| Thinking hard, or|
|Major trains of thought|
|The good, the bad|
and the brain fart
|Come to think of it|
Philosophical intuition is what philosophers use to work out what assumptions to start their arguments with and to judge the result at the end. It's basically introspecting and checking your feelings to see if what you just said makes sense to you and hence (since humans are, more or less, cognitively similar) your readers.
It has been conventional for centuries to assume philosophical intuition is a reliable black box. However, the trouble with treating the output of introspection as a reliable source of useful knowledge is that it doesn't work. The ancient Greek philosophers were quite sure that knowledge was better if unsoiled by real-world interactions; this is why they didn't invent science, which involves a step where you test your idea against the real world. Introspection leads to things like asserting that the philosophical zombie thought experiment proves dualism.
With the burgeoning study of cognitive biases, it turns out we can (a) analyse philosophical intuitions (b) see that they're frequently provably full of shit. Our intuition remains what we have to perceive philosophical relationships; we do, however, need to apply due caution and correction to the output of our stupid brains.
This has particular application to apologetics, which tends to try to prove the existence of a god with arguments that reduce to either god of the gaps or stimulating philosophical intuition with an intuition pump. The latter frequently reduces to coming up with arguments that feed their own cognitive biases.
A development over the last few years of the realization that philosophical intuition is crap is in the creation of "experimental philosophy". Basically, an individual philosopher will go "it seems like the reason I should save a child drowning in a kiddie pool in the yard that I happen to be walking by is because I have a moral obligation to those near me". The philosopher will then use this as a premise leading up to a philosophical argument in favor of a social contract theory of ethics. But, do people actually feel this way? A person delving in experimental philosophy would through some means (generally surveys) try to find out if people actually accept that we would save the child because of a moral obligation. If not, you may have found a theory of ethics that no one agrees with or is willing to follow.
 See also
- One More Take On the Role of Intuition in Philosophy, Massimo Pigliucci