"Any relation satisfying 1-3 is a partial order,..."

If the order does not satisfy the total property, then $~$S$~$ is a partially ordered set, and $~$\\le$~$ is a partial order on that set, in which case certain elements might be incomparable\.

Any relation satisfying 1-3 is a partial order, and the corresponding set is a poset. A total order is a special kind of partial order defined by also satisfying 4.