1337:
1255:
1000:
2254:
The value of modal companions and the BlokāEsakia theorem as a tool for investigation of intermediate logics comes from the fact that many interesting properties of logics are preserved by some or all of the mappings
1741:
1912:
2157:
753:
2772:
2226:
2016:
1854:
2065:
1536:
1382:
915:
830:
1590:
2664:
1671:
621:
399:
252:
2455:
487:
321:
183:
2589:
2400:
1261:
120:
428:
2495:
669:
776:
2526:
2105:
1776:
532:
1185:
143:
2591:; in particular, a logic has a continuum of modal companions if it has a continuum of extensions (this holds, for instance, for all intermediate logics below
1393:
1482:. An important corollary to the BlokāEsakia theorem is a simple syntactic description of largest modal companions: for every superintuitionistic logic
942:
2307:
1683:
1862:
1421:
2110:
688:
2675:
2185:
1922:. The point of the skeleton construction is that it preserves validity modulo Gƶdel translation: for any intuitionistic formula
1975:
1784:
1412:, join-complete and meet-complete lattice homomorphisms respectively. The cornerstone of the theory of modal companions is the
2024:
1492:
1467:
1349:
861:
797:
1549:
2608:
1620:
567:
327:
2321:
1409:
189:
2413:
436:
258:
43:
by a certain canonical translation, described below. Modal companions share various properties of the original
2541:
2289:
1085:
2407:
2076:
1332:{\displaystyle \tau ,\sigma \colon \mathrm {Ext} \,\mathbf {IPC} \to \mathrm {NExt} \,\mathbf {S4} .}
150:
2021:
The largest modal companions also have a semantic description. For any intuitionistic general frame
2555:
2366:
84:
2080:
514:. The translation is sometimes presented in slightly different ways: for example, one may insert
407:
123:
60:
2464:
645:
1250:{\displaystyle \rho \colon \mathrm {NExt} \,\mathbf {S4} \to \mathrm {Ext} \,\mathbf {IPC} ,}
761:
2854:
2504:
2271:
2090:
1761:
1612:
517:
63:
8:
1593:
848:
778:
denotes normal closure. It can be shown that every superintuitionistic logic also has a
2817:. The converse is not true in general, but it holds for the largest modal companion of
1597:
1343:
128:
44:
33:
25:
2599:
2335:
1755:
505:
47:, which enables to study intermediate logics using tools developed for modal logic.
1438:
1389:
1144:
535:
2841:, vol. 136 of Studies in Logic and the Foundations of Mathematics, Elsevier, 1997.
2782:
2848:
2537:
1747:
1601:
995:{\displaystyle \mathbf {Triv} =\mathbf {K} \oplus (A\leftrightarrow \Box A).}
509:
501:
2360:
2232:
1744:
1442:
1417:
929:
36:
1736:{\displaystyle \langle \rho F,\leq \rangle =\langle F,R\rangle /{\sim }}
534:
before every subformula. All such variants are provably equivalent in
2549:
1907:{\displaystyle \rho \mathbf {F} =\langle \rho F,\leq ,\rho V\rangle }
2152:{\displaystyle \sigma \mathbf {F} =\langle F,\leq ,\sigma V\rangle }
1605:
748:{\displaystyle \tau L=\mathbf {S4} \oplus \{T(A)\mid L\vdash A\},}
2834:, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.
2174:
is a superintuitionistic logic complete with respect to a class
839:
itself is the smallest modal companion of intuitionistic logic (
2767:{\displaystyle T(R)={\frac {T(A_{1}),\dots ,T(A_{n})}{T(B)}}.}
1680:, which identifies points belonging to the same cluster. Let
1546:
The Gƶdel translation has a frame-theoretic counterpart. Let
17:
2221:{\displaystyle \{\sigma \mathbf {F} ;\,\mathbf {F} \in C\}}
1151:. Similarly, the set of normal extensions of a modal logic
674:
Every superintuitionistic logic has modal companions. The
2011:{\displaystyle \{\rho \mathbf {F} ;\,\mathbf {F} \in C\}}
1849:{\displaystyle \rho V=\{A/{\sim }\mid A\in V,A=\Box A\}.}
2060:{\displaystyle \mathbf {F} =\langle F,\leq ,V\rangle }
1171:
can be considered as mappings between the lattices Ext
2678:
2611:
2558:
2532:-element cluster. The set of modal companions of any
2507:
2467:
2416:
2369:
2188:
2113:
2093:
2027:
1978:
1865:
1787:
1764:
1686:
1623:
1552:
1531:{\displaystyle \sigma L=\tau L\oplus \mathbf {Grz} .}
1495:
1352:
1264:
1188:
1139:
The set of extensions of a superintuitionistic logic
945:
864:
800:
764:
691:
648:
570:
520:
439:
410:
330:
261:
192:
153:
131:
87:
2595:). It is unknown whether the converse is also true.
2178:
of general frames, then its largest modal companion
50:
1377:{\displaystyle \rho \circ \tau =\rho \circ \sigma }
936:, whereas its largest modal companion is the logic
924:. The smallest modal companion of classical logic (
910:{\displaystyle \Box (\Box (A\to \Box A)\to A)\to A}
825:{\displaystyle \tau L\subseteq M\subseteq \sigma L}
2766:
2658:
2583:
2520:
2489:
2449:
2394:
2363:number of modal companions, and moreover, the set
2220:
2151:
2099:
2059:
2010:
1906:
1848:
1770:
1735:
1665:
1585:{\displaystyle \mathbf {F} =\langle F,R,V\rangle }
1584:
1530:
1376:
1331:
1249:
994:
909:
824:
770:
747:
663:
615:
526:
481:
422:
404:As negation is in intuitionistic logic defined by
393:
315:
246:
177:
137:
114:
2659:{\displaystyle R={\frac {A_{1},\dots ,A_{n}}{B}}}
2846:
1666:{\displaystyle x\sim y\iff x\,R\,y\land y\,R\,x}
2602:as well as formulas: the translation of a rule
1914:is an intuitionistic general frame, called the
74:) is defined by induction on the complexity of
2338:definability on Kripke frames is preserved by
630:is a superintuitionistic logic. A modal logic
616:{\displaystyle \rho M=\{A\mid M\vdash T(A)\}.}
394:{\displaystyle T(A\to B)=\Box (T(A)\to T(B)).}
2830:Alexander Chagrov and Michael Zakharyaschev,
2801:is admissible in a superintuitionistic logic
2235:is itself a Kripke frame. On the other hand,
1968:of transitive reflexive general frames, then
2215:
2189:
2146:
2125:
2054:
2036:
2005:
1979:
1956:Therefore, the si-fragment of a modal logic
1901:
1877:
1840:
1797:
1720:
1708:
1702:
1687:
1579:
1561:
739:
712:
607:
580:
626:The si-fragment of any normal extension of
247:{\displaystyle T(A\land B)=T(A)\land T(B),}
2450:{\displaystyle \rho ^{-1}(\mathbf {CPC} )}
2159:is a general modal frame. The skeleton of
1637:
1633:
1134:
2203:
1993:
1659:
1655:
1645:
1641:
1317:
1288:
1232:
1209:
482:{\displaystyle T(\neg A)=\Box \neg T(A).}
316:{\displaystyle T(A\lor B)=T(A)\lor T(B),}
2839:Admissibility of Logical Inference Rules
2813:) is admissible in a modal companion of
2598:The Gƶdel translation can be applied to
2544:. Rybakov has shown that the lattice Ext
2249:
1541:
2847:
1972:is complete with respect to the class
2246:is a Kripke frame of infinite depth.
1342:It is easy to see that all three are
1964:is complete with respect to a class
2350:
541:
13:
1313:
1310:
1307:
1304:
1284:
1281:
1278:
1228:
1225:
1222:
1205:
1202:
1199:
1196:
843:). The largest modal companion of
461:
446:
417:
169:
160:
14:
2866:
2075:under Boolean operations (binary
51:GƶdelāMcKinseyāTarski translation
2440:
2437:
2434:
2205:
2196:
2118:
2029:
1995:
1986:
1960:can be defined semantically: if
1870:
1554:
1521:
1518:
1515:
1322:
1319:
1296:
1293:
1290:
1240:
1237:
1234:
1214:
1211:
964:
956:
953:
950:
947:
705:
702:
1384:is the identity function on Ext
638:of a superintuitionistic logic
178:{\displaystyle T(\bot )=\bot ,}
2755:
2749:
2741:
2728:
2713:
2700:
2688:
2682:
2578:
2572:
2484:
2471:
2444:
2430:
2389:
2383:
1634:
1300:
1218:
986:
977:
971:
901:
898:
892:
889:
880:
874:
868:
724:
718:
604:
598:
473:
467:
452:
443:
414:
385:
382:
376:
370:
367:
361:
355:
346:
340:
334:
307:
301:
292:
286:
277:
265:
238:
232:
223:
217:
208:
196:
163:
157:
97:
91:
1:
2824:
2584:{\displaystyle \rho ^{-1}(L)}
2395:{\displaystyle \rho ^{-1}(L)}
1143:ordered by inclusion forms a
2542:cardinality of the continuum
2182:is complete with respect to
115:{\displaystyle T(p)=\Box p,}
7:
2497:for every positive integer
2242:is never a Kripke frame if
855:, axiomatized by the axiom
546:For any normal modal logic
10:
2871:
2789:if the set of theorems of
1416:, proved independently by
1159:. The companion operators
1155:is a complete lattice NExt
423:{\displaystyle A\to \bot }
2797:. It is easy to see that
2408:infinite descending chain
2355:Every intermediate logic
66:formula. A modal formula
2490:{\displaystyle L(C_{n})}
2083:). It can be shown that
676:smallest modal companion
664:{\displaystyle L=\rho M}
2402:of modal companions of
1480:BlokāEsakia isomorphism
1135:BlokāEsakia isomorphism
780:largest modal companion
771:{\displaystyle \oplus }
2768:
2660:
2585:
2522:
2491:
2451:
2396:
2222:
2153:
2101:
2061:
2012:
1908:
1850:
1772:
1737:
1667:
1586:
1532:
1378:
1333:
1251:
996:
911:
826:
782:, which is denoted by
772:
749:
665:
617:
528:
483:
424:
395:
317:
248:
179:
139:
124:propositional variable
116:
2837:Vladimir V. Rybakov,
2769:
2661:
2586:
2523:
2521:{\displaystyle C_{n}}
2492:
2452:
2397:
2290:finite model property
2250:Preservation theorems
2223:
2154:
2102:
2100:{\displaystyle \Box }
2062:
2013:
1909:
1851:
1773:
1771:{\displaystyle \sim }
1738:
1668:
1587:
1533:
1379:
1334:
1252:
997:
912:
827:
773:
750:
666:
618:
529:
527:{\displaystyle \Box }
484:
425:
396:
318:
249:
180:
140:
117:
28:(intermediate) logic
2676:
2609:
2556:
2505:
2465:
2414:
2367:
2186:
2111:
2091:
2025:
1976:
1863:
1785:
1762:
1684:
1621:
1613:equivalence relation
1550:
1542:Semantic description
1493:
1350:
1262:
1186:
1026:other companions of
943:
862:
798:
762:
689:
646:
568:
518:
437:
408:
328:
259:
190:
151:
129:
85:
2322:Kripke completeness
1756:equivalence classes
1414:BlokāEsakia theorem
26:superintuitionistic
2764:
2656:
2581:
2518:
2487:
2447:
2392:
2231:The skeleton of a
2218:
2149:
2097:
2071:be the closure of
2057:
2008:
1904:
1846:
1768:
1733:
1663:
1582:
1528:
1374:
1329:
1247:
992:
907:
822:
790:is a companion of
768:
745:
661:
613:
524:
479:
420:
391:
313:
244:
175:
135:
112:
45:intermediate logic
2759:
2654:
2461:, and the logics
2166:is isomorphic to
1132:
1131:
497:Gƶdel translation
138:{\displaystyle p}
2862:
2793:is closed under
2773:
2771:
2770:
2765:
2760:
2758:
2744:
2740:
2739:
2712:
2711:
2695:
2665:
2663:
2662:
2657:
2655:
2650:
2649:
2648:
2630:
2629:
2619:
2590:
2588:
2587:
2582:
2571:
2570:
2540:, or it has the
2527:
2525:
2524:
2519:
2517:
2516:
2496:
2494:
2493:
2488:
2483:
2482:
2456:
2454:
2453:
2448:
2443:
2429:
2428:
2401:
2399:
2398:
2393:
2382:
2381:
2351:Other properties
2324:is preserved by
2310:is preserved by
2292:is preserved by
2274:is preserved by
2227:
2225:
2224:
2219:
2208:
2199:
2158:
2156:
2155:
2150:
2121:
2106:
2104:
2103:
2098:
2087:is closed under
2066:
2064:
2063:
2058:
2032:
2017:
2015:
2014:
2009:
1998:
1989:
1913:
1911:
1910:
1905:
1873:
1855:
1853:
1852:
1847:
1812:
1807:
1777:
1775:
1774:
1769:
1742:
1740:
1739:
1734:
1732:
1727:
1672:
1670:
1669:
1664:
1591:
1589:
1588:
1583:
1557:
1537:
1535:
1534:
1529:
1524:
1396:have shown that
1383:
1381:
1380:
1375:
1338:
1336:
1335:
1330:
1325:
1316:
1299:
1287:
1256:
1254:
1253:
1248:
1243:
1231:
1217:
1208:
1145:complete lattice
1008:
1007:
1001:
999:
998:
993:
967:
959:
916:
914:
913:
908:
831:
829:
828:
823:
786:. A modal logic
777:
775:
774:
769:
754:
752:
751:
746:
708:
670:
668:
667:
662:
622:
620:
619:
614:
554:, we define its
542:Modal companions
533:
531:
530:
525:
488:
486:
485:
480:
429:
427:
426:
421:
400:
398:
397:
392:
322:
320:
319:
314:
253:
251:
250:
245:
184:
182:
181:
176:
144:
142:
141:
136:
121:
119:
118:
113:
39:that interprets
2870:
2869:
2865:
2864:
2863:
2861:
2860:
2859:
2845:
2844:
2827:
2745:
2735:
2731:
2707:
2703:
2696:
2694:
2677:
2674:
2673:
2644:
2640:
2625:
2621:
2620:
2618:
2610:
2607:
2606:
2563:
2559:
2557:
2554:
2553:
2512:
2508:
2506:
2503:
2502:
2478:
2474:
2466:
2463:
2462:
2433:
2421:
2417:
2415:
2412:
2411:
2410:. For example,
2374:
2370:
2368:
2365:
2364:
2353:
2267:. For example,
2252:
2204:
2195:
2187:
2184:
2183:
2117:
2112:
2109:
2108:
2092:
2089:
2088:
2028:
2026:
2023:
2022:
1994:
1985:
1977:
1974:
1973:
1940:if and only if
1869:
1864:
1861:
1860:
1808:
1803:
1786:
1783:
1782:
1763:
1760:
1759:
1743:be the induced
1728:
1723:
1685:
1682:
1681:
1622:
1619:
1618:
1553:
1551:
1548:
1547:
1544:
1514:
1494:
1491:
1490:
1478:are called the
1351:
1348:
1347:
1318:
1303:
1289:
1277:
1263:
1260:
1259:
1233:
1221:
1210:
1195:
1187:
1184:
1183:
1137:
1005:More examples:
963:
946:
944:
941:
940:
863:
860:
859:
799:
796:
795:
794:if and only if
763:
760:
759:
701:
690:
687:
686:
647:
644:
643:
636:modal companion
569:
566:
565:
544:
519:
516:
515:
438:
435:
434:
430:, we also have
409:
406:
405:
329:
326:
325:
260:
257:
256:
191:
188:
187:
152:
149:
148:
130:
127:
126:
86:
83:
82:
53:
22:modal companion
12:
11:
5:
2868:
2858:
2857:
2843:
2842:
2835:
2826:
2823:
2775:
2774:
2763:
2757:
2754:
2751:
2748:
2743:
2738:
2734:
2730:
2727:
2724:
2721:
2718:
2715:
2710:
2706:
2702:
2699:
2693:
2690:
2687:
2684:
2681:
2667:
2666:
2653:
2647:
2643:
2639:
2636:
2633:
2628:
2624:
2617:
2614:
2580:
2577:
2574:
2569:
2566:
2562:
2515:
2511:
2486:
2481:
2477:
2473:
2470:
2446:
2442:
2439:
2436:
2432:
2427:
2424:
2420:
2391:
2388:
2385:
2380:
2377:
2373:
2352:
2349:
2348:
2347:
2333:
2319:
2305:
2287:
2251:
2248:
2217:
2214:
2211:
2207:
2202:
2198:
2194:
2191:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2120:
2116:
2096:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2035:
2031:
2007:
2004:
2001:
1997:
1992:
1988:
1984:
1981:
1954:
1953:
1948:) is valid in
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1882:
1879:
1876:
1872:
1868:
1857:
1856:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1811:
1806:
1802:
1799:
1796:
1793:
1790:
1767:
1754:is the set of
1731:
1726:
1722:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1674:
1673:
1662:
1658:
1654:
1651:
1648:
1644:
1640:
1636:
1632:
1629:
1626:
1581:
1578:
1575:
1572:
1569:
1566:
1563:
1560:
1556:
1543:
1540:
1539:
1538:
1527:
1523:
1520:
1517:
1513:
1510:
1507:
1504:
1501:
1498:
1460:
1459:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1340:
1339:
1328:
1324:
1321:
1315:
1312:
1309:
1306:
1302:
1298:
1295:
1292:
1286:
1283:
1280:
1276:
1273:
1270:
1267:
1257:
1246:
1242:
1239:
1236:
1230:
1227:
1224:
1220:
1216:
1213:
1207:
1204:
1201:
1198:
1194:
1191:
1136:
1133:
1130:
1129:
1126:
1121:
1116:
1110:
1109:
1099:
1094:
1089:
1081:
1080:
1074:
1069:
1064:
1058:
1057:
1047:
1042:
1037:
1031:
1030:
1024:
1019:
1014:
1003:
1002:
991:
988:
985:
982:
979:
976:
973:
970:
966:
962:
958:
955:
952:
949:
918:
917:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
821:
818:
815:
812:
809:
806:
803:
767:
756:
755:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
707:
704:
700:
697:
694:
660:
657:
654:
651:
624:
623:
612:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
543:
540:
523:
495:is called the
490:
489:
478:
475:
472:
469:
466:
463:
460:
457:
454:
451:
448:
445:
442:
419:
416:
413:
402:
401:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
323:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
254:
243:
240:
237:
234:
231:
228:
225:
222:
219:
216:
213:
210:
207:
204:
201:
198:
195:
185:
174:
171:
168:
165:
162:
159:
156:
146:
134:
111:
108:
105:
102:
99:
96:
93:
90:
64:intuitionistic
52:
49:
9:
6:
4:
3:
2:
2867:
2856:
2853:
2852:
2850:
2840:
2836:
2833:
2829:
2828:
2822:
2820:
2816:
2812:
2808:
2804:
2800:
2796:
2792:
2788:
2784:
2780:
2761:
2752:
2746:
2736:
2732:
2725:
2722:
2719:
2716:
2708:
2704:
2697:
2691:
2685:
2679:
2672:
2671:
2670:
2651:
2645:
2641:
2637:
2634:
2631:
2626:
2622:
2615:
2612:
2605:
2604:
2603:
2601:
2596:
2594:
2575:
2567:
2564:
2560:
2551:
2547:
2543:
2539:
2535:
2531:
2513:
2509:
2500:
2479:
2475:
2468:
2460:
2425:
2422:
2418:
2409:
2405:
2386:
2378:
2375:
2371:
2362:
2358:
2345:
2341:
2337:
2334:
2331:
2327:
2323:
2320:
2317:
2313:
2309:
2306:
2303:
2299:
2295:
2291:
2288:
2285:
2281:
2277:
2273:
2270:
2269:
2268:
2266:
2262:
2258:
2247:
2245:
2241:
2238:
2234:
2229:
2212:
2209:
2200:
2192:
2181:
2177:
2173:
2169:
2165:
2162:
2143:
2140:
2137:
2134:
2131:
2128:
2122:
2114:
2094:
2086:
2082:
2078:
2074:
2070:
2051:
2048:
2045:
2042:
2039:
2033:
2019:
2002:
1999:
1990:
1982:
1971:
1967:
1963:
1959:
1951:
1947:
1943:
1939:
1936:
1932:
1929:
1928:
1927:
1925:
1921:
1917:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1874:
1866:
1843:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1809:
1804:
1800:
1794:
1791:
1788:
1781:
1780:
1779:
1765:
1757:
1753:
1749:
1748:partial order
1746:
1729:
1724:
1717:
1714:
1711:
1705:
1699:
1696:
1693:
1690:
1679:
1660:
1656:
1652:
1649:
1646:
1642:
1638:
1630:
1627:
1624:
1617:
1616:
1615:
1614:
1610:
1607:
1603:
1602:general frame
1599:
1595:
1576:
1573:
1570:
1567:
1564:
1558:
1525:
1511:
1508:
1505:
1502:
1499:
1496:
1489:
1488:
1487:
1485:
1481:
1477:
1473:
1469:
1465:
1462:Accordingly,
1457:
1453:
1450:
1446:
1444:
1440:
1437:are mutually
1434:
1430:
1429:The mappings
1427:
1426:
1425:
1423:
1419:
1415:
1411:
1408:are actually
1407:
1403:
1399:
1395:
1391:
1387:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1345:
1326:
1274:
1271:
1268:
1265:
1258:
1244:
1192:
1189:
1182:
1181:
1180:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1147:, denoted Ext
1146:
1142:
1127:
1125:
1122:
1120:
1117:
1115:
1112:
1111:
1107:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1087:
1083:
1082:
1078:
1075:
1073:
1070:
1068:
1065:
1063:
1060:
1059:
1055:
1051:
1048:
1046:
1043:
1041:
1038:
1036:
1033:
1032:
1029:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1009:
1006:
989:
983:
980:
974:
968:
960:
939:
938:
937:
935:
931:
927:
923:
904:
895:
886:
883:
877:
871:
865:
858:
857:
856:
854:
850:
846:
842:
838:
835:For example,
833:
819:
816:
813:
810:
807:
804:
801:
793:
789:
785:
781:
765:
742:
736:
733:
730:
727:
721:
715:
709:
698:
695:
692:
685:
684:
683:
681:
677:
672:
658:
655:
652:
649:
641:
637:
633:
629:
610:
601:
595:
592:
589:
586:
583:
577:
574:
571:
564:
563:
562:
560:
557:
553:
550:that extends
549:
539:
537:
521:
513:
511:
507:
503:
498:
494:
476:
470:
464:
458:
455:
449:
440:
433:
432:
431:
411:
388:
379:
373:
364:
358:
352:
349:
343:
337:
331:
324:
310:
304:
298:
295:
289:
283:
280:
274:
271:
268:
262:
255:
241:
235:
229:
226:
220:
214:
211:
205:
202:
199:
193:
186:
172:
166:
154:
147:
132:
125:
109:
106:
103:
100:
94:
88:
81:
80:
79:
77:
73:
69:
65:
62:
61:propositional
58:
48:
46:
42:
38:
35:
31:
27:
23:
19:
2838:
2831:
2818:
2814:
2810:
2806:
2802:
2798:
2794:
2790:
2786:
2778:
2776:
2669:is the rule
2668:
2597:
2592:
2545:
2533:
2529:
2498:
2458:
2457:consists of
2406:contains an
2403:
2356:
2354:
2343:
2339:
2329:
2325:
2315:
2311:
2301:
2297:
2293:
2283:
2279:
2275:
2272:decidability
2264:
2260:
2256:
2253:
2243:
2239:
2236:
2233:Kripke frame
2230:
2179:
2175:
2171:
2167:
2163:
2160:
2084:
2077:intersection
2072:
2068:
2020:
1969:
1965:
1961:
1957:
1955:
1949:
1945:
1941:
1937:
1934:
1933:is valid in
1930:
1923:
1919:
1915:
1858:
1751:
1677:
1675:
1611:induces the
1608:
1545:
1483:
1479:
1475:
1471:
1463:
1461:
1455:
1451:
1448:
1443:isomorphisms
1436:
1432:
1428:
1424:. It states
1413:
1405:
1401:
1397:
1390:L. Maksimova
1385:
1341:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1148:
1140:
1138:
1123:
1118:
1113:
1105:
1101:
1096:
1091:
1084:
1076:
1071:
1066:
1061:
1053:
1049:
1044:
1039:
1034:
1027:
1021:
1016:
1011:
1004:
933:
925:
921:
919:
852:
844:
840:
836:
834:
791:
787:
783:
779:
757:
679:
675:
673:
639:
635:
631:
627:
625:
558:
555:
551:
547:
545:
500:
496:
492:
491:
403:
75:
71:
67:
56:
54:
40:
29:
21:
15:
2855:Modal logic
2832:Modal Logic
2785:in a logic
2336:first-order
1778:), and put
1468:restriction
849:Grzegorczyk
556:si-fragment
512:translation
37:modal logic
2825:References
2783:admissible
2536:is either
2308:tabularity
2081:complement
1594:transitive
1422:Leo Esakia
1394:V. Rybakov
1128:see below
2805:whenever
2720:…
2635:…
2565:−
2561:ρ
2538:countable
2423:−
2419:ρ
2376:−
2372:ρ
2210:∈
2193:σ
2147:⟩
2141:σ
2135:≤
2126:⟨
2115:σ
2095:◻
2055:⟩
2046:≤
2037:⟨
2000:∈
1983:ρ
1902:⟩
1896:ρ
1890:≤
1881:ρ
1878:⟨
1867:ρ
1835:◻
1820:∈
1814:∣
1810:∼
1789:ρ
1766:∼
1730:∼
1721:⟩
1709:⟨
1703:⟩
1700:≤
1691:ρ
1688:⟨
1650:∧
1635:⟺
1628:∼
1598:reflexive
1580:⟩
1562:⟨
1512:⊕
1506:τ
1497:σ
1372:σ
1369:∘
1366:ρ
1360:τ
1357:∘
1354:ρ
1301:→
1275::
1272:σ
1266:τ
1219:→
1193::
1190:ρ
981:◻
978:↔
969:⊕
902:→
893:→
884:◻
881:→
872:◻
866:◻
817:σ
814:⊆
808:⊆
802:τ
766:⊕
734:⊢
728:∣
710:⊕
693:τ
656:ρ
593:⊢
587:∣
572:ρ
522:◻
462:¬
459:◻
447:¬
418:⊥
415:→
371:→
353:◻
341:→
296:∨
272:∨
227:∧
203:∧
170:⊥
161:⊥
104:◻
2849:Category
2550:embedded
2501:, where
2361:infinite
1916:skeleton
1745:quotient
1606:preorder
1466:and the
1441:lattice
1418:Wim Blok
1410:complete
1344:monotone
1175:and NExt
506:McKinsey
122:for any
2777:A rule
2548:can be
2528:is the
2359:has an
2107:, thus
1970:ρM
1750:(i.e.,
1474:to NExt
1439:inverse
1106:S4.3Dum
847:is the
2300:, and
2282:, and
2263:, and
2067:, let
1604:. The
1600:modal
1404:, and
1346:, and
1167:, and
1108:, ...
1102:S4.1.3
1079:, ...
1077:S4.1.2
1056:, ...
851:logic
758:where
510:Tarski
34:normal
2600:rules
2170:. If
1859:Then
1592:be a
1097:Grz.3
1072:Grz.2
930:Lewis
928:) is
920:over
634:is a
502:Gƶdel
59:be a
32:is a
24:of a
18:logic
2342:and
2328:and
2314:and
2079:and
1596:and
1454:NExt
1433:and
1420:and
1392:and
1124:Triv
1092:S4.3
1067:S4.2
1050:S4.1
55:Let
20:, a
2781:is
2552:in
1918:of
1758:of
1676:on
1476:Grz
1470:of
1456:Grz
1452:and
1449:IPC
1447:Ext
1386:IPC
1173:IPC
1114:CPC
1054:Dum
1045:Grz
1035:IPC
926:CPC
853:Grz
845:IPC
841:IPC
682:is
678:of
642:if
561:as
499:or
16:In
2851::
2821:.
2593:KC
2459:S5
2296:,
2278:,
2259:,
2228:.
2180:ĻL
2085:ĻV
2069:ĻV
2018:.
1926:,
1752:ĻF
1486:,
1445:of
1400:,
1388:.
1179::
1177:S4
1169:ĻL
1165:ĻL
1163:,
1161:ĻM
1119:S5
1104:,
1086:LC
1062:KC
1052:,
1040:S4
1022:ĻL
1017:ĻL
934:S5
932:'
837:S4
832:.
784:ĻL
671:.
628:S4
559:ĻM
552:S4
538:.
536:S4
78::
2819:L
2815:L
2811:R
2809:(
2807:T
2803:L
2799:R
2795:R
2791:L
2787:L
2779:R
2762:.
2756:)
2753:B
2750:(
2747:T
2742:)
2737:n
2733:A
2729:(
2726:T
2723:,
2717:,
2714:)
2709:1
2705:A
2701:(
2698:T
2692:=
2689:)
2686:R
2683:(
2680:T
2652:B
2646:n
2642:A
2638:,
2632:,
2627:1
2623:A
2616:=
2613:R
2579:)
2576:L
2573:(
2568:1
2546:L
2534:L
2530:n
2514:n
2510:C
2499:n
2485:)
2480:n
2476:C
2472:(
2469:L
2445:)
2441:C
2438:P
2435:C
2431:(
2426:1
2404:L
2390:)
2387:L
2384:(
2379:1
2357:L
2346:.
2344:Ļ
2340:Ļ
2332:,
2330:Ļ
2326:Ļ
2318:,
2316:Ļ
2312:Ļ
2304:,
2302:Ļ
2298:Ļ
2294:Ļ
2286:,
2284:Ļ
2280:Ļ
2276:Ļ
2265:Ļ
2261:Ļ
2257:Ļ
2244:F
2240:F
2237:Ļ
2216:}
2213:C
2206:F
2201:;
2197:F
2190:{
2176:C
2172:L
2168:F
2164:F
2161:Ļ
2144:V
2138:,
2132:,
2129:F
2123:=
2119:F
2073:V
2052:V
2049:,
2043:,
2040:F
2034:=
2030:F
2006:}
2003:C
1996:F
1991:;
1987:F
1980:{
1966:C
1962:M
1958:M
1952:.
1950:F
1946:A
1944:(
1942:T
1938:F
1935:Ļ
1931:A
1924:A
1920:F
1899:V
1893:,
1887:,
1884:F
1875:=
1871:F
1844:.
1841:}
1838:A
1832:=
1829:A
1826:,
1823:V
1817:A
1805:/
1801:A
1798:{
1795:=
1792:V
1725:/
1718:R
1715:,
1712:F
1706:=
1697:,
1694:F
1678:F
1661:x
1657:R
1653:y
1647:y
1643:R
1639:x
1631:y
1625:x
1609:R
1577:V
1574:,
1571:R
1568:,
1565:F
1559:=
1555:F
1526:.
1522:z
1519:r
1516:G
1509:L
1503:=
1500:L
1484:L
1472:Ļ
1464:Ļ
1458:.
1435:Ļ
1431:Ļ
1406:Ļ
1402:Ļ
1398:Ļ
1363:=
1327:.
1323:4
1320:S
1314:t
1311:x
1308:E
1305:N
1297:C
1294:P
1291:I
1285:t
1282:x
1279:E
1269:,
1245:,
1241:C
1238:P
1235:I
1229:t
1226:x
1223:E
1215:4
1212:S
1206:t
1203:x
1200:E
1197:N
1157:M
1153:M
1149:L
1141:L
1028:L
1012:L
990:.
987:)
984:A
975:A
972:(
965:K
961:=
957:v
954:i
951:r
948:T
922:K
905:A
899:)
896:A
890:)
887:A
878:A
875:(
869:(
820:L
811:M
805:L
792:L
788:M
743:,
740:}
737:A
731:L
725:)
722:A
719:(
716:T
713:{
706:4
703:S
699:=
696:L
680:L
659:M
653:=
650:L
640:L
632:M
611:.
608:}
605:)
602:A
599:(
596:T
590:M
584:A
581:{
578:=
575:M
548:M
508:ā
504:ā
493:T
477:.
474:)
471:A
468:(
465:T
456:=
453:)
450:A
444:(
441:T
412:A
389:.
386:)
383:)
380:B
377:(
374:T
368:)
365:A
362:(
359:T
356:(
350:=
347:)
344:B
338:A
335:(
332:T
311:,
308:)
305:B
302:(
299:T
293:)
290:A
287:(
284:T
281:=
278:)
275:B
269:A
266:(
263:T
242:,
239:)
236:B
233:(
230:T
224:)
221:A
218:(
215:T
212:=
209:)
206:B
200:A
197:(
194:T
173:,
167:=
164:)
158:(
155:T
145:,
133:p
110:,
107:p
101:=
98:)
95:p
92:(
89:T
76:A
72:A
70:(
68:T
57:A
41:L
30:L
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