A Galois connection approach to superposition and inaccessibility (Q1325675)

The authors consider ``superposition'' and ``accessibility'' in terms of the lattices generated by these relations over the power set of states. Thus superposition is the relation between states of finding wffs certain in that state, and this relation induces a Galois connection which in turn induces a closure map on the power set of states. Similarly inaccessibility is understood as the relation between states of making opposite predictions about a wff, and this too induces a Galois connection and hence induces a closure map on the power set of states. The purpose of this paper is to use the theory of Galois connections to provide a sufficient condition for the complete lattices induced by these two closure maps to coincide. A short introduction to Galois connections is included in the paper, proofs are indicated, though not detailed, and there is a useful section relating this work to that of others. According to the authors, this paper is in the tradition of Varadarajan, among others, but is novel in that it uses Galois connections to establish this result. Perhaps more importantly, the authors do not assume that the lattice \(L\) of ``events'' (the ``yes-no'' propositions) is orthomodular, nor that states are probability-measures on \(L\), or that the set of states be \(\sigma\)- convex.