Proof theory for minimal quantum logic: A remark

1995 ◽  
Vol 34 (4) ◽  
pp. 649-654 ◽  
Author(s):  
Mitio Takano
Keyword(s):  
Quantum Logic ◽  
Proof Theory

Proof theory for minimal quantum logic II

1994 ◽  
Vol 33 (7) ◽  
pp. 1427-1443 ◽  
Author(s):  
Hirokazu Nishimura
Keyword(s):  
Quantum Logic ◽  
Proof Theory

Proof Theory of Paraconsistent Quantum Logic

2017 ◽  
Vol 47 (2) ◽  
pp. 301-324 ◽  
Author(s):  
Norihiro Kamide
Keyword(s):  
Quantum Logic ◽  
Proof Theory

Proof theory for minimal quantum logic I

1994 ◽  
Vol 33 (1) ◽  
pp. 103-113 ◽  
Author(s):  
Hirokazu Nishimura
Keyword(s):  
Quantum Logic ◽  
Proof Theory

Sequential method in quantum logic

1980 ◽  
Vol 45 (2) ◽  
pp. 339-352 ◽  
Author(s):  
Hirokazu Nishimura
Keyword(s):  
Quantum Logic ◽  
Proof Theory ◽  
Axiomatic System ◽  
Orthomodular Lattices ◽  
Predominant Role ◽  
Sequential Method ◽  
Von Neumann ◽  
Material Implication ◽  
Close Relationship ◽  
Logical Study

Since Birkhoff and von Neumann [2] a new area of logical investigation has grown up under the name of quantum logic. However it seems to me that many authors have been inclined to discuss algebraic semantics as such (mainly lattices of a certain kind) almost directly without presenting any axiomatic system, far from developing any proof theory of quantum logic. See, e.g., Gunson [9], Jauch [10], Varadarajan [15], Zeirler [16], etc. In this sense many works presented under the name of quantum logic are algebraic in essence rather than genuinely logical, though it is absurd to doubt the close relationship between algebraic and logical study on quantum mechanics.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.It is true that several axiomatizations of quantum logic are present now. Several authors have investigated the so-called material implication α → β ( = ¬α∨(α ∧ β)) very closely with due regard to its importance. See, e.g., Finch [5], Piziak [11], etc. Indeed material implication plays a predominant role in any axiomatization of a logic in Hilbert-style. Clark [4] has presented an axiomatization of quantum logic with negation ¬ and material implication → as primitive connectives. In this paper we do not follow this approach. First of all, this approach is greatly complicated because orthomodular lattices are only locally distributive.

Structural Proof Theory

2001 ◽  
Author(s):  
Sara Negri ◽  
Jan von Plato ◽  
Aarne Ranta
Keyword(s):  
Proof Theory ◽  
Structural Proof Theory

Basic Proof Theory

2000 ◽  
Author(s):  
A. S. Troelstra ◽  
H. Schwichtenberg
Keyword(s):  
Proof Theory

Ternary quantum logic

2000 ◽  
Author(s):  
Normen Giesecke
Keyword(s):  
Quantum Logic

Turing Machines Based on Quantum Logic and Their Universality

2012 ◽  
Vol 35 (7) ◽  
pp. 1407 ◽  
Author(s):  
Yong-Ming LI ◽  
Ping LI
Keyword(s):  
Quantum Logic ◽  
Turing Machines

Proof Theory of the Cut Rule

2018 ◽  
Author(s):  
J. R. B. Cockett ◽  
R. A. G. Seely
Keyword(s):  
Proof Theory ◽  
Graphical Representation ◽  
Basic Component ◽  
Graphical Representations ◽  
Mathematical Language ◽  
Basic Logic ◽  
Cut Rule ◽  
Structural Rules ◽  
The One ◽  
Categorical Semantics

This chapter describes the categorical proof theory of the cut rule, a very basic component of any sequent-style presentation of a logic, assuming a minimum of structural rules and connectives, in fact, starting with none. It is shown how logical features can be added to this basic logic in a modular fashion, at each stage showing the appropriate corresponding categorical semantics of the proof theory, starting with multicategories, and moving to linearly distributive categories and *-autonomous categories. A key tool is the use of graphical representations of proofs (“proof circuits”) to represent formal derivations in these logics. This is a powerful symbolism, which on the one hand is a formal mathematical language, but crucially, at the same time, has an intuitive graphical representation.

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