{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,8]],"date-time":"2026-05-08T00:46:29Z","timestamp":1778201189615,"version":"3.51.4"},"reference-count":63,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2008,3,19]],"date-time":"2008-03-19T00:00:00Z","timestamp":1205884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The main aim of this report is to inform the quantum information community about investigations on the problem of probabilistic compatibility of a family of random variables: a possibility to realize such a family on the basis of a single probability measure (to construct a single Kolmogorov probability space). These investigations were started hundred of years ago by J. Boole (who invented Boolean algebras). The complete solution of the problem was obtained by Soviet mathematician Vorobjev in 60th. Surprisingly probabilists and statisticians obtained inequalities for probabilities and correlations among which one can find the famous Bell\u2019s inequality and its generalizations. Such inequalities appeared simply as constraints for probabilistic compatibility. In this framework one can not see a priori any link to such problems as nonlocality and \u201cdeath of reality\u201d which are typically linked to Bell\u2019s type inequalities in physical literature. We analyze the difference between positions of mathematicians and quantum physicists. In particular, we found that one of the most reasonable explanations of probabilistic incompatibility is mixing in Bell\u2019s type inequalities statistical data from a number of experiments performed under different experimental contexts.<\/jats:p>","DOI":"10.3390\/entropy-e10020019","type":"journal-article","created":{"date-parts":[[2008,3,29]],"date-time":"2008-03-29T14:57:12Z","timestamp":1206802632000},"page":"19-32","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":52,"title":["Bell-Boole Inequality: Nonlocality or Probabilistic Incompatibility of Random Variables?"],"prefix":"10.3390","volume":"10","author":[{"given":"Andrei","family":"Khrennikov","sequence":"first","affiliation":[{"name":"International Center for Mathematical Modeling in Physics, Engineering and Cognitive Science, V\u00e4xj\u00f6 University, S-35195, Sweden"}]}],"member":"1968","published-online":{"date-parts":[[2008,3,19]]},"reference":[{"key":"ref_1","unstructured":"Bell, J.S. (1987). Speakable and unspeakable in quantum mechanics, Cambridge Univ. Press."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"d\u2019Espagnat, B. (1995). Veiled Reality. An anlysis of present-day quantum mechanical concepts, Addison-Wesley.","DOI":"10.1063\/1.2808101"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Shimony, A. (1993). Search for a naturalistic world view, Cambridge Univ. Press.","DOI":"10.1017\/CBO9781139172196"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"880","DOI":"10.1103\/PhysRevLett.23.880","article-title":"Proposed Experiment to Test Local Hidden-Variable Theories","volume":"23","author":"Clauser","year":"1969","journal-title":"Phys. Rev. Letters"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1881","DOI":"10.1088\/0034-4885\/41\/12\/002","article-title":"Bells theorem: experimental facts and implications","volume":"41","author":"Clauser","year":"1978","journal-title":"Rep. Progr. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1119\/1.1976526","article-title":"On Hidden Variables and Quantum Mechanical Properties","volume":"38","author":"Wigner","year":"1970","journal-title":"Am. J. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1804","DOI":"10.1103\/PhysRevLett.49.1804","article-title":"Experimental test of Bell\u2019s inequalities using variable analyzers","volume":"49","author":"Aspect","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","first-page":"2","article-title":"Bell\u2019s theorem and the EPR paradox","volume":"14","author":"Home","year":"1991","journal-title":"Nuovo Cim. Rivista"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1303","DOI":"10.1103\/PhysRevD.3.1303","article-title":"S-Matrix Interpretation of Quantum Theory","volume":"3","author":"Stapp","year":"1971","journal-title":"Phys. Rev. D"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/BF02726212","article-title":"Bell\u2019s Theorem Without Hidden Variables","volume":"38","author":"Eberhard","year":"1977","journal-title":"Il Nuovo Cimento B"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1007\/BF02728628","article-title":"Bells theorem and the different concepts of locality","volume":"46","author":"Eberhard","year":"1978","journal-title":"Il Nuovo Cimento B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1474","DOI":"10.1103\/PhysRevLett.49.1474","article-title":"Constraints of Determinism and of Bell\u2019s Inequalities Are Not Equivalent","volume":"49","author":"Eberhard","year":"1982","journal-title":"Phys. Rev. Letters"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1119\/1.11393","article-title":"Unperformed experiments have no results","volume":"46","author":"Peres","year":"1978","journal-title":"Am. J. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5039","DOI":"10.1103\/PhysRevLett.81.5039","article-title":"Violation of Bell\u2019s Inequality under Strict Einstein Locality Conditions","volume":"81","author":"Weihs","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1063\/1.2713464","article-title":"A test of Bell\u2019s inequality with spacelike separation","volume":"Volume 889","author":"Weihs","year":"2007","journal-title":"Proc. Conf. Foundations of Probability and Physics - 4"},{"key":"ref_16","unstructured":"Khrennikov, A. (2001). Proc. Conf. Foundations of Probability and Physics, WSP."},{"key":"ref_17","unstructured":"Khrennikov, A. (2002). Proc. Conf. Quantum Theory: Reconsideration of Foundations, V\u00e4xj\u00f6 Univ. Press."},{"key":"ref_18","unstructured":"Khrennikov, A. (2003). Proc. Conf. Foundations of Probability and Physics - 2, V\u00e4xj\u00f6 Univ. Press."},{"key":"ref_19","unstructured":"Khrennikov, A. (2005). Proc. Conf. Foundations of Probability and Physics - 3, American Institute of Physics."},{"key":"ref_20","unstructured":"Adenier, G., Khrennikov, A., and Nieuwenhuizen, T. (2006). Proc. Conf. Quantum Theory: Reconsideration of Foundations - 3, American Institute of Physics."},{"key":"ref_21","unstructured":"Adenier, G., Fuchs, C., and Khrennikov, A. (2007). Proc. Conf. Foundations of Probability and Physics - 3, American Institute of Physics."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Kolmogoroff\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t              A.N.\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t            \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          Grundbegriffe der Wahrscheinlichkeitsrechnung\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          Springer Verlag\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          Berlin, Germany\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          1933\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t           reprinted: \n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          Foundations of the Probability Theory\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          Chelsea Publ. Comp.\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          New York, USA\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t          1956","DOI":"10.1007\/978-3-642-49888-6"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Gnedenko, B.V. (1962). The theory of probability, Chelsea Publ. Comp.","DOI":"10.1063\/1.3057804"},{"key":"ref_24","unstructured":"Accardi, L. (1970). Proc. Conf. The wave\u2013particle dualism. A tribute to Louis de Broglie on his 90th Birthday, Reidel Publ. Company."},{"key":"ref_25","unstructured":"Accardi, L. (1997). Urne e Camaleoni: Dialogo sulla realta, le leggi del caso e la teoria quantistica, Il Saggiatore."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1063\/1.2158707","article-title":"Could one now convince Einstein?","volume":"Volume 810","author":"Accardi","year":"2006","journal-title":"Proc. Conf. Quantum Theory: Reconsideration of Foundations - 3"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1103\/PhysRevLett.48.291","article-title":"Hidden Variables, Joint Probability, and the Bell Inequalities","volume":"48","author":"Fine","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1299","DOI":"10.1103\/PhysRevLett.48.1299","article-title":"Resolution of the Einstein-Podolsky-Rosen and Bell Paradoxes","volume":"48","author":"Pitowsky","year":"1982","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2316","DOI":"10.1103\/PhysRevD.27.2316","article-title":"Deterministic Model of Spin and Statistics","volume":"27","author":"Pitowsky","year":"1983","journal-title":"Phys. Rev. D"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1007\/BF00730098","article-title":"The Bell inequalities","volume":"13","author":"Rastall","year":"1983","journal-title":"Found. Phys."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"14224","DOI":"10.1073\/pnas.251524998","article-title":"A possible loophole in the theorem of Bell","volume":"98","author":"Hess","year":"2001","journal-title":"Proc. Natl. Acad. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1209\/epl\/i2002-00578-y","article-title":"Exclusion of time in the theorem of Bell","volume":"57","author":"Hess","year":"2002","journal-title":"Europhys. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1063\/1.1874568","article-title":"Bell\u2019s theorem: critique of proofs with and without inequalities","volume":"Volume 750","author":"Hess","year":"2005","journal-title":"Proc. Conf. Foundations of Probability and Physics - 3"},{"key":"ref_34","first-page":"179","article-title":"Statistical measure of ensemble nonreproducibility and correction to Bell\u2019s inequality","volume":"115","author":"Khrennikov","year":"1999","journal-title":"Il Nuovo Cimento B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1768","DOI":"10.1063\/1.533210","article-title":"Non-Kolmogorov probability models and modified Bell\u2019s inequality","volume":"41","author":"Khrennikov","year":"2000","journal-title":"J. Math. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"5934","DOI":"10.1063\/1.1287917","article-title":"A perturbation of CHSH inequality induced by fluctuations of ensemble distributions","volume":"41","author":"Khrennikov","year":"2000","journal-title":"J. Math. Phys."},{"key":"ref_37","unstructured":"Pitowsky, I. (1989). Proc. Conf. Bells Theorem, Quantum Theory and Conceptions of the Universe, Kluwer."},{"key":"ref_38","unstructured":"Pitowsky, I. (2002). Proc. Conf. Quantum Theory: Reconsideration of Foundations, Vaxjo University Press."},{"key":"ref_39","doi-asserted-by":"crossref","unstructured":"Vorob\u2019ev, N.N. (1962). Theory of Probability and its Applications, 7, 147\u2013162.","DOI":"10.1137\/1107014"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (2004). Interpretations of Probability, VSP Int. Sc. Publishers. [2nd Ed.].","DOI":"10.1515\/9783110917772"},{"key":"ref_41","first-page":"338","article-title":"Can a negative quantity be deemed a probability?","volume":"133","author":"Muckenheim","year":"1986","journal-title":"Phys. Rep."},{"key":"ref_42","first-page":"1075","article-title":"Axiomatics of the p-adic theory of probability","volume":"326","author":"Khrennikov","year":"1992","journal-title":"Dokl. Akad. Nauk SSSR Ser. Matem."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"932","DOI":"10.1063\/1.529353","article-title":"p-adic quantum mechanics with p-adic valued functions","volume":"32","author":"Khrennikov","year":"1991","journal-title":"J. Math. Phys."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/0375-9601(95)00163-W","article-title":"p-adic probability interpretation of Bell\u2019s inequality paradoxes","volume":"200","author":"Khrennikov","year":"1995","journal-title":"Phys. Lett. A"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1016\/0378-4371(95)00036-7","article-title":"p-adic probability distribution of hidden variables","volume":"215","author":"Khrennikov","year":"1995","journal-title":"Physica A"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"2423","DOI":"10.1007\/BF00670776","article-title":"p-adic probability description of Dirac\u2018s hypothetical world","volume":"34","author":"Khrennikov","year":"1995","journal-title":"Int. J. Theor. Phys."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"6625","DOI":"10.1063\/1.531342","article-title":"Statistical interpretation of p-adic quantum theories with p-adic valued wave functions","volume":"36","author":"Khrennikov","year":"1995","journal-title":"J. Math. Phys."},{"key":"ref_48","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (1994). p-Adic Valued Distributions in Mathematical Physics, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-015-8356-5"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Khrennikov, A. (1997). Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers.","DOI":"10.1007\/978-94-009-1483-4"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/S0065-2539(08)60855-0","article-title":"EPR paradox and Bell inequalities","volume":"68","year":"1986","journal-title":"Advances in Electronics and Electron Physics"},{"key":"ref_51","first-page":"95","article-title":"Interpretations of quantum mechanics, and interpretations of violations of Bell\u2019s inequality","volume":"Volume 13","year":"2001","journal-title":"Proc. Conf. Foundations of Probability and Physics"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1007\/BF02054787","article-title":"Marten Interpretations of quantum mechanics, joint measurement of incompatible observables, and counterfactual definiteness","volume":"24","year":"1994","journal-title":"Found. Phys."},{"key":"ref_53","first-page":"222","article-title":"On the Significance of the Bell Inequalities for the Locality Problem in Different Realistic Interpretations of Quantum Mechanics","volume":"45","author":"Stekelenborg","year":"1988","journal-title":"Annalen der Physik"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"9051","DOI":"10.1088\/0305-4470\/38\/41\/015","article-title":"A pre-quantum classical statistical model with infinite-dimensional phase space","volume":"38","author":"Khrennikov","year":"2005","journal-title":"J. Phys. A: Math. Gen."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"637","DOI":"10.1007\/s10702-005-1317-y","article-title":"Generalizations of Quantum Mechanics Induced by Classical Statistical Field Theory","volume":"18","author":"Khrennikov","year":"2006","journal-title":"Found. Phys. Lett."},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.physleta.2006.04.046","article-title":"Nonlinear Schrdinger equations from prequantum classical statistical field theory","volume":"357","author":"Khrennikov","year":"2006","journal-title":"Phys.\u00a0Lett. A"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"777","DOI":"10.1103\/PhysRev.47.777","article-title":"Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?","volume":"47","author":"Einstein","year":"1935","journal-title":"Phys. Rev."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1007\/s00500-005-0528-2","article-title":"Local realistic representation for correlations in the original EPR-model for position and momentum","volume":"10","author":"Khrennikov","year":"2005","journal-title":"Soft Computing"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1418","DOI":"10.1103\/PhysRevD.2.1418","article-title":"Hidden-Variable Example Based upon Data Rejection","volume":"2","author":"Pearle","year":"1970","journal-title":"Phys. Rev. D"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/S0375-9601(99)00519-8","article-title":"Bell inequality for arbitrary many settings of the analyzers","volume":"260","author":"Gisin","year":"1999","journal-title":"Phys. Lett. A"},{"key":"ref_61","unstructured":"Larsson, J. (2000). Quantum Paradoxes, Probability Theory, and Change of Ensemble, Link\u00f6ping Univ. Press."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1063\/1.2158730","article-title":"Anomalies in EPR-Bell experiments","volume":"Volume 810","author":"Adenier","year":"2006","journal-title":"Proc. Conf. Quantum Theory: Reconsideration of Foundations - 3"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1088\/0953-4075\/40\/1\/012","article-title":"Is the fair sampling assumption supported by EPR experiments?","volume":"40","author":"Adenier","year":"2007","journal-title":"J. Phys. B: Atomic, Molecular and Optical Physics"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/2\/19\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:20:50Z","timestamp":1760221250000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/10\/2\/19"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,3,19]]},"references-count":63,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2008,6]]}},"alternative-id":["e10020019"],"URL":"https:\/\/doi.org\/10.3390\/entropy-e10020019","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,3,19]]}}}