Notes on quantum logics and involutive bounded posets
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In this paper, we study the exclusion problem concerning the classes of involutive bounded lattices, logics, and quantum logics (i.e., orthomodular lattices). We also obtain that a logic is a quantum logic if and only if it is a paraconsistent logic. Moreover, we give some considerations on an open question to find sufficient conditions for the existence of an orthomodular orthocomplementation on lattices. Furthermore, we revisit the Dedekind–MacNeille completion of involutive bounded posets and correct a widely cited error in quantum logics.
KeywordsInvolutive bounded poset Ortholattice Orthomodular lattice (quantum logic) Dedekind–MacNeille completion
- Beran L (1985) Orthomodular lattices: algebraic approach. Erschienen Dordrecht: ReidelGoogle Scholar
- Birkhoff G (1967) Lattice theory, 3nd edn, vol 25. AMS Colloquium PublicationsGoogle Scholar
- Chiara ML, Dalla GR (2002) Quantum logics. In: Gabbay D, Guenthner F (eds) Handbook of philosophical logic. Kluwer, Dordrecht, pp 129–228Google Scholar
- de Vries A (2009) Algebraic hierarchy of logics unifying fuzzy logic and quantum logic. Lecture Notes, pp 1–65Google Scholar