Incorrigibility is the undeniable property of a belief-thing - by virtue of that thing *being* a belief. We have Rene Descartes to thank for this idea, or at least its popularity. This idea is an extension of the famous cogito argument - whatever I am thinking, because I am thinking it, must be. Of course this is a little simplistic, but it works for many, many examples, the most salient one being:

I think/believe I am in pain.
Therefore I must be in pain.
Prove me wrong.

(Yes, Yes, Yes, we could go on and on talking about this little gem - it leads naturally into discussions about phenomenology, epistemology, solipsism, existentialism and so and so forth. For now, lets just move on . . .)

Descartes as much as he doubted himself used this principal of incorrigibility very often in is several proofs for the existence of God -- usually with the corollary that you can trust your beliefs and thoughts as long as they are composed in a "clear and distinct"** mind. His most simple proof uses the idea of incorrigibility like this:

Personally, it seems that incorrigibilty could easily be the justification for any faith or any belief. It simply says that what is personal is undeniable - and that whether there is a God, or gods, or isn't a God or gods is really up to you. Which is kinda cool.

** Yes, Houston, we have a problem. Since Descartes, western philosophy has been trying to pin down exactly what "clear" and "distinct" are. It hasn't gone that well.