 Dave Ripley [2013] provides a multi-sequent calculus version of “classical logic” that he argues solves some of the paradoxes. Notably, he claims it solves at least the Sorites and Liar Paradoxes (see the entries on the  sorites paradox  and  liar Paradox).  The system conservatively extends classical logic. Ripley claims that this is what makes it classical. However, the system is not transitive, and does not have a Cut rule.