Algebraic quantum field theory, or AQFT for short, is a rigorous analysis of
    the structure of relativistic quantum mechanics. It is formulated in terms of a
    net of operator algebras indexed by regions of a Lorentzian manifold. In
    several cases the mentioned net is represented by a family of von Neumann
    algebras, concretely, type III factors. Local quantum field logic arises as a
    logical system that captures the propositional structure encoded in the
    algebras of the net. In this framework, this work contributes to the solution
    of a family of open problems, emerged since the 30s, about the characterization
    of those logical systems which can be identified with the lattice of projectors
    arising from the Murray-von Neumann classification of factors. More precisely,
    based on physical requirements formally described in AQFT, an equational theory
    able to characterizethe type III condition in a factor is provided. This
    equational system motivates the study of a variety of algebras having an
    underlying orthomodular lattice structure. A Hilbert style calculus,
    algebraizable in the mentioned variety, is also introduced and a corresponding
    completeness theorem is established.

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