The metalogical intractability of orthomodularity
Orthomodularity is not elementary (Goldblatt, 1984). In other words, there is no way to express the orthomodular property of the ortholattice Pr in an orthoframe 〈I, R〉 as an elementary (first-order) property.
It is not known whether every orthomodular lattice is embeddable into a complete orthomodular lattice.
It is an open question whether OQL is characterized by the class of all standard orthomodular Kripkean realizations.
It is not known whether OQL admits a standard canonical model. If we try to construct a canonical realization for OQL by taking Pr as the set of all possible propositions as in the OL-case (call such a realization a pseudo canonical realization), do we obtain an OQL-realization, satisfying the orthomodular property? In other words, is the pseudo canonical realization a model of OQL?
KeywordsNull Vector Canonical Model Orthomodular Lattice Elementary Class Elementary Substructure
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