Quantum Implication Algebras

Megill, Norman D.; Pavičić, Mladen

doi:10.1023/b:ijtp.0000006007.58191.da

Cited by 11 publications

(12 citation statements)

References 13 publications

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“…For example, as proved by Norman Megill and Mladen Pavičić [31] [32] Moreover, we can express any of such expressions by means of every appropriate other in a huge although definite number of equivalence classes. [32] For example, a shortest expression for ∪ expressed by means of quantum implications is a [31,32,33,34] For such a "weird" model, the question emerged as to whether it is possible to formulate a proper deductive quantum logic as a general theory of inference and how independent of its model this logic can be. In other words, can such a logic be more general than its orthomodular model?…”

Section: Introductionmentioning

confidence: 99%

Is Quantum Logic a Logic?

Pavičić¹,

Megill²

2009

Handbook of Quantum Logic and Quantum Structures

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Section: Introductionmentioning

confidence: 99%

Is Quantum Logic a Logic?

Pavičić¹,

Megill²

2009

Handbook of Quantum Logic and Quantum Structures

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“…The operation ⋅ is usually called Sasaki product, 2 and as usual we will often abbreviate x ⋅ y by xy. The operation + is well-known as a candidate for an implication-like operation in OMLs (see, e.g., [15]), and has been called Sasaki hook in this context. However, in our more general setting, + will not behave as an implication.…”

Section: From Orthomodular Lattices To Residuated Ortholatticesmentioning

confidence: 99%

Negative Translations of Orthomodular Lattices and Their Logic

Fussner,

John

2021

Preprint

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We introduce residuated ortholattices as a generalization of-and environment for the investigation of-orthomodular lattices. We establish a number of basic algebraic facts regarding these structures, characterize orthomodular lattices as those residuated ortholattices whose residual operation is term-definable in the involutive lattice signature, and demonstrate that residuated ortholattices are the equivalent algebraic semantics of an algebraizable propositional logic. We also show that orthomodular lattices may be interpreted in residuated ortholattices via a translation in the spirit of the double-negation translation of Boolean algebras into Heyting algebras, and conclude with some remarks about decidability. * W. Fussner received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No. 670624).† G. St. John acknowledges the support of MIUR within the project PRIN 2017: "Logic and cognition. Theory, experiments, and applications", CUP: 2013YP4N3, sub-project "Quantum structures and substructural logics: a unifying approach".

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“…Before we close, we should mention that there are five implications in an orthomodular lattice (quantum lattice) (Megill and Pavičić 2003). Here there are:…”

Section: Literature Review and Criticismmentioning

confidence: 99%

“…Later many algebraic structures were considered as quantum logic but none of them provides a solid basis for a logical system that can be used for reasoning; because those logics still have problems with the basic reasoning tools like implication. Although there are five QL implication operators in the literature, none of them addressed the difference between compatible and incompatible events (Dalla Chiara et al 2013) (Megill and Pavičić 2003).…”

Section: Introductionmentioning

confidence: 99%

On quantum implication

Younes

Schmitt

2019

Quantum Mach. Intell.

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Logic is an algebraic structure that defines a set of abstract rules which govern an area of interest. The abstraction property of the rules makes them reusable tools to model different problems and to reason with them. The proliferation of quantum theory brought attention to quantum logic which is a lattice of projectors and it is of importance to quantum computing. Unfortunately, basic tools like implication are not sufficiently studied in that logic, which prevents us from exploiting the power of quantum mechanics in reasoning. This note investigates the implication issue in quantum logic and defines a quantum implication operator for compatible events as well as for incompatible events. The suggested operator depends both on the angle between the vector sub-spaces of the involved events and the angles between the system state and the vector sub-spaces. It differentiates between three cases depending on the angle between the events' sub-spaces. The article further shows through an example that some classical reasoning rules such as Modus Ponens and Modus Tollens hold given the suggested implication.

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Quantum Implication Algebras

Abstract: Quantum implication algebras without complementation are formulated with the same axioms for all five quantum implications. Previous formulations of orthoimplication, orthomodular implication, and quasi-implication algebras are analysed and put in perspective to each other and our results.

Cited by 11 publications

References 13 publications

Is Quantum Logic a Logic?

Is Quantum Logic a Logic?

Negative Translations of Orthomodular Lattices and Their Logic

On quantum implication

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