Systems of Quantum Logic

Studia Logica 101 (1):193-217 (2013)
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Abstract

Logical implications are closely related to modal operators. Lattice-valued logic LL and quantum logic QL were formulated in Titani S (1999) Lattice Valued Set Theory. Arch Math Logic 38:395–421, Titani S (2009) A Completeness Theorem of Quantum Set Theory. In: Engesser K, Gabbay DM, Lehmann D (eds) Handbook of Quantum Logic and Quantum Structures: Quantum Logic. Elsevier Science Ltd., pp. 661–702, by introducing the basic implication → which represents the lattice order. In this paper, we fomulate a predicate orthologic provided with the basic implication, which corresponds to complete ortholattices, and then formulate a quantum logic which is equivalent to QL, by using a modal operator instead of the basic implication

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A Lattice-Valued Set Theory.Satoko Titani - 1999 - Archive for Mathematical Logic 38 (6):395-421.
Strong Completeness of Lattice-Valued Logic.Mitio Takano - 2002 - Archive for Mathematical Logic 41 (5):497-505.

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