595:
496:
118:
169:
253:
224:
195:
414:
636:
64:
629:
291:
660:
655:
385:
622:
302:
The following table lists several common normal modal systems. The notation refers to the table at
287:
136:
229:
263:. Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g.
516:
51:
606:
377:
303:
268:
203:
174:
58:
553:
280:
491:{\displaystyle \forall w\,\exists u\,(w\,R\,u\land \forall v\,(u\,R\,v\Rightarrow u=v))}
398:
649:
540:
276:
526:
508:
129:
264:
34:
17:
283:, for example, are non-normal, often because they give up the Kripke schema.
306:. Frame conditions for some of the systems were simplified: the logics are
594:
370:
310:
with respect to the frame classes given in the table, but they may
602:
28:
582:, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.
259:
The smallest logic satisfying the above conditions is called
610:
417:
232:
206:
177:
139:
67:
113:{\displaystyle \Box (A\to B)\to (\Box A\to \Box B)}
490:
247:
218:
189:
163:
112:
304:Kripke semantics § Common modal axiom schemata
630:
578:Alexander Chagrov and Michael Zakharyaschev,
8:
637:
623:
271:, are normal (and hence are extensions of
466:
462:
455:
442:
438:
431:
424:
416:
231:
205:
176:
138:
66:
316:
7:
591:
589:
609:. You can help Knowledge (XXG) by
566:transitive, serial, and Euclidean
449:
425:
418:
25:
593:
485:
482:
470:
456:
432:
143:
107:
98:
89:
86:
83:
77:
71:
1:
314:to a larger class of frames.
164:{\displaystyle A\to B,A\in L}
286:Every normal modal logic is
42:of modal formulas such that
248:{\displaystyle \Box A\in L}
677:
588:
298:Common normal modal logics
123:and it is closed under:
275:). However a number of
605:-related article is a
492:
249:
220:
219:{\displaystyle A\in L}
191:
190:{\displaystyle B\in L}
165:
114:
493:
250:
221:
192:
166:
115:
57:All instances of the
527:strict partial order
415:
386:equivalence relation
230:
204:
200:Necessitation rule:
175:
137:
65:
488:
308:sound and complete
245:
216:
187:
161:
110:
50:All propositional
618:
617:
570:
569:
536:Grz or T, 4, Grz
127:Detachment rule (
16:(Redirected from
668:
639:
632:
625:
597:
590:
497:
495:
494:
489:
382:T, 5 or D, B, 4
326:Frame condition
317:
281:epistemic logics
254:
252:
251:
246:
225:
223:
222:
217:
196:
194:
193:
188:
170:
168:
167:
162:
119:
117:
116:
111:
21:
18:S4 (modal logic)
676:
675:
671:
670:
669:
667:
666:
665:
646:
645:
644:
643:
586:
575:
413:
412:
300:
228:
227:
202:
201:
173:
172:
135:
134:
63:
62:
23:
22:
15:
12:
11:
5:
674:
672:
664:
663:
658:
648:
647:
642:
641:
634:
627:
619:
616:
615:
598:
584:
583:
574:
571:
568:
567:
564:
561:
557:
556:
551:
548:
544:
543:
537:
534:
530:
529:
523:
520:
513:
512:
506:
503:
499:
498:
487:
484:
481:
478:
475:
472:
469:
465:
461:
458:
454:
451:
448:
445:
441:
437:
434:
430:
427:
423:
420:
409:
406:
402:
401:
399:total preorder
396:
393:
389:
388:
383:
380:
374:
373:
368:
365:
361:
360:
357:
354:
350:
349:
346:
343:
339:
338:
335:
332:
328:
327:
324:
321:
299:
296:
257:
256:
244:
241:
238:
235:
215:
212:
209:
198:
186:
183:
180:
160:
157:
154:
151:
148:
145:
142:
121:
120:
109:
106:
103:
100:
97:
94:
91:
88:
85:
82:
79:
76:
73:
70:
55:
24:
14:
13:
10:
9:
6:
4:
3:
2:
673:
662:
659:
657:
654:
653:
651:
640:
635:
633:
628:
626:
621:
620:
614:
612:
608:
604:
599:
596:
592:
587:
581:
577:
576:
572:
565:
562:
559:
558:
555:
552:
549:
546:
545:
542:
541:partial order
538:
535:
532:
531:
528:
524:
521:
518:
515:
514:
510:
507:
504:
501:
500:
479:
476:
473:
467:
463:
459:
452:
446:
443:
439:
435:
428:
421:
410:
407:
404:
403:
400:
397:
394:
391:
390:
387:
384:
381:
379:
376:
375:
372:
369:
366:
363:
362:
358:
355:
352:
351:
347:
344:
341:
340:
336:
333:
330:
329:
325:
322:
319:
318:
315:
313:
309:
305:
297:
295:
293:
289:
284:
282:
278:
274:
270:
266:
262:
242:
239:
236:
233:
213:
210:
207:
199:
184:
181:
178:
158:
155:
152:
149:
146:
140:
132:
131:
126:
125:
124:
104:
101:
95:
92:
80:
74:
68:
60:
56:
53:
49:
48:
47:
45:
41:
37:
36:
30:
19:
611:expanding it
600:
585:
579:
522:GL or 4, GL
311:
307:
301:
285:
272:
260:
258:
130:modus ponens
128:
122:
43:
39:
32:
26:
661:Logic stubs
656:Modal logic
580:Modal Logic
533:Grz, S4Grz
359:transitive
337:all frames
265:C. I. Lewis
52:tautologies
35:modal logic
650:Categories
573:References
411:preorder,
348:reflexive
312:correspond
290:and hence
267:'s S4 and
46:contains:
511:preorder
471:⇒
450:∀
447:∧
426:∃
419:∀
292:classical
240:∈
234:◻
211:∈
182:∈
156:∈
144:→
102:◻
99:→
93:◻
87:→
78:→
69:◻
38:is a set
563:D, 4, 5
509:directed
505:T, 4, G
408:T, 4, M
395:T, 4, H
371:preorder
226:implies
171:implies
61:schema:
539:finite
525:finite
288:regular
277:deontic
33:normal
554:serial
519:, K4W
323:Axioms
59:Kripke
603:logic
601:This
502:S4.2
405:S4.1
392:S4.3
367:T, 4
29:logic
607:stub
560:D45
320:Name
279:and
31:, a
364:S4
353:K4
133:):
27:In
652::
550:D
547:D
517:GL
378:S5
356:4
345:T
342:T
334:—
331:K
294:.
269:S5
638:e
631:t
624:v
613:.
486:)
483:)
480:v
477:=
474:u
468:v
464:R
460:u
457:(
453:v
444:u
440:R
436:w
433:(
429:u
422:w
273:K
261:K
255:.
243:L
237:A
214:L
208:A
197:;
185:L
179:B
159:L
153:A
150:,
147:B
141:A
108:)
105:B
96:A
90:(
84:)
81:B
75:A
72:(
54:;
44:L
40:L
20:)
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