| Title:
|
Considering uncertainty and dependence in Boolean, quantum and fuzzy logics (English) |
| Author:
|
Navara, Mirko |
| Author:
|
Pták, Pavel |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
34 |
| Issue:
|
1 |
| Year:
|
1998 |
| Pages:
|
[121]-134 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A degree of probabilistic dependence is introduced in the classical logic using the Frank family of $t$-norms known from fuzzy logics. In the quantum logic a degree of quantum dependence is added corresponding to the level of noncompatibility. Further, in the case of the fuzzy logic with $P$-states, (resp. $T$-states) the consideration turned out to be fully analogous to (resp. considerably different from) the classical situation. (English) |
| Keyword:
|
degree of probabilistic dependence |
| Keyword:
|
$t$-norm |
| Keyword:
|
fuzzy logic |
| MSC:
|
03B48 |
| MSC:
|
03B52 |
| MSC:
|
03G12 |
| idZBL:
|
Zbl 1274.03101 |
| idMR:
|
MR1619059 |
| . |
| Date available:
|
2009-09-24T19:14:16Z |
| Last updated:
|
2015-03-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/135190 |
| . |
| Reference:
|
[1] Beran L.: Orthomodular Lattices.Algebraic Approach. Academia, Praha 1984 Zbl 0677.06003, MR 0785005 |
| Reference:
|
[2] Butnariu D., Klement E. P.: Triangular Norm–Based Measures and Games with Fuzzy Coalitions.Kluwer, Dordrecht 1993 Zbl 0804.90145, MR 2867321 |
| Reference:
|
[3] Lucia P. de, Pták P.: Quantum probability spaces that are nearly classical.Bull. Polish Acad. Sci. Math. 40 (1992), 163–173 Zbl 0765.60001, MR 1401868 |
| Reference:
|
[4] Dubois D.: Generalized probabilistic independence and its implications for utility.Oper. Res. Lett. 5 (1986), 255–260 Zbl 0611.90013, MR 0874728, 10.1016/0167-6377(86)90017-9 |
| Reference:
|
[5] Dvurečenskij A., Riečan B.: On joint distribution of observables for F-quantum spaces.Fuzzy Sets and Systems 39 (1991), 65–73 MR 1089012, 10.1016/0165-0114(91)90066-Y |
| Reference:
|
[6] Frank M. J.: On the simultaneous associativity of $F(x,y)$ and $x+y-F(x,y)$.Aequationes Math. 19 (1979), 194–226 Zbl 0444.39003, MR 0556722, 10.1007/BF02189866 |
| Reference:
|
[7] Kalmbach G.: Orthomodular Lattices.Academic Press, London 1983 Zbl 0554.06009, MR 0716496 |
| Reference:
|
[8] Kläy M. P., Foulis D. J.: Maximum likelihood estimation on generalized sample spaces: an alternative resolution of Simpson’s paradox.Found. Phys. 20 (1990), 777–799 MR 1008686, 10.1007/BF01889691 |
| Reference:
|
[9] Klement E. P., Mesiar R., Navara M.: Extensions of Boolean functions to $T$-tribes of fuzzy sets.BUSEFAL 63 (1995), 16–21 |
| Reference:
|
[10] Klement E. P., Navara M.: A characterization of tribes with respect to the Łukasiewicz $t$-norm.Czechoslovak Math. J. 47 (122) (1997), 689–700 Zbl 0902.28015, MR 1479313, 10.1023/A:1022822719086 |
| Reference:
|
[11] Majerník V., Pulmannová S.: Bell inequalities on quantum logics.J. Math. Phys. 33 (1992), 2173–2178 10.1063/1.529638 |
| Reference:
|
[12] Mesiar R.: Fundamental triangular norm based tribes and measures.J. Math. Anal. Appl. 177 (1993), 633–640 Zbl 0816.28014, MR 1231507, 10.1006/jmaa.1993.1283 |
| Reference:
|
[13] Mesiar R.: On the structure of $T_s$-tribes.Tatra Mountains Math. Publ. 3 (1993), 167–172 MR 1278531 |
| Reference:
|
[14] Mesiar R.: Do fuzzy quantum structures exist? Internat.J. Theoret. Physics 34 (1995), 1609–1614 MR 1353705, 10.1007/BF00676273 |
| Reference:
|
[15] Mesiar R., Navara M.: $T_s$-tribes and $T_s$-measures.J. Math. Anal. Appl. 201 (1996), 91–102 MR 1397888, 10.1006/jmaa.1996.0243 |
| Reference:
|
[16] Mundici D.: Interpretation of AF C$^*$-algebras in Łukasiewicz sentential calculus.J. Funct. Anal. 65 (1986), 15–63 Zbl 0597.46059, MR 0819173, 10.1016/0022-1236(86)90015-7 |
| Reference:
|
[17] Navara M.: A characterization of triangular norm based tribes.Tatra Mountains Math. Publ. 3 (1993), 161–166 Zbl 0799.28013, MR 1278530 |
| Reference:
|
[18] Navara M.: Algebraic approach to fuzzy quantum spaces.Demonstratio Math. 27 (1994), 589–600 Zbl 0830.03032, MR 1319404 |
| Reference:
|
[19] Navara M.: On generating finite orthomodular sublattices.Tatra Mountains Math. Publ. 10 (1997), 109–117 Zbl 0915.06004, MR 1469286 |
| Reference:
|
[20] Navara M., Pták P.: P-measures on soft fuzzy $\sigma $-algebras.Fuzzy Sets and Systems 56 (1993), 123–126 Zbl 0816.28011, MR 1223202, 10.1016/0165-0114(93)90193-L |
| Reference:
|
[21] Navara M., Pták P.: Uncertainty and dependence in classical and quantum logic – the role of triangular norms.To appear Zbl 0988.03096 |
| Reference:
|
[22] Piasecki K.: Probability of fuzzy events defined as denumerable additivity measure.Fuzzy Sets and Systems 17 (1985), 271–284 Zbl 0604.60005, MR 0819364, 10.1016/0165-0114(85)90093-4 |
| Reference:
|
[23] Pták P., Pulmannová S.: Orthomodular Structures as Quantum Logics.Kluwer Academic Publishers, Dordrecht – Boston – London 1991 MR 1176314 |
| Reference:
|
[24] Pták P., Pulmannová S.: A measure–theoretic characterization of Boolean algebras among orthomodular lattices.Comment. Math. Univ. Carolin. 35 (1994), 205–208 Zbl 0805.06010, MR 1292596 |
| Reference:
|
[25] Pykacz J.: Fuzzy set ideas in quantum logics.Internat. J. Theoret. Phys. 31 (1992), 1765–1781 Zbl 0789.03049, MR 1183522, 10.1007/BF00671785 |
| Reference:
|
[26] Salvati S.: A characterization of Boolean algebras.Ricerche Mat. 43 (1994), 357–363 Zbl 0915.06005, MR 1324757 |
| Reference:
|
[27] Schweizer B., Sklar A.: Probabilistic Metric Spaces.North–Holland, New York 1983 Zbl 0546.60010, MR 0790314 |
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