This should be some kind of paraconsistent logic. LInk to Stanford Encyclopedia entry here: http://plato.stanford.edu/entries/logic-paraconsistent/
Would be nice if someone could reference examples of any projects on this.The article points to research on something called inconsistent mathematics.
A nice overview of the issue is to be found in Priest's article , Mathematical Pluralism, Logic Jnl IGPL (2012) doi: 10.1093/jigpal/jzs018.