Quantum Logic

1. Front Matter

   Pages i-viii

   PDF

2. Introduction

   • Peter Mittelstaedt

   Pages 1-5

3. The Hilbert Space Formulation of Quantum Physics

   • Peter Mittelstaedt

   Pages 6-26

4. The Logical Interpretation of the Lattice L _q

   • Peter Mittelstaedt

   Pages 27-47

5. The Material Propositions of Quantum Physics

   • Peter Mittelstaedt

   Pages 48-71

6. The Calculus of Effective Quantum Logic

   • Peter Mittelstaedt

   Pages 72-98

7. The Lattice of Effective Quantum Logic

   • Peter Mittelstaedt

   Pages 99-118

8. The Calculus of Full Quantum Logic

   • Peter Mittelstaedt

   Pages 119-139

9. Concluding Remarks: Classical Logic and Quantum Logic

   • Peter Mittelstaedt

   Pages 140-143

10. Back Matter

    Pages 144-160

    PDF

In 1936, G. Birkhoff and J. v. Neumann published an article with the title The logic of quantum mechanics'. In this paper, the authors demonstrated that in quantum mechanics the most simple observables which correspond to yes-no propositions about a quantum physical system constitute an algebraic structure, the most important proper­ ties of which are given by an orthocomplemented and quasimodular lattice Lq. Furthermore, this lattice of quantum mechanical proposi­ tions has, from a formal point of view, many similarities with a Boolean lattice L8 which is known to be the lattice of classical propositional logic. Therefore, one could conjecture that due to the algebraic structure of quantum mechanical observables a logical calculus Q of quantum mechanical propositions is established, which is slightly different from the calculus L of classical propositional logic but which is applicable to all quantum mechanical propositions (C. F. v. Weizsacker, 1955). This calculus has sometimes been called 'quan­ tum logic'. However, the statement that propositions about quantum physical systems are governed by the laws of quantum logic, which differ from ordinary classical logic and which are based on the empirically well-established quantum theory, is exposed to two serious objec­ tions: (a) Logic is a theory which deals with those relationships between various propositions that are valid independent of the content of the respective propositions. Thus, the validity of logical relationships is not restricted to a special type of proposition, e. g. to propositions about classical physical systems.

• Interpretation
• logic
• proposition
• quantum mechanics

• University of Cologne, Germany

  Peter Mittelstaedt

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