Completeness of certain bimodal logics for subset spaces
The bimodal logic of subset spaces was introduced in a paper by \textit{A. Dabrowski, L. Moss} and \textit{R.~Parikh} [Ann. Pure Appl. Logic 78, No.~1-3, 73-110 (1996; Zbl 0861.68092)]. Several possible axioms for the logic of intersection-closed subset spaces were discussed there, too, but no complete axiomatization could be provided. A partial solution to that problem is given in the present paper. The authors call a subset frame \((X,{\mathcal O}),\) where \(X\) is a non-empty set and \({\mathcal O}\) a set of subsets of \(X,\) directed iff for all \(x\in X\) and \(U,V\in{\mathcal O}\) such that \(x\in U\cap V\) there exists \(W\in{\mathcal O}\) such that \(x\in W\subseteq (U\cap V).\) Then, a corresponding logical system is proposed and proved to be sound and complete for the class of directed spaces. The system contains a non-finitary axiom schema. This cannot be avoided, as is additionally shown in the paper. A theorem whose proof is contained in a forthcoming article by the first author will give a complete solution to the problem mentioned above.
scientific article; zbMATH DE number 1785606