There is apparently considerable discussion at present on the role of intuitions in analytic philosophy. Good examples of this discussion can be found at the philosophy weblog, "Experimental Philosophy," here. It's an interesting sort of debate; I find myself a bit perplexed by aspects of it, though. Here's why.
When I say I do history of philosophy, that's what I do. That is, that's where I start; and I start floating around various texts in the history of philosophy (usually early modern and medieval) and look at analytic philosophy in light of that background. As far as I can tell, this is fairly rare; most people who would say they do history of philosophy start with the more modern stuff, and examine the historical stuff in light of it. What this difference means in my case is that, despite a considerable interest in many of the things analytic philosophers do, especially these days, I am often puzzled by the assumptions made by analytic philosophers, and by the methods they sometimes employ. This whole 'intuition' thing is a case in point.
There was a time when 'intuition' would have conveyed something fairly precise. Intuitive cognition would be, roughly, thinking about something that was present. It would be distinguished from abstractive cognition, thinking about something absent. When Cartesians use 'intuitio' or its cognates, they are probably heavily influenced by this usage; we have an intuition of our being because when we think, well, there we are. Here and there you can find analytic philosophers using the term in something like this way, but it's not the common usage.
More common is something like what Berkeley and the Scottish common sense philosophers would call 'the plain dictates of common sense'. But it seems to me that the Scottish common sense philosophers were far more sophisticated in their appeal to common sense than analytic philosophers ever are in their appeals to intuition. For one thing, there was a clear criterion, derived from Berkeley, of what got to count as a 'plain dictate of common sense', namely, the principle under consideration had to be essential for practical life, i.e., for making any sense of life at all. Thus, 'Bodies exist' is a dictate of common sense. 'I exist' is a dictate of common sense. 'Memory under normal conditions is at least roughly reliable' would be a plain dictate of common sense. And so forth. It was a very pragmatic notion. And the Scottish common sense philosophers were not satisfied with vague appeals to common sense; rather, they attempted to formulate, as exactly as they could, the principles that common sense embodied, and to show (Beattie, for instance, devotes considerable time to this) that rejecting those principles, or throwing doubt on them, leads to a sort of practical absurdity. Skepticism about these particular principles, in other words, falls prey to the old apraxia objection, that it would make practical life impossible.
Analytic philosophers do not do this, or if they do, do so much more rarely. And indeed, while 'intuitions' as used by analytic philosophers sometimes seem a lot like 'plain dictates of common sense', in many cases they are very different. Sometimes it seems to mean just 'spontaneous judgment'. Sometimes it seems to mean 'common platitude'. Sometimes, indeed, it seems almost to mean the oracular pronouncements of some magical faculty that just knows the way things are. Sometimes it seems to mean just 'my considered opinion'. Sometimes it means 'what people would ordinarily say'. Perhaps I'm just missing something, but there seems no commonality here. 'Intuition', in effect, just means, 'I want this taken for granted (in some relevant way) because I don't want to be bothered with actually arguing for it right now.' Or so it seems.
I remember one night after a class I was hanging with some of my students at the GSU pub and the subject of analytic philosophy came up in the discussion. I don't remember all the details, but I do remember part of the exchange. One of my students, a top-notch student with a strong interest in analytic philosophy happened to say in the midst of the discussion:
I think one of the differences between the analytic approach and other approaches is that the analytic approach involves fewer assumptions.
Oh! I said, dropping my jaw (half-tongue-in-cheek, half-genuinely-surprised). And you actually believe that?
I always think of that when I hear anyone doing analytic philosophy appeal to intuitions.