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Distribution in the Logic of Meaning Containment and in Quantum Mechanics

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Book cover Paraconsistency: Logic and Applications

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 26))

Abstract

The paper examines the sentential and quantificational forms of distribution in classical, intuitionistic and relevant logics, and in relation to quantum mechanics. Particular attention is also paid to the first author’s logic MCQ of meaning containment. The ultimate aim is to determine whether distribution should be included and, if so, in what forms. We conclude that the sentential and existential rule-forms, both derivable from the two-premise meta-rules of MCQ, are the only two forms that generally hold.

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Notes

  1. 1.

    The lattice properties are the meet and join properties of conjunction and disjunction which follow from the following axioms: A&B → A, A&B → B, (A → B)&(A → C) → (A → B&C), A → A ∨ B, B → A ∨ B, (A → C)&(B → C) → . (A ∨ B) → C. With respect to an ordering based on the ‘ → ’, conjunction represents the greatest lower bound and disjunction represents the least upper bound.

  2. 2.

    To derive (A&B) ∨ (A&C) from A&(B ∨ C) using the Anderson and Belnap version of the & ∨ rule, first derive A, (A&B) ∨ C, C ∨ (A&B), A&(C ∨ (A&B)) and then apply & ∨ again. The derivation of (A&B) ∨ C a from A&(B ∨ C) a using our version of & ∨ follows from (A&B) ∨ (A&C) a , A&B → (A&B) ∨ C ∅ , and A&C → (A&B) ∨ C ∅ , by applying ∨ E.

  3. 3.

    The logic DW is MC without A11. (A → B)&(B → C) → (A → C).

  4. 4.

    Thanks to Bob Meyer and Greg Restall for pointing out this lattice to the first author.

  5. 5.

    This is shown as follows: \(\neg A,A \vee {B}^{a}{/}_{x} \Rightarrow {B}^{a}{/}_{x}\), where a does not occur in B, by the DS, and hence ¬A,  ∀x(A ∨ B) ⇒  ∀xB. By the two-premise metarule (*) and the LEM, A ∨  ∀x(A ∨ B) ⇒ A ∨  ∀xB, and hence ∀x(A ∨ B) ⇒ A ∨  ∀xB.

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Acknowledgements

We acknowledge the help of Prof. John Bigelow of Monash University for initiating the current enterprise, by pointing out his concerns about distribution from the point of view of quantum mechanics. We also acknowledge help from the audiences at the Australasian Association for Logic conference at Auckland and the World Congress on Paraconsistency at Melbourne, 2008, especially Greg Restall, Bob Meyer, Francesco Paoli and Graham Priest. We also thank the referee for diligent work on the paper, which found quite a number of omissions and inaccuracies. The first author also acknowledges the financial assistance of the Australian Research Council grant DP0556114 in carrying out this research.

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Authors and Affiliations

  1. Department of Philosophy, La Trobe University, Melbourne, Australia

    Ross T. Brady

  2. Departments of Philosophy and Physics, University of Helsinki, Helsinki, Finland

    Andrea Meinander

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  1. Ross T. Brady
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  2. Andrea Meinander
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Correspondence to Ross T. Brady .

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Editors and Affiliations

  1. , Department of Philosophy, University of Auckland, Arts 2, 18 Symonds Street, Auckland, 1142, New Zealand

    Koji Tanaka

  2. , Department of Philosophy, University of Aberdeen, King's College, High Street, Aberdeen, AB24 3UB, United Kingdom

    Francesco Berto

  3. , Department of Philosophy, Victoria University of Wellington, Murphy 518, Kelburn Parade, Wellington, New Zealand

    Edwin Mares

  4. , Department of Education, University of Cagliari, Via Is Mirrionis 1, Cagliari, 09123, Italy

    Francesco Paoli

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Brady, R.T., Meinander, A. (2013). Distribution in the Logic of Meaning Containment and in Quantum Mechanics. In: Tanaka, K., Berto, F., Mares, E., Paoli, F. (eds) Paraconsistency: Logic and Applications. Logic, Epistemology, and the Unity of Science, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4438-7_13

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