## Gentzen-like methods in quantum logic (1999)

Venue: | IN TABLEAUX’99 |

Citations: | 1 - 0 self |

### BibTeX

@INPROCEEDINGS{Egly99gentzen-likemethods,

author = {Uwe Egly and Hans Tompits},

title = {Gentzen-like methods in quantum logic},

booktitle = {IN TABLEAUX’99},

year = {1999},

publisher = {}

}

### OpenURL

### Abstract

Quantum logic generally refers to the logical structure characterized by the class of orthomodular lattices. It originated from certain postulates advocated in the Hilbert-space formalism of modern quantum mechanics. In this paper, we investigate issues related to the proof theory of minimal quantum logic, i.e., quantum logic where the axiom of modularity is not stipulated. Based on a sequent-type calculus introduced by Nishimura, we will show that a modification of this system will result in a much more concise system. Moreover, a corresponding tableau system for minimal quantum logic will be proposed. An example from quantum mechanics will be used to illustrate how the principle of modularity can be encoded within our framework.

### Citations

278 |
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(Show Context)
Citation Context ... Γ ⊢ B Γ ⊢ A ∧ B, ∆ Fig. 3. The inference rules of LMQ t. (∧ ⊢)2 ( ⊢ ∨)2 ( ⊢ ∧)t Γ ⊢ ∆, A Γ ⊢ ∆, A ′′ ( ⊢ ′′ ) It is common practice to specify tableau systems in terms of Smullyan‘s uniform notation =-=[20]-=-, which allows an elegant formulation of the tableau rules and reduces the number of different cases. Therefore, let us fix some notation. By a signed formula we understand an expression of the form T... |

271 |
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- 1983
(Show Context)
Citation Context ...est of our knowledge, this will constitute the first tableau-based account for any kind of quantum logic. The tableau system will be constructed in a style resembling the approach taken by Fitting in =-=[9]-=- for various modal logics—to wit, by exploiting the technique of branch-modification rules. Furthermore, a formalization of the quantum-mechanical principle known as Lüders’ Rule will be used to illus... |

265 |
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(Show Context)
Citation Context ...nding quantum-mechanical system. Later on, in a joint paper with Garrett Birkhoff, this “logic of projection operators” has been given a more elaborated treatment, characterizing its algebraic nature =-=[4]-=-. As it turned out, the resulting logical structure can be described in terms of non-distributive, orthomodular lattices, significantly different from classical Boolean algebra. Although controversial... |

162 | Basic Proof Theory
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(Show Context)
Citation Context ...A ⊢ A. Now, the point is that the same result holds if the cut-rule is replaced by double-negation elimination rules like the ones given in Lemma 5 below. (For more details on non-logical axioms, see =-=[24]-=-.) We commence with our result on the redundancy of a large number of GMQLrules by using the new rule alt. Afterwards, soundness and completeness of the calculus LMQ will be established. We conclude t... |

118 |
Orthomodular Lattices
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- 1983
(Show Context)
Citation Context ...others. For a thorough introduction to the subject of quantum logic, we refer the reader to several sources information: [6] provides a survey of quantum logic with emphasis on its logical structure; =-=[13]-=- and [19] discuss the logic of orthomodular structures; and [21] is a recent textbook on quantum logic with respect to physics. A sequent-type account of orthologic independent from the series of pape... |

79 | Automaton logic
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(Show Context)
Citation Context ...gic, we refer the reader to several sources information: [6] provides a survey of quantum logic with emphasis on its logical structure; [13] and [19] discuss the logic of orthomodular structures; and =-=[21]-=- is a recent textbook on quantum logic with respect to physics. A sequent-type account of orthologic independent from the series of papers mentioned above—but related to a system described in [13]—is ... |

69 |
Orthomodular structures as quantum logics
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(Show Context)
Citation Context ...or a thorough introduction to the subject of quantum logic, we refer the reader to several sources information: [6] provides a survey of quantum logic with emphasis on its logical structure; [13] and =-=[19]-=- discuss the logic of orthomodular structures; and [21] is a recent textbook on quantum logic with respect to physics. A sequent-type account of orthologic independent from the series of papers mentio... |

52 | Unsharp quantum logics
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- 1994
(Show Context)
Citation Context ...n by Goldblatt [11], Nishimura [16], Cutland and Gibbins [5], Tamura [23], and others. For a thorough introduction to the subject of quantum logic, we refer the reader to several sources information: =-=[6]-=- provides a survey of quantum logic with emphasis on its logical structure; [13] and [19] discuss the logic of orthomodular structures; and [21] is a recent textbook on quantum logic with respect to p... |

37 |
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(Show Context)
Citation Context ...n terms of ortholattices. Indeed, according to some standard results from abstract algebra, closed subsets of an O-frame form an ortholattice under the partial ordering of set inclusion (cf. Birkhoff =-=[3]-=- for more details). An important role in the context of sequent-type calculi plays the well-known cut-rule. However, as pointed out by Cutland and Gibbins [5] (with reference to an unpublished paper b... |

33 |
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- 1951
(Show Context)
Citation Context ...re are several possibilities to define an implication. In our case, “ → ” is defined by (A → B) := A ′ ∨ (A ∧ B), for any formulae A, B. Informally, the sequent (2) encodes the so-called Lüders’ Rule =-=[14]-=-. What does this rule mean? Let P correspond to the state of a quantum-mechanical system, and let Q correspond to the state of the system after a measurement has been performed. According to Lüders’ R... |

33 |
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- 1968
(Show Context)
Citation Context ...an be encoded within our framework. 1 Introduction Quantum logic has been introduced in the early 1930s by John von Neumann in his famous treatise on the mathematical foundations of quantum mechanics =-=[15]-=-. In that work, he proposed to regard projection operators over a given Hilbert space to represent certain propositions of a corresponding quantum-mechanical system. Later on, in a joint paper with Ga... |

29 |
Semantics analysis of orthologic
- Goldblatt
- 1974
(Show Context)
Citation Context ...l review the basic facts about minimal quantum logic. In particular, we will introduce the sequent calculus GMQL, due to Nishimura [17]. This system traces back to earlier accounts given by Goldblatt =-=[11]-=-, Nishimura [16], Cutland and Gibbins [5], Tamura [23], and others. For a thorough introduction to the subject of quantum logic, we refer the reader to several sources information: [6] provides a surv... |

11 |
Basic logic and the cube of its extensions
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- 1996
(Show Context)
Citation Context ...used as formalizations of context-dependent logics [12]. Also, quantum logic is a useful device if one is interested in the relation of different axiom systems weaker than classical logic (see, e.g., =-=[1, 2]-=-). We already mentioned the open problem of whether there is a cut-free axiomatization of (modular) quantum logic, or even a (semi-)analytic proof procedure. Related to this problem is the question of... |

9 | From basic logic to quantum logics with cut elimination
- C, Sambin
- 1997
(Show Context)
Citation Context ...ent textbook on quantum logic with respect to physics. A sequent-type account of orthologic independent from the series of papers mentioned above—but related to a system described in [13]—is given in =-=[2]-=-. We will use a propositional language with the unary operator ′ (“negation”), and the two binary operators ∧ (“conjunction”) and ∨ (“disjunction”) as primitive logical connectives. We use P, Q, R, . ... |

8 |
Sequential method in quantum logic
- Nishimura
- 1980
(Show Context)
Citation Context ...ic facts about minimal quantum logic. In particular, we will introduce the sequent calculus GMQL, due to Nishimura [17]. This system traces back to earlier accounts given by Goldblatt [11], Nishimura =-=[16]-=-, Cutland and Gibbins [5], Tamura [23], and others. For a thorough introduction to the subject of quantum logic, we refer the reader to several sources information: [6] provides a survey of quantum lo... |

7 |
A regular sequent calculus for quantum logic in which ∧ and ∨ are dual, Logique et Analyse - Nouvelle Serie
- Cutland, Gibbins
- 1982
(Show Context)
Citation Context ...ntum logic. In particular, we will introduce the sequent calculus GMQL, due to Nishimura [17]. This system traces back to earlier accounts given by Goldblatt [11], Nishimura [16], Cutland and Gibbins =-=[5]-=-, Tamura [23], and others. For a thorough introduction to the subject of quantum logic, we refer the reader to several sources information: [6] provides a survey of quantum logic with emphasis on its ... |

7 |
Particles and Paradoxes - The Limits of Quantum Logic
- Gibbins
- 1987
(Show Context)
Citation Context ...ed. In this section, we will describe how modularity can be included within our sequent-type formalism, by way of an example from quantum mechanics. Consider the following sequent, taken from Gibbins =-=[10]-=-: P ⊢ Q → (Q ∧ (P ∨ Q ′ )). (2) In quantum logic, there are several possibilities to define an implication. In our case, “ → ” is defined by (A → B) := A ′ ∨ (A ∧ B), for any formulae A, B. Informally... |

5 |
A Gentzen formulation without the cut rule for ortholattices, Kobe
- Tamura
- 1988
(Show Context)
Citation Context ...In particular, we will introduce the sequent calculus GMQL, due to Nishimura [17]. This system traces back to earlier accounts given by Goldblatt [11], Nishimura [16], Cutland and Gibbins [5], Tamura =-=[23]-=-, and others. For a thorough introduction to the subject of quantum logic, we refer the reader to several sources information: [6] provides a survey of quantum logic with emphasis on its logical struc... |

4 |
Introduction to quantum logic, unpublished
- Dummett
- 1976
(Show Context)
Citation Context ...ails). An important role in the context of sequent-type calculi plays the well-known cut-rule. However, as pointed out by Cutland and Gibbins [5] (with reference to an unpublished paper by M. Dummett =-=[7]-=-), the inclusion of an unrestricted cut rule in a system like GMQL would imply a collapse to classical logic. So, in order to remain in the realm of quantum logic, only restricted forms of cut can be ... |

3 |
Proof Theory for Minimal Quantum Logic I
- Nishimura
- 1994
(Show Context)
Citation Context ...logic which turns out to be more concise than an earlier ⋆ The authors would like to thank Markus Moschner for pointing out some of the relevant bibliographic references.ssystem proposed by Nishimura =-=[17]-=-. Although Takano [22] noted that one rule of Nishimura’s calculus is superfluous, instead of eliminating this rule, a slight modification of it actually yields a large number of other rules redundant... |

1 |
Proof Theory for Minimal Quantum Logic Revisited
- Egly, Tompits
- 1998
(Show Context)
Citation Context ...tableau systems for (modular) quantum logic must include some restricted forms of cut as well. A detailed mathematical proof of this informal discussion can be found in the full version of this paper =-=[8]-=-. As shown by Nishimura in [16], instead of non-logical axioms one can also use the following inference rule: G ′ ⊢ F ′ F ′ , G ⊢ F ′ ⊢ G ′ OM More precisely, one can always “expand” occurrences of no... |

1 |
Quantum Logic and Classical Logic: Their Respective Roles
- Heelan
- 1974
(Show Context)
Citation Context ...oved to be of interest also in other application domains. We like to mention the research initiated by Heelan showing that quantum logic can also be used as formalizations of context-dependent logics =-=[12]-=-. Also, quantum logic is a useful device if one is interested in the relation of different axiom systems weaker than classical logic (see, e.g., [1, 2]). We already mentioned the open problem of wheth... |

1 | Proof Theory for Minimal Quantum Logic II - Nishimura - 1994 |

1 |
Proof Theory for Minimal Quantum Logic: A Remark
- Takano
- 1995
(Show Context)
Citation Context ...to be more concise than an earlier ⋆ The authors would like to thank Markus Moschner for pointing out some of the relevant bibliographic references.ssystem proposed by Nishimura [17]. Although Takano =-=[22]-=- noted that one rule of Nishimura’s calculus is superfluous, instead of eliminating this rule, a slight modification of it actually yields a large number of other rules redundant. Based on the system ... |