We show that the violation of time reversal invariance (T) observed in meson decay may have a profound effect on the direction of time. Our starting point is a universe without a presumed direction of time. Time evolution is modeled by taking an equal superposition of steps in both directions of time. This gives rise to multiple paths through time. The presence of T violating processes is shown to give rise to destructive interference between all paths except for the two that comprise continuously forwards and continuously backwards steps. We show that this leads to a new kind of irreversibility that is quite unlike that found in thermodynamics.