Abstract
An orthoalgebra, which is a natural generalization of an orthomodular lattice or poset, may be viewed as a “logic” or “proposition system” and, under a welldefined set of circumstances, its elements may be classified according to the Aristotelian modalities: necessary, impossible, possible, and contingent. The necessary propositions band together to form a local filter, that is, a set that intersects every Boolean subalgebra in a filter. In this paper, we give a coherent account of the basic theory of Orthoalgebras, define and study filters, local filters, and associated structures, and prove a version of the compactness theorem in classical algebraic logic.
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Foulis, D.J., Greechie, R.J. & Rüttimann, G.T. Filters and supports in orthoalgebras. Int J Theor Phys 31, 789–807 (1992). https://doi.org/10.1007/BF00678545
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DOI: https://doi.org/10.1007/BF00678545