Strawson, elder son of Oxford philosopher P. F. Strawson, was educated at the Dragon School, Oxford (1959-65), from where he won a scholarship to Winchester College (1965-68).
Strawson taught at the University of Oxford from 1979 to 2000, first as a Stipendiary Lecturer at several different colleges, and then, from 1987 on, as Fellow and Tutor of Jesus College, Oxford.