Cordner was born in Ireland, where his family in Newry, belonged to a politically radical sect of the Presbyterian Church in Ireland. Cordner studied at the Royal Belfast Academical Institution, then worked in commerce. He was ordained in 1843 as a Unitarian minister. That year he emigrated to Montreal to become minister to the first Unitarian congregation in Canada.
Cordner's ministry was successful and popular, especially among the upper-middle classes and the business elite. He wrote and edited several magazines. In sermons and speeches, he championed progressive issues; world peace, woman's rights, anti-slavery in the USA among others. He was succeeded by William Sullivan Barnes.
He retired to Boston for the warmer climate where he died in 1894; still writing on issues important to the Unitarian movement.