In this paper, we propose a paradigm shift in representing and optimizing logic by using only majority (MAJ) and inversion (INV) functions as basic operations. We represent logic functions by majority-inverter graph (MIG): a directed acyclic graph consisting of three-input majority nodes and regular/complemented edges. We optimize MIGs via a new Boolean algebra, based exclusively on majority and inversion operations, that we formally axiomatize in this paper. As a complement to MIG algebraic optimization, we develop powerful Boolean methods exploiting global properties of MIGs, such as bit-error masking. MIG algebraic and Boolean methods together attain very high optimization quality. Considering the set of IWLS'05 benchmarks, our MIG optimizer (MIGhty) enables a 7% depth reduction in LUT-6 circuits mapped by ABC while also reducing size and power activity, with respect to similar and-inverter graph (AIG) optimization. Focusing on arithmetic intensive benchmarks instead, MIGhty enables a 16% depth reduction in LUT-6 circuits mapped by ABC, again with respect to similar AIG optimization. Employed as front-end to a delay-critical 22-nm application-specified integrated circuit flow (logic synthesis + physical design) MIGhty reduces the average delay/area/power by 13%/4%/3%, respectively, over 31 academic and industrial benchmarks. We also demonstrate delay/area/power improvements by 10%/10%/5% for a commercial FPGA flow.