1263:
198:
261:
758:
712:
324:
928:
884:
1151:
1107:
1051:
1005:
787:
951:
1194:
1174:
1025:
975:
827:
807:
123:
103:
83:
63:
137:
1333:
521:
1304:
204:
829:" can be placed on a subsequent line by itself. The above example in English is an application of the first sub-rule.
723:
677:
267:
497:
490:
483:
535:
1328:
895:
851:
572:
376:
1118:
1074:
576:
382:
1297:
563:
402:
389:
1278:
567:
408:
587:
421:
343:
35:
1030:
984:
766:
936:
516:
464:
455:
415:
1323:
428:
8:
1290:
978:
631:
615:
528:
511:
473:
434:
1179:
1159:
1010:
960:
812:
792:
441:
108:
88:
68:
48:
1061:
623:
554:
547:
359:
350:
336:
25:
653:
612:
668:
504:
193:{\displaystyle {\frac {P\land Q}{\therefore P}},{\frac {P\land Q}{\therefore Q}}}
1274:
1317:
1197:
1054:
619:
447:
646:
435:
365:
649:
by deriving one of the conjuncts of a conjunction on a line by itself.
667:
The rule consists of two separate sub-rules, which can be expressed in
429:
954:
627:
409:
1065:
842:
1262:
763:
The two sub-rules together mean that, whenever an instance of "
1270:
498:
456:
422:
377:
448:
416:
529:
484:
390:
366:
645:
is true. The rule makes it possible to shorten longer
1182:
1162:
1121:
1077:
1033:
1013:
987:
963:
939:
898:
854:
815:
795:
769:
726:
680:
270:
256:{\displaystyle (P\land Q)\vdash P,(P\land Q)\vdash Q}
207:
140:
111:
91:
71:
51:
1188:
1168:
1145:
1101:
1045:
1019:
999:
969:
945:
922:
878:
821:
801:
781:
752:
706:
318:
255:
192:
117:
97:
77:
57:
442:
1315:
753:{\displaystyle {\frac {P\land Q}{\therefore Q}}}
707:{\displaystyle {\frac {P\land Q}{\therefore P}}}
522:
491:
383:
319:{\displaystyle (P\land Q)\to P,(P\land Q)\to Q}
1216:
1298:
536:
505:
403:
396:
1305:
1291:
789:" appears on a line of a proof, either "
1219:Principles of Automated Theorem Proving
1316:
1257:
1196:are propositions expressed in some
1027:is also a syntactic consequence of
13:
1060:and expressed as truth-functional
923:{\displaystyle (P\land Q)\vdash Q}
879:{\displaystyle (P\land Q)\vdash P}
832:
14:
1345:
1261:
1334:Theorems in propositional logic
1146:{\displaystyle (P\land Q)\to Q}
1102:{\displaystyle (P\land Q)\to P}
1244:
1235:
1226:
1210:
1137:
1134:
1122:
1093:
1090:
1078:
911:
899:
867:
855:
660:It's raining and it's pouring.
310:
307:
295:
286:
283:
271:
244:
232:
220:
208:
1:
1203:
1277:. You can help Knowledge by
841:sub-rules may be written in
7:
10:
1350:
1256:
573:Existential generalization
378:Biconditional introduction
129:
41:
31:
21:
1068:of propositional logic:
1046:{\displaystyle P\land Q}
1000:{\displaystyle P\land Q}
782:{\displaystyle P\land Q}
564:Universal generalization
404:Disjunction introduction
391:Conjunction introduction
361:Implication introduction
1217:David A. Duffy (1991).
946:{\displaystyle \vdash }
839:conjunction elimination
663:Therefore it's raining.
592:conjunction elimination
17:Conjunction elimination
1273:-related article is a
1190:
1170:
1147:
1103:
1047:
1021:
1001:
971:
947:
924:
880:
823:
803:
783:
754:
708:
423:hypothetical syllogism
344:Propositional calculus
320:
257:
194:
119:
99:
79:
59:
36:Propositional calculus
1191:
1171:
1148:
1104:
1048:
1022:
1002:
979:syntactic consequence
972:
948:
925:
881:
824:
804:
784:
755:
709:
465:Negation introduction
458:modus ponendo tollens
321:
258:
195:
120:
100:
80:
60:
1180:
1160:
1119:
1075:
1031:
1011:
985:
961:
957:symbol meaning that
937:
896:
852:
813:
793:
767:
724:
678:
523:Material implication
474:Rules of replacement
337:Transformation rules
268:
205:
138:
109:
89:
69:
49:
616:immediate inference
588:propositional logic
436:destructive dilemma
45:If the conjunction
18:
1329:Rules of inference
1223:Sect.3.1.2.1, p.46
1221:. New York: Wiley.
1186:
1166:
1143:
1099:
1043:
1017:
997:
967:
943:
920:
876:
819:
799:
779:
750:
704:
555:Rules of inference
351:Rules of inference
316:
253:
190:
130:Symbolic statement
115:
95:
75:
55:
16:
1286:
1285:
1189:{\displaystyle Q}
1169:{\displaystyle P}
1020:{\displaystyle Q}
970:{\displaystyle P}
822:{\displaystyle Q}
802:{\displaystyle P}
748:
702:
624:rule of inference
584:
583:
331:
330:
188:
162:
118:{\displaystyle B}
98:{\displaystyle A}
78:{\displaystyle B}
58:{\displaystyle A}
26:Rule of inference
1341:
1307:
1300:
1293:
1265:
1258:
1251:
1248:
1242:
1241:Moore and Parker
1239:
1233:
1230:
1224:
1222:
1214:
1195:
1193:
1192:
1187:
1175:
1173:
1172:
1167:
1152:
1150:
1149:
1144:
1108:
1106:
1105:
1100:
1052:
1050:
1049:
1044:
1026:
1024:
1023:
1018:
1006:
1004:
1003:
998:
976:
974:
973:
968:
952:
950:
949:
944:
929:
927:
926:
921:
885:
883:
882:
877:
828:
826:
825:
820:
808:
806:
805:
800:
788:
786:
785:
780:
759:
757:
756:
751:
749:
747:
739:
728:
713:
711:
710:
705:
703:
701:
693:
682:
626:which makes the
538:
531:
524:
512:De Morgan's laws
507:
500:
493:
486:
460:
452:
444:
437:
431:
424:
418:
411:
405:
398:
392:
385:
379:
372:
362:
333:
332:
325:
323:
322:
317:
262:
260:
259:
254:
199:
197:
196:
191:
189:
187:
179:
168:
163:
161:
153:
142:
124:
122:
121:
116:
104:
102:
101:
96:
84:
82:
81:
76:
64:
62:
61:
56:
19:
15:
1349:
1348:
1344:
1343:
1342:
1340:
1339:
1338:
1314:
1313:
1312:
1311:
1255:
1254:
1249:
1245:
1240:
1236:
1231:
1227:
1215:
1211:
1206:
1181:
1178:
1177:
1161:
1158:
1157:
1120:
1117:
1116:
1076:
1073:
1072:
1032:
1029:
1028:
1012:
1009:
1008:
986:
983:
982:
962:
959:
958:
938:
935:
934:
897:
894:
893:
853:
850:
849:
835:
833:Formal notation
814:
811:
810:
794:
791:
790:
768:
765:
764:
740:
729:
727:
725:
722:
721:
694:
683:
681:
679:
676:
675:
669:formal language
548:Predicate logic
542:
506:Double negation
360:
269:
266:
265:
206:
203:
202:
180:
169:
167:
154:
143:
141:
139:
136:
135:
110:
107:
106:
90:
87:
86:
70:
67:
66:
50:
47:
46:
12:
11:
5:
1347:
1337:
1336:
1331:
1326:
1310:
1309:
1302:
1295:
1287:
1284:
1283:
1266:
1253:
1252:
1243:
1234:
1232:Copi and Cohen
1225:
1208:
1207:
1205:
1202:
1185:
1165:
1154:
1153:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1110:
1109:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1055:logical system
1042:
1039:
1036:
1016:
996:
993:
990:
966:
942:
931:
930:
919:
916:
913:
910:
907:
904:
901:
887:
886:
875:
872:
869:
866:
863:
860:
857:
834:
831:
818:
798:
778:
775:
772:
761:
760:
746:
743:
738:
735:
732:
715:
714:
700:
697:
692:
689:
686:
665:
664:
661:
652:An example in
637:is true, then
609:simplification
582:
581:
580:
579:
570:
558:
557:
551:
550:
544:
543:
541:
540:
533:
526:
519:
514:
509:
502:
499:Distributivity
495:
488:
480:
477:
476:
470:
469:
468:
467:
462:
439:
426:
413:
400:
387:
374:
354:
353:
347:
346:
340:
339:
329:
328:
327:
326:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
263:
252:
249:
246:
243:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
200:
186:
183:
178:
175:
172:
166:
160:
157:
152:
149:
146:
131:
127:
126:
114:
94:
85:is true, then
74:
54:
43:
39:
38:
33:
29:
28:
23:
9:
6:
4:
3:
2:
1346:
1335:
1332:
1330:
1327:
1325:
1322:
1321:
1319:
1308:
1303:
1301:
1296:
1294:
1289:
1288:
1282:
1280:
1276:
1272:
1267:
1264:
1260:
1259:
1247:
1238:
1229:
1220:
1213:
1209:
1201:
1199:
1198:formal system
1183:
1163:
1140:
1131:
1128:
1125:
1115:
1114:
1113:
1096:
1087:
1084:
1081:
1071:
1070:
1069:
1067:
1063:
1058:
1056:
1040:
1037:
1034:
1014:
994:
991:
988:
980:
964:
956:
940:
917:
914:
908:
905:
902:
892:
891:
890:
873:
870:
864:
861:
858:
848:
847:
846:
844:
840:
830:
816:
796:
776:
773:
770:
744:
741:
736:
733:
730:
720:
719:
718:
698:
695:
690:
687:
684:
674:
673:
672:
670:
662:
659:
658:
657:
655:
650:
648:
644:
641:is true, and
640:
636:
633:
630:that, if the
629:
625:
621:
620:argument form
617:
614:
610:
606:
605:∧ elimination
602:
599:
598:
594:(also called
593:
589:
578:
577:instantiation
574:
571:
569:
568:instantiation
565:
562:
561:
560:
559:
556:
553:
552:
549:
546:
545:
539:
534:
532:
527:
525:
520:
518:
517:Transposition
515:
513:
510:
508:
503:
501:
496:
494:
492:Commutativity
489:
487:
485:Associativity
482:
481:
479:
478:
475:
472:
471:
466:
463:
461:
459:
453:
451:
450:modus tollens
445:
440:
438:
432:
427:
425:
419:
414:
412:
406:
401:
399:
393:
388:
386:
380:
375:
373:
370:
367:elimination (
363:
358:
357:
356:
355:
352:
349:
348:
345:
342:
341:
338:
335:
334:
313:
304:
301:
298:
292:
289:
280:
277:
274:
264:
250:
247:
241:
238:
235:
229:
226:
223:
217:
214:
211:
201:
184:
181:
176:
173:
170:
164:
158:
155:
150:
147:
144:
134:
133:
132:
128:
112:
105:is true, and
92:
72:
52:
44:
40:
37:
34:
30:
27:
24:
20:
1279:expanding it
1268:
1246:
1237:
1228:
1218:
1212:
1155:
1111:
1059:
932:
888:
838:
836:
762:
716:
666:
651:
642:
638:
634:
608:
604:
600:
596:
595:
591:
585:
575: /
566: /
457:
454: /
449:
446: /
433: /
430:Constructive
420: /
407: /
395:
394: /
381: /
369:modus ponens
368:
364: /
1324:Logic stubs
1062:tautologies
955:metalogical
632:conjunction
601:elimination
530:Exportation
417:Disjunctive
410:elimination
397:elimination
384:elimination
1318:Categories
1204:References
845:notation:
443:Absorption
1138:→
1129:∧
1094:→
1085:∧
1038:∧
992:∧
941:⊢
915:⊢
906:∧
871:⊢
862:∧
774:∧
742:∴
734:∧
696:∴
688:∧
628:inference
537:Tautology
311:→
302:∧
287:→
278:∧
248:⊢
239:∧
224:⊢
215:∧
182:∴
174:∧
156:∴
148:∧
42:Statement
1066:theorems
125:is true.
843:sequent
654:English
635:A and B
611:) is a
1250:Hurley
1156:where
933:where
809:" or "
647:proofs
1271:logic
1269:This
977:is a
953:is a
613:valid
607:, or
32:Field
1275:stub
1176:and
1112:and
1007:and
889:and
837:The
717:and
671:as:
622:and
65:and
22:Type
1064:or
1053:in
981:of
597:and
586:In
1320::
1200:.
1057:;
656::
618:,
603:,
590:,
1306:e
1299:t
1292:v
1281:.
1184:Q
1164:P
1141:Q
1135:)
1132:Q
1126:P
1123:(
1097:P
1091:)
1088:Q
1082:P
1079:(
1041:Q
1035:P
1015:Q
995:Q
989:P
965:P
918:Q
912:)
909:Q
903:P
900:(
874:P
868:)
865:Q
859:P
856:(
817:Q
797:P
777:Q
771:P
745:Q
737:Q
731:P
699:P
691:Q
685:P
643:B
639:A
371:)
314:Q
308:)
305:Q
299:P
296:(
293:,
290:P
284:)
281:Q
275:P
272:(
251:Q
245:)
242:Q
236:P
233:(
230:,
227:P
221:)
218:Q
212:P
209:(
185:Q
177:Q
171:P
165:,
159:P
151:Q
145:P
113:B
93:A
73:B
53:A
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