586:
636:
746:
822:
157:
1151:
361:
1186:
527:
390:
309:
234:
280:
1219:
332:
110:
1083:
1109:
1056:
656:
492:
472:
452:
410:
254:
205:
185:
80:
958:
532:
926:
903:
866:
434:, each separate occurrence of a variable, either in inverse or uncomplemented form, is a literal. For example, if
1253:
686:
591:
696:
759:
130:
1248:
124:
1114:
882:
337:
1156:
497:
638:
would also be said to contain four literals, because although two of the literals are identical (
420:
44:
366:
285:
210:
120:
of a literal is positive or negative depending on whether it is a positive or negative literal.
1258:
916:
753:
671:
667:
259:
1191:
314:
92:
1061:
749:
413:
8:
1088:
938:
1041:
678:
641:
477:
457:
437:
395:
239:
190:
170:
65:
48:
20:
954:
922:
899:
893:
862:
854:
946:
431:
40:
682:
28:
950:
1242:
995:. As in propositional logic, prime formulas and their negations are called
690:
36:
889:
878:
824:
is a negative literal with the constant symbol 2, the variable symbols
983:, p. 57): "The formulas procured by (F1) and (F2) are said to be
412:. Double negation elimination occurs in classical logics but not in
32:
187:
can be defined as the literal corresponding to the negation of
427:
if the literal's complement does not appear in the formula.
658:
appears twice) these qualify as two separate occurrences.
16:
In mathematical logic, an atomic formula or its negation
752:
starting from constant symbols, variable symbols, and
1224:
1194:
1159:
1117:
1091:
1064:
1044:
762:
699:
644:
594:
535:
500:
480:
460:
440:
398:
369:
340:
317:
288:
262:
242:
213:
193:
173:
133:
95:
68:
1213:
1180:
1145:
1103:
1077:
1050:
1018:is an atom or a negation of an atom. An atom is a
816:
740:
650:
630:
580:
521:
486:
466:
446:
404:
384:
355:
326:
303:
274:
248:
228:
199:
179:
151:
104:
74:
1240:
588:contains four literals. However, the expression
581:{\displaystyle {\bar {A}}C+{\bar {B}}{\bar {C}}}
31:(also known as an atom or prime formula) or its
685:or its negation, where an atomic formula is a
943:A Concise Introduction to Mathematical Logic
529:contains three literals and the expression
1230:
1007:
1005:
980:
937:
915:Godse, Atul P.; Godse, Deepali A. (2008).
914:
945:. Universitext (3rd ed.). Springer.
54:Literals can be divided into two types:
1035:
1011:
1002:
859:Mathematical Logic for Computer Science
853:
631:{\displaystyle {\bar {A}}C+{\bar {B}}C}
236:to denote the complementary literal of
1241:
1085:is its complement. This means that if
898:. Amsterdam: Elsevier. pp. 1–78.
741:{\displaystyle P(t_{1},\ldots ,t_{n})}
817:{\displaystyle \neg Q(f(g(x),y,2),x)}
152:{\displaystyle \lnot \lnot x\equiv x}
877:
419:In the context of a formula in the
35:. The definition mostly appears in
13:
763:
494:are variables then the expression
347:
318:
137:
134:
96:
89:is the negation of an atom (e.g.,
14:
1270:
1022:and the negation of an atom is a
883:"An Introduction to Proof Theory"
1146:{\displaystyle l^{c}={\bar {p}}}
356:{\displaystyle l\equiv \lnot x}
1172:
1137:
1029:
974:
811:
802:
787:
781:
775:
769:
735:
703:
619:
601:
572:
560:
542:
507:
376:
295:
220:
1:
847:
1181:{\displaystyle l={\bar {p}}}
522:{\displaystyle {\bar {A}}BC}
7:
991:formulas, or simply called
840:, and the predicate symbol
661:
125:double negation elimination
10:
1275:
921:. Technical Publications.
861:(2nd ed.). Springer.
385:{\displaystyle {\bar {l}}}
304:{\displaystyle {\bar {l}}}
229:{\displaystyle {\bar {l}}}
951:10.1007/978-1-4419-1221-3
275:{\displaystyle l\equiv x}
968:
895:Handbook of Proof Theory
1214:{\displaystyle l^{c}=p}
832:, the function symbols
689:symbol applied to some
421:conjunctive normal form
327:{\displaystyle \lnot x}
105:{\displaystyle \lnot x}
62:is just an atom (e.g.,
45:conjunctive normal form
1254:Propositional calculus
1231:Godse & Godse 2008
1215:
1182:
1147:
1105:
1079:
1052:
918:Digital Logic Circuits
818:
756:symbols. For example,
742:
672:propositional variable
670:a literal is simply a
668:propositional calculus
652:
632:
582:
523:
488:
468:
448:
406:
386:
357:
328:
305:
276:
250:
230:
201:
181:
153:
106:
76:
1216:
1183:
1148:
1106:
1080:
1078:{\displaystyle l^{c}}
1053:
819:
743:
653:
633:
583:
524:
489:
469:
449:
407:
387:
358:
329:
306:
277:
256:. More precisely, if
251:
231:
202:
182:
161:complementary literal
154:
107:
77:
1192:
1157:
1115:
1089:
1062:
1042:
939:Rautenberg, Wolfgang
760:
697:
642:
592:
533:
498:
478:
458:
438:
414:intuitionistic logic
396:
367:
338:
315:
286:
260:
240:
211:
191:
171:
131:
93:
66:
1104:{\displaystyle l=p}
1038:, p. 69): "If
750:recursively defined
1249:Mathematical logic
1211:
1178:
1143:
1101:
1075:
1048:
1014:, p. 30): "A
855:Ben-Ari, Mordechai
814:
738:
679:predicate calculus
648:
628:
578:
519:
484:
464:
444:
402:
382:
353:
324:
301:
272:
246:
226:
197:
177:
149:
102:
72:
47:and the method of
21:mathematical logic
1175:
1140:
1051:{\displaystyle l}
960:978-1-4419-1220-6
674:or its negation.
651:{\displaystyle C}
622:
604:
575:
563:
545:
510:
487:{\displaystyle C}
467:{\displaystyle B}
447:{\displaystyle A}
432:Boolean functions
405:{\displaystyle x}
379:
298:
249:{\displaystyle l}
223:
200:{\displaystyle l}
180:{\displaystyle l}
75:{\displaystyle x}
1266:
1234:
1228:
1222:
1220:
1218:
1217:
1212:
1204:
1203:
1187:
1185:
1184:
1179:
1177:
1176:
1168:
1152:
1150:
1149:
1144:
1142:
1141:
1133:
1127:
1126:
1110:
1108:
1107:
1102:
1084:
1082:
1081:
1076:
1074:
1073:
1057:
1055:
1054:
1049:
1033:
1027:
1024:negative literal
1020:positive literal
1009:
1000:
981:Rautenberg (2010
978:
964:
932:
909:
887:
872:
823:
821:
820:
815:
747:
745:
744:
739:
734:
733:
715:
714:
681:a literal is an
657:
655:
654:
649:
637:
635:
634:
629:
624:
623:
615:
606:
605:
597:
587:
585:
584:
579:
577:
576:
568:
565:
564:
556:
547:
546:
538:
528:
526:
525:
520:
512:
511:
503:
493:
491:
490:
485:
473:
471:
470:
465:
453:
451:
450:
445:
411:
409:
408:
403:
391:
389:
388:
383:
381:
380:
372:
362:
360:
359:
354:
333:
331:
330:
325:
310:
308:
307:
302:
300:
299:
291:
281:
279:
278:
273:
255:
253:
252:
247:
235:
233:
232:
227:
225:
224:
216:
206:
204:
203:
198:
186:
184:
183:
178:
158:
156:
155:
150:
111:
109:
108:
103:
87:negative literal
81:
79:
78:
73:
60:positive literal
1274:
1273:
1269:
1268:
1267:
1265:
1264:
1263:
1239:
1238:
1237:
1229:
1225:
1199:
1195:
1193:
1190:
1189:
1167:
1166:
1158:
1155:
1154:
1132:
1131:
1122:
1118:
1116:
1113:
1112:
1090:
1087:
1086:
1069:
1065:
1063:
1060:
1059:
1043:
1040:
1039:
1034:
1030:
1010:
1003:
979:
975:
971:
961:
929:
906:
890:Buss, Samuel R.
885:
879:Buss, Samuel R.
869:
850:
761:
758:
757:
748:with the terms
729:
725:
710:
706:
698:
695:
694:
664:
643:
640:
639:
614:
613:
596:
595:
593:
590:
589:
567:
566:
555:
554:
537:
536:
534:
531:
530:
502:
501:
499:
496:
495:
479:
476:
475:
459:
456:
455:
439:
436:
435:
423:, a literal is
397:
394:
393:
371:
370:
368:
365:
364:
339:
336:
335:
316:
313:
312:
290:
289:
287:
284:
283:
261:
258:
257:
241:
238:
237:
215:
214:
212:
209:
208:
207:. We can write
192:
189:
188:
172:
169:
168:
132:
129:
128:
123:In logics with
94:
91:
90:
67:
64:
63:
41:classical logic
17:
12:
11:
5:
1272:
1262:
1261:
1256:
1251:
1236:
1235:
1223:
1210:
1207:
1202:
1198:
1174:
1171:
1165:
1162:
1139:
1136:
1130:
1125:
1121:
1100:
1097:
1094:
1072:
1068:
1058:is a literal,
1047:
1028:
1001:
972:
970:
967:
966:
965:
959:
934:
933:
927:
911:
910:
904:
874:
873:
867:
849:
846:
813:
810:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
737:
732:
728:
724:
721:
718:
713:
709:
705:
702:
683:atomic formula
663:
660:
647:
627:
621:
618:
612:
609:
603:
600:
574:
571:
562:
559:
553:
550:
544:
541:
518:
515:
509:
506:
483:
463:
443:
401:
378:
375:
352:
349:
346:
343:
323:
320:
297:
294:
271:
268:
265:
245:
222:
219:
196:
176:
148:
145:
142:
139:
136:
114:
113:
101:
98:
83:
71:
29:atomic formula
15:
9:
6:
4:
3:
2:
1271:
1260:
1259:Logic symbols
1257:
1255:
1252:
1250:
1247:
1246:
1244:
1232:
1227:
1208:
1205:
1200:
1196:
1169:
1163:
1160:
1134:
1128:
1123:
1119:
1098:
1095:
1092:
1070:
1066:
1045:
1037:
1036:Ben-Ari (2001
1032:
1025:
1021:
1017:
1013:
1012:Ben-Ari (2001
1008:
1006:
998:
994:
990:
986:
982:
977:
973:
962:
956:
952:
948:
944:
940:
936:
935:
930:
928:9788184314250
924:
920:
919:
913:
912:
907:
905:0-444-89840-9
901:
897:
896:
891:
884:
880:
876:
875:
870:
868:1-85233-319-7
864:
860:
856:
852:
851:
845:
843:
839:
835:
831:
827:
808:
805:
799:
796:
793:
790:
784:
778:
772:
766:
755:
751:
730:
726:
722:
719:
716:
711:
707:
700:
692:
688:
684:
680:
675:
673:
669:
659:
645:
625:
616:
610:
607:
598:
569:
557:
551:
548:
539:
516:
513:
504:
481:
461:
441:
433:
428:
426:
422:
417:
415:
399:
373:
350:
344:
341:
321:
292:
269:
266:
263:
243:
217:
194:
174:
167:of a literal
166:
162:
146:
143:
140:
126:
121:
119:
99:
88:
84:
69:
61:
57:
56:
55:
52:
50:
46:
42:
38:
34:
30:
26:
22:
1226:
1031:
1023:
1019:
1015:
996:
992:
988:
984:
976:
942:
917:
894:
858:
841:
837:
833:
829:
825:
676:
665:
429:
424:
418:
164:
160:
122:
117:
115:
86:
59:
53:
37:proof theory
24:
18:
43:), e.g. in
1243:Categories
848:References
165:complement
49:resolution
1173:¯
1138:¯
764:¬
720:…
687:predicate
620:¯
602:¯
573:¯
561:¯
543:¯
508:¯
377:¯
348:¬
345:≡
319:¬
296:¯
267:≡
221:¯
144:≡
138:¬
135:¬
97:¬
1111:, then,
997:literals
941:(2010).
881:(1998).
857:(2001).
754:function
662:Examples
118:polarity
33:negation
1153:and if
1016:literal
892:(ed.).
334:and if
127:(where
25:literal
989:atomic
957:
925:
902:
865:
159:) the
27:is an
1188:then
993:prime
985:prime
969:Notes
888:. In
886:(PDF)
691:terms
363:then
282:then
955:ISBN
923:ISBN
900:ISBN
863:ISBN
474:and
425:pure
116:The
39:(of
23:, a
987:or
947:doi
677:In
666:In
430:In
392:is
311:is
163:or
19:In
1245::
1221:."
1026:."
1004:^
999:."
953:.
844:.
836:,
828:,
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58:A
51:.
1233:.
1209:p
1206:=
1201:c
1197:l
1170:p
1164:=
1161:l
1135:p
1129:=
1124:c
1120:l
1099:p
1096:=
1093:l
1071:c
1067:l
1046:l
963:.
949::
931:.
908:.
871:.
842:Q
838:g
834:f
830:y
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812:)
809:x
806:,
803:)
800:2
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794:y
791:,
788:)
785:x
782:(
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776:(
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770:(
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626:C
617:B
611:+
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599:A
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400:x
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