Reasoning in Quantum Theory pp 185-191 | Cite as

# The metalogical intractability of orthomodularity

## Abstract

*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**?

## Keywords

Null Vector Canonical Model Orthomodular Lattice Elementary Class Elementary Substructure## Preview

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