A Reasoning System for Graded Modal Logic

In this paper, we propose a deduction system for the modal logic $\mathbf{K}$ with graded modalities, which is obtained by adding numerical constraints to the modal operators. Graded modalities can be seen as a generalization of traditional modal operators since they include numerical bounds in the modalities, thus enabling the formal treatment of more detailed expressions, such as “there are at least $n+1$ individuals satisfying a certain property”, where $n$ is a natural number. This kind of properties commonly arise in contexts such as concurrent systems: “if at least 3 processors are free, then proceed to compute the next task”, or epistemic logic:”if more than 2 people know $P$, then $P$ is known”. The system is based on the notion of tree-hypersequents. We show the system is correct, i.e. sound and complete.