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The British Journal for the Philosophy of Science

George Boole's ‘Conditions of Possible Experience’ and the Quantum Puzzle

Itamar Pitowsky

Published Mar 1, 1994• 180 citations
Publisher

Abstract

In the mid-nineteenth century George Boole formulated his ‘conditions of possible experience’. These are equations and ineqaulities that the relative frequencies of (logically connected) events must satisfy. Some of Boole's conditions have been rediscovered in more recent years by physicists, including Bell inequalities, Clauser Horne inequalities, and many others. In this paper, the nature of Boole's conditions and their relation to propositional logic is explained, and the puzzle associated with their violation by quantum frequencies is investigated in relation to a variety of approaches to the interpretation of quantum mechanics.

DOI
10.1093/bjps/45.1.95

Related papers

Logical Views of Bell Inequalities

Zeno Toffano

Bell's Inequalites structure can be considered in a logical-probabilistic framework, as outlined by Pitowsky [1], where the origin of the inequality was attributed to George Boole’s “conditions of possible experience”. The inequality has then been often renamed the Bell-Boole inequality. An interesting fact is that G. Boole’s method used to establish his probability inequalities stems directly from his mathematical treatment of logic using idempotent symbols (which he called “elective symbols”). The “Boolean Algebra” we consider today bears some major differences with Boole’s original method as thoroughly discussed by Hailperin [2].We proposed recently the “Eigenlogic” approach [3] where we translate Boole’s original logical method to linear algebra using idempotent and also unitary operators and outline the bridges to quantum theories in particular to quantum information [4]. Eigenlogic can thus be a framework for transposing logic in a quantum context, the probabilistic aspects are then simply derived by measurements on logical operators using the Born rule.Bell Inequalities (BI) can be expressed in an operator form where the measurement observables of an experiment are combined to form Bell operators, the outcomes are then derived by using the quantum mechanical mean value (again the Born rule). The popular form adapted for this method is the CHSH form of the BI for spin 1/2 particles or polarized photons with outcomes ±1. Using the isomorphism between idempotent and self-inverse unitary operators (Householder transform) the CHSH Bell operator transforms in Eigenlogic in a combination of idempotent operators from which by using the Born rule one gets the Fine inequality on probabilities.As stated by Khrennikov [5] the BI violation occurs only when incompatible (non-commuting) observables are used in the BI associated experiments. Particularly in the CHSH-BI experiments on an entangled state, mutually unbiased bases (MUB), representing maximally dissimilar contexts, lead to the maximum quantum violation represented by the number 2√2>2. This is the case for the couple of MUB (z,x) associated to Pauli spin operators σ_x and σ_z. In Eigenlogic these (z,x) contexts represent dual logical systems which are at interplay and have an original syntax-semantic interpretation [4]. It has to be outlined that in many quantum algorithms the MUB (z,x), linked by the Hadamard gate which represents the Fourier Transformation between the two bases, provides a quantum algorithmic advantage.The situations of the BI in the so called post-quantum regime are also analyzed by generalizing the concept of PR-Box which relates outputs (a,b) to inputs (x,y) in a two-party correlation by means of the logical equation: a⊕b=x∧y. Using Eigenlogic, families of boxes with all possible combinations of logical functions are analyzed through the equations: f(a,b)=g(x,y), where f and g correspond to the 16 different Boolean functions, giving 256 possible logical equations. The logical equations are transformed in logical operators and used to define the associated Bell operator. 16 cases of the PR-Box type (considering all the possible complementations of the variables a,b,x,y) violate the CHSH-BI maximally with the number 4>2√2 (Tsirelson’s bound). Another family of 32 logical cases gives a CHSH-BI violation of 3.33 also beyond the quantum limit; this is the case for the logical equation: a∨b=x∧y.[1] Pitowsky, I., George Boole’s ‘conditions of possible experience’ and the quantum puzzle. Brit. J. Phil. Sci., 45:95–125 (1994).[2] Hailperin, T. Boole’s Algebra isn’t Boolean algebra. A description using modern algebra, of what Boole really did create. Math. Mag. 54(4): 172–184 (1981).[3] Toffano, Z. Eigenlogic in the Spirit of George Boole. Log. Univers. 14, 175–207 (2020). https://doi.org/10.1007/s11787-020-00252-3[4] Toffano, Z.; Dubois, F. Adapting Logic to Physics: The Quantum-Like Eigenlogic Program. Entropy 2020, 22, 139. https://doi.org/10.3390/e22020139[5] Khrennikov, A. Contextuality, Complementarity, Signaling, and Bell Tests. Entropy 2022, 24, 1380. https://doi.org/10.3390/e24101380

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Elemér Elad Rosinger

As shown by Pitowsky, the Bell inequalities are related to certain classes of probabilistic inequalities dealt with by George Boole, back in the 1850s. Here a short presentation of this relationship is given. Consequently, the Bell inequalities can be obtained without any assumptions of physical nature, and merely through mathematical argument.

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John Bell's inequalities have already been considered by Boole in 1862. Boole established a one-to-one correspondence between experimental outcomes and mathematical abstractions of his probability theory. His abstractions are two-valued functions that permit the logical operations AND, OR and NOT and are the elements of an algebra. Violation of the inequalities indicated to Boole an inconsistency of definition of the abstractions and/or the necessity to revise the algebra. It is demonstrated in this paper, that a violation of Bell's inequality by Einstein-Podolsky-Rosen type of experiments can be explained by Boole's ideas. Violations of Bell's inequality also call for a revision of the mathematical abstractions and corresponding algebra. It will be shown that this particular view of Bell's inequalities points toward an incompleteness of quantum mechanics, rather than to any superluminal propagation or influences at a distance.

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Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables. Copyright (C) EPLA, 2009

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Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables.

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In the mid-nineteenth century George Boole formulated his ‘conditions of possible experience’. These are equations and ineqaulities that the relative frequencies of (logically connected) events must satisfy. Some of Boole's conditions have been rediscovered in more recent years by physicists, including Bell inequalities, Clauser Horne inequalities, and many others. In this paper, the nature of Boole's conditions and their relation to propositional logic is explained, and the puzzle associated with their violation by quantum frequencies is investigated in relation to a variety of approaches to the interpretation of quantum mechanics.
Bell's Inequalites structure can be considered in a logical-probabilistic framework, as outlined by Pitowsky [1], where the origin of the inequality was attributed to George Boole’s “conditions of possible experience”. The inequality has then been often renamed the Bell-Boole inequality. An interesting fact is that G. Boole’s method used to establish his probability inequalities stems directly from his mathematical treatment of logic using idempotent symbols (which he called “elective symbols”). The “Boolean Algebra” we consider today bears some major differences with Boole’s original method as thoroughly discussed by Hailperin [2].We proposed recently the “Eigenlogic” approach [3] where we translate Boole’s original logical method to linear algebra using idempotent and also unitary operators and outline the bridges to quantum theories in particular to quantum information [4]. Eigenlogic can thus be a framework for transposing logic in a quantum context, the probabilistic aspects are then simply derived by measurements on logical operators using the Born rule.Bell Inequalities (BI) can be expressed in an operator form where the measurement observables of an experiment are combined to form Bell operators, the outcomes are then derived by using the quantum mechanical mean value (again the Born rule). The popular form adapted for this method is the CHSH form of the BI for spin 1/2 particles or polarized photons with outcomes ±1. Using the isomorphism between idempotent and self-inverse unitary operators (Householder transform) the CHSH Bell operator transforms in Eigenlogic in a combination of idempotent operators from which by using the Born rule one gets the Fine inequality on probabilities.As stated by Khrennikov [5] the BI violation occurs only when incompatible (non-commuting) observables are used in the BI associated experiments. Particularly in the CHSH-BI experiments on an entangled state, mutually unbiased bases (MUB), representing maximally dissimilar contexts, lead to the maximum quantum violation represented by the number 2√2>2. This is the case for the couple of MUB (z,x) associated to Pauli spin operators σ_x and σ_z. In Eigenlogic these (z,x) contexts represent dual logical systems which are at interplay and have an original syntax-semantic interpretation [4]. It has to be outlined that in many quantum algorithms the MUB (z,x), linked by the Hadamard gate which represents the Fourier Transformation between the two bases, provides a quantum algorithmic advantage.The situations of the BI in the so called post-quantum regime are also analyzed by generalizing the concept of PR-Box which relates outputs (a,b) to inputs (x,y) in a two-party correlation by means of the logical equation: a⊕b=x∧y. Using Eigenlogic, families of boxes with all possible combinations of logical functions are analyzed through the equations: f(a,b)=g(x,y), where f and g correspond to the 16 different Boolean functions, giving 256 possible logical equations. The logical equations are transformed in logical operators and used to define the associated Bell operator. 16 cases of the PR-Box type (considering all the possible complementations of the variables a,b,x,y) violate the CHSH-BI maximally with the number 4>2√2 (Tsirelson’s bound). Another family of 32 logical cases gives a CHSH-BI violation of 3.33 also beyond the quantum limit; this is the case for the logical equation: a∨b=x∧y.[1] Pitowsky, I., George Boole’s ‘conditions of possible experience’ and the quantum puzzle. Brit. J. Phil. Sci., 45:95–125 (1994).[2] Hailperin, T. Boole’s Algebra isn’t Boolean algebra. A description using modern algebra, of what Boole really did create. Math. Mag. 54(4): 172–184 (1981).[3] Toffano, Z. Eigenlogic in the Spirit of George Boole. Log. Univers. 14, 175–207 (2020). https://doi.org/10.1007/s11787-020-00252-3[4] Toffano, Z.; Dubois, F. Adapting Logic to Physics: The Quantum-Like Eigenlogic Program. Entropy 2020, 22, 139. https://doi.org/10.3390/e22020139[5] Khrennikov, A. Contextuality, Complementarity, Signaling, and Bell Tests. Entropy 2022, 24, 1380. https://doi.org/10.3390/e24101380
John Bell's inequalities have already been considered by Boole in 1862. Boole established a one-to-one correspondence between experimental outcomes and mathematical abstractions of his probability theory. His abstractions are two-valued functions that permit the logical operations AND, OR and NOT and are the elements of an algebra. Violation of the inequalities indicated to Boole an inconsistency of definition of the abstractions and/or the necessity to revise the algebra. It is demonstrated in this paper, that a violation of Bell's inequality by Einstein-Podolsky-Rosen type of experiments can be explained by Boole's ideas. Violations of Bell's inequality also call for a revision of the mathematical abstractions and corresponding algebra. It will be shown that this particular view of Bell's inequalities points toward an incompleteness of quantum mechanics, rather than to any superluminal propagation or influences at a distance.
John Bell's inequalities have already been considered by Boole in 1862. Boole established a one-to-one correspondence between experimental outcomes and mathematical abstractions of his probability theory. His abstractions are two-valued functions that permit the logical operations AND, OR and NOT and are the elements of an algebra. Violation of the inequalities indicated to Boole an inconsistency of definition of the abstractions and/or the necessity to revise the algebra. It is demonstrated in this paper, that a violation of Bell's inequality by Einstein-Podolsky-Rosen type of experiments can be explained by Boole's ideas. Violations of Bell's inequality also call for a revision of the mathematical abstractions and corresponding algebra. It will be shown that this particular view of Bell's inequalities points toward an incompleteness of quantum mechanics, rather than to any superluminal propagation or influences at a distance.
Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables. Copyright (C) EPLA, 2009
Mainstream interpretations of quantum theory maintain that violations of the Bell inequalities deny at least either realism or Einstein locality. Here we investigate the premises of the Bell-type inequalities by returning to earlier inequalities presented by Boole and the findings of Vorob'ev as related to these inequalities. These findings together with a space-time generalization of Boole's elements of logic lead us to a completely transparent Einstein local counterexample from everyday life that violates certain variations of the Bell inequalities. We show that the counterexample suggests an interpretation of the Born rule as a pre-measure of probability that can be transformed into a Kolmogorov probability measure by certain Einstein local space-time characterizations of the involved random variables.