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If a predicate P is positively derived from a predicate Q (i.e., P is the head of a rule, and Q occurs positively in the body of the same rule), then the stratification number of P must be greater than or equal to the stratification number of Q, in short
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If a predicate P is derived from a negated predicate Q (i.e., P is the head of a rule, and Q occurs negatively in the body of the same rule), then the stratification number of P must be greater than the stratification number of Q, in short
720:, with fixed displacements from the values of each of their (bound) arguments in a (weakly) stratified formula. Defined term constructions are neatly handled by (possibly merely implicitly) using the theory of descriptions: a term
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The notion of stratified negation leads to a very effective operational semantics for stratified programs in terms of the stratified least fixpoint, that is obtained by iteratively applying the fixpoint operator to each
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A formula is stratified if and only if it is possible to assign types to all variables appearing in the formula in such a way that it will make sense in a version TST of the theory of types described in the
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generalizes readily to languages with more predicates and with term constructions. Each primitive predicate needs to have specified required displacements between values of
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It turns out that it is sufficient to require that these conditions be satisfied only when both variables in an atomic formula are bound in the set abstract
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at its (bound) arguments in a (weakly) stratified formula. In a language with term constructions, terms themselves need to be assigned values under
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445:(considered as an item of syntax) to a natural number (this works equally well if all integers are used) in such a way that any atomic formula
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of the program, from the lowest one up. Stratification is not only useful for guaranteeing unique interpretation of
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252:{\displaystyle Q_{1}\wedge \dots \wedge Q_{n}\wedge \neg Q_{n+1}\wedge \dots \wedge \neg Q_{n+m}\rightarrow P}
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is stratified if and only if there is a stratification assignment S that fulfills the following conditions:
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incorrectly led you here, you may wish to change the link to point directly to the intended article.
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of the topological space). This is not a useful notion when unrestricted; but when the various
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under consideration. A set abstract satisfying this weaker condition is said to be
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in the language of first-order logic with equality and membership is said to be
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article, and this is probably the best way to understand the stratification of
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includes a list of related items that share the same name (or similar names).
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895:), and fit together manageably, this idea is often applied in geometry.
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are defined by some recognisable set of conditions (for example being
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of a logical theory exists. Specifically, we say that a set of
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first defined formal conditions for stratification. See
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A motivation for the use of stratification is to address
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The notion of stratification can be extended to the
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2691:Index of articles associated with the same name
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821:, the antinomy considered to have undermined
814:; this is found in papers of Randall Holmes.
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145:is any consistent assignment of numbers to
50:. Unsourced material may be challenged and
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149:symbols guaranteeing that a unique formal
875:into disjoint subsets each of which is a
381:(NF) and related set theories, a formula
114:Learn how and when to remove this message
775:) must be assigned the same value under
132:
534:{\displaystyle \sigma (x)+1=\sigma (y)}
425:which sends each variable appearing in
2711:
955:
929:
832:
627:{\displaystyle \sigma (x)=\sigma (y)}
48:adding citations to reliable sources
15:
405:if and only if there is a function
129:has several usages in mathematics.
13:
833:Quine, Willard Van Orman (1963) .
224:
196:
14:
2750:
2719:Set index articles on mathematics
63:"Stratification" mathematics
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2667:
914:
20:
748:{\displaystyle (\iota x.\phi )}
662:{\displaystyle \{x\mid \phi \}}
909:topologically stratified space
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1:
2628:History of mathematical logic
301:{\displaystyle S(P)\geq S(Q)}
2553:Primitive recursive function
835:From a Logical Point of View
349:{\displaystyle S(P)>S(Q)}
7:
828:Grundgesetze der Arithmetik
10:
2755:
1617:SchröderâBernstein theorem
1344:Monadic predicate calculus
1003:Foundations of mathematics
879:(so that in particular a
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837:(2nd ed.). New York:
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2650:Philosophy of mathematics
2599:Automated theorem proving
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1724:Von NeumannâBernaysâGödel
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2729:Mathematical terminology
373:In a specific set theory
2300:Self-verifying theories
2121:Tarski's axiomatization
1072:Tarski's undefinability
1067:incompleteness theorems
788:{\displaystyle \sigma }
713:{\displaystyle \sigma }
693:{\displaystyle \sigma }
418:{\displaystyle \sigma }
2674:Mathematics portal
2285:Proof of impossibility
1933:propositional variable
1243:Propositional calculus
905:Whitney stratification
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676:The stratification of
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628:
584:
564:
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465:
464:{\displaystyle x\in y}
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2543:Kolmogorov complexity
2496:Computably enumerable
2396:Model complete theory
2188:Principia Mathematica
1248:Propositional formula
1077:BanachâTarski paradox
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770:
768:{\displaystyle \phi }
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715:
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583:{\displaystyle \phi }
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484:{\displaystyle \phi }
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438:{\displaystyle \phi }
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394:{\displaystyle \phi }
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133:In mathematical logic
2491:ChurchâTuring thesis
2478:Computability theory
1687:continuum hypothesis
1205:Square of opposition
1063:Gödel's completeness
877:topological manifold
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44:improve this article
2645:Mathematical object
2536:P versus NP problem
2501:Computable function
2295:Reverse mathematics
2221:Logical consequence
2098:primitive recursive
2093:elementary function
1866:Free/bound variable
1719:TarskiâGrothendieck
1238:Logical connectives
1168:Logical equivalence
1018:Logical consequence
921:stratified sampling
795:as the variable x.
563:{\displaystyle x=y}
2724:Mathematical logic
2443:Transfer principle
2406:Semantics of logic
2391:Categorical theory
2367:Non-standard model
1881:Logical connective
1008:Information theory
957:Mathematical logic
866:singularity theory
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139:mathematical logic
2695:set index article
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2613:Abstract category
2416:Theories of truth
2226:Rule of inference
2216:Natural deduction
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1742:
1741:
1447:Cartesian product
1352:
1351:
1258:Many-valued logic
1233:Boolean functions
1116:Russell's paradox
1091:diagonal argument
988:First-order logic
870:topological space
819:Russell's paradox
755:(the x such that
671:weakly stratified
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2623:History of logic
2618:Category of sets
2511:Decision problem
2290:Ordinal analysis
2231:Sequent calculus
2129:Boolean algebras
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2014:logical/constant
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1677:ZermeloâFraenkel
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1111:LöwenheimâSkolem
998:Formal semantics
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860:Stratified space
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839:Harper & Row
825:'s central work
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2739:Stratifications
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2604:Category theory
2594:Algebraic logic
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2548:Lambda calculus
2486:Church encoding
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2448:Truth predicate
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2270:Complete theory
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1761: and
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1714:New Foundations
1682:axiom of choice
1665:
1627:Gödel numbering
1567: and
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1228:Boolean algebra
1214:
1178:Equiconsistency
1143:Classical logic
1120:
1101:Halting problem
1089: and
1065: and
1053: and
1052:
1047:Theorems (
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959:
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897:Hassler Whitney
862:
856:
812:lambda calculus
805:New Foundations
801:New Foundations
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678:New Foundations
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379:New Foundations
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2589:Abstract logic
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2568:Turing machine
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2506:Computable set
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2401:Satisfiability
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2318:Interpretation
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2139:minimal axioms
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1829:Formation rule
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1748:Formal systems
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1637:Large cardinal
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1402:Extensionality
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1397:Ordinal number
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1081:Cantor's
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893:locally closed
881:stratification
858:Main article:
855:
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841:. p. 90.
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543:atomic formula
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217:
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209:
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198:
195:
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186:
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176:
171:
167:
151:interpretation
143:stratification
134:
131:
127:Stratification
122:
121:
28:
26:
19:
9:
6:
4:
3:
2:
2751:
2740:
2737:
2735:
2732:
2730:
2727:
2725:
2722:
2720:
2717:
2716:
2714:
2703:
2702:internal link
2696:
2687:
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2675:
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2662:
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2646:
2643:
2641:
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2607:
2605:
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2600:
2597:
2595:
2592:
2590:
2587:
2586:
2584:
2580:
2574:
2571:
2569:
2566:
2564:
2563:Recursive set
2561:
2559:
2556:
2554:
2551:
2549:
2546:
2544:
2541:
2537:
2534:
2532:
2529:
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2524:
2522:
2519:
2517:
2514:
2513:
2512:
2509:
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2502:
2499:
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2402:
2399:
2397:
2394:
2392:
2389:
2385:
2382:
2381:
2380:
2377:
2373:
2372:of arithmetic
2370:
2369:
2368:
2365:
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2351:
2348:
2346:
2343:
2341:
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2320:
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2316:
2315:
2313:
2311:
2307:
2301:
2298:
2296:
2293:
2291:
2288:
2286:
2283:
2280:
2279:from ZFC
2276:
2273:
2271:
2268:
2262:
2259:
2258:
2257:
2254:
2252:
2249:
2247:
2244:
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2242:
2239:
2237:
2234:
2232:
2229:
2227:
2224:
2222:
2219:
2217:
2214:
2212:
2209:
2208:
2206:
2204:
2200:
2190:
2189:
2185:
2184:
2179:
2178:non-Euclidean
2176:
2172:
2169:
2167:
2164:
2162:
2161:
2157:
2156:
2154:
2151:
2150:
2148:
2144:
2140:
2137:
2135:
2132:
2131:
2130:
2126:
2122:
2119:
2118:
2117:
2113:
2109:
2106:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2084:
2081:
2080:
2078:
2074:
2073:
2071:
2066:
2060:
2055:Example
2052:
2044:
2039:
2038:
2037:
2034:
2032:
2029:
2025:
2022:
2020:
2017:
2015:
2012:
2010:
2007:
2006:
2005:
2002:
2000:
1997:
1995:
1992:
1990:
1987:
1983:
1980:
1978:
1975:
1974:
1973:
1970:
1966:
1963:
1961:
1958:
1956:
1953:
1951:
1948:
1947:
1946:
1943:
1941:
1938:
1934:
1931:
1929:
1926:
1924:
1921:
1920:
1919:
1916:
1912:
1909:
1907:
1904:
1902:
1899:
1897:
1894:
1892:
1889:
1887:
1884:
1883:
1882:
1879:
1877:
1874:
1872:
1869:
1867:
1864:
1860:
1857:
1855:
1852:
1850:
1847:
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1842:
1841:
1840:
1837:
1835:
1832:
1830:
1827:
1825:
1822:
1818:
1815:
1813:
1812:by definition
1810:
1809:
1808:
1805:
1801:
1798:
1797:
1796:
1793:
1791:
1788:
1786:
1783:
1781:
1778:
1776:
1773:
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1769:
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1755:
1749:
1745:
1735:
1732:
1730:
1727:
1725:
1722:
1720:
1717:
1715:
1712:
1710:
1707:
1705:
1702:
1700:
1699:KripkeâPlatek
1697:
1695:
1692:
1688:
1685:
1683:
1680:
1679:
1678:
1675:
1674:
1672:
1668:
1660:
1657:
1656:
1655:
1652:
1650:
1647:
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1640:
1639:
1638:
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1628:
1625:
1623:
1620:
1618:
1615:
1612:
1608:
1604:
1601:
1597:
1594:
1592:
1589:
1587:
1584:
1583:
1582:
1578:
1575:
1574:
1572:
1570:
1566:
1562:
1554:
1551:
1549:
1546:
1544:
1543:constructible
1541:
1540:
1539:
1536:
1534:
1531:
1529:
1526:
1524:
1521:
1519:
1516:
1514:
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1509:
1506:
1504:
1501:
1499:
1496:
1494:
1491:
1489:
1486:
1484:
1481:
1479:
1476:
1475:
1473:
1471:
1466:
1458:
1455:
1453:
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1433:
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1423:
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1414:
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1410:
1408:
1405:
1403:
1400:
1398:
1395:
1393:
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1385:
1383:
1380:
1376:
1373:
1372:
1371:
1368:
1367:
1364:
1361:
1359:
1355:
1345:
1342:
1340:
1337:
1335:
1332:
1330:
1327:
1325:
1322:
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1317:
1313:
1310:
1309:
1308:
1305:
1301:
1296:
1295:
1294:
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1290:
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1274:
1271:
1269:
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1264:
1261:
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1259:
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1254:
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1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1225:
1223:
1221:
1220:Propositional
1217:
1211:
1208:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1188:
1186:
1183:
1179:
1176:
1175:
1174:
1171:
1169:
1166:
1164:
1161:
1159:
1156:
1154:
1151:
1149:
1148:Logical truth
1146:
1144:
1141:
1140:
1138:
1136:
1132:
1129:
1127:
1123:
1117:
1114:
1112:
1109:
1107:
1104:
1102:
1099:
1097:
1094:
1092:
1088:
1084:
1080:
1078:
1075:
1073:
1070:
1068:
1064:
1061:
1060:
1058:
1056:
1050:
1045:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1019:
1016:
1014:
1011:
1009:
1006:
1004:
1001:
999:
996:
994:
991:
989:
986:
984:
981:
977:
974:
973:
972:
969:
968:
966:
962:
958:
951:
946:
944:
939:
937:
932:
931:
928:
924:
922:
915:In statistics
912:
910:
906:
902:
898:
894:
890:
886:
882:
878:
874:
871:
867:
861:
851:
848:
844:
840:
836:
830:
829:
824:
820:
815:
813:
808:
807:in practice.
806:
802:
796:
782:
762:
739:
736:
733:
730:
707:
687:
679:
674:
672:
653:
650:
647:
635:
618:
612:
609:
603:
597:
577:
570:appearing in
557:
554:
551:
544:
525:
519:
516:
513:
510:
504:
498:
478:
471:appearing in
458:
455:
452:
432:
412:
404:
388:
380:
370:
368:
364:
340:
334:
331:
325:
319:
310:
292:
286:
283:
277:
271:
262:
261:
260:
246:
238:
235:
232:
228:
221:
218:
215:
210:
207:
204:
200:
193:
188:
184:
180:
177:
174:
169:
165:
156:
152:
148:
144:
140:
130:
128:
118:
115:
107:
104:December 2009
96:
93:
89:
86:
82:
79:
75:
72:
68:
65: â
64:
60:
59:Find sources:
53:
49:
45:
39:
38:
34:
29:This article
27:
23:
18:
17:
2665:
2463:Ultraproduct
2310:Model theory
2275:Independence
2211:Formal proof
2203:Proof theory
2186:
2159:
2116:real numbers
2088:second-order
1999:Substitution
1876:Metalanguage
1817:conservative
1790:Axiom schema
1734:Constructive
1704:MorseâKelley
1670:Set theories
1649:Aleph number
1642:inaccessible
1548:Grothendieck
1432:intersection
1319:Higher-order
1307:Second-order
1253:Truth tables
1210:Venn diagram
993:Formal proof
918:
888:
880:
872:
863:
834:
826:
816:
809:
797:
675:
670:
636:
402:
376:
362:
359:
157:of the form
142:
136:
126:
125:
110:
101:
91:
84:
77:
70:
58:
42:Please help
30:
2573:Type theory
2521:undecidable
2453:Truth value
2340:equivalence
2019:non-logical
1632:Enumeration
1622:Isomorphism
1569:cardinality
1553:Von Neumann
1518:Ultrafilter
1483:Uncountable
1417:equivalence
1334:Quantifiers
1324:Fixed-point
1293:First-order
1173:Consistency
1158:Proposition
1135:Traditional
1106:Lindström's
1096:Compactness
1038:Type theory
983:Cardinality
883:defines a
854:In topology
369:theories.
367:Horn clause
2734:Set theory
2713:Categories
2384:elementary
2077:arithmetic
1945:Quantifier
1923:functional
1795:Expression
1513:Transitive
1457:identities
1442:complement
1375:hereditary
1358:Set theory
590:satisfies
491:satisfies
403:stratified
74:newspapers
2655:Supertask
2558:Recursion
2516:decidable
2350:saturated
2328:of models
2251:deductive
2246:axiomatic
2166:Hilbert's
2153:Euclidean
2134:canonical
2057:axiomatic
1989:Signature
1918:Predicate
1807:Extension
1729:Ackermann
1654:Operation
1533:Universal
1523:Recursive
1498:Singleton
1493:Inhabited
1478:Countable
1468:Types of
1452:power set
1422:partition
1339:Predicate
1285:Predicate
1200:Syllogism
1190:Soundness
1163:Inference
1153:Tautology
1055:paradoxes
901:René Thom
885:partition
783:σ
763:ϕ
740:ϕ
731:ι
708:σ
688:σ
654:ϕ
651:∣
613:σ
598:σ
578:ϕ
520:σ
499:σ
479:ϕ
456:∈
433:ϕ
413:σ
389:ϕ
284:≥
244:→
225:¬
222:∧
219:⋯
216:∧
197:¬
194:∧
181:∧
178:⋯
175:∧
147:predicate
31:does not
2640:Logicism
2633:timeline
2609:Concrete
2468:Validity
2438:T-schema
2431:Kripke's
2426:Tarski's
2421:semantic
2411:Strength
2360:submodel
2355:spectrum
2323:function
2171:Tarski's
2160:Elements
2147:geometry
2103:Robinson
2024:variable
2009:function
1982:spectrum
1972:Sentence
1928:variable
1871:Language
1824:Relation
1785:Automata
1775:Alphabet
1759:language
1613:-jection
1591:codomain
1577:Function
1538:Universe
1508:Infinite
1412:Relation
1195:Validity
1185:Argument
1083:theorem,
847:61-15277
831:(1902).
541:and any
2582:Related
2379:Diagram
2277: (
2256:Hilbert
2241:Systems
2236:Theorem
2114:of the
2059:systems
1839:Formula
1834:Grammar
1750: (
1694:General
1407:Forcing
1392:Element
1312:Monadic
1087:paradox
1028:Theorem
964:General
363:stratum
155:clauses
88:scholar
52:removed
37:sources
2700:If an
2345:finite
2108:Skolem
2061:
2036:Theory
2004:Symbol
1994:String
1977:atomic
1854:ground
1849:closed
1844:atomic
1800:ground
1763:syntax
1659:binary
1586:domain
1503:Finite
1268:finite
1126:Logics
1085:
1033:Theory
889:strata
845:
90:
83:
76:
69:
61:
2693:This
2335:Model
2083:Peano
1940:Proof
1780:Arity
1709:Naive
1596:image
1528:Fuzzy
1488:Empty
1437:union
1382:Class
1023:Model
1013:Lemma
971:Axiom
823:Frege
95:JSTOR
81:books
2458:Type
2261:list
2065:list
2042:list
2031:Term
1965:rank
1859:open
1753:list
1565:Maps
1470:sets
1329:Free
1299:list
1049:list
976:list
919:See
907:and
899:and
843:LCCN
332:>
67:news
35:any
33:cite
2145:of
2127:of
2075:of
1607:Sur
1581:Map
1388:Ur-
1370:Set
864:In
377:In
137:In
46:by
2715::
2531:NP
2155::
2149::
2079::
1756:),
1611:Bi
1603:In
923:.
911:.
673:.
634:.
141:,
2611:/
2526:P
2281:)
2067:)
2063:(
1960:â
1955:!
1950:â
1911:=
1906:â
1901:â
1896:â§
1891:âš
1886:ÂŹ
1609:/
1605:/
1579:/
1390:)
1386:(
1273:â
1263:3
1051:)
949:e
942:t
935:v
873:X
849:.
743:)
737:.
734:x
728:(
657:}
648:x
645:{
622:)
619:y
616:(
610:=
607:)
604:x
601:(
558:y
555:=
552:x
529:)
526:y
523:(
517:=
514:1
511:+
508:)
505:x
502:(
459:y
453:x
356:.
344:)
341:Q
338:(
335:S
329:)
326:P
323:(
320:S
308:.
296:)
293:Q
290:(
287:S
281:)
278:P
275:(
272:S
247:P
239:m
236:+
233:n
229:Q
211:1
208:+
205:n
201:Q
189:n
185:Q
170:1
166:Q
117:)
111:(
106:)
102:(
92:·
85:·
78:·
71:·
54:.
40:.
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