He became the first person named to the Chair of Philosophy at the University of Birmingham in 1900.
Born in Glasgow, Scotland, Muirhead was educated at Gilbertfield House School, the Glasgow Academy (1866–70), and proceeded to Glasgow University, where he was deeply influenced by the Hegelianism of Edward Caird, the Professor of Moral Philosophy.
The same year he won a Snell exhibition at Balliol College, Oxford, to which he went up in Trinity Term 1875.
His Library was originally published by Allen & Unwin and continued through to the 1970s.
His Library is seen as a crucial landmark in the history of modern philosophy, publishing a number of prominent 20th century philosophers including Brand Blanshard, Francis Herbert Bradley, Axel Hagerstrom, Henri Bergson, Edmund Husserl, Bernard Bosanquet, Irving Thalberg, Jr., Georg Wilhelm Hegel, Bertrand Russell and George Edward Moore.