2673:
433:
2381:
141:
2668:{\displaystyle {\frac {\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Sigma \Rightarrow A\qquad \Omega _{1}\Rightarrow \Theta _{1}\mid \dots \mid \Omega _{m}\Rightarrow \Theta _{m}\mid \Pi \Rightarrow B}{\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Omega _{1}\Rightarrow \Theta _{1}\mid \dots \mid \Omega _{m}\Rightarrow \Theta _{m}\mid \Sigma \Rightarrow B\mid \Pi \Rightarrow A}}}
428:{\displaystyle {\frac {\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Sigma \Rightarrow A\qquad \Omega _{1}\Rightarrow \Theta _{1}\mid \dots \mid \Omega _{m}\Rightarrow \Theta _{m}\mid \Pi \Rightarrow B}{\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Omega _{1}\Rightarrow \Theta _{1}\mid \dots \mid \Omega _{m}\Rightarrow \Theta _{m}\mid \Sigma \Rightarrow B\mid \Pi \Rightarrow A}}}
641:
451:
1015:
2156:
636:{\displaystyle {\frac {\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Box \Sigma ,\Theta \Rightarrow \Box \Pi ,\Omega }{\Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}\mid \Box \Sigma \Rightarrow \Box \Pi \mid \Theta \Rightarrow \Omega }}}
752:
The sequents making up a hypersequent consist of pairs of multisets of formulae, and are called the components of the hypersequent. Variants defining hypersequents and sequents in terms of sets or lists instead of multisets are also considered, and depending on the considered logic the sequents can
1480:
1693:
1188:
2366:
Most hypersequent calculi for intermediate logics include the single-succedent versions of the propositional rules given above and a selection of the structural rules. The characteristics of a particular intermediate logic are mostly captured using a number of additional
1104:
2056:
1353:
1271:
3091:
1976:
913:
2062:
2171:
of the calculus without the cut rule directly using the semantics of S5. In line with the importance of modal logic S5, a number of alternative calculi have been formulated. Hypersequent calculi have also been proposed for many other modal logics.
121:
1585:
747:
2361:
2265:
2714:. It seems to have been developed independently in, also for treating modal logics, and in the influential, where calculi for modal, intermediate and substructural logics are considered, and the term hypersequent is introduced.
1903:. In a standard hypersequent calculus for this logic the formula interpretation is as above, and the propositional and structural rules are the ones from the previous section. Additionally, the calculus contains the modal rules
1404:
658:. Hypersequents usually have a formula interpretation, i.e., are interpreted by a formula in the object language, nearly always as some kind of disjunction. The precise formula interpretation depends on the considered logic.
905:
853:
757:. The rules for the propositional connectives usually are adaptions of the corresponding standard sequent rules with an additional side hypersequent, also called hypersequent context. E.g., a common set of rules for the
1410:
126:
The sequents making up a hypersequent are called components. The added expressivity of the hypersequent framework is provided by rules manipulating different components, such as the communication rule for the
1596:
1118:
1112:
are considered in their internal and external variants. The internal weakening and internal contraction rules are the adaptions of the corresponding sequent rules with an added hypersequent context:
1022:
1983:
1277:
1195:
1909:
1877:
1010:{\displaystyle {\frac {{\mathcal {G}}\mid \Gamma ,B\Rightarrow \Delta \qquad {\mathcal {G}}\mid \Gamma \Rightarrow A,\Delta }{{\mathcal {G}}\mid \Gamma ,A\to B\Rightarrow \Delta }}}
2151:{\displaystyle {\frac {{\mathcal {G}}\mid \Box \Gamma ,\Sigma \Rightarrow \Box \Delta ,\Pi }{{\mathcal {G}}\mid \Box \Gamma \Rightarrow \Box \Delta \mid \Sigma \Rightarrow \Pi }}}
1762:
791:
1785:
54:
1518:
680:
2276:
1805:
1716:
1897:
1825:
2862:
Restall, Greg (2007). Dimitracopoulos, Costas; Newelski, Ludomir; Normann, Dag; Steel, John R (eds.). "Proofnets for S5: Sequents and circuits for modal logic".
3089:; Maffezioli, Paolo; Spendier, Lara (2013). Galmiche, Didier; Larchey-Wendling, Dominique (eds.). "Hypersequent and Labelled Calculi for Intermediate Logics".
2198:
1362:
1475:{\displaystyle {\frac {{\mathcal {G}}\mid \Gamma \Rightarrow \Delta \mid \Gamma \Rightarrow \Delta }{{\mathcal {G}}\mid \Gamma \Rightarrow \Delta }}}
861:
803:
2951:
1899:
is interpreted as a disjunction of boxes. The prime example of a modal logic for which hypersequents provide an analytic calculus is the logic
1357:
The external weakening and external contraction rules are the corresponding rules on the level of hypersequent components instead of formulae:
3181:
2968:
2929:
2755:
1688:{\displaystyle \Box (\bigwedge \Gamma _{1}\to \bigvee \Delta _{1})\lor \dots \lor \Box (\bigwedge \Gamma _{n}\to \bigvee \Delta _{n})}
2893:
1879:. Note that the single components are interpreted using the standard formula interpretation for sequents, and the hypersequent bar
1183:{\displaystyle {\frac {{\mathcal {G}}\mid \Gamma \Rightarrow \Delta }{{\mathcal {G}}\mid \Gamma ,\Sigma \Rightarrow \Delta ,\Pi }}}
1099:{\displaystyle {\frac {{\mathcal {G}}\mid \Gamma ,A\Rightarrow B,\Delta }{{\mathcal {G}}\mid \Gamma \Rightarrow A\to B,\Delta }}}
2051:{\displaystyle {\frac {{\mathcal {G}}\mid \Gamma ,A\Rightarrow \Delta }{{\mathcal {G}}\mid \Gamma ,\Box A\Rightarrow \Delta }}}
1487:
of these rules is closely connected to the formula interpretation of the hypersequent structure, nearly always as some form of
2994:
1348:{\displaystyle {\frac {{\mathcal {G}}\mid \Gamma \Rightarrow A,A,\Delta }{{\mathcal {G}}\mid \Gamma \Rightarrow A,\Delta }}}
1266:{\displaystyle {\frac {{\mathcal {G}}\mid \Gamma ,A,A\Rightarrow \Delta }{{\mathcal {G}}\mid \Gamma ,A\Rightarrow \Delta }}}
794:
1971:{\displaystyle {\frac {{\mathcal {G}}\mid \Box \Gamma \Rightarrow A}{{\mathcal {G}}\mid \Box \Gamma \Rightarrow \Box A}}}
2678:
Hypersequent calculi for many other intermediate logics have been introduced, and there are very general results about
2992:
Indrzejczak, Andrzej (2015). "Eliminability of cut in hypersequent calculi for some modal logics of linear frames".
2372:
132:
1830:
2192:. Since the hypersequents in this setting are based on single-succedent sequents, they have the following form:
2181:
33:
2820:
3159:
3120:
40:
for logics that are not captured in the sequent framework. A hypersequent is usually taken to be a finite
3058:; Ferrari, Mauro (2001). "Hypersequent calculi for some intermediate logics with bounded Kripke models".
3229:
1721:
758:
3154:; Galatos, Nikolaos; Terui, Kazushige (2008). "From Axioms to Analytic Rules in Nonclassical Logics".
1512:
proved elusive. In the context of modal logics the standard formula interpretation of a hypersequent
2691:
2375:, sometimes also called Gödel–Dummett logic, contains additionally the so-called communication rule:
2189:
2168:
754:
655:
116:{\displaystyle \Gamma _{1}\Rightarrow \Delta _{1}\mid \cdots \mid \Gamma _{n}\Rightarrow \Delta _{n}}
3164:
1580:{\displaystyle \Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}}
764:
742:{\displaystyle \Gamma _{1}\Rightarrow \Delta _{1}\mid \dots \mid \Gamma _{n}\Rightarrow \Delta _{n}}
3027:
2185:
1491:. The precise formula interpretation depends on the considered logic, see below for some examples.
651:
3125:
1767:
3187:
2974:
2844:
2746:(1996). "The method of hypersequents in the proof theory of propositional non-classical logics".
128:
17:
2356:{\displaystyle (\bigwedge \Gamma _{1}\to A_{1})\lor \dots \lor (\bigwedge \Gamma _{n}\to A_{n})}
3177:
2964:
2925:
2889:
2751:
2690:
As for intermediate logics, hypersequents have been used to obtain analytic calculi for many
1790:
1701:
3169:
3151:
3130:
3108:
3086:
3068:
3055:
3035:
3003:
2956:
2917:
2879:
2871:
2836:
2711:
1900:
1509:
442:
29:
1882:
1810:
3060:
2828:
2679:
2368:
1109:
2799:
Pottinger, Garrell (1983). "Uniform, cut-free formulations of T, S4 and S5 (abstract)".
2260:{\displaystyle \Gamma _{1}\Rightarrow A_{1}\mid \dots \mid \Gamma _{n}\Rightarrow A_{n}}
2160:
37:
3111:; Fermüller, Christian G. (2003). "Hypersequent Calculi for Gödel Logics – A Survey".
3223:
2801:
2774:
1399:{\displaystyle {\frac {\mathcal {G}}{{\mathcal {G}}\mid \Gamma \Rightarrow \Delta }}}
2848:
3191:
2695:
2167:
can be shown by a syntactic argument on the structure of derivations or by showing
25:
2978:
2948:
Lahav, Ori (2013). "From Frame
Properties to Hypersequent Rules in Modal Logics".
2912:
Kurokawa, Hidenori (2014). "Hypersequent
Calculi for Modal Logics Extending S4".
2875:
2921:
2743:
1505:
1488:
647:
439:
900:{\displaystyle {\frac {}{{\mathcal {G}}\mid \Gamma ,\bot \Rightarrow \Delta }}}
3206:
3134:
3072:
3040:
3022:
3007:
2840:
2706:
The hypersequent structure seem to have appeared first in under the name of
848:{\displaystyle {\frac {}{{\mathcal {G}}\mid \Gamma ,p\Rightarrow p,\Delta }}}
3023:"Hypersequent rules with restricted contexts for propositional modal logics"
1484:
3173:
2960:
2164:
667:
41:
2884:
671:
45:
2916:. Lecture Notes in Computer Science. Vol. 8417. pp. 51–68.
1108:
Due to the additional structure in the hypersequent setting, the
2270:
The standard formula interpretation for such an hypersequent is
3156:
2008 23rd Annual IEEE Symposium on Logic in
Computer Science
2110:
2071:
2019:
1992:
1942:
1918:
1504:
Hypersequents have been used to obtain analytic calculi for
1452:
1419:
1376:
1369:
1319:
1286:
1237:
1204:
1148:
1127:
1064:
1031:
975:
948:
922:
871:
813:
666:
Formally, a hypersequent is usually taken to be a finite
2184:
have been used successfully to capture a large class of
2821:"A cut-free simple sequent calculus for modal logic S5"
2384:
2371:. E.g., the standard calculus for intermediate logic
2279:
2201:
2065:
1986:
1912:
1885:
1833:
1813:
1793:
1770:
1724:
1704:
1599:
1521:
1413:
1365:
1280:
1198:
1121:
1025:
916:
864:
806:
767:
683:
454:
144:
57:
2667:
2355:
2259:
2150:
2050:
1970:
1891:
1871:
1819:
1799:
1779:
1756:
1710:
1687:
1579:
1474:
1398:
1347:
1265:
1182:
1098:
1009:
899:
847:
785:
741:
635:
427:
115:
2180:Hypersequent calculi based on intuitionistic or
1787:for the result of prefixing every formula in
646:Hypersequent calculi have been used to treat
8:
1872:{\displaystyle \Box A_{1},\dots ,\Box A_{n}}
780:
768:
2710:, to obtain a calculus for the modal logic
2777:(1971). "On some calculi of modal logic".
662:Formal definitions and propositional rules
3163:
3124:
3039:
2883:
2632:
2619:
2600:
2587:
2574:
2561:
2542:
2529:
2505:
2492:
2473:
2460:
2437:
2424:
2405:
2392:
2385:
2383:
2344:
2331:
2303:
2290:
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2251:
2238:
2219:
2206:
2200:
2109:
2108:
2070:
2069:
2066:
2064:
2018:
2017:
1991:
1990:
1987:
1985:
1941:
1940:
1917:
1916:
1913:
1911:
1884:
1863:
1841:
1832:
1812:
1792:
1769:
1748:
1729:
1723:
1703:
1676:
1660:
1629:
1613:
1598:
1571:
1558:
1539:
1526:
1520:
1451:
1450:
1418:
1417:
1414:
1412:
1375:
1374:
1368:
1366:
1364:
1318:
1317:
1285:
1284:
1281:
1279:
1236:
1235:
1203:
1202:
1199:
1197:
1147:
1146:
1126:
1125:
1122:
1120:
1063:
1062:
1030:
1029:
1026:
1024:
974:
973:
947:
946:
921:
920:
917:
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870:
869:
865:
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812:
811:
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733:
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594:
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321:
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184:
165:
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145:
143:
107:
94:
75:
62:
56:
2914:New Frontiers in Artificial Intelligence
2163:of a suitably formulated version of the
2748:Logic: From Foundations to Applications
2722:
2952:Symposium on Logic in Computer Science
797:is given by the following four rules:
3146:
3144:
7:
3205:Metcalfe, George; Olivetti, Nicola;
2943:
2941:
2907:
2905:
2794:
2792:
2769:
2767:
2738:
2736:
2734:
2732:
2730:
2728:
2726:
438:or the modal splitting rule for the
2779:Proc. Steklov Inst. Of Mathematics
2653:
2641:
2629:
2616:
2597:
2584:
2571:
2558:
2539:
2526:
2514:
2502:
2489:
2470:
2457:
2446:
2434:
2421:
2402:
2389:
2328:
2287:
2235:
2203:
2190:intuitionistic propositional logic
2142:
2136:
2130:
2121:
2103:
2097:
2088:
2082:
2042:
2027:
2012:
2000:
1953:
1929:
1794:
1774:
1757:{\displaystyle A_{1},\dots ,A_{n}}
1705:
1673:
1657:
1626:
1610:
1568:
1555:
1536:
1523:
1466:
1460:
1445:
1439:
1433:
1427:
1390:
1384:
1339:
1327:
1312:
1294:
1257:
1245:
1230:
1212:
1174:
1168:
1162:
1156:
1141:
1135:
1090:
1072:
1057:
1039:
1001:
983:
968:
956:
942:
930:
891:
885:
879:
839:
821:
771:
730:
717:
698:
685:
627:
621:
615:
606:
591:
578:
559:
546:
540:
534:
525:
519:
504:
491:
472:
459:
413:
401:
389:
376:
357:
344:
331:
318:
299:
286:
274:
262:
249:
230:
217:
206:
194:
181:
162:
149:
104:
91:
72:
59:
14:
24:framework is an extension of the
2455:
945:
215:
2995:Information Processing Letters
2819:Poggiolesi, Francesca (2008).
2656:
2644:
2625:
2593:
2567:
2535:
2517:
2498:
2466:
2449:
2430:
2398:
2350:
2337:
2321:
2309:
2296:
2280:
2244:
2212:
2139:
2124:
2091:
2039:
2009:
1956:
1932:
1682:
1666:
1650:
1635:
1619:
1603:
1564:
1532:
1463:
1442:
1430:
1387:
1330:
1297:
1254:
1227:
1165:
1138:
1081:
1075:
1048:
998:
992:
959:
939:
888:
830:
786:{\displaystyle \{\bot ,\to \}}
777:
726:
694:
624:
609:
587:
555:
528:
500:
468:
416:
404:
385:
353:
327:
295:
277:
258:
226:
209:
190:
158:
100:
68:
1:
3211:Proof theory for fuzzy logics
795:classical propositional logic
2876:10.1017/CBO9780511546464.012
1780:{\displaystyle \Box \Gamma }
2922:10.1007/978-3-319-10061-6_4
3246:
2950:2013 28th Annual ACM–IEEE
2866:. Lecture Notes in Logic.
3041:10.1016/j.tcs.2016.10.004
3008:10.1016/j.ipl.2014.07.002
2841:10.1017/S1755020308080040
2182:single-succedent sequents
3021:Lellmann, Björn (2016).
3135:10.1093/logcom/13.6.835
3073:10.1093/logcom/11.2.283
1800:{\displaystyle \Gamma }
1711:{\displaystyle \Gamma }
34:structural proof theory
2669:
2357:
2261:
2188:, i.e., extensions of
2152:
2052:
1972:
1893:
1873:
1821:
1801:
1781:
1758:
1712:
1689:
1581:
1476:
1400:
1349:
1267:
1184:
1100:
1011:
901:
849:
787:
743:
637:
429:
117:
2864:Logic Colloquium 2005
2670:
2358:
2262:
2153:
2053:
1973:
1894:
1892:{\displaystyle \mid }
1874:
1827:, i.e., the multiset
1822:
1820:{\displaystyle \Box }
1802:
1782:
1759:
1713:
1690:
1582:
1508:, for which analytic
1477:
1401:
1350:
1268:
1185:
1101:
1012:
902:
850:
788:
759:functionally complete
744:
638:
430:
118:
3174:10.1109/LICS.2008.39
3158:. pp. 229–240.
2961:10.1109/LICS.2013.47
2955:. pp. 408–417.
2692:substructural logics
2686:Substructural logics
2382:
2277:
2199:
2063:
1984:
1910:
1883:
1831:
1811:
1791:
1768:
1722:
1702:
1597:
1519:
1411:
1363:
1278:
1196:
1119:
1023:
914:
862:
804:
765:
681:
656:substructural logics
452:
142:
55:
3213:. Springer, Berlin.
3028:Theor. Comput. Sci.
2186:intermediate logics
2176:Intermediate logics
761:set of connectives
652:intermediate logics
133:Gödel–Dummett logic
2665:
2353:
2257:
2148:
2048:
1968:
1889:
1869:
1817:
1797:
1777:
1754:
1708:
1685:
1577:
1472:
1396:
1345:
1263:
1180:
1096:
1007:
897:
845:
783:
739:
633:
425:
129:intermediate logic
113:
18:mathematical logic
3183:978-0-7695-3183-0
3152:Ciabattoni, Agata
3109:Ciabattoni, Agata
3087:Ciabattoni, Agata
3056:Ciabattoni, Agata
2970:978-1-4799-0413-6
2931:978-3-319-10060-9
2757:978-0-19-853862-2
2750:. pp. 1–32.
2682:in such calculi.
2663:
2146:
2046:
1966:
1470:
1394:
1343:
1261:
1178:
1094:
1005:
895:
867:
843:
809:
631:
423:
26:proof-theoretical
3237:
3215:
3214:
3202:
3196:
3195:
3167:
3148:
3139:
3138:
3128:
3107:Baaz, Matthias;
3104:
3098:
3097:
3083:
3077:
3076:
3052:
3046:
3045:
3043:
3018:
3012:
3011:
2989:
2983:
2982:
2945:
2936:
2935:
2909:
2900:
2899:
2887:
2859:
2853:
2852:
2825:
2816:
2810:
2809:
2796:
2787:
2786:
2771:
2762:
2761:
2740:
2674:
2672:
2671:
2666:
2664:
2662:
2637:
2636:
2624:
2623:
2605:
2604:
2592:
2591:
2579:
2578:
2566:
2565:
2547:
2546:
2534:
2533:
2523:
2510:
2509:
2497:
2496:
2478:
2477:
2465:
2464:
2442:
2441:
2429:
2428:
2410:
2409:
2397:
2396:
2386:
2369:structural rules
2362:
2360:
2359:
2354:
2349:
2348:
2336:
2335:
2308:
2307:
2295:
2294:
2266:
2264:
2263:
2258:
2256:
2255:
2243:
2242:
2224:
2223:
2211:
2210:
2157:
2155:
2154:
2149:
2147:
2145:
2114:
2113:
2106:
2075:
2074:
2067:
2057:
2055:
2054:
2049:
2047:
2045:
2023:
2022:
2015:
1996:
1995:
1988:
1977:
1975:
1974:
1969:
1967:
1965:
1946:
1945:
1938:
1922:
1921:
1914:
1898:
1896:
1895:
1890:
1878:
1876:
1875:
1870:
1868:
1867:
1846:
1845:
1826:
1824:
1823:
1818:
1806:
1804:
1803:
1798:
1786:
1784:
1783:
1778:
1763:
1761:
1760:
1755:
1753:
1752:
1734:
1733:
1718:is the multiset
1717:
1715:
1714:
1709:
1694:
1692:
1691:
1686:
1681:
1680:
1665:
1664:
1634:
1633:
1618:
1617:
1586:
1584:
1583:
1578:
1576:
1575:
1563:
1562:
1544:
1543:
1531:
1530:
1481:
1479:
1478:
1473:
1471:
1469:
1456:
1455:
1448:
1423:
1422:
1415:
1405:
1403:
1402:
1397:
1395:
1393:
1380:
1379:
1372:
1367:
1354:
1352:
1351:
1346:
1344:
1342:
1323:
1322:
1315:
1290:
1289:
1282:
1272:
1270:
1269:
1264:
1262:
1260:
1241:
1240:
1233:
1208:
1207:
1200:
1189:
1187:
1186:
1181:
1179:
1177:
1152:
1151:
1144:
1131:
1130:
1123:
1110:structural rules
1105:
1103:
1102:
1097:
1095:
1093:
1068:
1067:
1060:
1035:
1034:
1027:
1016:
1014:
1013:
1008:
1006:
1004:
979:
978:
971:
952:
951:
926:
925:
918:
906:
904:
903:
898:
896:
894:
875:
874:
866:
854:
852:
851:
846:
844:
842:
817:
816:
808:
792:
790:
789:
784:
753:be classical or
748:
746:
745:
740:
738:
737:
725:
724:
706:
705:
693:
692:
642:
640:
639:
634:
632:
630:
599:
598:
586:
585:
567:
566:
554:
553:
543:
512:
511:
499:
498:
480:
479:
467:
466:
456:
434:
432:
431:
426:
424:
422:
397:
396:
384:
383:
365:
364:
352:
351:
339:
338:
326:
325:
307:
306:
294:
293:
283:
270:
269:
257:
256:
238:
237:
225:
224:
202:
201:
189:
188:
170:
169:
157:
156:
146:
122:
120:
119:
114:
112:
111:
99:
98:
80:
79:
67:
66:
38:analytic calculi
3245:
3244:
3240:
3239:
3238:
3236:
3235:
3234:
3220:
3219:
3218:
3204:
3203:
3199:
3184:
3165:10.1.1.405.8176
3150:
3149:
3142:
3106:
3105:
3101:
3085:
3084:
3080:
3061:J. Log. Comput.
3054:
3053:
3049:
3020:
3019:
3015:
2991:
2990:
2986:
2971:
2947:
2946:
2939:
2932:
2911:
2910:
2903:
2896:
2861:
2860:
2856:
2829:Rev. Symb. Log.
2823:
2818:
2817:
2813:
2798:
2797:
2790:
2773:
2772:
2765:
2758:
2742:
2741:
2724:
2720:
2704:
2688:
2680:cut elimination
2628:
2615:
2596:
2583:
2570:
2557:
2538:
2525:
2524:
2501:
2488:
2469:
2456:
2433:
2420:
2401:
2388:
2387:
2380:
2379:
2340:
2327:
2299:
2286:
2275:
2274:
2247:
2234:
2215:
2202:
2197:
2196:
2178:
2107:
2068:
2061:
2060:
2016:
1989:
1982:
1981:
1939:
1915:
1908:
1907:
1881:
1880:
1859:
1837:
1829:
1828:
1809:
1808:
1789:
1788:
1766:
1765:
1744:
1725:
1720:
1719:
1700:
1699:
1672:
1656:
1625:
1609:
1595:
1594:
1590:is the formula
1567:
1554:
1535:
1522:
1517:
1516:
1510:sequent calculi
1502:
1497:
1449:
1416:
1409:
1408:
1373:
1361:
1360:
1316:
1283:
1276:
1275:
1234:
1201:
1194:
1193:
1145:
1124:
1117:
1116:
1061:
1028:
1021:
1020:
972:
919:
912:
911:
868:
860:
859:
810:
802:
801:
763:
762:
729:
716:
697:
684:
679:
678:
664:
590:
577:
558:
545:
544:
503:
490:
471:
458:
457:
450:
449:
388:
375:
356:
343:
330:
317:
298:
285:
284:
261:
248:
229:
216:
193:
180:
161:
148:
147:
140:
139:
103:
90:
71:
58:
53:
52:
30:sequent calculi
12:
11:
5:
3243:
3241:
3233:
3232:
3222:
3221:
3217:
3216:
3197:
3182:
3140:
3119:(6): 835–861.
3113:J. Log. Comput
3099:
3078:
3067:(2): 283–294.
3047:
3013:
2984:
2969:
2937:
2930:
2901:
2894:
2854:
2811:
2788:
2775:Mints, Grigori
2763:
2756:
2721:
2719:
2716:
2703:
2700:
2687:
2684:
2676:
2675:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2635:
2631:
2627:
2622:
2618:
2614:
2611:
2608:
2603:
2599:
2595:
2590:
2586:
2582:
2577:
2573:
2569:
2564:
2560:
2556:
2553:
2550:
2545:
2541:
2537:
2532:
2528:
2522:
2519:
2516:
2513:
2508:
2504:
2500:
2495:
2491:
2487:
2484:
2481:
2476:
2472:
2468:
2463:
2459:
2454:
2451:
2448:
2445:
2440:
2436:
2432:
2427:
2423:
2419:
2416:
2413:
2408:
2404:
2400:
2395:
2391:
2364:
2363:
2352:
2347:
2343:
2339:
2334:
2330:
2326:
2323:
2320:
2317:
2314:
2311:
2306:
2302:
2298:
2293:
2289:
2285:
2282:
2268:
2267:
2254:
2250:
2246:
2241:
2237:
2233:
2230:
2227:
2222:
2218:
2214:
2209:
2205:
2177:
2174:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2112:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2073:
2044:
2041:
2038:
2035:
2032:
2029:
2026:
2021:
2014:
2011:
2008:
2005:
2002:
1999:
1994:
1979:
1978:
1964:
1961:
1958:
1955:
1952:
1949:
1944:
1937:
1934:
1931:
1928:
1925:
1920:
1888:
1866:
1862:
1858:
1855:
1852:
1849:
1844:
1840:
1836:
1816:
1796:
1776:
1773:
1751:
1747:
1743:
1740:
1737:
1732:
1728:
1707:
1696:
1695:
1684:
1679:
1675:
1671:
1668:
1663:
1659:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1632:
1628:
1624:
1621:
1616:
1612:
1608:
1605:
1602:
1588:
1587:
1574:
1570:
1566:
1561:
1557:
1553:
1550:
1547:
1542:
1538:
1534:
1529:
1525:
1501:
1498:
1496:
1493:
1468:
1465:
1462:
1459:
1454:
1447:
1444:
1441:
1438:
1435:
1432:
1429:
1426:
1421:
1392:
1389:
1386:
1383:
1378:
1371:
1341:
1338:
1335:
1332:
1329:
1326:
1321:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1288:
1259:
1256:
1253:
1250:
1247:
1244:
1239:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1206:
1191:
1190:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1150:
1143:
1140:
1137:
1134:
1129:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1066:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1033:
1018:
1017:
1003:
1000:
997:
994:
991:
988:
985:
982:
977:
970:
967:
964:
961:
958:
955:
950:
944:
941:
938:
935:
932:
929:
924:
908:
907:
893:
890:
887:
884:
881:
878:
873:
856:
855:
841:
838:
835:
832:
829:
826:
823:
820:
815:
782:
779:
776:
773:
770:
755:intuitionistic
750:
749:
736:
732:
728:
723:
719:
715:
712:
709:
704:
700:
696:
691:
687:
663:
660:
644:
643:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
597:
593:
589:
584:
580:
576:
573:
570:
565:
561:
557:
552:
548:
542:
539:
536:
533:
530:
527:
524:
521:
518:
515:
510:
506:
502:
497:
493:
489:
486:
483:
478:
474:
470:
465:
461:
436:
435:
421:
418:
415:
412:
409:
406:
403:
400:
395:
391:
387:
382:
378:
374:
371:
368:
363:
359:
355:
350:
346:
342:
337:
333:
329:
324:
320:
316:
313:
310:
305:
301:
297:
292:
288:
282:
279:
276:
273:
268:
264:
260:
255:
251:
247:
244:
241:
236:
232:
228:
223:
219:
214:
211:
208:
205:
200:
196:
192:
187:
183:
179:
176:
173:
168:
164:
160:
155:
151:
124:
123:
110:
106:
102:
97:
93:
89:
86:
83:
78:
74:
70:
65:
61:
13:
10:
9:
6:
4:
3:
2:
3242:
3231:
3228:
3227:
3225:
3212:
3208:
3201:
3198:
3193:
3189:
3185:
3179:
3175:
3171:
3166:
3161:
3157:
3153:
3147:
3145:
3141:
3136:
3132:
3127:
3126:10.1.1.8.5319
3122:
3118:
3114:
3110:
3103:
3100:
3095:
3093:
3088:
3082:
3079:
3074:
3070:
3066:
3063:
3062:
3057:
3051:
3048:
3042:
3037:
3033:
3030:
3029:
3024:
3017:
3014:
3009:
3005:
3001:
2997:
2996:
2988:
2985:
2980:
2976:
2972:
2966:
2962:
2958:
2954:
2953:
2944:
2942:
2938:
2933:
2927:
2923:
2919:
2915:
2908:
2906:
2902:
2897:
2895:9780511546464
2891:
2886:
2881:
2877:
2873:
2869:
2865:
2858:
2855:
2850:
2846:
2842:
2838:
2834:
2831:
2830:
2822:
2815:
2812:
2807:
2804:
2803:
2802:J. Symb. Log.
2795:
2793:
2789:
2784:
2780:
2776:
2770:
2768:
2764:
2759:
2753:
2749:
2745:
2739:
2737:
2735:
2733:
2731:
2729:
2727:
2723:
2717:
2715:
2713:
2709:
2701:
2699:
2697:
2693:
2685:
2683:
2681:
2659:
2650:
2647:
2638:
2633:
2620:
2612:
2609:
2606:
2601:
2588:
2580:
2575:
2562:
2554:
2551:
2548:
2543:
2530:
2520:
2511:
2506:
2493:
2485:
2482:
2479:
2474:
2461:
2452:
2443:
2438:
2425:
2417:
2414:
2411:
2406:
2393:
2378:
2377:
2376:
2374:
2370:
2345:
2341:
2332:
2324:
2318:
2315:
2312:
2304:
2300:
2291:
2283:
2273:
2272:
2271:
2252:
2248:
2239:
2231:
2228:
2225:
2220:
2216:
2207:
2195:
2194:
2193:
2191:
2187:
2183:
2175:
2173:
2170:
2166:
2162:
2161:Admissibility
2158:
2133:
2127:
2118:
2115:
2100:
2094:
2085:
2079:
2076:
2058:
2036:
2033:
2030:
2024:
2006:
2003:
1997:
1962:
1959:
1950:
1947:
1935:
1926:
1923:
1906:
1905:
1904:
1902:
1886:
1864:
1860:
1856:
1853:
1850:
1847:
1842:
1838:
1834:
1814:
1771:
1749:
1745:
1741:
1738:
1735:
1730:
1726:
1677:
1669:
1661:
1653:
1647:
1644:
1641:
1638:
1630:
1622:
1614:
1606:
1600:
1593:
1592:
1591:
1572:
1559:
1551:
1548:
1545:
1540:
1527:
1515:
1514:
1513:
1511:
1507:
1499:
1495:Main examples
1494:
1492:
1490:
1486:
1482:
1457:
1436:
1424:
1406:
1381:
1358:
1355:
1336:
1333:
1324:
1309:
1306:
1303:
1300:
1291:
1273:
1251:
1248:
1242:
1224:
1221:
1218:
1215:
1209:
1171:
1159:
1153:
1132:
1115:
1114:
1113:
1111:
1106:
1087:
1084:
1078:
1069:
1054:
1051:
1045:
1042:
1036:
995:
989:
986:
980:
965:
962:
953:
936:
933:
927:
910:
909:
882:
876:
858:
857:
836:
833:
827:
824:
818:
800:
799:
798:
796:
774:
760:
756:
734:
721:
713:
710:
707:
702:
689:
677:
676:
675:
673:
669:
661:
659:
657:
653:
649:
618:
612:
603:
600:
595:
582:
574:
571:
568:
563:
550:
537:
531:
522:
516:
513:
508:
495:
487:
484:
481:
476:
463:
448:
447:
446:
444:
441:
419:
410:
407:
398:
393:
380:
372:
369:
366:
361:
348:
340:
335:
322:
314:
311:
308:
303:
290:
280:
271:
266:
253:
245:
242:
239:
234:
221:
212:
203:
198:
185:
177:
174:
171:
166:
153:
138:
137:
136:
134:
130:
108:
95:
87:
84:
81:
76:
63:
51:
50:
49:
47:
43:
39:
35:
31:
28:framework of
27:
23:
19:
3230:Proof theory
3210:
3200:
3155:
3116:
3112:
3102:
3090:
3081:
3064:
3059:
3050:
3031:
3026:
3016:
3002:(2): 75–81.
2999:
2993:
2987:
2949:
2913:
2867:
2863:
2857:
2832:
2827:
2814:
2805:
2800:
2782:
2778:
2747:
2744:Avron, Arnon
2707:
2705:
2696:fuzzy logics
2689:
2677:
2365:
2269:
2179:
2169:completeness
2159:
2059:
1980:
1697:
1589:
1506:modal logics
1503:
1500:Modal logics
1483:
1407:
1359:
1356:
1274:
1192:
1107:
1019:
751:
670:of ordinary
665:
648:modal logics
645:
437:
125:
44:of ordinary
22:hypersequent
21:
15:
3207:Gabbay, Dov
2885:11343/31712
2870:: 151–172.
1489:disjunction
440:modal logic
36:to provide
3034:: 76–105.
2718:References
674:, written
48:, written
3160:CiteSeerX
3121:CiteSeerX
2808:(3): 900.
2785:: 97–122.
2657:⇒
2654:Π
2651:∣
2645:⇒
2642:Σ
2639:∣
2630:Θ
2626:⇒
2617:Ω
2613:∣
2610:⋯
2607:∣
2598:Θ
2594:⇒
2585:Ω
2581:∣
2572:Δ
2568:⇒
2559:Γ
2555:∣
2552:⋯
2549:∣
2540:Δ
2536:⇒
2527:Γ
2518:⇒
2515:Π
2512:∣
2503:Θ
2499:⇒
2490:Ω
2486:∣
2483:⋯
2480:∣
2471:Θ
2467:⇒
2458:Ω
2450:⇒
2447:Σ
2444:∣
2435:Δ
2431:⇒
2422:Γ
2418:∣
2415:⋯
2412:∣
2403:Δ
2399:⇒
2390:Γ
2338:→
2329:Γ
2325:⋀
2319:∨
2316:⋯
2313:∨
2297:→
2288:Γ
2284:⋀
2245:⇒
2236:Γ
2232:∣
2229:⋯
2226:∣
2213:⇒
2204:Γ
2143:Π
2140:⇒
2137:Σ
2134:∣
2131:Δ
2128:◻
2125:⇒
2122:Γ
2119:◻
2116:∣
2104:Π
2098:Δ
2095:◻
2092:⇒
2089:Σ
2083:Γ
2080:◻
2077:∣
2043:Δ
2040:⇒
2034:◻
2028:Γ
2025:∣
2013:Δ
2010:⇒
2001:Γ
1998:∣
1960:◻
1957:⇒
1954:Γ
1951:◻
1948:∣
1933:⇒
1930:Γ
1927:◻
1924:∣
1887:∣
1857:◻
1851:…
1835:◻
1815:◻
1795:Γ
1775:Γ
1772:◻
1764:we write
1739:…
1706:Γ
1674:Δ
1670:⋁
1667:→
1658:Γ
1654:⋀
1648:◻
1645:∨
1642:⋯
1639:∨
1627:Δ
1623:⋁
1620:→
1611:Γ
1607:⋀
1601:◻
1569:Δ
1565:⇒
1556:Γ
1552:∣
1549:⋯
1546:∣
1537:Δ
1533:⇒
1524:Γ
1485:Soundness
1467:Δ
1464:⇒
1461:Γ
1458:∣
1446:Δ
1443:⇒
1440:Γ
1437:∣
1434:Δ
1431:⇒
1428:Γ
1425:∣
1391:Δ
1388:⇒
1385:Γ
1382:∣
1340:Δ
1331:⇒
1328:Γ
1325:∣
1313:Δ
1298:⇒
1295:Γ
1292:∣
1258:Δ
1255:⇒
1246:Γ
1243:∣
1231:Δ
1228:⇒
1213:Γ
1210:∣
1175:Π
1169:Δ
1166:⇒
1163:Σ
1157:Γ
1154:∣
1142:Δ
1139:⇒
1136:Γ
1133:∣
1091:Δ
1082:→
1076:⇒
1073:Γ
1070:∣
1058:Δ
1049:⇒
1040:Γ
1037:∣
1002:Δ
999:⇒
993:→
984:Γ
981:∣
969:Δ
960:⇒
957:Γ
954:∣
943:Δ
940:⇒
931:Γ
928:∣
892:Δ
889:⇒
886:⊥
880:Γ
877:∣
840:Δ
831:⇒
822:Γ
819:∣
778:→
772:⊥
731:Δ
727:⇒
718:Γ
714:∣
711:⋯
708:∣
699:Δ
695:⇒
686:Γ
628:Ω
625:⇒
622:Θ
619:∣
616:Π
613:◻
610:⇒
607:Σ
604:◻
601:∣
592:Δ
588:⇒
579:Γ
575:∣
572:⋯
569:∣
560:Δ
556:⇒
547:Γ
541:Ω
535:Π
532:◻
529:⇒
526:Θ
520:Σ
517:◻
514:∣
505:Δ
501:⇒
492:Γ
488:∣
485:⋯
482:∣
473:Δ
469:⇒
460:Γ
417:⇒
414:Π
411:∣
405:⇒
402:Σ
399:∣
390:Θ
386:⇒
377:Ω
373:∣
370:⋯
367:∣
358:Θ
354:⇒
345:Ω
341:∣
332:Δ
328:⇒
319:Γ
315:∣
312:⋯
309:∣
300:Δ
296:⇒
287:Γ
278:⇒
275:Π
272:∣
263:Θ
259:⇒
250:Ω
246:∣
243:⋯
240:∣
231:Θ
227:⇒
218:Ω
210:⇒
207:Σ
204:∣
195:Δ
191:⇒
182:Γ
178:∣
175:⋯
172:∣
163:Δ
159:⇒
150:Γ
105:Δ
101:⇒
92:Γ
88:∣
85:⋯
82:∣
73:Δ
69:⇒
60:Γ
3224:Category
3209:(2008).
3096:: 81–96.
3092:Tableaux
2849:37437016
2835:: 3–15.
2165:cut rule
1698:Here if
672:sequents
668:multiset
46:sequents
42:multiset
32:used in
3192:7456109
2708:cortege
2702:History
3190:
3180:
3162:
3123:
2979:221813
2977:
2967:
2928:
2892:
2847:
2754:
654:, and
20:, the
3188:S2CID
2975:S2CID
2845:S2CID
2824:(PDF)
1807:with
3178:ISBN
3094:2013
2965:ISBN
2926:ISBN
2890:ISBN
2752:ISBN
2694:and
793:for
131:LC (
3170:doi
3131:doi
3069:doi
3036:doi
3032:656
3004:doi
3000:115
2957:doi
2918:doi
2880:hdl
2872:doi
2837:doi
135:)
16:In
3226::
3186:.
3176:.
3168:.
3143:^
3129:.
3117:13
3115:.
3065:11
3025:.
2998:.
2973:.
2963:.
2940:^
2924:.
2904:^
2888:.
2878:.
2868:28
2843:.
2826:.
2806:48
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2783:98
2781:.
2766:^
2725:^
2712:S5
2698:.
2373:LC
1901:S5
650:,
445::
443:S5
3194:.
3172::
3137:.
3133::
3075:.
3071::
3044:.
3038::
3010:.
3006::
2981:.
2959::
2934:.
2920::
2898:.
2882::
2874::
2851:.
2839::
2833:1
2760:.
2660:A
2648:B
2634:m
2621:m
2602:1
2589:1
2576:n
2563:n
2544:1
2531:1
2521:B
2507:m
2494:m
2475:1
2462:1
2453:A
2439:n
2426:n
2407:1
2394:1
2351:)
2346:n
2342:A
2333:n
2322:(
2310:)
2305:1
2301:A
2292:1
2281:(
2253:n
2249:A
2240:n
2221:1
2217:A
2208:1
2111:G
2101:,
2086:,
2072:G
2037:A
2031:,
2020:G
2007:A
2004:,
1993:G
1963:A
1943:G
1936:A
1919:G
1865:n
1861:A
1854:,
1848:,
1843:1
1839:A
1750:n
1746:A
1742:,
1736:,
1731:1
1727:A
1683:)
1678:n
1662:n
1651:(
1636:)
1631:1
1615:1
1604:(
1573:n
1560:n
1541:1
1528:1
1453:G
1420:G
1377:G
1370:G
1337:,
1334:A
1320:G
1310:,
1307:A
1304:,
1301:A
1287:G
1252:A
1249:,
1238:G
1225:A
1222:,
1219:A
1216:,
1205:G
1172:,
1160:,
1149:G
1128:G
1088:,
1085:B
1079:A
1065:G
1055:,
1052:B
1046:A
1043:,
1032:G
996:B
990:A
987:,
976:G
966:,
963:A
949:G
937:B
934:,
923:G
883:,
872:G
837:,
834:p
828:p
825:,
814:G
781:}
775:,
769:{
735:n
722:n
703:1
690:1
596:n
583:n
564:1
551:1
538:,
523:,
509:n
496:n
477:1
464:1
420:A
408:B
394:m
381:m
362:1
349:1
336:n
323:n
304:1
291:1
281:B
267:m
254:m
235:1
222:1
213:A
199:n
186:n
167:1
154:1
109:n
96:n
77:1
64:1
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