We introduce two distinguishing chromatic numbers of partially ordered sets, one based on incomparability and the other on comparability. This study has given us the opportunity to apply classic results in poset theory to obtain results about these parameters. We use Dilworth's Theorem to obtain a bound for the parameter based on incomparability. In distinguishing chromatic bounds based on comparability, we use the proof of Birkoff's Fundamental Theorem of Distributive Lattices. In contrast, we provide a bound for the Boolean lattice, Bn, based on its high degree of symmetry.