Considering uncertainty and dependence in Boolean, quantum and fuzzy logics

Mirko Navara; Pavel Pták

Kybernetika (1998)

  • Volume: 34, Issue: 1, page [121]-134
  • ISSN: 0023-5954

Abstract

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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.

How to cite

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Navara, Mirko, and Pták, Pavel. "Considering uncertainty and dependence in Boolean, quantum and fuzzy logics." Kybernetika 34.1 (1998): [121]-134. <http://eudml.org/doc/33339>.

@article{Navara1998,
abstract = {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.},
author = {Navara, Mirko, Pták, Pavel},
journal = {Kybernetika},
keywords = {degree of probabilistic dependence; $t$-norm; fuzzy logic; degree of probabilistic dependence; t-norm; fuzzy logic},
language = {eng},
number = {1},
pages = {[121]-134},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Considering uncertainty and dependence in Boolean, quantum and fuzzy logics},
url = {http://eudml.org/doc/33339},
volume = {34},
year = {1998},
}

TY - JOUR
AU - Navara, Mirko
AU - Pták, Pavel
TI - Considering uncertainty and dependence in Boolean, quantum and fuzzy logics
JO - Kybernetika
PY - 1998
PB - Institute of Information Theory and Automation AS CR
VL - 34
IS - 1
SP - [121]
EP - 134
AB - 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.
LA - eng
KW - degree of probabilistic dependence; $t$-norm; fuzzy logic; degree of probabilistic dependence; t-norm; fuzzy logic
UR - http://eudml.org/doc/33339
ER -

References

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  1. Beran L., Orthomodular Lattices, Algebraic Approach. Academia, Praha 1984 Zbl0677.06003MR0785005
  2. Butnariu D., Klement E. P., Triangular Norm–Based Measures and Games with Fuzzy Coalitions, Kluwer, Dordrecht 1993 Zbl0804.90145MR2867321
  3. Lucia P. de, Pták P., Quantum probability spaces that are nearly classical, Bull. Polish Acad. Sci. Math. 40 (1992), 163–173 (1992) Zbl0765.60001MR1401868
  4. Dubois D., 10.1016/0167-6377(86)90017-9, Oper. Res. Lett. 5 (1986), 255–260 (1986) Zbl0611.90013MR0874728DOI10.1016/0167-6377(86)90017-9
  5. Dvurečenskij A., Riečan B., 10.1016/0165-0114(91)90066-Y, Fuzzy Sets and Systems 39 (1991), 65–73 (1991) MR1089012DOI10.1016/0165-0114(91)90066-Y
  6. Frank M. J., 10.1007/BF02189866, Aequationes Math. 19 (1979), 194–226 (1979) Zbl0444.39003MR0556722DOI10.1007/BF02189866
  7. Kalmbach G., Orthomodular Lattices, Academic Press, London 1983 Zbl0554.06009MR0716496
  8. Kläy M. P., Foulis D. J., 10.1007/BF01889691, Found. Phys. 20 (1990), 777–799 (1990) MR1008686DOI10.1007/BF01889691
  9. Klement E. P., Mesiar R., Navara M., Extensions of Boolean functions to T -tribes of fuzzy sets, BUSEFAL 63 (1995), 16–21 (1995) 
  10. Klement E. P., Navara M., 10.1023/A:1022822719086, Czechoslovak Math. J. 47 (122) (1997), 689–700 (1997) Zbl0902.28015MR1479313DOI10.1023/A:1022822719086
  11. Majerník V., Pulmannová S., 10.1063/1.529638, J. Math. Phys. 33 (1992), 2173–2178 (1992) DOI10.1063/1.529638
  12. Mesiar R., 10.1006/jmaa.1993.1283, J. Math. Anal. Appl. 177 (1993), 633–640 (1993) Zbl0816.28014MR1231507DOI10.1006/jmaa.1993.1283
  13. Mesiar R., On the structure of T s -tribes, Tatra Mountains Math. Publ. 3 (1993), 167–172 (1993) MR1278531
  14. Mesiar R., 10.1007/BF00676273, J. Theoret. Physics 34 (1995), 1609–1614 (1995) MR1353705DOI10.1007/BF00676273
  15. Mesiar R., Navara M., 10.1006/jmaa.1996.0243, J. Math. Anal. Appl. 201 (1996), 91–102 (1996) MR1397888DOI10.1006/jmaa.1996.0243
  16. Mundici D., 10.1016/0022-1236(86)90015-7, J. Funct. Anal. 65 (1986), 15–63 (1986) Zbl0597.46059MR0819173DOI10.1016/0022-1236(86)90015-7
  17. Navara M., A characterization of triangular norm based tribes, Tatra Mountains Math. Publ. 3 (1993), 161–166 (1993) Zbl0799.28013MR1278530
  18. Navara M., Algebraic approach to fuzzy quantum spaces, Demonstratio Math. 27 (1994), 589–600 (1994) Zbl0830.03032MR1319404
  19. Navara M., On generating finite orthomodular sublattices, Tatra Mountains Math. Publ. 10 (1997), 109–117 (1997) Zbl0915.06004MR1469286
  20. Navara M., Pták P., 10.1016/0165-0114(93)90193-L, Fuzzy Sets and Systems 56 (1993), 123–126 (1993) Zbl0816.28011MR1223202DOI10.1016/0165-0114(93)90193-L
  21. Navara M., Pták P., Uncertainty and dependence in classical and quantum logic – the role of triangular norms, To appear Zbl0988.03096
  22. Piasecki K., 10.1016/0165-0114(85)90093-4, Fuzzy Sets and Systems 17 (1985), 271–284 (1985) Zbl0604.60005MR0819364DOI10.1016/0165-0114(85)90093-4
  23. Pták P., Pulmannová S., Orthomodular Structures as Quantum Logics, Kluwer Academic Publishers, Dordrecht – Boston – London 1991 MR1176314
  24. Pták P., Pulmannová S., A measure–theoretic characterization of Boolean algebras among orthomodular lattices, Comment. Math. Univ. Carolin. 35 (1994), 205–208 (1994) Zbl0805.06010MR1292596
  25. Pykacz J., 10.1007/BF00671785, Internat. J. Theoret. Phys. 31 (1992), 1765–1781 (1992) Zbl0789.03049MR1183522DOI10.1007/BF00671785
  26. Salvati S., A characterization of Boolean algebras, Ricerche Mat. 43 (1994), 357–363 (1994) Zbl0915.06005MR1324757
  27. Schweizer B., Sklar A., Probabilistic Metric Spaces, North–Holland, New York 1983 Zbl0546.60010MR0790314

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