3518:
1367:
40:
1273:
120:
636:
1761:
1174:
1188:
557:
909:
1028:
signified by "XOR" or "â". Naturally, these variations in usage have caused some failures to communicate between mathematicians and switching engineers over the years. At any rate, one has the following array of corresponding forms for the symbols associated with logical inequality:
1138:
566:, like those in the left column, which they interpret as an application of a function to a pair of arguments â and thus a mere indication that the value of the compound expression depends on the values of the component expressions â and an
455:
1161:
operation; "NXOR" is a less commonly used alternative. Another rationalization of the admittedly circuitous name "XNOR" is that one begins with the "both false" operator NOR and then adds the eXception "or both true".
747:
186:
258:
1035:
1040:
460:
570:, like those in the right column, which they interpret as an assertion that the arguments have equal values, in other words, that the functional value of the compound expression is
1552:
956:
298:
1581:
1657:
1682:
1457:
1628:
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1403:
1711:
107:
1748:
1523:
1486:
1357:
73:
374:
1897:
552:{\displaystyle {\begin{aligned}x&\leftrightarrow y&x&\Leftrightarrow y&\mathrm {E} xy\\x&\mathrm {~EQ~} y&x&=y\end{aligned}}}
904:{\displaystyle (x=y)=\lnot (x\oplus y)=\lnot x\oplus y=x\oplus \lnot y=(x\land y)\lor (\lnot x\land \lnot y)=(\lnot x\lor y)\land (x\lor \lnot y)}
2572:
1311:
2655:
1796:
1024:
signified by "âš" but is actually equivalent to the logical inequality operator signified by "â ", or what amounts to the same thing, the
418:, we say two formulas are equal when their truth values are equal for all possible resolutions of free variables. It corresponds to
367:
139:
199:
2969:
1133:{\displaystyle {\begin{aligned}x&+y&x&\not \equiv y&Jxy\\x&\mathrm {~XOR~} y&x&\neq y\end{aligned}}}
3127:
1915:
2982:
2305:
1010:, the plus sign "+" almost invariably indicates an operation that satisfies the axioms assigned to addition in the type of
360:
3552:
2987:
2977:
2714:
2567:
1920:
415:
1911:
3123:
1304:
1277:
1252:
2465:
3220:
2964:
1789:
437:
It is customary practice in various applications, if not always technically precise, to indicate the operation of
2525:
2218:
1959:
3481:
3183:
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2187:
1871:
3476:
3259:
3176:
2889:
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1939:
3542:
3401:
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2913:
2547:
2146:
1537:
1297:
3279:
3274:
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1881:
1782:
935:
3208:
2798:
2192:
2160:
1851:
271:
1566:
1020:. For boolean algebra, this means that the logical operation signified by "+" is not the same as the
3498:
3447:
3344:
2842:
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1925:
1954:
39:
3339:
3269:
2808:
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1442:
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1413:
423:
192:
132:
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1242:
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2535:
2434:
2401:
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1733:
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1432:
1342:
264:
8:
3493:
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2444:
2116:
2016:
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1935:
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1211:
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1016:
1011:
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52:
17:
3522:
3291:
3254:
3239:
3232:
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3019:
3001:
2867:
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2542:
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2182:
2167:
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1321:
329:
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3517:
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2429:
2413:
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2141:
2106:
2001:
1836:
1765:
1585:
1332:
1248:
1193:
1179:
586:
304:
1366:
3471:
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3316:
3138:
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3079:
2905:
2862:
2557:
2507:
2081:
2043:
1407:
1201:
391:
3452:
3442:
3396:
3379:
3334:
3296:
3198:
3118:
2925:
2852:
2825:
2813:
2719:
2633:
2607:
2562:
2530:
2331:
2133:
2076:
2026:
1991:
1949:
3437:
3416:
3374:
3354:
3249:
3104:
2702:
2692:
2682:
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2611:
2485:
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2250:
2245:
2223:
1824:
1589:
1378:
1206:
3536:
3411:
3089:
2596:
2381:
2371:
2341:
2326:
1996:
1727:
1723:
1336:
1289:
590:
410:
3311:
3158:
3059:
3051:
2931:
2879:
2788:
2724:
2707:
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2356:
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1841:
1527:
640:
395:
3421:
3301:
2480:
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2417:
2101:
2021:
2006:
1886:
1831:
1636:
1603:
1284:
1007:
922:
605:
594:
79:
2351:
2206:
2177:
1983:
1436:
113:
3503:
3406:
2459:
2376:
2336:
2300:
2236:
2048:
2038:
2011:
1774:
1560:
1382:
1144:
21:
3488:
3286:
2734:
2439:
2033:
1690:
1607:
1531:
1498:
601:
if and only if both operands are false or both operands are true.
3084:
1876:
1465:
1272:
2628:
1974:
1819:
404:
181:{\displaystyle x\cdot y+{\overline {x}}\cdot {\overline {y}}}
253:{\displaystyle ({\overline {x}}+y)\cdot (x+{\overline {y}})}
119:
635:
562:
Some logicians, however, draw a firm distinction between a
1157:
1241:
Keeton, Brian; Cavaness, Chuck; Friesen, Geoff (2001),
414:
if they are different. In the case where formulas have
1736:
1699:
1670:
1645:
1616:
1569:
1540:
1511:
1474:
1445:
1416:
1391:
1345:
1240:
1038:
938:
750:
458:
274:
202:
142:
89:
55:
1169:
1742:
1705:
1676:
1651:
1622:
1575:
1546:
1517:
1480:
1451:
1422:
1397:
1351:
1132:
950:
903:
551:
292:
252:
180:
101:
67:
408:if both arguments have the same truth value, and
3534:
925:of the logical equality operator is as follows:
1319:
1790:
1305:
368:
713:
1982:
1797:
1783:
1312:
1298:
375:
361:
1143:This explains why "EQ" is often called "
634:
3535:
1804:
1151:of circuit engineers, since it is the
1778:
1547:{\displaystyle \not \leftrightarrow }
1293:
932:
13:
1737:
1512:
1346:
1101:
1098:
1095:
951:{\displaystyle x\leftrightarrow y}
892:
865:
850:
841:
811:
790:
769:
520:
517:
494:
14:
3564:
1265:
293:{\displaystyle 1\oplus x\oplus y}
16:For the corresponding concept in
3516:
1759:
1576:{\displaystyle \leftrightarrow }
1365:
1271:
1186:
1172:
118:
38:
1247:, Que Publishing, p. 112,
449:by any of the following forms:
402:, such that it gives the value
1617:
1570:
1446:
1417:
1392:
1234:
942:
898:
883:
877:
862:
856:
838:
832:
820:
784:
772:
763:
751:
593:, typically the values of two
485:
470:
247:
228:
222:
203:
96:
90:
1:
3477:History of mathematical logic
1652:{\displaystyle \nrightarrow }
1227:
1001:
726:) is equivalent to the form (
577:
3402:Primitive recursive function
1677:{\displaystyle \nleftarrow }
1452:{\displaystyle \rightarrow }
643:of A EQ B (red part is true)
242:
211:
173:
160:
7:
1623:{\displaystyle \downarrow }
1423:{\displaystyle \leftarrow }
1165:
597:, that produces a value of
10:
3569:
2466:SchröderâBernstein theorem
2193:Monadic predicate calculus
1852:Foundations of mathematics
15:
3553:Equivalence (mathematics)
3512:
3499:Philosophy of mathematics
3448:Automated theorem proving
3430:
3325:
3157:
3050:
2902:
2619:
2595:
2573:Von NeumannâBernaysâGödel
2518:
2412:
2316:
2214:
2205:
2132:
2067:
1973:
1895:
1812:
1756:
1719:
1599:
1494:
1398:{\displaystyle \uparrow }
1374:
1363:
1328:
972:
959:
398:, or more generally, two
356:
348:
338:
328:
320:
312:
303:
263:
191:
131:
126:
112:
78:
46:
37:
32:
714:Alternative descriptions
441:on the logical operands
3149:Self-verifying theories
2970:Tarski's axiomatization
1921:Tarski's undefinability
1916:incompleteness theorems
1662:Converse nonimplication
3523:Mathematics portal
3134:Proof of impossibility
2782:propositional variable
2092:Propositional calculus
1744:
1707:
1706:{\displaystyle \land }
1678:
1653:
1624:
1577:
1548:
1519:
1482:
1453:
1424:
1399:
1353:
1222:Propositional calculus
1134:
952:
905:
650:Logical equality
644:
553:
432:propositional calculus
294:
254:
182:
103:
102:{\displaystyle (1001)}
69:
3392:Kolmogorov complexity
3345:Computably enumerable
3245:Model complete theory
3037:Principia Mathematica
2097:Propositional formula
1926:BanachâTarski paradox
1766:Philosophy portal
1745:
1743:{\displaystyle \bot }
1708:
1679:
1654:
1625:
1578:
1549:
1520:
1518:{\displaystyle \neg }
1483:
1481:{\displaystyle \lor }
1454:
1425:
1400:
1354:
1352:{\displaystyle \top }
1217:Logical biconditional
1135:
1026:exclusive disjunction
1022:inclusive disjunction
953:
906:
638:
554:
428:logical biconditional
295:
255:
183:
104:
70:
3340:ChurchâTuring thesis
3327:Computability theory
2536:continuum hypothesis
2054:Square of opposition
1912:Gödel's completeness
1734:
1697:
1668:
1643:
1614:
1567:
1538:
1509:
1472:
1443:
1414:
1408:Converse implication
1389:
1343:
1280:at Wikimedia Commons
1036:
936:
748:
456:
272:
265:Zhegalkin polynomial
200:
140:
87:
53:
3543:Logical connectives
3494:Mathematical object
3385:P versus NP problem
3350:Computable function
3144:Reverse mathematics
3070:Logical consequence
2947:primitive recursive
2942:elementary function
2715:Free/bound variable
2568:TarskiâGrothendieck
2087:Logical connectives
2017:Logical equivalence
1867:Logical consequence
1322:logical connectives
1212:Logical equivalence
1149:combinational logic
1014:that is known as a
1012:algebraic structure
651:
68:{\displaystyle x=y}
29:
18:combinational logic
3292:Transfer principle
3255:Semantics of logic
3240:Categorical theory
3216:Non-standard model
2730:Logical connective
1857:Information theory
1806:Mathematical logic
1740:
1703:
1674:
1649:
1620:
1573:
1544:
1515:
1478:
1449:
1420:
1395:
1379:Alternative denial
1349:
1130:
1128:
948:
901:
649:
645:
549:
547:
394:that compares two
290:
250:
178:
99:
65:
27:
3530:
3529:
3462:Abstract category
3265:Theories of truth
3075:Rule of inference
3065:Natural deduction
3046:
3045:
2591:
2590:
2296:Cartesian product
2201:
2200:
2107:Many-valued logic
2082:Boolean functions
1965:Russell's paradox
1940:diagonal argument
1837:First-order logic
1772:
1771:
1276:Media related to
1194:Psychology portal
1180:Philosophy portal
1106:
1094:
997:
996:
913:For the operands
709:
708:
632:) is as follows:
612:(also written as
525:
516:
385:
384:
245:
214:
176:
163:
3560:
3521:
3520:
3472:History of logic
3467:Category of sets
3360:Decision problem
3139:Ordinal analysis
3080:Sequent calculus
2978:Boolean algebras
2918:
2917:
2892:
2863:logical/constant
2617:
2616:
2603:
2526:ZermeloâFraenkel
2277:Set operations:
2212:
2211:
2149:
1980:
1979:
1960:LöwenheimâSkolem
1847:Formal semantics
1799:
1792:
1785:
1776:
1775:
1764:
1763:
1762:
1749:
1747:
1746:
1741:
1712:
1710:
1709:
1704:
1683:
1681:
1680:
1675:
1658:
1656:
1655:
1650:
1629:
1627:
1626:
1621:
1582:
1580:
1579:
1574:
1553:
1551:
1550:
1545:
1524:
1522:
1521:
1516:
1487:
1485:
1484:
1479:
1458:
1456:
1455:
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1429:
1427:
1426:
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1404:
1402:
1401:
1396:
1369:
1358:
1356:
1355:
1350:
1314:
1307:
1300:
1291:
1290:
1278:Logical equality
1275:
1259:
1257:
1238:
1202:Boolean function
1196:
1191:
1190:
1189:
1182:
1177:
1176:
1175:
1139:
1137:
1136:
1131:
1129:
1107:
1104:
1092:
957:
955:
954:
949:
930:
929:
910:
908:
907:
902:
652:
648:
583:Logical equality
558:
556:
555:
550:
548:
526:
523:
514:
497:
439:logical equality
392:logical operator
388:Logical equality
377:
370:
363:
307:
299:
297:
296:
291:
259:
257:
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238:
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207:
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169:
164:
156:
122:
108:
106:
105:
100:
74:
72:
71:
66:
42:
30:
28:Logical equality
26:
3568:
3567:
3563:
3562:
3561:
3559:
3558:
3557:
3533:
3532:
3531:
3526:
3515:
3508:
3453:Category theory
3443:Algebraic logic
3426:
3397:Lambda calculus
3335:Church encoding
3321:
3297:Truth predicate
3153:
3119:Complete theory
3042:
2911:
2907:
2903:
2898:
2890:
2610: and
2606:
2601:
2587:
2563:New Foundations
2531:axiom of choice
2514:
2476:Gödel numbering
2416: and
2408:
2312:
2197:
2147:
2128:
2077:Boolean algebra
2063:
2027:Equiconsistency
1992:Classical logic
1969:
1950:Halting problem
1938: and
1914: and
1902: and
1901:
1896:Theorems (
1891:
1808:
1803:
1773:
1768:
1760:
1758:
1752:
1735:
1732:
1731:
1715:
1698:
1695:
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1644:
1641:
1640:
1615:
1612:
1611:
1595:
1568:
1565:
1564:
1539:
1536:
1535:
1510:
1507:
1506:
1490:
1473:
1470:
1469:
1444:
1441:
1440:
1415:
1412:
1411:
1390:
1387:
1386:
1370:
1361:
1344:
1341:
1340:
1324:
1318:
1268:
1263:
1262:
1255:
1239:
1235:
1230:
1192:
1187:
1185:
1178:
1173:
1171:
1168:
1127:
1126:
1116:
1111:
1091:
1089:
1083:
1082:
1071:
1061:
1056:
1046:
1039:
1037:
1034:
1033:
1004:
937:
934:
933:
749:
746:
745:
716:
580:
568:equational form
564:functional form
546:
545:
535:
530:
513:
511:
505:
504:
493:
491:
481:
476:
466:
459:
457:
454:
453:
424:Boolean algebra
381:
306:Post's lattices
305:
273:
270:
269:
237:
206:
201:
198:
197:
168:
155:
141:
138:
137:
88:
85:
84:
54:
51:
50:
25:
12:
11:
5:
3566:
3556:
3555:
3550:
3545:
3528:
3527:
3513:
3510:
3509:
3507:
3506:
3501:
3496:
3491:
3486:
3485:
3484:
3474:
3469:
3464:
3455:
3450:
3445:
3440:
3438:Abstract logic
3434:
3432:
3428:
3427:
3425:
3424:
3419:
3417:Turing machine
3414:
3409:
3404:
3399:
3394:
3389:
3388:
3387:
3382:
3377:
3372:
3367:
3357:
3355:Computable set
3352:
3347:
3342:
3337:
3331:
3329:
3323:
3322:
3320:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3284:
3283:
3282:
3277:
3272:
3262:
3257:
3252:
3250:Satisfiability
3247:
3242:
3237:
3236:
3235:
3225:
3224:
3223:
3213:
3212:
3211:
3206:
3201:
3196:
3191:
3181:
3180:
3179:
3174:
3167:Interpretation
3163:
3161:
3155:
3154:
3152:
3151:
3146:
3141:
3136:
3131:
3121:
3116:
3115:
3114:
3113:
3112:
3102:
3097:
3087:
3082:
3077:
3072:
3067:
3062:
3056:
3054:
3048:
3047:
3044:
3043:
3041:
3040:
3032:
3031:
3030:
3029:
3024:
3023:
3022:
3017:
3012:
2992:
2991:
2990:
2988:minimal axioms
2985:
2974:
2973:
2972:
2961:
2960:
2959:
2954:
2949:
2944:
2939:
2934:
2921:
2919:
2900:
2899:
2897:
2896:
2895:
2894:
2882:
2877:
2876:
2875:
2870:
2865:
2860:
2850:
2845:
2840:
2835:
2834:
2833:
2828:
2818:
2817:
2816:
2811:
2806:
2801:
2791:
2786:
2785:
2784:
2779:
2774:
2764:
2763:
2762:
2757:
2752:
2747:
2742:
2737:
2727:
2722:
2717:
2712:
2711:
2710:
2705:
2700:
2695:
2685:
2680:
2678:Formation rule
2675:
2670:
2669:
2668:
2663:
2653:
2652:
2651:
2641:
2636:
2631:
2626:
2620:
2614:
2597:Formal systems
2593:
2592:
2589:
2588:
2586:
2585:
2580:
2575:
2570:
2565:
2560:
2555:
2550:
2545:
2540:
2539:
2538:
2533:
2522:
2520:
2516:
2515:
2513:
2512:
2511:
2510:
2500:
2495:
2494:
2493:
2486:Large cardinal
2483:
2478:
2473:
2468:
2463:
2449:
2448:
2447:
2442:
2437:
2422:
2420:
2410:
2409:
2407:
2406:
2405:
2404:
2399:
2394:
2384:
2379:
2374:
2369:
2364:
2359:
2354:
2349:
2344:
2339:
2334:
2329:
2323:
2321:
2314:
2313:
2311:
2310:
2309:
2308:
2303:
2298:
2293:
2288:
2283:
2275:
2274:
2273:
2268:
2258:
2253:
2251:Extensionality
2248:
2246:Ordinal number
2243:
2233:
2228:
2227:
2226:
2215:
2209:
2203:
2202:
2199:
2198:
2196:
2195:
2190:
2185:
2180:
2175:
2170:
2165:
2164:
2163:
2153:
2152:
2151:
2138:
2136:
2130:
2129:
2127:
2126:
2125:
2124:
2119:
2114:
2104:
2099:
2094:
2089:
2084:
2079:
2073:
2071:
2065:
2064:
2062:
2061:
2056:
2051:
2046:
2041:
2036:
2031:
2030:
2029:
2019:
2014:
2009:
2004:
1999:
1994:
1988:
1986:
1977:
1971:
1970:
1968:
1967:
1962:
1957:
1952:
1947:
1942:
1930:Cantor's
1928:
1923:
1918:
1908:
1906:
1893:
1892:
1890:
1889:
1884:
1879:
1874:
1869:
1864:
1859:
1854:
1849:
1844:
1839:
1834:
1829:
1828:
1827:
1816:
1814:
1810:
1809:
1802:
1801:
1794:
1787:
1779:
1770:
1769:
1757:
1754:
1753:
1751:
1750:
1739:
1720:
1717:
1716:
1714:
1713:
1702:
1684:
1673:
1659:
1648:
1633:Nonimplication
1630:
1619:
1600:
1597:
1596:
1594:
1593:
1590:Digital buffer
1583:
1572:
1554:
1543:
1525:
1514:
1495:
1492:
1491:
1489:
1488:
1477:
1459:
1448:
1430:
1419:
1405:
1394:
1375:
1372:
1371:
1364:
1362:
1360:
1359:
1348:
1329:
1326:
1325:
1317:
1316:
1309:
1302:
1294:
1288:
1287:
1281:
1267:
1266:External links
1264:
1261:
1260:
1253:
1232:
1231:
1229:
1226:
1225:
1224:
1219:
1214:
1209:
1207:If and only if
1204:
1198:
1197:
1183:
1167:
1164:
1141:
1140:
1125:
1122:
1119:
1117:
1115:
1112:
1110:
1103:
1100:
1097:
1090:
1088:
1085:
1084:
1081:
1078:
1075:
1072:
1070:
1067:
1064:
1062:
1060:
1057:
1055:
1052:
1049:
1047:
1045:
1042:
1041:
1003:
1000:
999:
998:
995:
994:
991:
988:
984:
983:
980:
977:
974:
970:
969:
966:
962:
961:
958:
947:
944:
941:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
849:
846:
843:
840:
837:
834:
831:
828:
825:
822:
819:
816:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
759:
756:
753:
715:
712:
711:
710:
707:
706:
703:
700:
696:
695:
692:
689:
685:
684:
681:
678:
674:
673:
670:
667:
663:
662:
659:
656:
591:logical values
579:
576:
560:
559:
544:
541:
538:
536:
534:
531:
529:
522:
519:
512:
510:
507:
506:
503:
500:
496:
492:
490:
487:
484:
482:
480:
477:
475:
472:
469:
467:
465:
462:
461:
416:free variables
383:
382:
380:
379:
372:
365:
357:
354:
353:
350:
346:
345:
342:
336:
335:
332:
326:
325:
322:
318:
317:
314:
310:
309:
301:
300:
289:
286:
283:
280:
277:
267:
261:
260:
249:
244:
241:
236:
233:
230:
227:
224:
221:
218:
213:
210:
205:
195:
189:
188:
175:
172:
167:
162:
159:
154:
151:
148:
145:
135:
129:
128:
124:
123:
116:
110:
109:
98:
95:
92:
82:
76:
75:
64:
61:
58:
48:
44:
43:
35:
34:
9:
6:
4:
3:
2:
3565:
3554:
3551:
3549:
3546:
3544:
3541:
3540:
3538:
3525:
3524:
3519:
3511:
3505:
3502:
3500:
3497:
3495:
3492:
3490:
3487:
3483:
3480:
3479:
3478:
3475:
3473:
3470:
3468:
3465:
3463:
3459:
3456:
3454:
3451:
3449:
3446:
3444:
3441:
3439:
3436:
3435:
3433:
3429:
3423:
3420:
3418:
3415:
3413:
3412:Recursive set
3410:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3386:
3383:
3381:
3378:
3376:
3373:
3371:
3368:
3366:
3363:
3362:
3361:
3358:
3356:
3353:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3333:
3332:
3330:
3328:
3324:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3281:
3278:
3276:
3273:
3271:
3268:
3267:
3266:
3263:
3261:
3258:
3256:
3253:
3251:
3248:
3246:
3243:
3241:
3238:
3234:
3231:
3230:
3229:
3226:
3222:
3221:of arithmetic
3219:
3218:
3217:
3214:
3210:
3207:
3205:
3202:
3200:
3197:
3195:
3192:
3190:
3187:
3186:
3185:
3182:
3178:
3175:
3173:
3170:
3169:
3168:
3165:
3164:
3162:
3160:
3156:
3150:
3147:
3145:
3142:
3140:
3137:
3135:
3132:
3129:
3128:from ZFC
3125:
3122:
3120:
3117:
3111:
3108:
3107:
3106:
3103:
3101:
3098:
3096:
3093:
3092:
3091:
3088:
3086:
3083:
3081:
3078:
3076:
3073:
3071:
3068:
3066:
3063:
3061:
3058:
3057:
3055:
3053:
3049:
3039:
3038:
3034:
3033:
3028:
3027:non-Euclidean
3025:
3021:
3018:
3016:
3013:
3011:
3010:
3006:
3005:
3003:
3000:
2999:
2997:
2993:
2989:
2986:
2984:
2981:
2980:
2979:
2975:
2971:
2968:
2967:
2966:
2962:
2958:
2955:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2929:
2927:
2923:
2922:
2920:
2915:
2909:
2904:Example
2901:
2893:
2888:
2887:
2886:
2883:
2881:
2878:
2874:
2871:
2869:
2866:
2864:
2861:
2859:
2856:
2855:
2854:
2851:
2849:
2846:
2844:
2841:
2839:
2836:
2832:
2829:
2827:
2824:
2823:
2822:
2819:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2796:
2795:
2792:
2790:
2787:
2783:
2780:
2778:
2775:
2773:
2770:
2769:
2768:
2765:
2761:
2758:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2732:
2731:
2728:
2726:
2723:
2721:
2718:
2716:
2713:
2709:
2706:
2704:
2701:
2699:
2696:
2694:
2691:
2690:
2689:
2686:
2684:
2681:
2679:
2676:
2674:
2671:
2667:
2664:
2662:
2661:by definition
2659:
2658:
2657:
2654:
2650:
2647:
2646:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2621:
2618:
2615:
2613:
2609:
2604:
2598:
2594:
2584:
2581:
2579:
2576:
2574:
2571:
2569:
2566:
2564:
2561:
2559:
2556:
2554:
2551:
2549:
2548:KripkeâPlatek
2546:
2544:
2541:
2537:
2534:
2532:
2529:
2528:
2527:
2524:
2523:
2521:
2517:
2509:
2506:
2505:
2504:
2501:
2499:
2496:
2492:
2489:
2488:
2487:
2484:
2482:
2479:
2477:
2474:
2472:
2469:
2467:
2464:
2461:
2457:
2453:
2450:
2446:
2443:
2441:
2438:
2436:
2433:
2432:
2431:
2427:
2424:
2423:
2421:
2419:
2415:
2411:
2403:
2400:
2398:
2395:
2393:
2392:constructible
2390:
2389:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2353:
2350:
2348:
2345:
2343:
2340:
2338:
2335:
2333:
2330:
2328:
2325:
2324:
2322:
2320:
2315:
2307:
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2278:
2276:
2272:
2269:
2267:
2264:
2263:
2262:
2259:
2257:
2254:
2252:
2249:
2247:
2244:
2242:
2238:
2234:
2232:
2229:
2225:
2222:
2221:
2220:
2217:
2216:
2213:
2210:
2208:
2204:
2194:
2191:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2162:
2159:
2158:
2157:
2154:
2150:
2145:
2144:
2143:
2140:
2139:
2137:
2135:
2131:
2123:
2120:
2118:
2115:
2113:
2110:
2109:
2108:
2105:
2103:
2100:
2098:
2095:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2074:
2072:
2070:
2069:Propositional
2066:
2060:
2057:
2055:
2052:
2050:
2047:
2045:
2042:
2040:
2037:
2035:
2032:
2028:
2025:
2024:
2023:
2020:
2018:
2015:
2013:
2010:
2008:
2005:
2003:
2000:
1998:
1997:Logical truth
1995:
1993:
1990:
1989:
1987:
1985:
1981:
1978:
1976:
1972:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1948:
1946:
1943:
1941:
1937:
1933:
1929:
1927:
1924:
1922:
1919:
1917:
1913:
1910:
1909:
1907:
1905:
1899:
1894:
1888:
1885:
1883:
1880:
1878:
1875:
1873:
1870:
1868:
1865:
1863:
1860:
1858:
1855:
1853:
1850:
1848:
1845:
1843:
1840:
1838:
1835:
1833:
1830:
1826:
1823:
1822:
1821:
1818:
1817:
1815:
1811:
1807:
1800:
1795:
1793:
1788:
1786:
1781:
1780:
1777:
1767:
1755:
1729:
1725:
1724:Contradiction
1722:
1721:
1718:
1700:
1692:
1688:
1685:
1671:
1663:
1660:
1646:
1638:
1634:
1631:
1609:
1605:
1602:
1601:
1598:
1591:
1587:
1584:
1562:
1558:
1557:Biconditional
1555:
1541:
1533:
1529:
1526:
1504:
1500:
1497:
1496:
1493:
1475:
1467:
1463:
1460:
1438:
1434:
1431:
1409:
1406:
1384:
1380:
1377:
1376:
1373:
1368:
1338:
1334:
1331:
1330:
1327:
1323:
1315:
1310:
1308:
1303:
1301:
1296:
1295:
1292:
1286:
1282:
1279:
1274:
1270:
1269:
1256:
1254:9780789724687
1250:
1246:
1245:
1237:
1233:
1223:
1220:
1218:
1215:
1213:
1210:
1208:
1205:
1203:
1200:
1199:
1195:
1184:
1181:
1170:
1163:
1160:
1159:
1154:
1150:
1146:
1123:
1120:
1118:
1113:
1108:
1086:
1079:
1076:
1073:
1068:
1065:
1063:
1058:
1053:
1050:
1048:
1043:
1032:
1031:
1030:
1027:
1023:
1019:
1018:
1013:
1009:
992:
989:
986:
985:
981:
978:
975:
971:
967:
964:
963:
945:
939:
931:
928:
927:
926:
924:
920:
916:
911:
895:
889:
886:
880:
874:
871:
868:
859:
853:
847:
844:
835:
829:
826:
823:
817:
814:
808:
805:
802:
799:
796:
793:
787:
781:
778:
775:
766:
760:
757:
754:
743:
741:
737:
733:
729:
725:
721:
704:
701:
698:
697:
693:
690:
687:
686:
682:
679:
676:
675:
671:
668:
665:
664:
660:
657:
654:
653:
647:
646:
642:
637:
633:
631:
627:
623:
619:
615:
611:
607:
602:
600:
596:
592:
588:
584:
575:
573:
569:
565:
542:
539:
537:
532:
527:
508:
501:
498:
488:
483:
478:
473:
468:
463:
452:
451:
450:
448:
444:
440:
435:
433:
429:
425:
421:
417:
413:
412:
407:
406:
401:
397:
393:
389:
378:
373:
371:
366:
364:
359:
358:
355:
351:
347:
343:
341:
337:
333:
331:
327:
323:
319:
315:
311:
308:
302:
287:
284:
281:
278:
275:
268:
266:
262:
239:
234:
231:
225:
219:
216:
208:
196:
194:
190:
170:
165:
157:
152:
149:
146:
143:
136:
134:
130:
125:
121:
117:
115:
111:
93:
83:
81:
77:
62:
59:
56:
49:
45:
41:
36:
31:
23:
19:
3514:
3312:Ultraproduct
3159:Model theory
3124:Independence
3060:Formal proof
3052:Proof theory
3035:
3008:
2965:real numbers
2937:second-order
2848:Substitution
2759:
2725:Metalanguage
2666:conservative
2639:Axiom schema
2583:Constructive
2553:MorseâKelley
2519:Set theories
2498:Aleph number
2491:inaccessible
2397:Grothendieck
2281:intersection
2168:Higher-order
2156:Second-order
2102:Truth tables
2059:Venn diagram
1842:Formal proof
1604:Joint denial
1528:Exclusive or
1244:Using Java 2
1243:
1236:
1156:
1152:
1142:
1015:
1005:
918:
914:
912:
744:
739:
735:
731:
727:
723:
719:
717:
641:Venn diagram
629:
625:
621:
617:
613:
609:
603:
598:
595:propositions
582:
581:
571:
567:
563:
561:
446:
442:
438:
436:
409:
403:
396:truth values
387:
386:
321:1-preserving
313:0-preserving
127:Normal forms
3548:Logic gates
3422:Type theory
3370:undecidable
3302:Truth value
3189:equivalence
2868:non-logical
2481:Enumeration
2471:Isomorphism
2418:cardinality
2402:Von Neumann
2367:Ultrafilter
2332:Uncountable
2266:equivalence
2183:Quantifiers
2173:Fixed-point
2142:First-order
2022:Consistency
2007:Proposition
1984:Traditional
1955:Lindström's
1945:Compactness
1887:Type theory
1832:Cardinality
1687:Conjunction
1637:NIMPLY gate
1462:Disjunction
1433:Implication
1283:Mathworld,
1008:mathematics
923:truth table
606:truth table
426:and to the
193:Conjunctive
133:Disjunctive
80:Truth table
3537:Categories
3233:elementary
2926:arithmetic
2794:Quantifier
2772:functional
2644:Expression
2362:Transitive
2306:identities
2291:complement
2224:hereditary
2207:Set theory
1437:IMPLY gate
1228:References
1002:Inequality
718:The form (
578:Definition
114:Logic gate
47:Definition
3504:Supertask
3407:Recursion
3365:decidable
3199:saturated
3177:of models
3100:deductive
3095:axiomatic
3015:Hilbert's
3002:Euclidean
2983:canonical
2906:axiomatic
2838:Signature
2767:Predicate
2656:Extension
2578:Ackermann
2503:Operation
2382:Universal
2372:Recursive
2347:Singleton
2342:Inhabited
2327:Countable
2317:Types of
2301:power set
2271:partition
2188:Predicate
2134:Predicate
2049:Syllogism
2039:Soundness
2012:Inference
2002:Tautology
1904:paradoxes
1738:⊥
1701:∧
1672:↚
1647:↛
1618:↓
1586:Statement
1571:↔
1561:XNOR gate
1513:¬
1476:∨
1447:→
1418:←
1393:↑
1383:NAND gate
1347:⊤
1333:Tautology
1147:" in the
1121:≠
943:↔
893:¬
890:∨
881:∧
872:∨
866:¬
851:¬
848:∧
842:¬
836:∨
827:∧
812:¬
809:⊕
797:⊕
791:¬
779:⊕
770:¬
587:operation
486:⇔
471:↔
349:Self-dual
285:⊕
279:⊕
243:¯
226:⋅
212:¯
174:¯
166:⋅
161:¯
147:⋅
22:XNOR gate
3489:Logicism
3482:timeline
3458:Concrete
3317:Validity
3287:T-schema
3280:Kripke's
3275:Tarski's
3270:semantic
3260:Strength
3209:submodel
3204:spectrum
3172:function
3020:Tarski's
3009:Elements
2996:geometry
2952:Robinson
2873:variable
2858:function
2831:spectrum
2821:Sentence
2777:variable
2720:Language
2673:Relation
2634:Automata
2624:Alphabet
2608:language
2462:-jection
2440:codomain
2426:Function
2387:Universe
2357:Infinite
2261:Relation
2044:Validity
2034:Argument
1932:theorem,
1691:AND gate
1608:NOR gate
1542:↮
1532:XOR gate
1503:NOT gate
1499:Negation
1166:See also
1153:negation
1066:≢
420:equality
400:formulas
330:Monotone
33:EQ, XNOR
3431:Related
3228:Diagram
3126: (
3105:Hilbert
3090:Systems
3085:Theorem
2963:of the
2908:systems
2688:Formula
2683:Grammar
2599: (
2543:General
2256:Forcing
2241:Element
2161:Monadic
1936:paradox
1877:Theorem
1813:General
1693:)
1689: (
1639:)
1635: (
1610:)
1606: (
1588: (
1563:)
1559: (
1534:)
1530: (
1505:)
1501: (
1468:)
1466:OR gate
1464: (
1439:)
1435: (
1385:)
1381: (
1320:Common
1155:of the
589:on two
3194:finite
2957:Skolem
2910:
2885:Theory
2853:Symbol
2843:String
2826:atomic
2703:ground
2698:closed
2693:atomic
2649:ground
2612:syntax
2508:binary
2435:domain
2352:Finite
2117:finite
1975:Logics
1934:
1882:Theory
1730:
1664:
1410:
1339:
1251:
1105:
1093:
921:, the
734:) âš (ÂŹ
661:p = q
630:p == q
610:p EQ q
585:is an
524:
515:
340:Affine
20:, see
3184:Model
2932:Peano
2789:Proof
2629:Arity
2558:Naive
2445:image
2377:Fuzzy
2337:Empty
2286:union
2231:Class
1872:Model
1862:Lemma
1820:Axiom
1728:False
1017:field
628:, or
626:p ⥠q
618:p â q
614:p = q
411:False
390:is a
3307:Type
3110:list
2914:list
2891:list
2880:Term
2814:rank
2708:open
2602:list
2414:Maps
2319:sets
2178:Free
2148:list
1898:list
1825:list
1337:True
1285:XNOR
1249:ISBN
1145:XNOR
917:and
639:The
604:The
599:true
572:true
445:and
405:True
94:1001
2994:of
2976:of
2924:of
2456:Sur
2430:Map
2237:Ur-
2219:Set
1158:XOR
1006:In
742:).
738:⧠
622:Epq
608:of
434:.
430:in
422:in
344:yes
324:yes
3539::
3380:NP
3004::
2998::
2928::
2605:),
2460:Bi
2452:In
993:T
990:F
987:F
982:F
979:T
976:T
968:F
960:y
730:â§
722:=
705:1
694:0
683:0
672:1
658:q
655:p
624:,
620:,
616:,
574:.
352:no
334:no
316:no
3460:/
3375:P
3130:)
2916:)
2912:(
2809:â
2804:!
2799:â
2760:=
2755:â
2750:â
2745:â§
2740:âš
2735:ÂŹ
2458:/
2454:/
2428:/
2239:)
2235:(
2122:â
2112:3
1900:)
1798:e
1791:t
1784:v
1726:/
1592:)
1335:/
1313:e
1306:t
1299:v
1258:.
1124:y
1114:x
1109:y
1102:R
1099:O
1096:X
1087:x
1080:y
1077:x
1074:J
1069:y
1059:x
1054:y
1051:+
1044:x
973:x
965:T
946:y
940:x
919:y
915:x
899:)
896:y
887:x
884:(
878:)
875:y
869:x
863:(
860:=
857:)
854:y
845:x
839:(
833:)
830:y
824:x
821:(
818:=
815:y
806:x
803:=
800:y
794:x
788:=
785:)
782:y
776:x
773:(
767:=
764:)
761:y
758:=
755:x
752:(
740:y
736:x
732:y
728:x
724:y
720:x
702:1
699:1
691:0
688:1
680:1
677:0
669:0
666:0
543:y
540:=
533:x
528:y
521:Q
518:E
509:x
502:y
499:x
495:E
489:y
479:x
474:y
464:x
447:y
443:x
376:e
369:t
362:v
288:y
282:x
276:1
248:)
240:y
235:+
232:x
229:(
223:)
220:y
217:+
209:x
204:(
171:y
158:x
153:+
150:y
144:x
97:)
91:(
63:y
60:=
57:x
24:.
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