1327:
1681:
508:
1393:
1833:
2325:
in the following: "In the more recent systems of philosophy, the universality and necessity of the axiom of Reason has, with other logical laws, been controverted and rejected by speculators on the absolute." (Hamilton
1226:
1899:
1754:
2252:
2020:
1219:
1953:
1133:
1056:
2181:
2068:
938:
897:
1146:
The latter requires a proof of rain, whereas the former merely requires a proof that rain would not be contradictory. This distinction also arises in natural language in the form of
106:
1484:
852:
810:
2287:
2139:
2103:
1519:
1018:
980:
1557:
1596:
745:
720:
610:
1422:
700:
681:
648:
629:
1604:
437:
1334:
1762:
1322:{\displaystyle \left(\phi \to \left(\psi \rightarrow \xi \right)\right)\to \left(\left(\phi \to \psi \right)\to \left(\phi \to \xi \right)\right)}
1841:
1689:
2189:
2436:
301:
1961:
1182:
1907:
2407:
2389:
2357:
1429:
1425:
1100:
1029:
277:
2148:
1564:
270:
263:
2028:
908:
867:
386:; this can be expressed by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign
315:
2335:
The of Kleene's formula *49 indicates "the demonstration is not valid for both systems ", Kleene 1952:101.
76:
1439:
1171:), and is rather a theorem. We describe a proof of this theorem in the system of three axioms proposed by
2446:
2302:
825:
783:
400:
352:
156:
2260:
2112:
2076:
1492:
991:
953:
356:
162:
2441:
1524:
758:
343:
182:
169:
1574:
1097:. Double negation introduction is a theorem of both intuitionistic logic and minimal logic, as is
730:
705:
595:
1168:
347:
188:
175:
66:
If a statement is true, then it is not the case that the statement is not true, and vice versa."
2412:
420:
412:
367:
201:
123:
48:
425:
296:
244:
235:
195:
1401:
1090:
1063:
685:
666:
633:
614:
538:
408:
383:
208:
8:
2362:
Lectures on
Metaphysics and Logic, Vol. II. Logic; Edited by Henry Mansel and John Veitch
1071:
379:
308:
291:
253:
214:
2384:, 6th reprinting with corrections 1971, North-Holland Publishing Company, Amsterdam NY,
1676:{\displaystyle (\neg \neg \varphi _{0}\to \neg \neg p)\to (\neg p\to \neg \varphi _{0})}
503:{\displaystyle \mathbf {*4\cdot 13} .\ \ \vdash .\ p\ \equiv \ \thicksim (\thicksim p)}
221:
2372:
Logic: The
Judgment, Concept, and Inference; Second Edition, Translated by Helen Dendy
1388:{\displaystyle \left(\lnot \phi \to \lnot \psi \right)\to \left(\psi \to \phi \right)}
1172:
2403:
2395:
2385:
2377:
2367:
944:
334:
327:
139:
130:
116:
2183: (instance of the first part of the theorem we have just proven)
2416:
554:
535:
416:
374:
of a statement states that "it is not the case that the statement is not true". In
2105: (from (6) and (8) by the hypothetical syllogism metatheorem)
1955: (from (4) and (5) by the hypothetical syllogism metatheorem)
1835: (from (2) and (3) by the hypothetical syllogism metatheorem)
1086:
751:
375:
53:
20:
1164:
858:
404:
1828:{\displaystyle (\neg \neg \varphi _{0}\to \neg \neg p)\to (\varphi _{0}\to p)}
1167:
for propositional logic, double negation is not always taken as an axiom (see
2430:
1094:
227:
1428:, which we refer to as (L1), and use the following additional lemma, proved
570:
215:
145:
19:
This article is about the logical concept. For the linguistic concept, see
1079:
565:
is true, respectively. The rule allows one to introduce or eliminate a
209:
1894:{\displaystyle \neg \neg p\to (\neg \neg \varphi _{0}\to \neg \neg p)}
748:
542:
189:
2423:, 2nd edition 1927, reprint 1962, Cambridge at the University Press.
1749:{\displaystyle (\neg p\to \neg \varphi _{0})\to (\varphi _{0}\to p)}
566:
393:
2247:{\displaystyle (\neg \neg \neg p\to \neg p)\to (p\to \neg \neg p)}
1147:
774:
35:
516:
a proposition is equivalent of the falsehood of its negation."
2322:
1158:
1138:
Because of their constructive character, a statement such as
2402:, Dover edition 2002, Dover Publications, Inc, Mineola N.Y.
1023:
These can be combined into a single biconditional formula:
278:
236:
202:
157:
2015:{\displaystyle \varphi _{0}\to ((\varphi _{0}\to p)\to p)}
228:
573:. The rule is based on the equivalence of, for example,
196:
2289: (from (1) and (2) by modus ponens)
2070: (from (1) and (7) by modus ponens)
309:
264:
170:
146:
2263:
2192:
2151:
2115:
2079:
2031:
1964:
1910:
1844:
1765:
1692:
1607:
1577:
1527:
1495:
1442:
1404:
1337:
1229:
1185:
1103:
1032:
994:
956:
911:
870:
828:
786:
733:
708:
688:
669:
636:
617:
598:
440:
79:
2344:
PM 1952 reprint of 2nd edition 1927 pp. 101–02, 117.
1214:{\displaystyle \phi \to \left(\psi \to \phi \right)}
2281:
2246:
2175:
2133:
2097:
2062:
2014:
1948:{\displaystyle \neg \neg p\to (\varphi _{0}\to p)}
1947:
1893:
1827:
1748:
1675:
1590:
1551:
1513:
1478:
1416:
1387:
1321:
1213:
1127:
1050:
1012:
974:
932:
891:
846:
804:
739:
714:
694:
675:
642:
623:
604:
502:
100:
222:
2428:
754:representing "can be replaced in a proof with."
382:to its double negation, but this is not true in
302:
757:In logics that have both rules, negation is an
522:
271:
163:
1128:{\displaystyle \neg \neg \neg A\vdash \neg A}
316:
285:
183:
176:
1085:Double negative elimination is a theorem of
1051:{\displaystyle \neg \neg P\leftrightarrow P}
407:in classical logic, but it is disallowed by
1563:. We also use repeatedly the method of the
512:"This is the principle of double negation,
411:. The principle was stated as a theorem of
2176:{\displaystyle \neg \neg \neg p\to \neg p}
1159:In classical propositional calculus system
947:(plain propositional calculus sentence):
2063:{\displaystyle (\varphi _{0}\to p)\to p}
1567:as a shorthand for several proof steps.
933:{\displaystyle {\frac {\neg \neg P}{P}}}
892:{\displaystyle {\frac {P}{\neg \neg P}}}
2254: (instance of (A3))
2022: (instance of (L2))
1901: (instance of (A1))
1756: (instance of (A3))
1683: (instance of (A3))
1598: (instance of (A1))
1140:It's not the case that it's not raining
403:, this principle is considered to be a
2429:
101:{\displaystyle A\equiv \sim (\sim A)}
1479:{\displaystyle p\to ((p\to q)\to q)}
1089:, but not of weaker logics such as
847:{\displaystyle \neg \neg P\vdash P}
805:{\displaystyle P\vdash \neg \neg P}
575:It is false that it is not raining.
13:
2303:Gödel–Gentzen negative translation
2273:
2270:
2235:
2232:
2211:
2202:
2199:
2196:
2167:
2158:
2155:
2152:
2125:
2122:
2083:
2080:
1914:
1911:
1882:
1879:
1863:
1860:
1848:
1845:
1791:
1788:
1772:
1769:
1705:
1696:
1657:
1648:
1633:
1630:
1614:
1611:
1565:hypothetical syllogism metatheorem
1499:
1496:
1352:
1343:
1119:
1110:
1107:
1104:
1036:
1033:
998:
995:
966:
963:
918:
915:
880:
877:
832:
829:
796:
793:
764:
689:
670:
637:
618:
14:
2458:
491:
485:
2282:{\displaystyle p\to \neg \neg p}
2134:{\displaystyle p\to \neg \neg p}
2098:{\displaystyle \neg \neg p\to p}
1514:{\displaystyle \neg \neg p\to p}
1013:{\displaystyle \neg \neg P\to P}
975:{\displaystyle P\to \neg \neg P}
451:
445:
442:
2437:Theorems in propositional logic
2382:Introduction to Metamathematics
2374:, Macmillan & Co. New York.
2351:
1165:Hilbert-style deductive systems
2338:
2329:
2315:
2267:
2241:
2229:
2223:
2220:
2217:
2208:
2193:
2164:
2119:
2089:
2054:
2051:
2045:
2032:
2009:
2003:
2000:
1994:
1981:
1978:
1975:
1942:
1936:
1923:
1920:
1888:
1876:
1857:
1854:
1822:
1816:
1803:
1800:
1797:
1785:
1766:
1743:
1737:
1724:
1721:
1718:
1702:
1693:
1670:
1654:
1645:
1642:
1639:
1627:
1608:
1546:
1540:
1534:
1531:
1505:
1473:
1467:
1464:
1458:
1452:
1449:
1446:
1408:
1374:
1363:
1349:
1303:
1292:
1281:
1265:
1249:
1238:
1200:
1189:
1042:
1004:
960:
734:
709:
599:
497:
488:
95:
86:
1:
2308:
1552:{\displaystyle q\to (r\to q)}
1062:Since biconditionality is an
2421:Principia Mathematica to *56
2364:, Boston, Gould and Lincoln.
1591:{\displaystyle \varphi _{0}}
1082:of the well-formed formula.
771:double negation introduction
740:{\displaystyle \Rightarrow }
715:{\displaystyle \Rightarrow }
605:{\displaystyle \Rightarrow }
585:double negation introduction
532:double negation introduction
523:Elimination and introduction
7:
2296:
2293:And the proof is complete.
1521:. For shortness, we denote
817:double negation elimination
657:double negation elimination
528:Double negation elimination
16:Propositional logic theorem
10:
2463:
401:law of the excluded middle
353:Existential generalization
158:Biconditional introduction
18:
1153:
70:
62:
41:
31:
1078:, leaving unchanged the
819:rule may be written as:
344:Universal generalization
184:Disjunction introduction
171:Conjunction introduction
141:Implication introduction
2321:Hamilton is discussing
1169:list of Hilbert systems
773:rule may be written in
2413:Alfred North Whitehead
2283:
2248:
2177:
2135:
2099:
2064:
2016:
1949:
1895:
1829:
1750:
1677:
1592:
1553:
1515:
1480:
1418:
1417:{\displaystyle p\to p}
1389:
1323:
1215:
1129:
1052:
1014:
976:
934:
893:
848:
806:
741:
716:
696:
677:
644:
625:
606:
504:
203:hypothetical syllogism
124:Propositional calculus
102:
49:Propositional calculus
2284:
2249:
2178:
2136:
2100:
2065:
2017:
1950:
1896:
1830:
1751:
1678:
1593:
1554:
1516:
1481:
1419:
1390:
1324:
1216:
1130:
1053:
1015:
977:
935:
894:
849:
807:
742:
717:
697:
695:{\displaystyle \neg }
678:
676:{\displaystyle \neg }
645:
643:{\displaystyle \neg }
626:
624:{\displaystyle \neg }
607:
505:
426:Principia Mathematica
378:, every statement is
245:Negation introduction
238:modus ponendo tollens
103:
2261:
2190:
2149:
2113:
2077:
2029:
1962:
1908:
1842:
1763:
1690:
1605:
1575:
1525:
1493:
1440:
1402:
1335:
1227:
1183:
1101:
1091:intuitionistic logic
1066:, any instance of ¬¬
1064:equivalence relation
1030:
992:
954:
909:
868:
826:
784:
731:
706:
686:
667:
634:
615:
596:
539:rules of replacement
438:
409:intuitionistic logic
384:intuitionistic logic
380:logically equivalent
303:Material implication
254:Rules of replacement
117:Transformation rules
77:
1074:can be replaced by
1072:well-formed formula
413:propositional logic
368:propositional logic
216:destructive dilemma
28:
2447:Rules of inference
2400:Mathematical Logic
2279:
2244:
2173:
2131:
2095:
2060:
2012:
1945:
1891:
1825:
1746:
1673:
1588:
1549:
1511:
1476:
1414:
1385:
1319:
1211:
1125:
1048:
1010:
972:
930:
889:
844:
802:
737:
712:
692:
673:
640:
621:
602:
500:
335:Rules of inference
131:Rules of inference
98:
84:≡ ∼
71:Symbolic statement
26:
2396:Stephen C. Kleene
2378:Stephen C. Kleene
2368:Christoph Sigwart
1398:We use the lemma
928:
887:
553:is true, and its
484:
478:
472:
463:
460:
364:
363:
111:
110:
2454:
2417:Bertrand Russell
2358:William Hamilton
2345:
2342:
2336:
2333:
2327:
2319:
2288:
2286:
2285:
2280:
2253:
2251:
2250:
2245:
2182:
2180:
2179:
2174:
2140:
2138:
2137:
2132:
2104:
2102:
2101:
2096:
2069:
2067:
2066:
2061:
2044:
2043:
2021:
2019:
2018:
2013:
1993:
1992:
1974:
1973:
1954:
1952:
1951:
1946:
1935:
1934:
1900:
1898:
1897:
1892:
1875:
1874:
1834:
1832:
1831:
1826:
1815:
1814:
1784:
1783:
1755:
1753:
1752:
1747:
1736:
1735:
1717:
1716:
1682:
1680:
1679:
1674:
1669:
1668:
1626:
1625:
1597:
1595:
1594:
1589:
1587:
1586:
1558:
1556:
1555:
1550:
1520:
1518:
1517:
1512:
1485:
1483:
1482:
1477:
1423:
1421:
1420:
1415:
1394:
1392:
1391:
1386:
1384:
1380:
1362:
1358:
1328:
1326:
1325:
1320:
1318:
1314:
1313:
1309:
1291:
1287:
1264:
1260:
1259:
1255:
1220:
1218:
1217:
1212:
1210:
1206:
1134:
1132:
1131:
1126:
1057:
1055:
1054:
1049:
1019:
1017:
1016:
1011:
981:
979:
978:
973:
939:
937:
936:
931:
929:
924:
913:
898:
896:
895:
890:
888:
886:
872:
853:
851:
850:
845:
811:
809:
808:
803:
746:
744:
743:
738:
721:
719:
718:
713:
701:
699:
698:
693:
682:
680:
679:
674:
649:
647:
646:
641:
630:
628:
627:
622:
611:
609:
608:
603:
509:
507:
506:
501:
482:
476:
470:
461:
458:
454:
391:
318:
311:
304:
292:De Morgan's laws
287:
280:
273:
266:
240:
232:
224:
217:
211:
204:
198:
191:
185:
178:
172:
165:
159:
152:
142:
113:
112:
107:
105:
104:
99:
29:
25:
2462:
2461:
2457:
2456:
2455:
2453:
2452:
2451:
2442:Classical logic
2427:
2426:
2354:
2349:
2348:
2343:
2339:
2334:
2330:
2320:
2316:
2311:
2299:
2262:
2259:
2258:
2191:
2188:
2187:
2150:
2147:
2146:
2114:
2111:
2110:
2078:
2075:
2074:
2039:
2035:
2030:
2027:
2026:
1988:
1984:
1969:
1965:
1963:
1960:
1959:
1930:
1926:
1909:
1906:
1905:
1870:
1866:
1843:
1840:
1839:
1810:
1806:
1779:
1775:
1764:
1761:
1760:
1731:
1727:
1712:
1708:
1691:
1688:
1687:
1664:
1660:
1621:
1617:
1606:
1603:
1602:
1582:
1578:
1576:
1573:
1572:
1562:
1526:
1523:
1522:
1494:
1491:
1490:
1489:We first prove
1441:
1438:
1437:
1403:
1400:
1399:
1370:
1366:
1342:
1338:
1336:
1333:
1332:
1299:
1295:
1277:
1273:
1272:
1268:
1245:
1241:
1234:
1230:
1228:
1225:
1224:
1196:
1192:
1184:
1181:
1180:
1173:Jan Łukasiewicz
1161:
1156:
1142:is weaker than
1102:
1099:
1098:
1087:classical logic
1031:
1028:
1027:
993:
990:
989:
955:
952:
951:
914:
912:
910:
907:
906:
876:
871:
869:
866:
865:
827:
824:
823:
785:
782:
781:
767:
765:Formal notation
732:
729:
728:
707:
704:
703:
687:
684:
683:
668:
665:
664:
635:
632:
631:
616:
613:
612:
597:
594:
593:
541:. They are the
525:
441:
439:
436:
435:
387:
376:classical logic
372:double negation
328:Predicate logic
322:
286:Double negation
140:
78:
75:
74:
58:
54:Classical logic
27:Double negation
24:
21:double negative
17:
12:
11:
5:
2460:
2450:
2449:
2444:
2439:
2425:
2424:
2410:
2393:
2375:
2365:
2353:
2350:
2347:
2346:
2337:
2328:
2313:
2312:
2310:
2307:
2306:
2305:
2298:
2295:
2291:
2290:
2278:
2275:
2272:
2269:
2266:
2255:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2184:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2130:
2127:
2124:
2121:
2118:
2107:
2106:
2094:
2091:
2088:
2085:
2082:
2071:
2059:
2056:
2053:
2050:
2047:
2042:
2038:
2034:
2023:
2011:
2008:
2005:
2002:
1999:
1996:
1991:
1987:
1983:
1980:
1977:
1972:
1968:
1956:
1944:
1941:
1938:
1933:
1929:
1925:
1922:
1919:
1916:
1913:
1902:
1890:
1887:
1884:
1881:
1878:
1873:
1869:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1836:
1824:
1821:
1818:
1813:
1809:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1782:
1778:
1774:
1771:
1768:
1757:
1745:
1742:
1739:
1734:
1730:
1726:
1723:
1720:
1715:
1711:
1707:
1704:
1701:
1698:
1695:
1684:
1672:
1667:
1663:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1635:
1632:
1629:
1624:
1620:
1616:
1613:
1610:
1599:
1585:
1581:
1560:
1548:
1545:
1542:
1539:
1536:
1533:
1530:
1510:
1507:
1504:
1501:
1498:
1487:
1486:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1413:
1410:
1407:
1396:
1395:
1383:
1379:
1376:
1373:
1369:
1365:
1361:
1357:
1354:
1351:
1348:
1345:
1341:
1329:
1317:
1312:
1308:
1305:
1302:
1298:
1294:
1290:
1286:
1283:
1280:
1276:
1271:
1267:
1263:
1258:
1254:
1251:
1248:
1244:
1240:
1237:
1233:
1221:
1209:
1205:
1202:
1199:
1195:
1191:
1188:
1160:
1157:
1155:
1152:
1124:
1121:
1118:
1115:
1112:
1109:
1106:
1060:
1059:
1047:
1044:
1041:
1038:
1035:
1021:
1020:
1009:
1006:
1003:
1000:
997:
983:
982:
971:
968:
965:
962:
959:
941:
940:
927:
923:
920:
917:
900:
899:
885:
882:
879:
875:
855:
854:
843:
840:
837:
834:
831:
813:
812:
801:
798:
795:
792:
789:
766:
763:
736:
725:
724:
711:
691:
672:
653:
652:
639:
620:
601:
579:It is raining.
561:is true, then
549:is true, then
524:
521:
520:
519:
518:
517:
510:
499:
496:
493:
490:
487:
481:
475:
469:
466:
457:
453:
450:
447:
444:
405:law of thought
362:
361:
360:
359:
350:
338:
337:
331:
330:
324:
323:
321:
320:
313:
306:
299:
294:
289:
282:
279:Distributivity
275:
268:
260:
257:
256:
250:
249:
248:
247:
242:
219:
206:
193:
180:
167:
154:
134:
133:
127:
126:
120:
119:
109:
108:
97:
94:
91:
88:
85:
82:
72:
68:
67:
64:
60:
59:
57:
56:
51:
45:
43:
39:
38:
33:
15:
9:
6:
4:
3:
2:
2459:
2448:
2445:
2443:
2440:
2438:
2435:
2434:
2432:
2422:
2418:
2414:
2411:
2409:
2408:0-486-42533-9
2405:
2401:
2397:
2394:
2391:
2390:0-7204-2103-9
2387:
2383:
2379:
2376:
2373:
2369:
2366:
2363:
2359:
2356:
2355:
2341:
2332:
2324:
2318:
2314:
2304:
2301:
2300:
2294:
2276:
2264:
2256:
2238:
2226:
2214:
2205:
2185:
2170:
2161:
2144:
2143:
2142:
2128:
2116:
2109:We now prove
2092:
2086:
2072:
2057:
2048:
2040:
2036:
2024:
2006:
1997:
1989:
1985:
1970:
1966:
1957:
1939:
1931:
1927:
1917:
1903:
1885:
1871:
1867:
1851:
1837:
1819:
1811:
1807:
1794:
1780:
1776:
1758:
1740:
1732:
1728:
1713:
1709:
1699:
1685:
1665:
1661:
1651:
1636:
1622:
1618:
1600:
1583:
1579:
1570:
1569:
1568:
1566:
1543:
1537:
1528:
1508:
1502:
1470:
1461:
1455:
1443:
1435:
1434:
1433:
1431:
1427:
1411:
1405:
1381:
1377:
1371:
1367:
1359:
1355:
1346:
1339:
1330:
1315:
1310:
1306:
1300:
1296:
1288:
1284:
1278:
1274:
1269:
1261:
1256:
1252:
1246:
1242:
1235:
1231:
1222:
1207:
1203:
1197:
1193:
1186:
1178:
1177:
1176:
1174:
1170:
1166:
1151:
1149:
1145:
1144:It's raining.
1141:
1136:
1122:
1116:
1113:
1096:
1095:minimal logic
1092:
1088:
1083:
1081:
1077:
1073:
1069:
1065:
1045:
1039:
1026:
1025:
1024:
1007:
1001:
988:
987:
986:
969:
957:
950:
949:
948:
946:
925:
921:
905:
904:
903:
883:
873:
864:
863:
862:
860:
841:
838:
835:
822:
821:
820:
818:
799:
790:
787:
780:
779:
778:
776:
772:
762:
760:
755:
753:
750:
723:
662:
661:
660:
658:
651:
590:
589:
588:
586:
581:
580:
576:
572:
568:
564:
560:
556:
552:
548:
544:
540:
537:
533:
529:
515:
511:
494:
479:
473:
467:
464:
455:
448:
434:
433:
432:
431:
430:
428:
427:
422:
418:
414:
410:
406:
402:
397:
395:
390:
385:
381:
377:
373:
369:
358:
357:instantiation
354:
351:
349:
348:instantiation
345:
342:
341:
340:
339:
336:
333:
332:
329:
326:
325:
319:
314:
312:
307:
305:
300:
298:
297:Transposition
295:
293:
290:
288:
283:
281:
276:
274:
272:Commutativity
269:
267:
265:Associativity
262:
261:
259:
258:
255:
252:
251:
246:
243:
241:
239:
233:
231:
230:modus tollens
225:
220:
218:
212:
207:
205:
199:
194:
192:
186:
181:
179:
173:
168:
166:
160:
155:
153:
150:
147:elimination (
143:
138:
137:
136:
135:
132:
129:
128:
125:
122:
121:
118:
115:
114:
92:
89:
83:
80:
73:
69:
65:
61:
55:
52:
50:
47:
46:
44:
40:
37:
34:
30:
22:
2420:
2399:
2381:
2371:
2361:
2352:Bibliography
2340:
2331:
2317:
2292:
2108:
1488:
1397:
1162:
1143:
1139:
1137:
1084:
1075:
1067:
1061:
1022:
984:
942:
901:
856:
816:
814:
770:
768:
756:
726:
663:
656:
654:
591:
584:
582:
578:
574:
571:formal proof
562:
558:
550:
546:
531:
527:
526:
513:
424:
398:
388:
371:
365:
355: /
346: /
284:
237:
234: /
229:
226: /
213: /
210:Constructive
200: /
187: /
174: /
161: /
149:modus ponens
148:
144: /
1080:truth-value
749:metalogical
557:, that, if
310:Exportation
197:Disjunctive
190:elimination
177:elimination
164:elimination
2431:Categories
2309:References
777:notation:
759:involution
543:inferences
392:expresses
223:Absorption
2274:¬
2271:¬
2268:→
2236:¬
2233:¬
2230:→
2221:→
2212:¬
2209:→
2203:¬
2200:¬
2197:¬
2168:¬
2165:→
2159:¬
2156:¬
2153:¬
2126:¬
2123:¬
2120:→
2090:→
2084:¬
2081:¬
2055:→
2046:→
2037:φ
2004:→
1995:→
1986:φ
1976:→
1967:φ
1937:→
1928:φ
1921:→
1915:¬
1912:¬
1883:¬
1880:¬
1877:→
1868:φ
1864:¬
1861:¬
1855:→
1849:¬
1846:¬
1817:→
1808:φ
1801:→
1792:¬
1789:¬
1786:→
1777:φ
1773:¬
1770:¬
1738:→
1729:φ
1722:→
1710:φ
1706:¬
1703:→
1697:¬
1662:φ
1658:¬
1655:→
1649:¬
1643:→
1634:¬
1631:¬
1628:→
1619:φ
1615:¬
1612:¬
1580:φ
1559:by φ
1541:→
1532:→
1506:→
1500:¬
1497:¬
1468:→
1459:→
1447:→
1409:→
1378:ϕ
1375:→
1372:ψ
1364:→
1356:ψ
1353:¬
1350:→
1347:ϕ
1344:¬
1307:ξ
1304:→
1301:ϕ
1293:→
1285:ψ
1282:→
1279:ϕ
1266:→
1253:ξ
1250:→
1247:ψ
1239:→
1236:ϕ
1204:ϕ
1201:→
1198:ψ
1190:→
1187:ϕ
1120:¬
1117:⊢
1111:¬
1108:¬
1105:¬
1043:↔
1037:¬
1034:¬
1005:→
999:¬
996:¬
967:¬
964:¬
961:→
945:tautology
919:¬
916:¬
881:¬
878:¬
859:rule form
839:⊢
833:¬
830:¬
797:¬
794:¬
791:⊢
735:⇒
710:⇒
690:¬
671:¬
659:rule is:
638:¬
619:¬
600:⇒
587:rule is:
563:not not-A
547:not not-A
545:that, if
492:∼
486:∼
480:≡
465:⊢
449:⋅
443:∗
421:Whitehead
399:Like the
317:Tautology
90:∼
63:Statement
2398:, 1967,
2380:, 1952,
2370:, 1895,
2360:, 1860,
2326:1860:68)
2297:See also
943:or as a
655:and the
567:negation
555:converse
534:are two
394:negation
1424:proved
1148:litotes
775:sequent
747:" is a
727:Where "
569:from a
417:Russell
36:Theorem
2406:
2388:
1154:Proofs
752:symbol
483:
477:
471:
462:
459:
370:, the
2323:Hegel
1436:(L2)
1070:in a
536:valid
42:Field
2415:and
2404:ISBN
2386:ISBN
2257:(3)
2186:(2)
2145:(1)
2073:(9)
2025:(8)
1958:(7)
1904:(6)
1838:(5)
1759:(4)
1686:(3)
1601:(2)
1571:(1)
1430:here
1426:here
1331:A3.
1223:A2.
1179:A1.
1093:and
985:and
902:and
815:The
769:The
583:The
577:and
530:and
514:i.e.
429:as:
419:and
32:Type
1163:In
857:In
423:in
415:by
366:In
2433::
2419:,
2141:.
1432::
1175::
1150:.
1135:.
861::
761:.
702:P
592:P
452:13
396:.
2392:.
2277:p
2265:p
2242:)
2239:p
2227:p
2224:(
2218:)
2215:p
2206:p
2194:(
2171:p
2162:p
2129:p
2117:p
2093:p
2087:p
2058:p
2052:)
2049:p
2041:0
2033:(
2010:)
2007:p
2001:)
1998:p
1990:0
1982:(
1979:(
1971:0
1943:)
1940:p
1932:0
1924:(
1918:p
1889:)
1886:p
1872:0
1858:(
1852:p
1823:)
1820:p
1812:0
1804:(
1798:)
1795:p
1781:0
1767:(
1744:)
1741:p
1733:0
1725:(
1719:)
1714:0
1700:p
1694:(
1671:)
1666:0
1652:p
1646:(
1640:)
1637:p
1623:0
1609:(
1584:0
1561:0
1547:)
1544:q
1538:r
1535:(
1529:q
1509:p
1503:p
1474:)
1471:q
1465:)
1462:q
1456:p
1453:(
1450:(
1444:p
1412:p
1406:p
1382:)
1368:(
1360:)
1340:(
1316:)
1311:)
1297:(
1289:)
1275:(
1270:(
1262:)
1257:)
1243:(
1232:(
1208:)
1194:(
1123:A
1114:A
1076:A
1068:A
1058:.
1046:P
1040:P
1008:P
1002:P
970:P
958:P
926:P
922:P
884:P
874:P
842:P
836:P
800:P
788:P
722:P
650:P
559:A
551:A
498:)
495:p
489:(
474:p
468:.
456:.
446:4
389:~
151:)
96:)
93:A
87:(
81:A
23:.
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