Knowledge on treelike spaces (Q1372362)

This paper deals with an extension of the familiar modal logic by introducing two modalities and by using a treelike space of subsets to represent an epistemic process of knowledge acquisition. The modal operators are \(K\) for ``is known'' and \(\square\) for spending of resources, with \(\lozenge\) for ``possible'', as a dual for \(\square\). Interpreting \(K\) as a universal quantifier offers a novel way of understanding the meaning of quantifiers in varying domains. The language and semantics of the proposed bimodal logic are defined in a natural way. A model induced by a tree space is called a treelike model. Treelike spaces are equivalent to Ockhamist frames introduced by Zanardo. For the bimodal logic two axiom systems denoted MP and \(\text{MP}^*\) are presented. The MP system had been proven to be sound and complete with respect to subset spaces. \(\text{MP}^*\) is MP plus two additional axiom schemes characterizing incestual frames and union. \(\text{MP}^*\) is proved to be sound and canonically complete with respect to subset spaces, which are complete lattices. Finally, it is proved that the theory of treelike spaces is decidable.