Implication connectives in orthomodular lattices.
Louis Herman, Edwin Marsden, Robert Piziak · Notre Dame Journal of Formal Logic · 1975
implication, the modal operators of "necessity" and "possibility" can be introduced.This is interesting since orthomodular lattices generalize Boolean algebras.As the latter model classical logic, it is natural to ask what logical analogue might correspond to the former.We shall present evidence that the analogue seems to be a system resembling the classical Lewis modal system S4.The culmination of this evidence is in the section of the paper where we prove a restricted deduction theorem of the type that is well known to hold in S4.The authors would like to express sincerest thanks to J. Jay Zeman for many interesting and stimulating conversations.Indeed, to understand our focus and our axioms, the reader would do well to consult Zeman's paper, "Quantum logic with implication" [38], The authors are grateful to Veronica K. Piziak for the many hours she spent typing several drafts of this paper.