### Abstract

In this paper we show a duality between two approaches to represent quantum structures abstractly and to model the logic and dynamics therein. One approach puts forward a “quantum dynamic frame” (Baltag et al. in Int J Theor Phys, 44(12):2267–2282, 2005), a labelled transition system whose transition relations are intended to represent projections and unitaries on a (generalized) Hilbert space. The other approach considers a “Piron lattice” (Piron in Foundations of Quantum Physics, 1976), which characterizes the algebra of closed linear subspaces of a (generalized) Hilbert space. We define categories of these two sorts of structures and show a duality between them. This result establishes, on one direction of the duality, that quantum dynamic frames represent quantum structures correctly; on the other direction, it gives rise to a representation of dynamics on a Piron lattice.

Original language | English (US) |
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Pages (from-to) | 781-805 |

Number of pages | 25 |

Journal | Studia Logica |

Volume | 103 |

Issue number | 4 |

DOIs | |

State | Published - Aug 28 2015 |

Externally published | Yes |

### Keywords

- Duality
- Labelled transition system
- Modal logic
- Orthomodular lattice
- Piron lattice
- Quantum logic

### ASJC Scopus subject areas

- Logic
- History and Philosophy of Science

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## Cite this

*Studia Logica*,

*103*(4), 781-805. https://doi.org/10.1007/s11225-014-9592-x