Edwards was born in Vienna in 1923 to assimilated Jewish parents, the youngest of three brothers.

According to Peter Singer, his upbringing was non-religious. He distinguished himself early on as a gifted student and was admitted to the Akademisches Gymnasium, a prestigious Viennese high school. When Germany annexed Austria in 1938, Edwards was sent by his family to Scotland, later joining them in Melbourne, Australia, where the family name was changed to Edwards. He attended Melbourne High School, graduating as dux of the school, then studied philosophy at the University of Melbourne, completing a B.A. and M.A.

He was awarded a scholarship to study in England in 1947, but on his way there, he stopped in New York and ended up staying there for the rest of his life, apart from a brief period teaching at the University of California in Berkeley. He finished his doctorate in New York, teaching at Columbia University, New York University, Brooklyn College, and the New School. While writing his doctoral thesis he contacted Bertrand Russell because he shared Russell's scepticism about religious belief. This led to a lasting friendship and a number of joint projects. Edwards collected Russell's writings on religion and published them 1957, with an appendix on "the Bertrand Russell case", under the title Why I am not a Christian.

Edwards was characterized by Michael Wreen as "mixed one part analytic philosopher to one part philosophe" with "a deep respect for science and common sense." His considerable influence on moral philosophy came from two works he edited, a very widely used introductory book he co-edited with Arthur Pap (A modern introduction to philosophy, 1965), and the famous Encyclopedia of Philosophy, an eight volume "massive Enlightenment work with notable analytic sensibility."

A friend wrote in an obituary: "Those who knew Edwards will always remember his erudition and his wicked sense of humour. [ ... ] Given Paul's own biting wit, it's not surprising that he so admired Voltaire and Russell. [ ... ] Never one to hide his own unbelief, he often commented that his two main goals were to demolish the influence of Heidegger and keep alive the memory of Wilhelm Reich, the much-reviled psychoanalyst whose critiques of religion Edwards felt remained valid.

Edwards was editor-in-chief of Macmillan's Encyclopedia of Philosophy, published in 1967. With eight volumes and nearly 1,500 entries by over 500 contributors it is one of the monumental works of twentieth century philosophy. Using his editorial prerogative, Edwards made sure that there were plentiful entries on atheism, materialism and related subjects. He always remained "a fervent advocate of clarity and rigour in philosophical argument." When, after four decades, the Encyclopedia was revised by other editors for a new edition, Edwards told Peter Singer that he was "distressed that the revisions had diluted the philosophical message and had been too gentle on a lot of postmodern thought."

Edwards related in a BBC talk that when he came to New York in 1947, "Wilhelm Reich was the talk of the town. Reich had at that time a large and enthusiastic following, especially among young intellectuals and people whose sympathies were clearly on the left but who, like Reich himself, had become totally disenchanted with communism as it had developed in Russia. The main source of Reich's attractiveness was not the orgone theory. [ ... ] It was first and foremost Reich's new therapy that seemed an exciting advance over the techniques of establishment psychiatry of the Freudian and other schools. There was also a widespread feeling that Reich had an original and penetrating insight into the troubles of the human race. ... For some years many of my friends and I regarded him as something akin to a messiah."

Twenty years later, as editor-in-chief of the Encyclopedia of Philosophy, Edwards wrote an article about Reich, comprising 11 pages as compared e.g. to the four pages devoted to Freud, the only serious text a noted philosopher ever wrote on Reich. He pointed out what is of interest to philosophers in Reich: his views concerning the origin of religious and metaphysical needs, the relation between the individual and society and the possibility of social progress, and, above all, the implications of his psychiatry for certain aspects of the mind-body problem. An abridged version of the article appeared in the Encyclopedia of Unbelief (ed. Gordon Stein, 1985).

As to Reich's orgone theory, Edwards omitted it from the Encyclopedia article because "it is of no philosophical interest." But in the BBC talk he said somewhat more: "I concede that Reich had no real competence as a physicist... At the same time I am quite convinced that the orgone theory cannot be complete nonsense. For a number of years, largely out of curiosity, I sat in an orgone accumulator once a day." Then he related some "facts", which in his opinion "make it impossible to dismiss the orgone theory." After Edwards' death, one of Reich's orgone accumulators was found in his apartment.