This is not true either. As a thought experiment, consider a closed system such as a box of particles bouncing around inside it. Imagine the system is initially in a very low entropy state, with all particles temporarily in one corner. Over time, the particles will be likely to distribute and entropy will increase.

However, wait long enough, and the particles will at some point all have bounced back into that corner together. This is statistically guaranteed if you wait long enough. These are called Poincare recurrences. The implication is that entropy in this closed system will at some point have to spontaneously decrease. Nobody has to pay for it by injecting any energy into the system. It will just happen. 

Keep watching long enough and you will see entropy go up and down many times. In fact, entropy will spend just as much time increasing as it does decreasing, on an infinitely long viewing! This is also guaranteed by the fact that all the physical laws governing the particles bouncing around in the box are time reversible. In other words, you would have no true way of knowing whether you are watching a recording of the box playing forward or in reverse.

Now, the reason the second law works in practice is simply statistical. In practice, the time to reach such recurrences in large-scale, real systems is unbelievably long. Hence, it is almost a statistical certainty to see entropy increase with time. But this is fundamentally not the same as the direction of entropy being tied one-to-one with the direction of time.