
4968:
4967:
4962:
4957:
4942:
4940:
4930:
4929:
4927:
4926:
4925:
4924:
4919:
4914:
4904:
4899:
4894:
4889:
4884:
4879:
4874:
4869:
4864:
4859:
4854:
4849:
4843:
4841:
4834:
4833:
4831:
4830:
4829:
4828:
4823:
4818:
4813:
4808:
4803:
4795:
4794:
4793:
4788:
4778:
4773:
4771:Extensionality
4768:
4766:Ordinal number
4763:
4753:
4748:
4747:
4746:
4735:
4729:
4723:
4722:
4719:
4718:
4716:
4715:
4710:
4705:
4700:
4695:
4690:
4685:
4684:
4683:
4673:
4672:
4671:
4658:
4656:
4650:
4649:
4647:
4646:
4645:
4644:
4639:
4634:
4624:
4619:
4614:
4609:
4604:
4599:
4593:
4591:
4585:
4584:
4582:
4581:
4576:
4571:
4566:
4561:
4556:
4551:
4550:
4549:
4539:
4534:
4529:
4524:
4519:
4514:
4508:
4506:
4497:
4491:
4490:
4488:
4487:
4482:
4477:
4472:
4467:
4462:
4450:Cantor's
4448:
4443:
4438:
4428:
4426:
4413:
4412:
4410:
4409:
4404:
4399:
4394:
4389:
4384:
4379:
4374:
4369:
4364:
4359:
4354:
4349:
4348:
4347:
4336:
4334:
4330:
4329:
4324:
4322:
4321:
4314:
4307:
4299:
4292:
4291:
4279:
4267:
4254:
4234:
4211:
4198:(3): 209–217,
4178:
4171:
4148:
4143:
4139:
4135:
4131:
4127:
4120:
4092:
4060:
4048:(4): 549–550,
4028:
3996:
3989:
3971:
3964:
3942:
3935:
3913:
3906:
3898:Academic Press
3883:
3881:
3878:
3877:
3876:
3870:
3864:
3858:
3853:
3847:
3841:
3835:
3827:
3824:
3788:
3785:
3746:
3743:
3742:
3741:
3694:
3693:
3692:
3691:
3657:
3623:
3602:
3601:
3600:
3566:
3532:
3509:
3506:
3501:
3500:
3489:
3486:
3483:
3480:
3477:
3474:
3471:
3468:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3268:
3267:Three elements
3265:
3254:
3251:
3248:
3245:
3242:
3239:
3219:
3216:
3213:
3210:
3207:
3187:
3184:
3181:
3178:
3175:
3155:
3152:
3149:
3146:
3143:
3123:
3120:
3117:
3114:
3111:
3091:
3088:
3085:
3082:
3079:
3059:
3056:
3053:
3050:
3047:
3027:
3024:
3021:
3018:
3015:
2995:
2992:
2989:
2986:
2983:
2963:
2960:
2957:
2954:
2951:
2931:
2928:
2925:
2922:
2919:
2899:
2896:
2893:
2890:
2887:
2867:
2864:
2861:
2858:
2855:
2835:
2832:
2829:
2826:
2823:
2803:
2800:
2797:
2794:
2791:
2771:
2768:
2765:
2762:
2759:
2739:
2736:
2733:
2730:
2727:
2707:
2704:
2701:
2698:
2695:
2685:
2682:
2679:
2620:
2617:
2613:Post's lattice
2589:
2588:
2576:
2573:
2570:
2567:
2564:
2561:
2558:
2555:
2552:
2529:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2467:
2436:
2425:de Morgan dual
2417:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2366:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2288:Post's lattice
2283:
2280:
2163:
2129:
2099:
2082:Boolean domain
2077:
2074:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1969:
1958:
1957:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1896:
1876:
1856:
1836:
1825:
1824:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1754:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1701:
1700:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1644:
1642:
1639:
1636:
1633:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1612:
1610:
1607:
1604:
1601:
1600:
1573:
1570:
1567:
1564:
1561:
1558:
1555:
1552:
1549:
1546:
1543:
1523:
1512:
1511:
1500:
1497:
1494:
1491:
1488:
1485:
1482:
1479:
1476:
1453:
1433:
1409:
1385:
1361:
1337:
1321:
1318:
1251:. Each of the
1207:
1206:
1204:
1203:
1196:
1189:
1181:
1178:
1177:
1166:
1165:
1163:
1162:
1157:
1152:
1147:
1141:
1138:
1137:
1133:
1132:
1130:
1129:
1124:
1119:
1114:
1112:Truth function
1109:
1104:
1099:
1094:
1088:
1085:
1084:
1080:
1079:
1076:
1075:
1064:
1061:
1058:
1038:
1035:
1032:
1012:
1009:
1006:
996:
990:
989:
978:
975:
972:
952:
947:
944:
939:
929:
923:
922:
911:
897:
887:
881:
880:
869:
866:
863:
843:
840:
837:
817:
814:
811:
791:
788:
785:
775:
769:
768:
757:
754:
732:
729:
707:
704:
684:
681:
671:
665:
664:
651:
647:
644:
641:
618:
615:
612:
592:
587:
584:
579:
569:
563:
562:
551:
548:
545:
525:
522:
519:
499:
496:
493:
483:
479:
478:
465:
461:
458:
455:
432:
429:
426:
406:
403:
400:
380:
375:
372:
367:
357:
351:
350:
339:
336:
333:
313:
310:
307:
287:
284:
281:
271:
265:
264:
253:
250:
247:
227:
224:
221:
201:
198:
195:
185:
179:
178:
167:
164:
161:
158:
138:
135:
132:
112:
109:
89:
86:
83:
63:
60:
57:
47:
37:
36:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6090:
6079:
6076:
6074:
6071:
6069:
6066:
6064:
6061:
6060:
6058:
6045:
6044:
6039:
6031:
6025:
6022:
6020:
6017:
6015:
6012:
6010:
6007:
6003:
6000:
5999:
5998:
5995:
5993:
5990:
5988:
5985:
5983:
5979:
5976:
5974:
5971:
5969:
5966:
5964:
5961:
5959:
5956:
5955:
5953:
5949:
5943:
5940:
5938:
5935:
5933:
5932:Recursive set
5930:
5928:
5925:
5923:
5920:
5918:
5915:
5913:
5910:
5906:
5903:
5901:
5898:
5896:
5893:
5891:
5888:
5886:
5883:
5882:
5881:
5878:
5876:
5873:
5871:
5868:
5866:
5863:
5861:
5858:
5856:
5853:
5852:
5850:
5848:
5844:
5838:
5835:
5833:
5830:
5828:
5825:
5823:
5820:
5818:
5815:
5813:
5810:
5808:
5805:
5801:
5798:
5796:
5793:
5791:
5788:
5787:
5786:
5783:
5781:
5778:
5776:
5773:
5771:
5768:
5766:
5763:
5761:
5758:
5754:
5751:
5750:
5749:
5746:
5742:
5741:of arithmetic
5739:
5738:
5737:
5734:
5730:
5727:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5706:
5705:
5702:
5698:
5695:
5693:
5690:
5689:
5688:
5685:
5684:
5682:
5680:
5676:
5670:
5667:
5665:
5662:
5660:
5657:
5655:
5652:
5649:
5648:from ZFC
5645:
5642:
5640:
5637:
5631:
5628:
5627:
5626:
5623:
5621:
5618:
5616:
5613:
5612:
5611:
5608:
5606:
5603:
5601:
5598:
5596:
5593:
5591:
5588:
5586:
5583:
5581:
5578:
5577:
5575:
5573:
5569:
5559:
5558:
5554:
5553:
5548:
5547:non-Euclidean
5545:
5541:
5538:
5536:
5533:
5531:
5530:
5526:
5525:
5523:
5520:
5519:
5517:
5513:
5509:
5506:
5504:
5501:
5500:
5499:
5495:
5491:
5488:
5487:
5486:
5482:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5458:
5455:
5453:
5450:
5449:
5447:
5443:
5442:
5440:
5435:
5429:
5424:Example
5421:
5413:
5408:
5407:
5406:
5403:
5401:
5398:
5394:
5391:
5389:
5386:
5384:
5381:
5379:
5376:
5375:
5374:
5371:
5369:
5366:
5364:
5361:
5359:
5356:
5352:
5349:
5347:
5344:
5343:
5342:
5339:
5335:
5332:
5330:
5327:
5325:
5322:
5320:
5317:
5316:
5315:
5312:
5310:
5307:
5303:
5300:
5298:
5295:
5293:
5290:
5289:
5288:
5285:
5281:
5278:
5276:
5273:
5271:
5268:
5266:
5263:
5261:
5258:
5256:
5253:
5252:
5251:
5248:
5246:
5243:
5241:
5238:
5236:
5233:
5229:
5226:
5224:
5221:
5219:
5216:
5214:
5211:
5210:
5209:
5206:
5204:
5201:
5199:
5196:
5194:
5191:
5187:
5184:
5182:
5181:by definition
5179:
5178:
5177:
5174:
5170:
5167:
5166:
5165:
5162:
5160:
5157:
5155:
5152:
5150:
5147:
5145:
5142:
5141:
5138:
5135:
5133:
5129:
5124:
5118:
5114:
5104:
5101:
5099:
5096:
5094:
5091:
5089:
5086:
5084:
5081:
5079:
5076:
5074:
5071:
5069:
5068:Kripke–Platek
5066:
5064:
5061:
5057:
5054:
5052:
5049:
5048:
5047:
5044:
5043:
5041:
5037:
5029:
5026:
5025:
5024:
5021:
5019:
5016:
5012:
5009:
5008:
5007:
5004:
5002:
4999:
4997:
4994:
4992:
4989:
4987:
4984:
4981:
4977:
4973:
4970:
4966:
4963:
4961:
4958:
4956:
4953:
4952:
4951:
4947:
4944:
4943:
4941:
4939:
4935:
4931:
4923:
4920:
4918:
4915:
4913:
4912:constructible
4910:
4909:
4908:
4905:
4903:
4900:
4898:
4895:
4893:
4890:
4888:
4885:
4883:
4880:
4878:
4875:
4873:
4870:
4868:
4865:
4863:
4860:
4858:
4855:
4853:
4850:
4848:
4845:
4844:
4842:
4840:
4835:
4827:
4824:
4822:
4819:
4817:
4814:
4812:
4809:
4807:
4804:
4802:
4799:
4798:
4796:
4792:
4789:
4787:
4784:
4783:
4782:
4779:
4777:
4774:
4772:
4769:
4767:
4764:
4762:
4758:
4754:
4752:
4749:
4745:
4742:
4741:
4740:
4737:
4736:
4733:
4730:
4728:
4724:
4714:
4711:
4709:
4706:
4704:
4701:
4699:
4696:
4694:
4691:
4689:
4686:
4682:
4679:
4678:
4677:
4674:
4670:
4665:
4664:
4663:
4660:
4659:
4657:
4655:
4651:
4643:
4640:
4638:
4635:
4633:
4630:
4629:
4628:
4625:
4623:
4620:
4618:
4615:
4613:
4610:
4608:
4605:
4603:
4600:
4598:
4595:
4594:
4592:
4590:
4589:Propositional
4586:
4580:
4577:
4575:
4572:
4570:
4567:
4565:
4562:
4560:
4557:
4555:
4552:
4548:
4545:
4544:
4543:
4540:
4538:
4535:
4533:
4530:
4528:
4525:
4523:
4520:
4518:
4517:Logical truth
4515:
4513:
4510:
4509:
4507:
4505:
4501:
4498:
4496:
4492:
4486:
4483:
4481:
4478:
4476:
4473:
4471:
4468:
4466:
4463:
4461:
4457:
4453:
4449:
4447:
4444:
4442:
4439:
4437:
4433:
4430:
4429:
4427:
4425:
4419:
4414:
4408:
4405:
4403:
4400:
4398:
4395:
4393:
4390:
4388:
4385:
4383:
4380:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4358:
4355:
4353:
4350:
4346:
4343:
4342:
4341:
4338:
4337:
4335:
4331:
4327:
4320:
4315:
4313:
4308:
4306:
4301:
4300:
4297:
4289:
4283:
4280:
4277:
4271:
4268:
4252:
4232:
4224:
4218:
4216:
4212:
4206:
4201:
4197:
4193:
4189:
4182:
4179:
4174:
4168:
4164:
4158:
4152:
4149:
4123:
4121:9781400882366
4117:
4113:
4109:
4105:
4104:
4096:
4093:
4088:
4083:
4079:
4075:
4071:
4064:
4061:
4056:
4051:
4047:
4043:
4039:
4032:
4029:
4024:
4019:
4015:
4011:
4007:
4000:
3997:
3992:
3986:
3982:
3975:
3972:
3967:
3961:
3957:
3953:
3946:
3943:
3938:
3932:
3928:
3924:
3917:
3914:
3909:
3903:
3899:
3895:
3888:
3885:
3879:
3874:
3871:
3868:
3865:
3862:
3859:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3829:
3825:
3823:
3821:
3818:
3817:universal set
3806:
3802:
3798:
3794:
3786:
3784:
3782:
3778:
3774:
3773:Hadamard gate
3770:
3765:
3763:
3759:
3754:
3752:
3744:
3740:
3736:
3732:
3728:
3724:
3720:
3716:
3713:
3712:
3711:
3704:
3700:
3689:
3685:
3681:
3677:
3673:
3669:
3665:
3661:
3658:
3655:
3651:
3647:
3643:
3639:
3635:
3631:
3627:
3624:
3622:
3618:
3614:
3610:
3609:
3603:
3598:
3594:
3590:
3586:
3582:
3578:
3574:
3570:
3567:
3564:
3560:
3556:
3552:
3548:
3544:
3540:
3536:
3533:
3531:
3527:
3523:
3519:
3518:
3512:
3511:
3507:
3505:
3487:
3478:
3475:
3472:
3469:
3443:
3440:
3434:
3431:
3402:
3396:
3393:
3364:
3361:
3358:
3355:
3329:
3326:
3320:
3317:
3288:
3282:
3279:
3269:
3266:
3252:
3243:
3240:
3211:
3208:
3179:
3176:
3147:
3144:
3115:
3112:
3083:
3080:
3054:
3051:
3022:
3019:
2990:
2987:
2958:
2955:
2926:
2923:
2894:
2891:
2859:
2827:
2795:
2763:
2731:
2728:
2699:
2696:
2686:
2683:
2680:
2677:
2676:
2675:
2673:
2668:
2666:
2662:
2658:
2654:
2650:
2646:
2642:
2638:
2634:
2630:
2626:
2618:
2616:
2614:
2608:
2604:
2598:
2594:
2574:
2571:
2568:
2565:
2559:
2556:
2553:
2550:
2542:
2538:
2534:
2530:
2512:
2506:
2500:
2497:
2494:
2491:
2483:
2479:
2476:
2472:
2468:
2463:
2459:
2455:
2451:
2426:
2422:
2418:
2403:
2400:
2394:
2388:
2382:
2371:
2367:
2349:
2343:
2340:
2337:
2334:
2326:
2322:
2318:
2314:
2310:
2306:
2302:
2298:
2297:
2296:
2294:
2289:
2281:
2279:
2276:
2272:
2268:
2264:
2260:
2254:
2250:
2244:
2240:
2236:
2232:
2228:
2222:
2217:
2215:
2211:
2207:
2201:
2195:
2191:
2187:
2182:
2177:
2175:
2171:
2166:
2162:
2156:
2151:
2146:
2142:
2138:
2132:
2128:
2124:
2120:
2116:
2111:
2107:
2102:
2098:
2093:
2087:
2083:
2075:
2073:
2053:
2047:
2044:
2041:
2038:
2035:
2021:
1999:
1996:
1993:
1990:
1944:
1941:
1935:
1929:
1926:
1923:
1920:
1910:
1909:
1908:
1874:
1854:
1834:
1811:
1805:
1799:
1796:
1784:
1781:
1778:
1775:
1768:
1767:
1766:
1752:
1744:
1725:
1722:
1719:
1716:
1683:
1677:
1671:
1665:
1659:
1653:
1647:
1645:
1640:
1634:
1627:
1624:
1621:
1615:
1613:
1608:
1602:
1591:
1590:
1589:
1587:
1565:
1559:
1556:
1553:
1550:
1547:
1498:
1492:
1489:
1483:
1480:
1474:
1467:
1466:
1465:
1423:
1422:biconditional
1399:
1375:
1359:
1351:
1335:
1327:
1319:
1317:
1315:
1311:
1307:
1303:
1298:
1296:
1292:
1288:
1283:
1280:
1276:
1268:
1260:
1254:
1248:
1244:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1202:
1197:
1195:
1190:
1188:
1183:
1182:
1180:
1179:
1176:
1168:
1167:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1145:Digital logic
1143:
1142:
1140:
1139:
1134:
1128:
1127:Scope (logic)
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1090:
1089:
1087:
1086:
1081:
1062:
1056:
1036:
1033:
1030:
1010:
1004:
997:
995:
992:
991:
976:
973:
970:
950:
945:
942:
937:
930:
928:
925:
924:
909:
895:
888:
886:
883:
882:
867:
864:
861:
841:
838:
835:
815:
812:
809:
789:
786:
783:
776:
774:
771:
770:
755:
752:
727:
705:
702:
682:
672:
670:
667:
666:
645:
642:
639:
616:
610:
590:
582:
577:
570:
568:
565:
564:
549:
546:
543:
523:
520:
517:
497:
494:
491:
484:
482:nonequivalent
481:
480:
459:
456:
453:
430:
427:
424:
404:
398:
378:
370:
365:
358:
356:
353:
352:
337:
331:
311:
308:
305:
285:
279:
272:
270:
267:
266:
251:
245:
225:
219:
199:
196:
193:
186:
184:
181:
180:
165:
156:
136:
130:
110:
107:
87:
84:
81:
61:
58:
55:
48:
46:
43:
42:
39:
38:
35:
31:
19:
6034:
5832:Ultraproduct
5679:Model theory
5644:Independence
5580:Formal proof
5572:Proof theory
5555:
5528:
5485:real numbers
5457:second-order
5368:Substitution
5245:Metalanguage
5186:conservative
5159:Axiom schema
5103:Constructive
5073:Morse–Kelley
5039:Set theories
5018:Aleph number
5011:inaccessible
4917:Grothendieck
4801:intersection
4688:Higher-order
4676:Second-order
4622:Truth tables
4579:Venn diagram
4362:Formal proof
4282:
4270:
4222:
4195:
4191:
4181:
4162:
4156:
4151:
4102:
4095:
4077:
4073:
4063:
4045:
4041:
4031:
4013:
4009:
3999:
3980:
3974:
3951:
3945:
3922:
3916:
3893:
3887:
3820:is forbidden
3813:{¬, ∪}
3809:{¬, ∩}
3795:between the
3791:There is an
3790:
3766:
3762:Toffoli gate
3758:Fredkin gate
3756:The 3-input
3755:
3748:
3738:
3734:
3730:
3726:
3722:
3718:
3714:
3702:
3698:
3695:
3687:
3683:
3679:
3675:
3671:
3667:
3663:
3659:
3653:
3649:
3645:
3641:
3637:
3633:
3629:
3625:
3620:
3616:
3612:
3596:
3592:
3588:
3584:
3580:
3576:
3572:
3568:
3562:
3558:
3554:
3550:
3546:
3542:
3538:
3534:
3529:
3525:
3521:
3502:
2684:Two elements
2669:
2643:, which are
2628:
2624:
2622:
2606:
2602:
2590:
2540:
2536:
2532:
2481:
2477:
2470:
2461:
2457:
2453:
2420:
2324:
2320:
2316:
2312:
2308:
2304:
2291:
2274:
2270:
2266:
2262:
2258:
2252:
2248:
2242:
2238:
2234:
2230:
2226:
2220:
2218:
2213:
2209:
2199:
2193:
2189:
2185:
2180:
2178:
2173:
2169:
2164:
2160:
2154:
2149:
2144:
2140:
2136:
2130:
2126:
2122:
2114:
2109:
2105:
2100:
2096:
2091:
2085:
2079:
1959:
1826:
1702:
1585:
1513:
1323:
1320:Introduction
1299:
1294:
1291:expressively
1290:
1284:
1281:
1229:truth tables
1216:
1210:
1136:Applications
1121:
5942:Type theory
5890:undecidable
5822:Truth value
5709:equivalence
5388:non-logical
5001:Enumeration
4991:Isomorphism
4938:cardinality
4922:Von Neumann
4887:Ultrafilter
4852:Uncountable
4786:equivalence
4703:Quantifiers
4693:Fixed-point
4662:First-order
4542:Consistency
4527:Proposition
4504:Traditional
4475:Lindström's
4465:Compactness
4407:Type theory
4352:Cardinality
3927:McGraw–Hill
3793:isomorphism
2678:One element
2475:truth value
1980:and one of
1350:disjunction
1326:conjunction
1107:Truth table
6057:Categories
5753:elementary
5446:arithmetic
5314:Quantifier
5292:functional
5164:Expression
4882:Transitive
4826:identities
4811:complement
4744:hereditary
4727:Set theory
4080:(4): 551,
3880:References
3861:NAND logic
3787:Set theory
3751:reversible
2674:≤ 2:
2148:, for all
2080:Given the
1310:NAND gates
1306:logic gate
183:equivalent
6024:Supertask
5927:Recursion
5885:decidable
5719:saturated
5697:of models
5620:deductive
5615:axiomatic
5535:Hilbert's
5522:Euclidean
5503:canonical
5426:axiomatic
5358:Signature
5287:Predicate
5176:Extension
5098:Ackermann
5023:Operation
4902:Universal
4892:Recursive
4867:Singleton
4862:Inhabited
4847:Countable
4837:Types of
4821:power set
4791:partition
4708:Predicate
4654:Predicate
4569:Syllogism
4559:Soundness
4532:Inference
4522:Tautology
4424:paradoxes
4253:↛
4233:↚
4016:: 86–88,
3867:NOR logic
3815:. If the
3805:structure
3482:⊤
3476:↮
3470:∧
3444:↮
3438:↔
3432:∧
3406:⊥
3400:↔
3394:∧
3368:⊤
3362:↮
3356:∨
3330:↮
3324:↔
3318:∨
3292:⊥
3286:↔
3280:∨
3247:↔
3241:↚
3215:↔
3209:↛
3183:⊤
3177:↚
3151:⊤
3145:↛
3119:¬
3113:↚
3087:¬
3081:↛
3055:↚
3049:←
3023:↛
3017:←
2991:↚
2985:→
2959:↛
2953:→
2927:↮
2921:←
2895:↮
2889:→
2863:⊥
2857:←
2831:⊥
2825:→
2799:¬
2793:←
2767:¬
2761:→
2735:¬
2729:∧
2703:¬
2697:∨
2681:{↑}, {↓}.
2657:NAND gate
2575:↮
2569:↛
2563:⊥
2557:∧
2551:∨
2516:↔
2510:→
2504:⊤
2498:∧
2492:∨
2435:¬
2421:self-dual
2404:↮
2398:↔
2392:⊥
2386:⊤
2380:¬
2353:⊥
2347:⊤
2341:∧
2335:∨
2301:monotonic
2293:Emil Post
2152:integers
2057:↔
2051:→
2045:∨
2039:∧
2033:¬
2003:→
1997:∨
1991:∧
1968:¬
1939:→
1933:¬
1924:∨
1895:→
1875:∨
1855:∨
1835:∧
1803:¬
1800:∧
1794:¬
1788:¬
1779:∨
1753:∨
1726:∨
1720:∧
1714:¬
1675:→
1666:∧
1657:→
1638:↔
1625:∨
1619:¬
1606:→
1569:↔
1563:→
1557:∨
1551:∧
1545:¬
1522:↑
1496:¬
1493:∧
1487:¬
1478:↓
1452:↓
1432:↔
1408:→
1384:¬
1360:∨
1336:∧
1314:NOR gates
1253:singleton
1060:←
1034:⊂
1008:⇐
974:⊕
946:_
943:∨
865:∥
839:∣
787:∨
753:∼
731:¯
703:−
680:¬
650:¯
614:↓
586:¯
583:∨
547:↮
464:¯
457:⋅
428:∣
402:↑
374:¯
371:∧
335:→
309:⊃
283:⇒
249:⇋
223:⇔
197:≡
163:&
160:&
134:&
85:⋅
59:∧
6009:Logicism
6002:timeline
5978:Concrete
5837:Validity
5807:T-schema
5800:Kripke's
5795:Tarski's
5790:semantic
5780:Strength
5729:submodel
5724:spectrum
5692:function
5540:Tarski's
5529:Elements
5516:geometry
5472:Robinson
5393:variable
5378:function
5351:spectrum
5341:Sentence
5297:variable
5240:Language
5193:Relation
5154:Automata
5144:Alphabet
5128:language
4982:-jection
4960:codomain
4946:Function
4907:Universe
4877:Infinite
4781:Relation
4564:Validity
4554:Argument
4452:theorem,
3826:See also
3799:and the
3775:and the
3729:∧
3701:∧
3662:∧
3628:∨
3571:∨
3537:∧
3508:Examples
2661:NOR gate
2188: :
2139: :
2104: :
2090:, a set
2088:= {0, 1}
1374:negation
1295:adequate
1175:Category
994:converse
521:⇎
495:≢
5951:Related
5748:Diagram
5646: (
5625:Hilbert
5610:Systems
5605:Theorem
5483:of the
5428:systems
5208:Formula
5203:Grammar
5119: (
5063:General
4776:Forcing
4761:Element
4681:Monadic
4456:paradox
4397:Theorem
4333:General
2595:of all
2593:lattice
2327:, e.g.
2206:nullary
2117:if the
1586:minimal
1273:{ AND,
1235:into a
1219:set of
269:implies
5714:finite
5477:Skolem
5430:
5405:Theory
5373:Symbol
5363:String
5346:atomic
5223:ground
5218:closed
5213:atomic
5169:ground
5132:syntax
5028:binary
4955:domain
4872:Finite
4637:finite
4495:Logics
4454:
4402:Theory
4169:
4157:binary
4118:
3987:
3962:
3933:
3904:
3777:T gate
3771:, the
3670:) ↓ (¬
3579:) ↑ (¬
2597:clones
2370:affine
2020:subset
1918:
907:
899:
5704:Model
5452:Peano
5309:Proof
5149:Arity
5078:Naive
4965:image
4897:Fuzzy
4857:Empty
4806:union
4751:Class
4392:Model
4382:Lemma
4340:Axiom
3708:A ↑ B
3682:) ↓ (
3674:) ≡ (
3648:) ↓ (
3640:) ≡ (
3591:) ↑ (
3583:) ≡ (
3557:) ↑ (
3549:) ≡ (
2672:arity
2633:unary
2119:clone
1255:sets
1213:logic
5827:Type
5630:list
5434:list
5411:list
5400:Term
5334:rank
5228:open
5122:list
4934:Maps
4839:sets
4698:Free
4668:list
4418:list
4345:list
4245:and
4167:ISBN
4116:ISBN
3985:ISBN
3960:ISBN
3931:ISBN
3902:ISBN
3811:and
3725:);
3666:≡ (¬
3632:≡ ¬(
3575:≡ (¬
3541:≡ ¬(
3515:NAND
2639:and
2637:NAND
2531:The
2469:The
2419:The
2368:The
2299:The
2273:) =
2256:and
2241:) =
1263:and
1259:NAND
1215:, a
903:XNOR
885:XNOR
355:NAND
5514:of
5496:of
5444:of
4976:Sur
4950:Map
4757:Ur-
4739:Set
4200:doi
4142:, D
4138:, C
4134:, C
4130:, L
4108:doi
4082:doi
4050:doi
4018:doi
3767:In
3717:≡ (
3710:",
3606:NOR
2641:NOR
2450:maj
2323:to
2315:to
2307:to
2246:if
2212:if
2202:≥ 0
2157:≥ 1
2121:on
2113:is
2022:of
1396:);
1372:);
1348:);
1267:NOR
1247:NOT
1243:AND
1233:set
1223:or
1211:In
927:XOR
669:NOT
567:NOR
45:AND
6059::
5900:NP
5524::
5518::
5448::
5125:),
4980:Bi
4972:In
4214:^
4194:,
4190:,
4114:.
4078:16
4076:,
4072:,
4046:16
4044:,
4040:,
4014:16
4012:,
4008:,
3958:,
3954:,
3929:,
3900:,
3764:.
3737:↑
3733:≡
3721:↑
3686:↓
3678:↓
3652:↓
3644:↓
3636:↓
3619:↓
3615:≡
3595:↑
3587:↑
3561:↑
3553:↑
3545:↑
3528:↑
3524:≡
3459:,
3421:,
3383:,
3345:,
3307:,
3230:,
3198:,
3166:,
3134:,
3102:,
3070:,
3038:,
3006:,
2974:,
2942:,
2910:,
2878:,
2846:,
2814:,
2782:,
2750:,
2718:,
2667:.
2605:,
2460:,
2456:,
2447:,
2269:,
2265:,
2251:=
2237:,
2233:,
2192:→
2176:.
2143:→
2108:→
2072:.
1930::=
1907::
1785::=
1648::=
1616::=
1484::=
1316:.
1297:.
1293:)
1275:OR
1265:{
1257:{
1245:,
1241:{
1049:,
1023:,
963:,
854:,
828:,
802:,
773:OR
745:,
718:,
695:,
629:,
603:,
536:,
510:,
443:,
417:,
391:,
324:,
298:,
238:,
212:,
149:,
123:,
100:,
74:,
5980:/
5895:P
5650:)
5436:)
5432:(
5329:∀
5324:!
5319:∃
5280:=
5275:↔
5270:→
5265:∧
5260:∨
5255:¬
4978:/
4974:/
4948:/
4759:)
4755:(
4642:∞
4632:3
4420:)
4318:e
4311:t
4304:v
4265:.
4209:.
4202::
4196:6
4176:.
4144:3
4140:3
4136:2
4132:1
4128:1
4124:.
4110::
4084::
4052::
4020::
3993:.
3739:X
3735:X
3731:B
3727:A
3723:B
3719:A
3715:X
3703:B
3699:A
3690:)
3688:B
3684:B
3680:A
3676:A
3672:B
3668:A
3664:B
3660:A
3656:)
3654:B
3650:A
3646:B
3642:A
3638:B
3634:A
3630:B
3626:A
3621:A
3617:A
3613:A
3611:¬
3599:)
3597:B
3593:B
3589:A
3585:A
3581:B
3577:A
3573:B
3569:A
3565:)
3563:B
3559:A
3555:B
3551:A
3547:B
3543:A
3539:B
3535:A
3530:A
3526:A
3522:A
3520:¬
3488:.
3485:}
3479:,
3473:,
3467:{
3447:}
3441:,
3435:,
3429:{
3409:}
3403:,
3397:,
3391:{
3371:}
3365:,
3359:,
3353:{
3333:}
3327:,
3321:,
3315:{
3295:}
3289:,
3283:,
3277:{
3253:.
3250:}
3244:,
3238:{
3218:}
3212:,
3206:{
3186:}
3180:,
3174:{
3154:}
3148:,
3142:{
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2060:}
2054:,
2048:,
2042:,
2036:,
2030:{
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1988:{
1945:.
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1791:(
1782:B
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1603:A
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896:A
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862:A
842:B
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816:B
813:+
810:A
790:B
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756:A
728:A
706:A
683:A
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643:+
640:A
617:B
611:A
591:B
578:A
550:B
544:A
524:B
518:A
498:B
492:A
460:B
454:A
431:B
425:A
405:B
399:A
379:B
366:A
338:B
332:A
312:B
306:A
286:B
280:A
252:B
246:A
226:B
220:A
200:B
194:A
166:B
157:A
137:B
131:A
111:B
108:A
88:B
82:A
62:B
56:A
20:)
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