A "public intellectual" is a lot like a "gourmet burger"--a fusion of two completely separate worlds, each with their own independent standards. Sometimes the addition of fancy toppings makes for a better burger, and sometimes it doesn't; but either way, it's a burger, and it'll be judged as one, regardless of the pedigree of the sauce. So it is with the public intellectual: nice though it can be for publicly engaged figures to have intellectual credentials, a glittering academic record is neither necessary nor sufficient for an outstanding contribution to the public sphere.
Take the man who appears to be everyone's favorite hall-of-fame public intellectual in recent discussions: George Orwell. Now, it's actually a bit odd to lionize Orwell, an itinerant journalist, novelist and essayist, specifically as an intellectual; his abstract ideas were not particularly original, nor did he make a lasting contribution to any particular academic discipline that I know of. Rather, his great achievement was part political, part artistic: he depicted through his writings, in a searingly vivid way, the character, structure and evolution of the political phenomenon known as totalitarianism.
His descriptive skill alone would not have made him a successful public intellectual, of course; after all, Hunter S. Thompson depicted in a searingly vivid way the character, structure and evolution of drug binges, and today he's a mostly-forgotten cult figure. Orwell's particular judgment and insight, on the other hand, was of immense value to Western society, which did mammoth battle with totalitarianism in his day, and has continued to do so, on and off, ever since. Intellectuals have to be judged brilliant and original by peers, but the acid test of a public intellectual like Orwell is to be judged correct and prescient by history. Those are two very different criteria, and it's not surprising that exceedingly few people come out as giants under both of them.
That's why Richard Posner's complaint about the decline of the gourmet burger--whoops, public intellectual--is as far off-base as the many counterarguments raised against him. Posner worries that modern public intellectuals, because they are less often distinguished academics than they used to be, are therefore less likely to be rigorous thinkers. But that just misses the point. Nobody denies that Martin Heidegger, to name just one famous example, was a paragon of rigor as a philosopher; as a public intellectual, though--well, it's hard to think of a worse blunder than becoming a Nazi apologist. The ranks of highly successful intellectuals--William Shockley, Noam Chomsky, Anthony Blunt and Paul de Man, just to name a few--who tripped up embarrassingly when trying to transfer the brilliant clarity of their thinking to the messy, muddy world of public affairs is long indeed; meanwhile, the giants of public affairs--King, Ghandi, Churchill, Walesa, Mandela, and so many more--are revered not for their intellectual contributions, but simply for being wise, brave and right when their societies needed them to be.
So before we answer Posner's question of whether formal academics are better or worse than they once were at making the leap into public life, we should first ask why we should care. After all, a burger with truffles and brie may be more chichi than a mushroom cheeseburger, but it isn't necessarily a better burger.