Arsen has a post that discusses worries about the implications of precognition, if one assumes it to be possible, on free will, and this started me thinking on the published discussion between C. D. Broad and H. H. Price on the implications of precognition, which in turn started me thinking about the role of twentieth century philosophers in parapsychology research. Let's take the most respectable organization devoted to issues in parapsychology, the Society of Psychical Research, and list the presidents of the Society who were philosophers. The years of their presidency is in parentheses.) For those who don't have a philosophy background, I've starred the ones who were Big Names in the sense that any philosophy student properly trained in an Anglo-American philosophy department would at least recognize the name, and put a plus by those who were founding members of the Society:
Henry Sidgwick (1882-1884 and 1888-1892)*+
Arthur Balfour (1893)
William James (1894-1895)*
Frederic William Henry Myers (1900)+
Henri Bergson (1913)*
Ferdinand Canning Scott Schiller (1914)
L. P. Jacks (1917-1918)
Charlie Dunbar Broad (1935-1936 and 1958-1960)*
Henry Habberley Price (1939-1941 and 1960-1961)*
Clement William Kennedy Mundle (1971-1974)
Analytic philosophers today would, as a rule, not be caught dead associated with parapsychology, but things weren't always so. At the first founding of SPR, the idea attracted skeptics as well as critics. What is more, psychology being relatively new, there was no sharp distinction between parapsychology and psychology at large. A number of phenomena whose early study would have been classified as psychical research early on have since become respectable topics of research in the discipline at large -- of the two big topics that have done this, hallucinatory states, because of its relations to chemistry, did this fairly early, while hypnotism did so much more slowly. Now hallucinations are pretty much never thought of as parapsychology despite its early associations, and studies of hypnotism rarely so. Psychical research was always controversial; but at its beginning it was as reputable as anything else in psychology, and much more popular than most of the rest of the field. (Despite the decline in its reputation, there are still a fair number of respected psychological researchers, especially in Britain, who have involved themselves with it or been members of the SPR at one point or another. It's possible, of course, to do these things as a skeptic of paranormal phenomena, and this has become more common over time. But it's still research into the same topics.)
A second reason for the involvement of philosophers was created by the carelessness of early critics of psychical research. The philosophers involved early on ranged from largely skeptical to largely accepting of parapsychological ideas, and they came at it from radically different philosophical perspectives, but one thing that united them all in the early days was the belief that the arguments of the critics of psychical research were very, very bad. William James has a number of works in which he is very coldly acidic (for James) about the scientific sloppiness of certain criticisms of psychical research. James's criticisms of the critics are some of the harshest things he ever wrote, and often make good points. One way to get the sympathy of philosophers is to have critics who repeatedly and vehemently give obviously bad arguments; it really doesn't matter what the position is.
Perhaps more important than this was the origin of philosophy as a professional field, which was beginning to happen precisely during an era of interest in psychical research. Philosophy departments in the Anglo-American world were usually founded in opposition to the rise of psychology departments. While it would vary from place to place, early philosophy departments were largely peopled by academics interested in research into the mind but vehemently opposed to the dominant fashions in the field of psychology at the time; this is why, out of apparently nowhere, the philosophy of mind became the dominant topic of so much of the philosophical discussions of the twentieth century.
And, perhaps less important, but certainly of some influence, is the fact that psychical research posed some interesting questions about the mind and the world in which it finds itself -- questions like free will, like what distinguishes one mind from another, like mental causation, and so forth. Broad himself discussed this aspect of it in his article Henry Sidgwick and Psychical Research. H. H. Price has an article, I forget the name of it, in which he proposes that psychical research's problematizing of the sharp separation we tend to assume to exist between minds as one of the great philosophical revolutions of the day. And while I've no doubt that there are analytic philosophers who would look down on this as a motive, it's worth pointing out that some of the paradigmatic methods of analytic philosophy almost certainly derive from precisely this strain of early twentieth-century philosophy. 'Thought experiments' in philosophy of mind are, over and over again, parapsychological; there are only two related differences between then and now. The first is that the experiments of moderns are taken as hypothetical and often unimplementable, whereas their predecessors were interested in experiments that could at least in principle (and sometimes that were actually in practice) tried, although they, too, sometimes considered them purely hypothetically; the second is that the earlier philosophers thought evidence was essential to drawing conclusions, whereas their successors just make things up. (It is astounding, truly astounding, that despite the contempt with which most analytic philosophers would regard parapsychology today, so much of what they discuss is not just parapsychology but explicitly science-fictional parapsychology, parapsychology not based - as in the earlier case - on what scientific evidence was thought to show directly - however wrong they sometimes were - but having only the most tenuous connection to any identifiable evidence at all.)
In any case, the discussion between Price and Broad on the implications of precognition (neither actually committed to its existence, but simply asked about its philosophical implications) does not seem to be available anywhere online. I do find, however, that C. D. Broad's The Relevance of Psychical Research to Philosophy is online. Since very few people read or remember it today, it would be inaccurate to call it a classic, but it's actually a very nice representative of the method and approach of analytic philosophy in the 1940s; Broad was arguably at the height of his powers. Of the papers that Broad and Price published on philosophy and psychical research, however, the most famous, and the one that is most likely to be read by philosophers today (since it is sometimes anthologized for philosophy of religion) is Price's "Survival and the Idea of 'Another World'." Unfortunately, it too doesn't seem to be available online, although if you can access it at Google Books, you can read parts of it. (Broad's less-known article Human Personality and the Possibility of Its Survival, however, is.)