Reducibility of an Ising Ferromagnet from Griffiths "Equalities"

For an Ising ferromagnet, it is shown that the correlation function between a pair of spins vanishes if and only if, irrespective of an external magnetic field, the pair decouples from each other. For the correlation functions between two- and two-body spins as well as one- and two-body spins, it is proved, in the presence of a suitable magnetic field, the vanishing of these implies the reducibility of the system.