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Editors’ Introduction: The Third Life of Quantum Logic: Quantum Logic Inspired by Quantum Computing

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Notes

  1. Boole [4] spoke somewhat abstractly of classes of “conceivable cases and conjunctures of circumstances,” whereas Boole [5] took a more concrete temporal interpretation, speaking of classes of “times” (calling these “durations”). See Kneale [19].

  2. That paper was anticipated by von Neumann’s [29] book on the mathematical foundations of quantum mechanics. There in Section 5, chapter 3, he observed that the projections defined on a Hilbert space could be regarded as representing experimental propositions concerning the properties of a quantum mechanical system. Projections correspond to closed subspaces.

  3. This enlarges the union in two distinct ways. First by adding all linear combinations (the “span”), and secondly by adding all limit points (the “closure”).

  4. This has the philosophically memorable equivalent: \(x\wedge (\sim x \vee (x \wedge y))\leq y\), which has led to regarding \(\sim x\vee(x\wedge y)\) as a conditional – the so-called “Sasakai hook,” named after its discoverer.

  5. Rédei [24] contains an interesting discussion of this, as do other publications by Rédei. See particularly Rédei [23] regarding the background correspondence from von Neumann to Birkhoff.

  6. Von Neumann seems to have gone back and forth on how he interpreted probability (frequency or logical), but about this time seemed to favor logical probability. See Rédei [23].

  7. This apparently was an unpublished idea of Alan Day, and the proof was first presented by Greechie [18]. See Dalla et al. [6]

  8. See Nishimura [21] for presentation and history of cut-free Gentzen systems for “minimal quantum logic” (what we are calling orthologic) and its history. See also Egly and Tompits [12]. Chiara and Giuntini [7] in sec. 17 (by G. Battilotti and C Faggian) discuss a Gentzen system for orthologic developed by Sambin, Battilotti, and Faggian that has a cut-free formulation, but they do not address orthomodular logic or modular orthologic.

  9. This is usually written in the Dirac notation as \(\alpha |0\rangle +\beta |1\rangle \), but we will not be so fussy in our motivating explanations here.

  10. We also note a quite opposing viewpoint, discussed in Dunn [9], which hearsay attributes to a lecture Saul Kripke gave at the University of Pittsburgh in 1974 (see [27]). Kripke apparently argued that given a logicist or set-theoretic understanding of numbers, it can be shown using Putnam’s views that \(2 + 2 > 4\) since the cartesian product of a 2-membered set with itself has more than 4 ordered pairs.

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Authors and Affiliations

  1. School of Informatics and Computing, Indiana University, Bloomington, IN, 47405, USA

    J. Michael Dunn

  2. Department of Mathematics, Indiana University, Bloomington, IN, 47405, USA

    Lawrence S. Moss

  3. Microsoft Station Q, Elings Hall 2237, University of California, Santa Barbara, CA, 93106, USA

    Zhenghan Wang

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  1. J. Michael Dunn
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  2. Lawrence S. Moss
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  3. Zhenghan Wang
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Correspondence to J. Michael Dunn.

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Dunn, J.M., Moss, L.S. & Wang, Z. Editors’ Introduction: The Third Life of Quantum Logic: Quantum Logic Inspired by Quantum Computing. J Philos Logic 42, 443–459 (2013). https://doi.org/10.1007/s10992-013-9273-7

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