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1348:
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322:
276:
230:
1252:
1635:{\displaystyle {\frac {\Gamma _{1}\vdash \Delta _{1}\mid \dots \mid \Gamma _{n}\vdash \Delta _{n}\mid \Omega \vdash A\qquad \Sigma _{1}\vdash \Pi _{1}\mid \dots \mid \Sigma _{m}\vdash \Pi _{m}\mid \Theta \vdash B}{\Gamma _{1}\vdash \Delta _{1}\mid \dots \mid \Gamma _{n}\vdash \Delta _{n}\mid \Sigma _{1}\vdash \Pi _{1}\mid \dots \mid \Sigma _{m}\vdash \Pi _{m}\mid \Omega \vdash B\mid \Theta \vdash A}}}
1062:
1050:
882:
194:
they are interpreted by in the sequent calculus: the structural operators are used in every rule of the calculus, and are not considered when asking whether the subformula property applies. Furthermore, the logical rules go one way only: logical structure is introduced by logical rules, and cannot be
198:
The idea of looking at the syntactic features of sequents as special, non-logical operators is not old, and was forced by innovations in proof theory: when the structural operators are as simple as in Getzen's original sequent calculus there is little need to analyse them, but proof calculi of
721:
1247:{\displaystyle {\frac {\Gamma _{1}\vdash \Delta _{1}\mid \dots \mid \Gamma _{n}\vdash \Delta _{n}\mid \Box \Sigma ,\Omega \vdash \Box \Pi ,\Theta }{\Gamma _{1}\vdash \Delta _{1}\mid \dots \mid \Gamma _{n}\vdash \Delta _{n}\mid \Box \Sigma \vdash \Box \Pi \mid \Omega \vdash \Theta }}}
615:
894:
189:
are operators normally interpreted as conjunctions, those to the right as disjunctions, whilst the turnstile symbol itself is interpreted as an implication. However, it is important to note that there is a fundamental difference in behaviour between these operators and the
740:
106:
in structural proof theory comes from a technical notion introduced in the sequent calculus: the sequent calculus represents the judgement made at any stage of an inference using special, extra-logical operators called structural operators: in
463:
624:
36:, a kind of proof whose semantic properties are exposed. When all the theorems of a logic formalised in a structural proof theory have analytic proofs, then the proof theory can be used to demonstrate such things as
1045:{\displaystyle {\frac {\Gamma _{1}\vdash \Delta _{1}\mid \dots \mid \Gamma _{n}\vdash \Delta _{n}\mid \Gamma _{n}\vdash \Delta _{n}}{\Gamma _{1}\vdash \Delta _{1}\mid \dots \mid \Gamma _{n}\vdash \Delta _{n}}}}
524:
183:
877:{\displaystyle {\frac {\Gamma _{1}\vdash \Delta _{1}\mid \dots \mid \Gamma _{n}\vdash \Delta _{n}}{\Gamma _{1}\vdash \Delta _{1}\mid \dots \mid \Gamma _{n}\vdash \Delta _{n}\mid \Sigma \vdash \Pi }}}
505:
1306:
1283:
519:
of the hypersequents depends on the logic under consideration, but is nearly always some form of disjunction. The most common interpretations are as a simple disjunction
1329:
511:
of the hypersequent. As for sequents, hypersequents can be based on sets, multisets, or sequences, and the components can be single-conclusion or multi-conclusion
1054:
The additional expressivity of the hypersequent framework is provided by rules manipulating the hypersequent structure. An important example is provided by the
396:
716:{\displaystyle \Box (\bigwedge \Gamma _{1}\rightarrow \bigvee \Delta _{1})\lor \dots \lor \Box (\bigwedge \Gamma _{n}\rightarrow \bigvee \Delta _{n})}
1763:
44:, and allow mathematical or computational witnesses to be extracted as counterparts to theorems, the kind of task that is more often given to
1899:
1873:
610:{\displaystyle (\bigwedge \Gamma _{1}\rightarrow \bigvee \Delta _{1})\lor \dots \lor (\bigwedge \Gamma _{n}\rightarrow \bigvee \Delta _{n})}
110:
87:
728:
In line with the disjunctive interpretation of the hypersequent bar, essentially all hypersequent calculi include the
195:
eliminated once created, while structural operators can be introduced and eliminated in the course of a derivation.
470:
75:
1814:
211:
in 1982) support structural operators as complex as the logical connectives, and demand sophisticated treatment.
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71:
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384:
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380:
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41:
17:
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The nested sequent calculus is a formalisation that resembles a 2-sided calculus of structures.
1895:
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266:
83:
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364:
67:
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458:{\displaystyle \Gamma _{1}\vdash \Delta _{1}\mid \dots \mid \Gamma _{n}\vdash \Delta _{n}}
375:) to separate different sequents. It has been used to provide analytic calculi for, e.g.,
312:
220:
63:
1792:"The method of hypersequents in the proof theory of propositional non-classical logics"
200:
91:
57:
33:
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321:
275:
229:
1912:
1843:
29:
1644:
Note that in the communication rule the components are single-conclusion sequents.
358:
45:
25:
1768:
The
Calculi of Symbolic Logic. Proceedings of the Steklov Institute of Mathematics
376:
208:
79:
37:
1861:
1812:
Pottinger, Garrel (1983). "Uniform, cut-free formulations of T, S4, and S5".
82:; the definition is slightly more complex—the analytic proofs are the
368:
1835:
512:
1721:
371:
of sequents, using an additional structural connective | (called the
1827:
1799:
Logic: From
Foundations to Applications: European Logic Colloquium
62:
The notion of analytic proof was introduced into proof theory by
1657:
316:
270:
224:
178:{\displaystyle A_{1},\dots ,A_{m}\vdash B_{1},\dots ,B_{n}}
261:
Natural deduction and the formulae-as-types correspondence
78:
also supports a notion of analytic proof, as was shown by
1674:
333:
287:
241:
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for intermediate logics, or as a disjunction of boxes
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897:
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1044:
876:
715:
609:
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457:
177:
363:The hypersequent framework extends the ordinary
500:{\displaystyle \Gamma _{i}\vdash \Delta _{i}}
8:
1894:(2nd ed.). Cambridge University Press.
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70:; the analytic proofs are those that are
1713:
215:Cut-elimination in the sequent calculus
86:, which are related to the notion of
7:
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14:
507:is an ordinary sequent, called a
1764:"On some calculi of modal logic"
1661:
1334:Another example is given by the
320:
274:
228:
185:, the commas to the left of the
1744:N. D. Belnap. "Display Logic."
1422:
1868:. Cambridge University Press.
1746:Journal of Philosophical Logic
710:
694:
678:
663:
647:
631:
604:
588:
572:
560:
544:
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1:
1301:{\displaystyle \Box \Sigma }
1285:means that every formula in
1278:{\displaystyle \Box \Sigma }
1338:for the intermediate logic
307:Logical duality and harmony
1935:
1697:
1651:
356:
310:
264:
218:
98:Structures and connectives
76:natural deduction calculus
55:
1815:Journal of Symbolic Logic
1722:"Structural Proof Theory"
888:external contraction rule
730:external structural rules
32:that support a notion of
1864:; Jan Von Plato (2001).
1801:. Clarendon Press: 1–32.
1056:modalised splitting rule
24:is the subdiscipline of
1866:Structural proof theory
1700:Nested sequent calculus
1694:Nested sequent calculus
734:external weakening rule
22:structural proof theory
1654:Calculus of structures
1648:Calculus of structures
1636:
1325:
1324:{\displaystyle \Box A}
1302:
1279:
1248:
1046:
878:
717:
611:
517:formula interpretation
501:
459:
179:
1888:Helmut Schwichtenberg
1790:Avron, Arnon (1996).
1637:
1326:
1303:
1280:
1249:
1047:
879:
718:
612:
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460:
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1884:Anne Sjerp Troelstra
1762:Minc, G.E. (1971) .
1349:
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1063:
895:
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732:, in particular the
625:
525:
471:
397:
111:
1752:(4), 375–417, 1982.
192:logical connectives
42:decision procedures
1892:Basic proof theory
1726:www.philpapers.org
1673:. You can help by
1632:
1336:communication rule
1321:
1298:
1275:
1244:
1042:
874:
725:for modal logics.
713:
607:
497:
455:
332:. You can help by
286:. You can help by
240:. You can help by
175:
18:mathematical logic
1901:978-0-521-77911-1
1875:978-0-521-79307-0
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365:sequent structure
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68:sequent calculus
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221:Cut-elimination
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207:(introduced by
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1774:. AMS: 97–124.
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1698:Main article:
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92:term rewriting
58:Analytic proof
56:Main article:
53:
52:Analytic proof
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34:analytic proof
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1682:December 2009
1676:
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1669:This section
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385:substructural
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366:
360:
353:Hypersequents
352:
344:
341:December 2009
335:
331:
328:This section
326:
323:
319:
318:
314:
306:
298:
295:December 2009
289:
285:
282:This section
280:
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272:
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260:
252:
249:December 2009
243:
239:
236:This section
234:
231:
227:
226:
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212:
210:
206:
205:display logic
202:
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170:
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59:
51:
49:
47:
43:
39:
35:
31:
30:proof calculi
28:that studies
27:
23:
19:
1919:Proof theory
1891:
1865:
1819:
1813:
1807:
1798:
1771:
1767:
1757:
1749:
1745:
1740:
1729:. Retrieved
1725:
1716:
1703:
1679:
1675:adding to it
1670:
1643:
1345:
1340:
1335:
1333:
1257:
1255:
1059:
1055:
1053:
891:
887:
885:
737:
733:
729:
727:
724:
621:
618:
521:
516:
508:
466:
393:
389:hypersequent
388:
381:intermediate
372:
362:
359:Hypersequent
338:
334:adding to it
329:
292:
288:adding to it
283:
246:
242:adding to it
237:
204:
197:
103:
101:
84:normal forms
61:
46:model theory
26:proof theory
21:
15:
467:where each
209:Nuel Belnap
88:normal form
80:Dag Prawitz
38:consistency
1862:Sara Negri
1855:References
1822:(3): 900.
1731:2024-08-18
40:, provide
1844:250346853
1624:⊢
1621:Θ
1618:∣
1612:⊢
1609:Ω
1606:∣
1597:Π
1593:⊢
1584:Σ
1580:∣
1577:⋯
1574:∣
1565:Π
1561:⊢
1552:Σ
1548:∣
1539:Δ
1535:⊢
1526:Γ
1522:∣
1519:⋯
1516:∣
1507:Δ
1503:⊢
1494:Γ
1485:⊢
1482:Θ
1479:∣
1470:Π
1466:⊢
1457:Σ
1453:∣
1450:⋯
1447:∣
1438:Π
1434:⊢
1425:Σ
1417:⊢
1414:Ω
1411:∣
1402:Δ
1398:⊢
1389:Γ
1385:∣
1382:⋯
1379:∣
1370:Δ
1366:⊢
1357:Γ
1316:◻
1296:Σ
1293:◻
1273:Σ
1270:◻
1239:Θ
1236:⊢
1233:Ω
1230:∣
1227:Π
1224:◻
1221:⊢
1218:Σ
1215:◻
1212:∣
1203:Δ
1199:⊢
1190:Γ
1186:∣
1183:⋯
1180:∣
1171:Δ
1167:⊢
1158:Γ
1152:Θ
1146:Π
1143:◻
1140:⊢
1137:Ω
1131:Σ
1128:◻
1125:∣
1116:Δ
1112:⊢
1103:Γ
1099:∣
1096:⋯
1093:∣
1084:Δ
1080:⊢
1071:Γ
1031:Δ
1027:⊢
1018:Γ
1014:∣
1011:⋯
1008:∣
999:Δ
995:⊢
986:Γ
974:Δ
970:⊢
961:Γ
957:∣
948:Δ
944:⊢
935:Γ
931:∣
928:⋯
925:∣
916:Δ
912:⊢
903:Γ
869:Π
866:⊢
863:Σ
860:∣
851:Δ
847:⊢
838:Γ
834:∣
831:⋯
828:∣
819:Δ
815:⊢
806:Γ
794:Δ
790:⊢
781:Γ
777:∣
774:⋯
771:∣
762:Δ
758:⊢
749:Γ
702:Δ
698:⋁
695:→
686:Γ
682:⋀
676:◻
673:∨
670:⋯
667:∨
655:Δ
651:⋁
648:→
639:Γ
635:⋀
629:◻
596:Δ
592:⋁
589:→
580:Γ
576:⋀
570:∨
567:⋯
564:∨
552:Δ
548:⋁
545:→
536:Γ
532:⋀
509:component
489:Δ
485:⊢
476:Γ
447:Δ
443:⊢
434:Γ
430:∣
427:⋯
424:∣
415:Δ
411:⊢
402:Γ
387:logics A
187:turnstile
160:…
144:⊢
128:…
104:structure
102:The term
1913:Category
1890:(2000).
1262:, where
886:and the
513:sequents
369:multiset
203:such as
72:cut-free
66:for the
1836:2273495
1898:
1872:
1842:
1834:
515:. The
74:. His
1840:S2CID
1832:JSTOR
1795:(PDF)
1708:Notes
377:modal
367:to a
1896:ISBN
1870:ISBN
383:and
1824:doi
1677:.
336:.
290:.
244:.
90:in
16:In
1915::
1886:;
1838:.
1830:.
1820:48
1818:.
1797:.
1780:^
1772:98
1770:.
1766:.
1750:11
1748:,
1724:.
1341:LC
1331:.
1259:S5
379:,
94:.
48:.
20:,
1904:.
1878:.
1846:.
1826::
1734:.
1684:)
1680:(
1627:A
1615:B
1601:m
1588:m
1569:1
1556:1
1543:n
1530:n
1511:1
1498:1
1488:B
1474:m
1461:m
1442:1
1429:1
1420:A
1406:n
1393:n
1374:1
1361:1
1319:A
1207:n
1194:n
1175:1
1162:1
1149:,
1134:,
1120:n
1107:n
1088:1
1075:1
1035:n
1022:n
1003:1
990:1
978:n
965:n
952:n
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920:1
907:1
855:n
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823:1
810:1
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690:n
679:(
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339:(
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293:(
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171:n
167:B
163:,
157:,
152:1
148:B
139:m
135:A
131:,
125:,
120:1
116:A
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