If I say that I am certain of something, this can mean that I personally do not have any doubt that it is true. Naturally, this does not ensure that the thing is in fact true. The fact that I do not doubt it, does not prevent it from being false, and people are frequently sure of such things.
But asserting that I am certain can also imply that the thing cannot fail to be true. As discussed in the previous post, this could mean that the thing cannot fail to be true on account of the objective nature of my conviction, or on account of its subjective nature.
As an example of the objective nature of the conviction, someone can say that he has a demonstrative argument for a conclusion, based on first principles. Given this kind of conviction, the thing cannot fail to be true, because something that actually follows from first principles will always be true. Thus I can prove in this way that 13 is a prime number. The objective nature of the conviction here is mathematical knowledge, and given that I have mathematical knowledge of a thing, the thing will always be true.
However, it would be either rare or impossible to have a subjective apprehension of my own knowledge such that I infallibly recognize my own possession of mathematical knowledge, and therefore can judge about the truth of the conclusion infallibly. I consider my knowledge and say, “This is a valid mathematical demonstration,” but my apprehension of this fact is not itself infallible. This was illustrated earlier with the example of a mathematician claiming that there is a flaw in a proof. If my apprehension of the fact that something is a demonstration is infallible, then I will know through this infallible knowledge that his claim is mistaken. But this does not happen in reality, and thus my knowledge is not subjectively infallible, not even when I have a valid mathematical demonstration for some conclusion.
To be continued…