The metalogical intractability of orthomodularity

Abstract

As we have seen, the proposition-ortholattice in a Kripkean realization k = (I, R, Pr, V) does not generally coincide with the (algebraically) complete ortholattice of all propositions of the orthoframe 〈I, R〉. When Pr is the set of all propositions, k will be called standard. Thus, a standard orthomodular Kripkean realization is a standard realization, where Pr is orthomodular. In the case of OL, every nonstandard Kripkean realization can be naturally extended to a standard one (see the proof of Theorem 8.1.13). In particular, Pr can always be embedded into the complete ortholattice of all propositions of the orthoframe at issue. Moreover, as we have learned from the completeness proof, the canonical model of OL is standard. In the case of OQL, however, there are various reasons that make significant the distinction between standard and nonstandard realizations:

• (i) 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.

• (ii) It is not known whether every orthomodular lattice is embeddable into a complete orthomodular lattice.

• (iii) It is an open question whether OQL is characterized by the class of all standard orthomodular Kripkean realizations.

• (iv) 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?