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We present a superconducting logic family whose operation relies on the availability of a current gain greater than one, based on the analogy to semiconductor complementary metal-oxide-semiconductor (CMOS) logic family. The Complementary Josephson Junction (CJJ) logic family utilizes two types of nonlatching devices: a conventional device and a complementary device. The conventional device has a finite critical current, and the complementary device has zero critical current with no input applied. When the input is high, the complementary device has a finite critical current, while the conventional device has zero critical current. The bias current can be steered between a branch with a complementary device and a branch with a conventional device performing logic (and memory) functions. We can also use a resistor as a load to a complementary device. We call this circuit topology the Resistor Complementary Josephson Junction (RCJJ) family. It is analogous to the semiconductor PMOS/resistor logic family. In this paper, we investigate methods of realizing complementary devices, and we present a preliminary analysis of speed, margins, and power dissipation in simple CJJ and RCJJ inverter circuits.