# Proof theory for minimal quantum logic I

@article{Nishimura1994ProofTF, title={Proof theory for minimal quantum logic I}, author={H. Nishimura}, journal={International Journal of Theoretical Physics}, year={1994}, volume={33}, pages={103-113} }

In this paper we give a sequential system of minimal quantum logic which enjoys cut-freeness naturally. The duality theorem, the cut-elimination theorem, and the completeness theorem with respect to the relational semantics of R. I. Goldblatt are presented. Due to severe limitations of space, technically heavy proofs of the first two theorems are relegated to a subsequent paper.

#### 16 Citations

Proof theory for minimal quantum logic: A remark

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It is remarked that the inference rule (‘ → ’) is superfluous for the sequential system GMQL introduced by H. Nishimura for the minimal quantum logic.

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This chapter was originally published in the book Handbook of Quantum Logic and Quantum Structures: Quantum Logic. The copy attached is provided by Elsevier for the author s benefit and for the… Expand

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A sequent system is used to give alternative proofs of two well known properties of free lattices: Whitman’s condition and semidistributivity. It demonstrates usefulness of such proof systems outside… Expand

Focusing in Orthologic

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New sequent calculus systems for orthologic which satisfy the cut elimination property and incorporate the notion of focusing to add constraints on proofs and to optimise proof search are proposed. Expand

Focusing in Orthologic

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New sequent calculus systems for orthologic which satisfy the cut elimination property and incorporate the notion of focusing to add constraints on proofs and thus to facilitate proof search are proposed. Expand

Logic and Quantum Physics

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It is the aim to provide the reader with some food for thought and to give some pointers to the literature that provide an easy access to this field of research, dealing in particular with the implication problem. Expand

#### References

SHOWING 1-10 OF 11 REFERENCES

A Cut-Free Sequential System for the Propositional Modal Logic of Finite Chains

- Mathematics
- 1983

The main purpose of this paper is to give a cut-free Gentzen-type sequential system for K4.3G of finite chains. The cut-elimination theorem is proved both modeltheoretically and proof-theoretically.

Sequential Method in Quantum Logic

- Mathematics, Computer Science
- J. Symb. Log.
- 1980

The main purpose of this paper is to alter this situation by presenting an axiomatization of quantum logic as natural and as elegant as possible, which further proof-theoretical study is to be based on. Expand

Semantics of the minimal logic of quantum mechanics

- Mathematics
- 1972

The question what is the logic of the atomic world, belongs to the empirical science. It can be solved only by ways of hypotheses framing and testing. Birkhoff and von Neumann [1] put forward a… Expand

Imbedding of the Quantum Logic in the Modal System of Brower

- Mathematics, Computer Science
- J. Symb. Log.
- 1977

The semantics of modal categories is broadened, admitting propositions about the possibility of results of experiments, and the usual variant of the logic of quantum mechanics is leaned upon. Expand

Orthomodularity is not Elementary

- Mathematics, Computer Science
- J. Symb. Log.
- 1984

It is shown that the property of orthomodularity of the lattice of orthoclosed subspaces of a pre-Hilbert space ?? is not determined by any first-order properties of the relation I of orthogonality between vectors in b, and (CA, I) is an elementary substructure of ( I, I). Expand

Untersuchungen über das logische Schließen. I

- Mathematics
- 1935

The present invention relates to a process for sizing cellulose fibers or cellulose fiber containing materials and to a composition for carrying out the process. More particularly the invention… Expand