Armstrong presents a response to Colin Radford's modified version of the "unconfident examinee" Example. A student is asked when Queen Elizabeth I died, and he hesitatingly answers "1603" and exhibits no confidence in his answer. He has forgotten that at some point previously, he studied English history. Radford presents this as an Example of knowledge without belief. But Armstrong differs on this: the unconfident examinee has a belief that Queen Elizabeth I died in 1603, he knows that she died in 1603, but he does not know that he knows. Armstrong rejects the KK Principle—that to know some thing p, one must know that one knows p. Armstrong's rejection of the KK Principle is consistent with his wider externalist project.
In 1950, Armstrong formed an Anti-Conscription Committee with David Stove and Eric Dowling, all three former students of John Anderson, the Australian Philosopher, who supported conscription and also believed that anti-conscription opinions ought to be suppressed.
After studying at the University of Sydney, Armstrong did 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.
In terms of the origin of Armstrong's view of universals, Armstrong says his view of universals is "relatively unexplored territory" but points to Hilary Putnam's 1970 paper 'On Properties' as a possible forerunner. He also says that "Plato in his later works, Aristotle and the Scholastic Realists were ahead of contemporary philosophy in this matter, although handicapped by the relative backwardness of the science and the scientific methodology of their day".
In 1974, when the University of Sydney's Philosophy department split into two departments—the Department for General Philosophy and the Department for Traditional and Modern Philosophy—Armstrong joined the latter along with David Stove and Keith Campbell, while the former department pursued more radical politics and taught courses on Marxism and feminism. The two departments were reunified in 2000.
Armstrong married Jennifer Mary de Bohun Clark in 1982 and had step children. He previously married Madeleine Annette Haydon in 1950. He also served in the Royal Australian Navy.
On the question of the relationship between beliefs and knowledge, Armstrong defends a "weak acceptance" of the belief condition, namely that if a person can be said to know some thing p, he or she believes p. In a paper for the Aristotelian Society, Armstrong rejects a series of linguistic arguments for a rejection of the belief condition which argue that one can have knowledge without having belief because a Common usage of the word 'belief' is to imply lack of knowledge—Armstrong gives the Example of if you asked a man on a railway station whether the train has just left and he said "I believe it has", you would take from this that he does not know that it has.
