Essay on modality across different logics

Abstract: In this paper, we deal with the problem of putting together modal worlds that operate in different logic systems. When evaluating a modal sentence $\Box \varphi$, we argue that it is not sufficient to inspect the truth of $\varphi$ in accessed worlds (possibly in different logics). Instead, ways of transferring more subtle semantic information between logical systems must be established. Thus, we will introduce modal structures that accommodate communication between logic systems by fixing a common lattice $L$ where different logics build their semantics. The semantics of each logic being considered in the modal structure is a sublattice of $L$. In this system, necessity and possibility of a statement should not solely rely on the satisfaction relation in each world and the accessibility relation. The value of a formula $\Box \varphi$ will be defined in terms of a comparison between the values of $\varphi$ in accessible worlds and the common lattice $L$. We will investigate natural instances where formulas $\varphi$ can be said to be necessary$/$possible even though all accessible world falsify $\varphi$. Finally, we will discuss frames that characterize dynamic relations between logic systems: classically increasing, classically decreasing and dialectic frames.