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I was reading that properly basic beliefs are beliefs that are either self-evidently true, evident to the senses or incorrigible. The general example given for incorrigible beliefs is "beliefs I cannot be wrong about, like reporting on what seems true to us, since we cannot be wrong about what seems true to us." An example would be, "I have a headache," or Descartes' statement, "I think, therefore, I am."

My question is twofold: What are other examples of incorrigible basic beliefs, and how are incorrigible basic beliefs justified in being true by default? It is obvious that we can be mistaken concerning what seems true to us, so how can something that seems to be true to us be automatically true when simultaneously it may be false, assuming we have not come to realize that our perception is false? In other words, suppose that at least one of our incorrigible beliefs is actually false prior to our realizing that it is false. Then prior to our realizing that it is false, the belief by definition is true. But in reality, the belief is actually false, so in that moment prior to our realizing that it is false, the belief has two truth values assigned to it, which is contradictory. How can we make sense of incorrigible beliefs then? Maybe it actually has only one truth value assigned to it as dictated from the reality that it is false, so even though it is perceived as being true, it is actually false. Regardless, how is it that incorrigible beliefs are immediately true? Is it true in a subjective sense then? I'm sorry for the rambling.

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(too long for a comment)

The logic in your second paragraph misses the point. Assume "I believe that ripe bananas are yellow". It is a properly basic belief that "I hold that 'I believe that ripe bananas are yellow'". Note the second level of indirection. The sensation of believing something just comes to oneself as a subjective experience. Since this "sensation of believing" is a subjective, internal experience, it is (or at least appears to be) on the same kind of footing as having as other internal, subjective experiences, e.g. perceptual ones.

Whether or not there are bananas out there in the world that aren't yellow when ripe doesn't change the fact that, in my current mental state, it is true that "I believe that ripe bananas are yellow".

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Assuming a belief is incorrigible means you can't be wrong about it, and so the supposition "that at least one of our incorrigible beliefs is actually false prior to our realizing that it is false" is a contradiction, since you are simultaneously both affirming it's truth through it's definition and denying it's truth by the additional comment of it being "actually false".

Now, incorrigible beliefs are justified insofar as we are justified in identifying them as incorrigible, since through this we can derive that they are true. Typically, that which you are subjectively aware of is taken as incorrigible, since if it was false, and you were aware of it, then it would be true, which is a contradiction. The contradiction here is generated from the inference of awareness of x to truth of x existing, but this makes assumptions about the nature of consciousness.

For instance, how do we know if that which we think we are aware of isn't just another belief? So instead of you being aware that you see blue, you really just believe that you are aware that you see blue? You might respond, "yes, but I have the phenomenal experience of seeing blue", but how do you know that is not just a belief?

This happens because awareness, in being taken as a primitive form of knowledge, must thereby include a primitive form of belief, and so in the extremely common case that belief and it's truthmaker (that which makes it true) are independent, one can decohere from the other, so that you can get false primitive beliefs and not know it. The only way to solve this that is apparent to me is to find a way to couple belief and it's truthmaker so they are one and the same and thus not independent so that decoherence is impossible, that is, a belief where having it is sufficient for it's being true. This would be literally incorrigible, since if it was false and you had the belief, since you had the belief it would also be true, which is a contradiction. This doesn't rely on assumptions about the nature of consciousness to generate a contradiction, which this time confines the contradiction to the nature of the belief only in the case of falsity.

The problem here is that the content believable this way is very limited, and there is a nasty gap between our experiences and these reflexive self-confirming types of beliefs. The cartesian "I believe" is one of them, for if you believe, then that you believe is contained therein.

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