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Boolean Descriptions

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Abstract

Chapters 2 and 3 outline the nature of Boolean versus non-Boolean descriptions. While a Boolean two-valued logic with truth values “true” and “false” is best characterized by the famous “rule of the excluded middle” (or tertium non datur), non-Boolean logic violates this rule. The consequence is incompatible descriptions, which are central to the notion of complementarity. Originally imported into quantum physics by Bohr, complementary descriptions are formally related to non-commutative algebras of observables.

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Notes

  1. 1.

    Banaschewski (1977) showed that Spencer Brown’s concept of a primary algebra is exactly the theory of join and addition (i.e., symmetric difference) of Boolean algebras.

  2. 2.

    Frege (1918/1919, p. 74, editor’s translation): “Neither logic nor mathematics has the task to inquire into the souls and conscious contents of individual human beings. One might rather consider it as their task to study the mind, not the minds.”

  3. 3.

    Hilbert (1931, p. 125, editor’s translation): “Through the tertium non datur logic acquires full harmony; its theorems attain such simplicity and the system of its concepts such completeness as it ought to conform to the significance of a discipline expressing the structure of all our thinking.”

  4. 4.

    See Padoa (1901, pp. 309–365). English translation quoted from van Heijenoort (1967, p. 121).

  5. 5.

    According to Bohr (1934, p. 53), our “interpretation of the experimental material rests essentially upon the classical concepts”. Bohr’s emphasis on the special role of classical physics is somewhat misleading. By a “classical description” Bohr means a description in terms of ordinary language. Important for Bohr’s arguments is only that facts have to be described in a Boolean language, but not necessarily in terms of classical physics.

  6. 6.

    Editor’s note: For the history of the concept of complementarity and its origin in psychology and philosophy see Holton (1970).

  7. 7.

    Editor’s note: In fact, complementarity is a maximal form of incompatibility, as will be discussed later in this monograph (see Sects. 3.4 and 3.5).

  8. 8.

    Strauss (1936a, 1936b, 1938, 1967, 1970, 1973) defined a logic of complementarity as a two-valued logic with restricted sentential connectivity whose propositional calculus is given by a partial Boolean algebra.

  9. 9.

    English translation by A. Stairs as The logic of propositions which are not simultaneously decidable, reprinted by Hooker (1975, pp. 135–140). A more recent translation is by M.P. Seevinck: The logic of non-simultaneously decidable propositions, arxiv.org/abs/1103.4537. Editor’s note: Specker presented a slightly revised version on March 27, 2000, in his seminar on Quantum Logic and Hidden Parameters at ETH Zurich. This version has not been translated into English so far.

  10. 10.

    Editor’s note: The particular version of Specker’s parable is a complication of the usual pairwise incompatibility of measurements in quantum physics. It expresses that compatible measurements are jointly possible pairwise but not triplewise (see Liang et al. 2011, see also Sect. 3.5.4).

  11. 11.

    Partial Boolean algebras have been used by Franz Kamber (1964) as the propositional calculus for quantum systems before Kochen and Specker (1965a, 1965b) proved that the partial Boolean structure arising in quantum theory cannot be embedded into a Boolean algebra. Compare Sect. 3.4 for more details.

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  1. (deceased), Lab. für Physikalische Chemie, Eidgen. Technische Hochschule Zürich, Zürich, Switzerland

    Hans Primas

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  1. Collegium Helveticum, Eidgen. Technische Hochschule Zürich, Zürich, Switzerland

    Harald Atmanspacher

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Primas, H. (2017). Boolean Descriptions. In: Atmanspacher, H. (eds) Knowledge and Time. Springer, Cham. https://doi.org/10.1007/978-3-319-47370-3_2

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