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