$\mathrm{FDE}$-Modalities and weak definability (joint work with H. Wansing)

The goal of this talk is to compare various modal logics based on Belnap and Dunn's paraconsistent four-valued logic $\mathrm{FDE}$. One of such logics is the modal logic $\mathrm{BK}$ defined by S. Odintsov and H. Wansing in 2010. Its extension $\mathrm{BS4}$ (a natural counterpart of $\mathrm{S4}$) relates to the paraconsistent Nelson's logic $\mathrm{N4}^\perp$ in the same way as $\mathrm{S4}$ relates to intuitionistic logic. Other versons of FDE-based modal logics are the paraconsistent modal logic $\mathrm{KN4}$ by L. Goble whose non-modal base coincides with R. Brady's $\mathrm{BD4}$ and the modal bilattice logic $\mathrm{MBL}$ introduced and investigated by A. Jung, U. Rivieccio, and R. Jansana. $\mathrm{MBL}$ is a generalization of $\mathrm{BK}$ that in its Kripke semantics makes use of a four-valued accessibility relation. On the way from $\mathrm{BK}$ to $\mathrm{MBL}$, the Fischer Servi–style modal logic $\mathrm{BK^{FS}}$ is defined as the set of all modal formulas valid under a modified standard translation into first-order $\mathrm{FDE}$. To compare the expressive power of these logics having the strong negation $\sim$ in the language we must weaken the notion of definitional equivalence in a suitable way. It is proved, e.g., that $\mathrm{BK^{FS}}$ is weakly definitionally equivalent to $\mathrm{BK\times BK}$ and that $\mathrm{MBL}$ is faithfully embedded into $\mathrm{BK\times BK}$ via a weakly structural translation.