Introducing Grades in Deontic Logics

In this paper we define a framework to introduce gradedness in Deontic logics through the use of fuzzy modalities. By way of example, we instantiate the framework to Standard Deontic logic (SDL) formulas. Given a deontic formula $\Phi\in SDL$, our language contains formulas of the form $\overline{r}\to N\Phi$ or $\overline{r} \to P\Phi$, where $r \in [0, 1]$, expressing that the preference or probability degree respectively of a norm $\Phi$ is at least $r$. We present sound and complete axiomatisations for these logics.