1208:
789:
257:
983:
700:
has to be true. The reasoning is simple: since at least one of the statements P and R is true, and since either of them would be sufficient to entail Q, Q is certainly true.
1112:
1006:
870:
1086:
1060:
844:
818:
1271:
1251:
1231:
1030:
890:
698:
678:
658:
638:
618:
598:
578:
186:
166:
146:
126:
106:
86:
66:
1341:
1127:
1370:
730:
452:
198:
1316:
914:
428:
421:
414:
466:
1365:
1291:
503:
307:
507:
313:
494:
333:
320:
498:
326:
352:
274:
38:
1091:
991:
849:
447:
395:
386:
346:
1065:
1039:
823:
797:
1345:
1296:
359:
8:
1033:
549:
518:
459:
442:
404:
365:
1256:
1236:
1216:
1015:
875:
683:
663:
643:
623:
603:
583:
563:
372:
171:
151:
131:
111:
91:
71:
51:
1118:
545:
485:
478:
290:
281:
267:
28:
704:
538:
435:
1320:
1359:
1274:
541:
378:
553:
366:
296:
1286:
360:
1009:
557:
340:
905:
1203:{\displaystyle (((P\to Q)\land (R\to Q))\land (P\lor R))\to Q}
784:{\displaystyle {\frac {P\to Q,R\to Q,P\lor R}{\therefore Q}}}
252:{\displaystyle {\frac {P\to Q,R\to Q,P\lor R}{\therefore Q}}}
429:
387:
353:
308:
379:
347:
460:
415:
321:
297:
1259:
1239:
1219:
1130:
1094:
1068:
1042:
1018:
994:
917:
878:
852:
826:
800:
733:
686:
666:
646:
626:
606:
586:
566:
201:
174:
154:
134:
114:
94:
74:
54:
978:{\displaystyle (P\to Q),(R\to Q),(P\lor R)\vdash Q}
1265:
1245:
1225:
1202:
1106:
1080:
1054:
1024:
1000:
977:
884:
864:
838:
812:
783:
692:
672:
652:
632:
612:
592:
572:
251:
180:
160:
140:
120:
100:
80:
60:
717:It is true that either I'm inside or I'm outside.
373:
1357:
453:
422:
314:
794:where the rule is that whenever instances of "
467:
436:
334:
327:
16:Rule of inference of propositional logic
714:If I'm outside, I have my wallet on me.
1358:
892:" can be placed on a subsequent line.
711:If I'm inside, I have my wallet on me.
1317:"Rule of Or-Elimination - ProofWiki"
1117:and expressed as a truth-functional
1273:are propositions expressed in some
1121:or theorem of propositional logic:
13:
895:
720:Therefore, I have my wallet on me.
14:
1382:
724:It is the rule can be stated as:
1371:Theorems in propositional logic
872:" appear on lines of a proof, "
548:that allows one to eliminate a
1334:
1309:
1194:
1191:
1188:
1176:
1170:
1167:
1161:
1155:
1149:
1143:
1137:
1134:
1131:
1072:
1046:
966:
954:
948:
942:
936:
930:
924:
918:
830:
804:
752:
740:
220:
208:
1:
1302:
7:
1292:Argument in the alternative
1280:
10:
1387:
504:Existential generalization
309:Biconditional introduction
192:
44:
34:
24:
1114:in some logical system;
495:Universal generalization
335:Disjunction introduction
322:Conjunction introduction
292:Implication introduction
1107:{\displaystyle P\lor R}
1001:{\displaystyle \vdash }
904:rule may be written in
902:disjunction elimination
865:{\displaystyle P\lor R}
523:disjunction elimination
20:Disjunction elimination
1267:
1247:
1227:
1204:
1108:
1082:
1081:{\displaystyle R\to Q}
1056:
1055:{\displaystyle P\to Q}
1026:
1002:
979:
886:
866:
840:
839:{\displaystyle R\to Q}
814:
813:{\displaystyle P\to Q}
785:
694:
674:
654:
634:
614:
594:
574:
354:hypothetical syllogism
275:Propositional calculus
253:
182:
162:
142:
122:
102:
82:
62:
39:Propositional calculus
1268:
1248:
1228:
1205:
1109:
1083:
1057:
1034:syntactic consequence
1027:
1003:
980:
887:
867:
841:
815:
786:
695:
675:
655:
635:
615:
595:
575:
560:that if a statement
550:disjunctive statement
396:Negation introduction
389:modus ponendo tollens
254:
183:
163:
143:
123:
103:
83:
63:
1297:Disjunct normal form
1257:
1237:
1217:
1128:
1092:
1066:
1040:
1016:
1012:symbol meaning that
992:
915:
876:
850:
824:
798:
731:
684:
664:
644:
624:
604:
584:
580:implies a statement
564:
454:Material implication
405:Rules of replacement
268:Transformation rules
199:
172:
152:
132:
112:
92:
72:
68:implies a statement
52:
519:propositional logic
367:destructive dilemma
21:
1366:Rules of inference
1263:
1243:
1223:
1200:
1104:
1078:
1052:
1022:
998:
975:
882:
862:
836:
810:
781:
690:
670:
650:
630:
610:
590:
570:
486:Rules of inference
282:Rules of inference
249:
193:Symbolic statement
178:
158:
138:
118:
98:
78:
58:
19:
1266:{\displaystyle R}
1246:{\displaystyle Q}
1226:{\displaystyle P}
1025:{\displaystyle Q}
885:{\displaystyle Q}
779:
693:{\displaystyle Q}
673:{\displaystyle R}
653:{\displaystyle P}
640:, then if either
633:{\displaystyle Q}
613:{\displaystyle R}
593:{\displaystyle Q}
573:{\displaystyle P}
546:rule of inference
525:(sometimes named
515:
514:
262:
261:
247:
181:{\displaystyle Q}
161:{\displaystyle R}
141:{\displaystyle P}
128:, then if either
121:{\displaystyle Q}
101:{\displaystyle R}
81:{\displaystyle Q}
61:{\displaystyle P}
29:Rule of inference
1378:
1350:
1349:
1344:. Archived from
1342:"Proof by cases"
1338:
1332:
1331:
1329:
1328:
1319:. Archived from
1313:
1272:
1270:
1269:
1264:
1252:
1250:
1249:
1244:
1232:
1230:
1229:
1224:
1209:
1207:
1206:
1201:
1113:
1111:
1110:
1105:
1087:
1085:
1084:
1079:
1061:
1059:
1058:
1053:
1031:
1029:
1028:
1023:
1007:
1005:
1004:
999:
984:
982:
981:
976:
891:
889:
888:
883:
871:
869:
868:
863:
845:
843:
842:
837:
819:
817:
816:
811:
790:
788:
787:
782:
780:
778:
770:
735:
699:
697:
696:
691:
679:
677:
676:
671:
659:
657:
656:
651:
639:
637:
636:
631:
619:
617:
616:
611:
600:and a statement
599:
597:
596:
591:
579:
577:
576:
571:
469:
462:
455:
443:De Morgan's laws
438:
431:
424:
417:
391:
383:
375:
368:
362:
355:
349:
342:
336:
329:
323:
316:
310:
303:
293:
264:
263:
258:
256:
255:
250:
248:
246:
238:
203:
187:
185:
184:
179:
167:
165:
164:
159:
147:
145:
144:
139:
127:
125:
124:
119:
107:
105:
104:
99:
88:and a statement
87:
85:
84:
79:
67:
65:
64:
59:
22:
18:
1386:
1385:
1381:
1380:
1379:
1377:
1376:
1375:
1356:
1355:
1354:
1353:
1340:
1339:
1335:
1326:
1324:
1315:
1314:
1310:
1305:
1283:
1258:
1255:
1254:
1238:
1235:
1234:
1218:
1215:
1214:
1129:
1126:
1125:
1093:
1090:
1089:
1067:
1064:
1063:
1041:
1038:
1037:
1017:
1014:
1013:
993:
990:
989:
916:
913:
912:
898:
896:Formal notation
877:
874:
873:
851:
848:
847:
825:
822:
821:
799:
796:
795:
771:
736:
734:
732:
729:
728:
685:
682:
681:
665:
662:
661:
645:
642:
641:
625:
622:
621:
605:
602:
601:
585:
582:
581:
565:
562:
561:
479:Predicate logic
473:
437:Double negation
291:
239:
204:
202:
200:
197:
196:
188:has to be true.
173:
170:
169:
153:
150:
149:
133:
130:
129:
113:
110:
109:
93:
90:
89:
73:
70:
69:
53:
50:
49:
48:If a statement
17:
12:
11:
5:
1384:
1374:
1373:
1368:
1352:
1351:
1348:on 2002-03-07.
1333:
1307:
1306:
1304:
1301:
1300:
1299:
1294:
1289:
1282:
1279:
1262:
1242:
1222:
1211:
1210:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1103:
1100:
1097:
1077:
1074:
1071:
1051:
1048:
1045:
1021:
997:
986:
985:
974:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
897:
894:
881:
861:
858:
855:
835:
832:
829:
809:
806:
803:
792:
791:
777:
774:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
722:
721:
718:
715:
712:
703:An example in
689:
680:is true, then
669:
649:
629:
609:
589:
569:
535:or elimination
527:proof by cases
513:
512:
511:
510:
501:
489:
488:
482:
481:
475:
474:
472:
471:
464:
457:
450:
445:
440:
433:
430:Distributivity
426:
419:
411:
408:
407:
401:
400:
399:
398:
393:
370:
357:
344:
331:
318:
305:
285:
284:
278:
277:
271:
270:
260:
259:
245:
242:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
207:
194:
190:
189:
177:
168:is true, then
157:
137:
117:
97:
77:
57:
46:
42:
41:
36:
32:
31:
26:
15:
9:
6:
4:
3:
2:
1383:
1372:
1369:
1367:
1364:
1363:
1361:
1347:
1343:
1337:
1323:on 2015-04-18
1322:
1318:
1312:
1308:
1298:
1295:
1293:
1290:
1288:
1285:
1284:
1278:
1276:
1275:formal system
1260:
1240:
1220:
1197:
1185:
1182:
1179:
1173:
1164:
1158:
1152:
1146:
1140:
1124:
1123:
1122:
1120:
1115:
1101:
1098:
1095:
1075:
1069:
1049:
1043:
1035:
1019:
1011:
995:
972:
969:
963:
960:
957:
951:
945:
939:
933:
927:
921:
911:
910:
909:
907:
903:
893:
879:
859:
856:
853:
833:
827:
807:
801:
775:
772:
767:
764:
761:
758:
755:
749:
746:
743:
737:
727:
726:
725:
719:
716:
713:
710:
709:
708:
706:
701:
687:
667:
647:
627:
620:also implies
607:
587:
567:
559:
555:
554:logical proof
551:
547:
543:
542:argument form
540:
536:
532:
531:case analysis
528:
524:
520:
509:
508:instantiation
505:
502:
500:
499:instantiation
496:
493:
492:
491:
490:
487:
484:
483:
480:
477:
476:
470:
465:
463:
458:
456:
451:
449:
448:Transposition
446:
444:
441:
439:
434:
432:
427:
425:
423:Commutativity
420:
418:
416:Associativity
413:
412:
410:
409:
406:
403:
402:
397:
394:
392:
390:
384:
382:
381:modus tollens
376:
371:
369:
363:
358:
356:
350:
345:
343:
337:
332:
330:
324:
319:
317:
311:
306:
304:
301:
298:elimination (
294:
289:
288:
287:
286:
283:
280:
279:
276:
273:
272:
269:
266:
265:
243:
240:
235:
232:
229:
226:
223:
217:
214:
211:
205:
195:
191:
175:
155:
135:
115:
108:also implies
95:
75:
55:
47:
43:
40:
37:
33:
30:
27:
23:
1346:the original
1336:
1325:. Retrieved
1321:the original
1311:
1212:
1116:
987:
901:
899:
793:
723:
702:
556:. It is the
534:
530:
526:
522:
516:
506: /
497: /
388:
385: /
380:
377: /
364: /
361:Constructive
351: /
339:
338: /
325: /
312: /
300:modus ponens
299:
295: /
1287:Disjunction
1010:metalogical
461:Exportation
348:Disjunctive
341:elimination
328:elimination
315:elimination
1360:Categories
1327:2015-04-09
1303:References
908:notation:
374:Absorption
1195:→
1183:∨
1174:∧
1162:→
1153:∧
1144:→
1119:tautology
1099:∨
1073:→
1047:→
996:⊢
970:⊢
961:∨
943:→
925:→
857:∨
831:→
805:→
773:∴
765:∨
753:→
741:→
558:inference
537:) is the
468:Tautology
241:∴
233:∨
221:→
209:→
45:Statement
1281:See also
820:", and "
906:sequent
846:" and "
705:English
552:from a
1253:, and
1213:where
1062:, and
988:where
1032:is a
1008:is a
539:valid
533:, or
35:Field
1088:and
900:The
544:and
25:Type
1036:of
660:or
517:In
148:or
1362::
1277:.
1233:,
707::
529:,
521:,
1330:.
1261:R
1241:Q
1221:P
1198:Q
1192:)
1189:)
1186:R
1180:P
1177:(
1171:)
1168:)
1165:Q
1159:R
1156:(
1150:)
1147:Q
1141:P
1138:(
1135:(
1132:(
1102:R
1096:P
1076:Q
1070:R
1050:Q
1044:P
1020:Q
973:Q
967:)
964:R
958:P
955:(
952:,
949:)
946:Q
940:R
937:(
934:,
931:)
928:Q
922:P
919:(
880:Q
860:R
854:P
834:Q
828:R
808:Q
802:P
776:Q
768:R
762:P
759:,
756:Q
750:R
747:,
744:Q
738:P
688:Q
668:R
648:P
628:Q
608:R
588:Q
568:P
302:)
244:Q
236:R
230:P
227:,
224:Q
218:R
215:,
212:Q
206:P
176:Q
156:R
136:P
116:Q
96:R
76:Q
56:P
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.