After studying at the University of Sydney, Armstrong undertook a B.Phil at the University of Oxford and a Ph.D at the University of Melbourne. He taught at Birkbeck College in 1954–55, then at the University of Melbourne from 1956–63. In 1964, he became Challis Professor of Philosophy at the University of Sydney, where he stayed until his retirement in 1991. During his career, he was a visiting lecturer at a number of institutions including Yale, Stanford, the University of Notre Dame, the University of Texas at Austin and Franklin and Marshall College.