Summary: We present $\mathsf{s}_1$, a family of logics that is useful to disprove propositional formulas by means of an anytime approximation process. The system follows the paradigm of a parametrized family of logics established by Schaerf’s and Cadoli’s system $S_1$. We show that $\mathsf{s}_1$ inherits several of the nice properties of $S_1$, while presenting several attractive new properties. The family $\mathsf{s}_1$ deals with the full propositional language, has a complete tableau proof system which provides for incremental approximations; furthermore, it constitutes a full approximation of classical logic from above, with an approximation process with better relevance and locality properties than $S_1$. When applied to clausal inference, $\mathsf{s}_1$ provides a strong simplification method. An application of $\mathsf{s}_1$ to model-based diagnosis is presented, demonstrating how the solution to this problem can benefit from the use of $\mathsf{s}_1$ approximations.