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.
