25:
2009:
Possibility theory: There is one bottle, either completely full or totally empty. The proposition "the possibility level that the bottle is full is 0.5" describes a degree of belief. One way to interpret 0.5 in that proposition is to define its meaning as: I am ready to bet that it's empty as long as
756:
924:
2276:
2005:
Fuzzy logic: When a bottle is half full, it can be said that the level of truth of the proposition "The bottle is full" is 0.5. The word "full" is seen as a fuzzy predicate describing the amount of liquid in the
2000:
Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator. The relationship with fuzzy theory can be explained with the following classic example.
575:
341:
465:
2128:
496:
Axiom 3 corresponds to the additivity axiom in probabilities. However, there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1–3 imply that:
1346:
2199:}. It is easy to see that the theories of such a logic are the generalized necessities and that the completely consistent theories coincide with the necessities (see for example Gerla 2001).
1024:
1262:
470:
Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω.
220:
637:
1162:
1055:
259:
823:
181:
1963:
1715:
1603:
1566:
1408:
54:
1890:
1095:
1928:
1821:
1678:
1491:
1443:
1784:
491:
154:
1983:
1861:
1841:
1755:
1735:
1643:
1623:
1531:
1511:
1373:
1286:
1182:
1115:
1075:
964:
783:
619:
599:
385:
365:
2019:
There is an extensive formal correspondence between probability and possibility theories, where the addition operator corresponds to the maximum operator.
100:. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor
832:
502:
268:
631:
with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally:
401:
76:
2155:
the dual of a generalized possibility. The generalized necessities are related to a very simple and interesting fuzzy logic called
2057:
493:
was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1.
938:
uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the
2327:, in Proceedings of the International Symposium on Multiple-Valued Logic, pp. 183-192, Bloomington, Indiana, May 13-16, 1975.
625:
Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is
2238:
2038:
626:
113:
1294:
37:
2340:
2027:
47:
41:
33:
2262:
Dubois, D.; Prade, H.: Possibility Theory: An
Approach to Computerized Processing of Uncertainty. Plenum Press, 1988
972:
2233:
2031:
2030:
of evidence. The operators of possibility theory can be seen as a hyper-cautious version of the operators of the
1193:
58:
2046:
2299:
2023:
751:{\displaystyle \Pi (U\cap V)\leq \min \left(\Pi (U),\Pi (V)\right)\leq \max \left(\Pi (U),\Pi (V)\right).}
190:
2228:
2136:
133:
1123:
2350:
2345:
2294:
2208:
1990:
1033:
229:
121:
788:
2223:
2213:
1985:. Because it allows for indeterminacy like this, possibility theory relates to the graduation of a
935:
159:
97:
2324:
1933:
1685:
1573:
1536:
1378:
1268:
Note that contrary to probability theory, possibility is not self-dual. That is, for any event
2160:
1986:
826:
392:
1866:
1080:
1898:
1791:
1648:
1461:
1413:
117:
2159:. In the deduction apparatus of necessity logic the logical axioms are the usual classical
2320:
1760:
476:
139:
116:
and Henri Prade further contributed to its development. Earlier, in the 1950s, economist
1994:
1968:
1846:
1826:
1740:
1720:
1628:
1608:
1516:
1496:
1358:
1271:
1167:
1100:
1060:
949:
768:
604:
584:
370:
350:
2334:
344:
2164:
2277:
Possibility Theory, Probability Theory and
Multiple-valued Logics: A Clarification
2010:
the odds are even (1:1) or better, and I would not bet at any rate that it's full.
919:{\displaystyle \Pi \left(\bigcup _{i\in I}U_{i}\right)=\sup _{i\in I}\Pi (U_{i}).}
2287:
2306:
2218:
109:
101:
93:
2042:
1448:
Accordingly, beliefs about an event can be represented by a number and a bit.
105:
104:
first introduced possibility theory in 1978 as an extension of his theory of
473:
Axiom 2 could be interpreted as the assumption that the evidence from which
395:, the possibility measure is determined by its behavior on singletons:
2163:. Also, there is only a fuzzy inference rule extending the usual
2135:
This allows one to study possibility theory using the tools of
570:{\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)}
336:{\displaystyle \Pi (U\cup V)=\max \left(\Pi (U),\Pi (V)\right)}
460:{\displaystyle \Pi (U)=\max _{\omega \in U}\Pi (\{\omega \}).}
18:
2123:{\displaystyle K=\{\,P\mid \forall S\ P(S)\leq \Pi (S)\,\}.}
92:
is a mathematical theory for dealing with certain types of
2309:, "Fuzzy Sets as the Basis for a Theory of Possibility",
2288:
Fuzzy logic: Mathematical Tools for
Approximate Reasoning
1456:
There are four cases that can be interpreted as follows:
2151:
every function satisfying Axiom 1 and Axiom 3. We call
136:Ω is a finite set. A possibility measure is a function
2295:
2060:
2015:
Possibility theory as an imprecise probability theory
1971:
1936:
1901:
1869:
1849:
1829:
1794:
1763:
1743:
1723:
1688:
1651:
1631:
1611:
1576:
1539:
1519:
1499:
1464:
1416:
1381:
1361:
1297:
1274:
1196:
1170:
1126:
1103:
1083:
1063:
1036:
975:
952:
835:
791:
771:
640:
607:
587:
505:
479:
404:
373:
353:
271:
232:
193:
162:
142:
1843:is unnecessary. I would not be surprised at all if
1341:{\displaystyle \Pi (U)+\Pi ({\overline {U}})\geq 1}
2122:
2034:, a modern development of the theory of evidence.
1977:
1957:
1922:
1884:
1855:
1835:
1815:
1778:
1749:
1729:
1709:
1672:
1637:
1617:
1597:
1560:
1525:
1505:
1485:
1437:
1402:
1367:
1340:
1280:
1256:
1176:
1156:
1109:
1089:
1069:
1049:
1018:
958:
918:
817:
777:
761:When Ω is not finite, Axiom 3 can be replaced by:
750:
613:
593:
569:
485:
459:
379:
359:
335:
253:
214:
175:
148:
2281:Annals of Mathematics and Artificial Intelligence
2022:A possibility measure can be seen as a consonant
2041:: any possibility distribution defines a unique
1737:is possible. I would not be surprised at all if
1218:
879:
705:
662:
527:
421:
293:
46:but its sources remain unclear because it lacks
2290:, Kluwer Academic Publishers, Dordrecht 2001.
8:
2114:
2067:
1965:meaning that I believe nothing at all about
1019:{\displaystyle N(U)=1-\Pi ({\overline {U}})}
448:
442:
391:It follows that, like probability on finite
16:Mathematical theory for handling uncertainty
1351:However, the following duality rule holds:
124:to describe degrees of potential surprise.
1895:The intersection of the last two cases is
1257:{\displaystyle N(U\cap V)=\min(N(U),N(V))}
2113:
2070:
2059:
1970:
1935:
1900:
1868:
1848:
1828:
1793:
1762:
1742:
1722:
1687:
1650:
1630:
1610:
1575:
1538:
1518:
1498:
1463:
1415:
1380:
1360:
1319:
1296:
1273:
1195:
1169:
1125:
1102:
1082:
1062:
1037:
1035:
1003:
974:
951:
904:
882:
864:
848:
834:
803:
796:
790:
770:
639:
606:
586:
504:
478:
424:
403:
372:
352:
270:
231:
192:
167:
161:
141:
77:Learn how and when to remove this message
2255:
200:
1117:. It is straightforward to show that:
966:, the necessity measure is defined by
7:
1645:is certainly false. It implies that
215:{\displaystyle \Pi (\varnothing )=0}
2187:, respectively, then we can assert
1533:is certainly true. It implies that
2275:Dubois, Didier and Prade, Henri, "
2101:
2077:
1937:
1870:
1689:
1577:
1540:
1382:
1313:
1298:
1142:
1084:
997:
894:
836:
728:
713:
685:
670:
641:
550:
535:
506:
480:
436:
405:
316:
301:
272:
239:
233:
194:
168:
143:
14:
1157:{\displaystyle N(U)\leq \Pi (U)}
132:For simplicity, assume that the
23:
1288:, we only have the inequality:
1050:{\displaystyle {\overline {U}}}
254:{\displaystyle \Pi (\Omega )=1}
2317:100 (Supplement): 9–34, 1999.)
2110:
2104:
2095:
2089:
2037:Possibility can be seen as an
1946:
1940:
1911:
1905:
1879:
1873:
1804:
1798:
1773:
1767:
1698:
1692:
1661:
1655:
1586:
1580:
1549:
1543:
1474:
1468:
1426:
1420:
1391:
1385:
1329:
1316:
1307:
1301:
1251:
1248:
1242:
1233:
1227:
1221:
1212:
1200:
1151:
1145:
1136:
1130:
1013:
1000:
985:
979:
910:
897:
818:{\displaystyle U_{i,\,i\in I}}
737:
731:
722:
716:
694:
688:
679:
673:
656:
644:
559:
553:
544:
538:
521:
509:
451:
439:
414:
408:
325:
319:
310:
304:
287:
275:
242:
236:
203:
197:
1:
2239:Upper and lower probabilities
2313:1:3–28, 1978. (Reprinted in
1993:, rather than the classical
1324:
1042:
1008:
128:Formalization of possibility
2167:. Such a rule says that if
176:{\displaystyle 2^{\Omega }}
2367:
1863:does not occur. It leaves
1077:, that is the elements of
1057:denotes the complement of
946:of the event. For any set
2234:Transferable belief model
2047:probability distributions
2032:transferable belief model
1958:{\displaystyle \Pi (U)=1}
1710:{\displaystyle \Pi (U)=1}
1598:{\displaystyle \Pi (U)=0}
1561:{\displaystyle \Pi (U)=1}
1403:{\displaystyle \Pi (U)=1}
96:and is an alternative to
2323:and Ladislav J. Kohout,
32:This article includes a
2149:generalized possibility
2137:imprecise probabilities
1885:{\displaystyle \Pi (U)}
1090:{\displaystyle \Omega }
61:more precise citations.
2315:Fuzzy Sets and Systems
2311:Fuzzy Sets and Systems
2300:Fuzzy Sets and Systems
2124:
2028:Dempster–Shafer theory
1979:
1959:
1924:
1923:{\displaystyle N(U)=0}
1886:
1857:
1837:
1817:
1816:{\displaystyle N(U)=0}
1780:
1751:
1731:
1711:
1674:
1673:{\displaystyle N(U)=0}
1639:
1619:
1599:
1562:
1527:
1507:
1487:
1486:{\displaystyle N(U)=1}
1439:
1438:{\displaystyle N(U)=0}
1404:
1369:
1342:
1282:
1258:
1178:
1158:
1111:
1097:that do not belong to
1091:
1071:
1051:
1030:In the above formula,
1020:
960:
920:
819:
779:
752:
615:
595:
571:
487:
461:
381:
361:
337:
255:
216:
177:
150:
2293:Ladislav J. Kohout, "
2229:Random-fuzzy variable
2179:are proved at degree
2153:generalized necessity
2125:
1980:
1960:
1925:
1887:
1858:
1838:
1818:
1781:
1752:
1732:
1712:
1675:
1640:
1620:
1600:
1563:
1528:
1508:
1488:
1440:
1405:
1370:
1343:
1283:
1259:
1179:
1159:
1112:
1092:
1072:
1052:
1021:
961:
921:
820:
780:
753:
616:
596:
572:
488:
462:
382:
362:
338:
256:
217:
178:
151:
134:universe of discourse
2209:Fuzzy measure theory
2058:
2024:plausibility measure
1991:intuitionistic logic
1969:
1934:
1899:
1867:
1847:
1827:
1792:
1779:{\displaystyle N(U)}
1761:
1741:
1721:
1686:
1649:
1629:
1609:
1574:
1537:
1517:
1497:
1462:
1414:
1379:
1359:
1295:
1272:
1194:
1168:
1124:
1101:
1081:
1061:
1034:
973:
950:
833:
789:
769:
638:
605:
585:
503:
486:{\displaystyle \Pi }
477:
402:
371:
351:
269:
230:
191:
160:
149:{\displaystyle \Pi }
140:
2325:"Possible Automata"
2286:Gerla Giangiacomo,
2224:Probabilistic logic
2214:Logical possibility
765:For all index sets
2341:Probability theory
2120:
1975:
1955:
1920:
1882:
1853:
1833:
1813:
1776:
1757:occurs. It leaves
1747:
1727:
1707:
1670:
1635:
1615:
1595:
1558:
1523:
1503:
1483:
1435:
1400:
1365:
1338:
1278:
1254:
1174:
1154:
1107:
1087:
1067:
1047:
1016:
956:
936:probability theory
916:
893:
859:
815:
775:
748:
611:
591:
567:
483:
457:
435:
393:probability spaces
377:
357:
333:
251:
212:
173:
146:
98:probability theory
90:Possibility theory
34:list of references
2303:25:357-367, 1988.
2085:
2039:upper probability
1987:many-valued logic
1978:{\displaystyle U}
1856:{\displaystyle U}
1836:{\displaystyle U}
1750:{\displaystyle U}
1730:{\displaystyle U}
1638:{\displaystyle U}
1618:{\displaystyle U}
1526:{\displaystyle U}
1506:{\displaystyle U}
1368:{\displaystyle U}
1327:
1281:{\displaystyle U}
1177:{\displaystyle U}
1110:{\displaystyle U}
1070:{\displaystyle U}
1045:
1011:
959:{\displaystyle U}
878:
844:
827:pairwise disjoint
785:, if the subsets
778:{\displaystyle I}
614:{\displaystyle V}
594:{\displaystyle U}
420:
380:{\displaystyle V}
360:{\displaystyle U}
87:
86:
79:
2358:
2263:
2260:
2129:
2127:
2126:
2121:
2083:
1995:two-valued logic
1984:
1982:
1981:
1976:
1964:
1962:
1961:
1956:
1929:
1927:
1926:
1921:
1891:
1889:
1888:
1883:
1862:
1860:
1859:
1854:
1842:
1840:
1839:
1834:
1822:
1820:
1819:
1814:
1785:
1783:
1782:
1777:
1756:
1754:
1753:
1748:
1736:
1734:
1733:
1728:
1716:
1714:
1713:
1708:
1679:
1677:
1676:
1671:
1644:
1642:
1641:
1636:
1624:
1622:
1621:
1616:
1604:
1602:
1601:
1596:
1567:
1565:
1564:
1559:
1532:
1530:
1529:
1524:
1512:
1510:
1509:
1504:
1492:
1490:
1489:
1484:
1444:
1442:
1441:
1436:
1409:
1407:
1406:
1401:
1374:
1372:
1371:
1366:
1347:
1345:
1344:
1339:
1328:
1320:
1287:
1285:
1284:
1279:
1263:
1261:
1260:
1255:
1183:
1181:
1180:
1175:
1163:
1161:
1160:
1155:
1116:
1114:
1113:
1108:
1096:
1094:
1093:
1088:
1076:
1074:
1073:
1068:
1056:
1054:
1053:
1048:
1046:
1038:
1025:
1023:
1022:
1017:
1012:
1004:
965:
963:
962:
957:
925:
923:
922:
917:
909:
908:
892:
874:
870:
869:
868:
858:
824:
822:
821:
816:
814:
813:
784:
782:
781:
776:
757:
755:
754:
749:
744:
740:
701:
697:
620:
618:
617:
612:
600:
598:
597:
592:
576:
574:
573:
568:
566:
562:
492:
490:
489:
484:
466:
464:
463:
458:
434:
386:
384:
383:
378:
366:
364:
363:
358:
342:
340:
339:
334:
332:
328:
260:
258:
257:
252:
221:
219:
218:
213:
182:
180:
179:
174:
172:
171:
155:
153:
152:
147:
118:G. L. S. Shackle
82:
75:
71:
68:
62:
57:this article by
48:inline citations
27:
26:
19:
2366:
2365:
2361:
2360:
2359:
2357:
2356:
2355:
2331:
2330:
2321:Brian R. Gaines
2283:32:35–66, 2002.
2272:
2267:
2266:
2261:
2257:
2252:
2247:
2205:
2157:necessity logic
2145:
2143:Necessity logic
2056:
2055:
2017:
1967:
1966:
1932:
1931:
1897:
1896:
1892:unconstrained.
1865:
1864:
1845:
1844:
1825:
1824:
1790:
1789:
1786:unconstrained.
1759:
1758:
1739:
1738:
1719:
1718:
1684:
1683:
1647:
1646:
1627:
1626:
1625:is impossible.
1607:
1606:
1572:
1571:
1535:
1534:
1515:
1514:
1495:
1494:
1460:
1459:
1454:
1412:
1411:
1377:
1376:
1357:
1356:
1293:
1292:
1270:
1269:
1192:
1191:
1166:
1165:
1122:
1121:
1099:
1098:
1079:
1078:
1059:
1058:
1032:
1031:
971:
970:
948:
947:
932:
900:
860:
843:
839:
831:
830:
792:
787:
786:
767:
766:
712:
708:
669:
665:
636:
635:
603:
602:
583:
582:
534:
530:
501:
500:
475:
474:
400:
399:
369:
368:
349:
348:
300:
296:
267:
266:
228:
227:
189:
188:
183:to such that:
163:
158:
157:
138:
137:
130:
122:min/max algebra
83:
72:
66:
63:
52:
38:related reading
28:
24:
17:
12:
11:
5:
2364:
2362:
2354:
2353:
2348:
2343:
2333:
2332:
2329:
2328:
2318:
2304:
2291:
2284:
2271:
2268:
2265:
2264:
2254:
2253:
2251:
2248:
2246:
2243:
2242:
2241:
2236:
2231:
2226:
2221:
2216:
2211:
2204:
2201:
2191:at degree min{
2144:
2141:
2133:
2132:
2131:
2130:
2119:
2116:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2082:
2079:
2076:
2073:
2069:
2066:
2063:
2045:of admissible
2016:
2013:
2012:
2011:
2007:
1974:
1954:
1951:
1948:
1945:
1942:
1939:
1919:
1916:
1913:
1910:
1907:
1904:
1881:
1878:
1875:
1872:
1852:
1832:
1812:
1809:
1806:
1803:
1800:
1797:
1775:
1772:
1769:
1766:
1746:
1726:
1706:
1703:
1700:
1697:
1694:
1691:
1669:
1666:
1663:
1660:
1657:
1654:
1634:
1614:
1594:
1591:
1588:
1585:
1582:
1579:
1557:
1554:
1551:
1548:
1545:
1542:
1522:
1513:is necessary.
1502:
1482:
1479:
1476:
1473:
1470:
1467:
1453:
1452:Interpretation
1450:
1446:
1445:
1434:
1431:
1428:
1425:
1422:
1419:
1399:
1396:
1393:
1390:
1387:
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119:
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67:February 2012
60:
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2307:Zadeh, Lotfi
2298:
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2165:modus ponens
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53:Please help
45:
2351:Possibility
2346:Fuzzy logic
2219:Modal logic
2161:tautologies
1823:means that
1717:means that
1605:means that
1493:means that
940:possibility
110:fuzzy logic
102:Lotfi Zadeh
94:uncertainty
59:introducing
2335:Categories
2245:References
2043:credal set
1989:, such as
1187:and that:
944:necessity
106:fuzzy sets
2250:Citations
2102:Π
2099:≤
2078:∀
2075:∣
1938:Π
1871:Π
1690:Π
1578:Π
1541:Π
1383:Π
1375:, either
1333:≥
1325:¯
1314:Π
1299:Π
1207:∩
1143:Π
1140:≤
1085:Ω
1043:¯
1009:¯
998:Π
995:−
930:Necessity
895:Π
887:∈
853:∈
846:⋃
837:Π
808:∈
729:Π
714:Π
703:≤
686:Π
671:Π
660:≤
651:∩
642:Π
551:Π
536:Π
516:∪
507:Π
481:Π
446:ω
437:Π
429:∈
426:ω
406:Π
317:Π
302:Π
282:∪
273:Π
265:Axiom 3:
240:Ω
234:Π
226:Axiom 2:
201:∅
195:Π
187:Axiom 1:
169:Ω
144:Π
2203:See also
2147:We call
1164:for any
942:and the
934:Whereas
581:subsets
347:subsets
345:disjoint
343:for any
2270:Sources
2026:in the
2006:bottle.
55:improve
2084:
1410:, or
156:from
40:, or
2183:and
2171:and
1930:and
825:are
601:and
577:for
367:and
108:and
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2049:by
1219:min
880:sup
706:max
663:min
579:any
528:max
422:max
294:max
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