I have a doubt about the basic nature of a priori knowledge: How do we say that '2 + 2 = 4' is an a priori knowledge. First thing: the number 2, and the operator + are both abstract entities, members of an algebraic world. We see that 2 men + 2 men = 4 men; 2 balls + 2 balls = 4 balls. Thus, we see physical instances of this fact happening in the concrete world. From this, perhaps, we deduce, or perhaps postulate, that 2 + 2 = 4. I wonder what would happen if we treated 2, 4, + and = just as symbols, wrote down a basic set of rules independent of our knowledge of numbers. What category of knowledge would that fall into? To generate a(n algebraic) world of our own, without regard to anything concrete...does that make the knowledge of that world a priori? I am not a philosopher, nor a mathematician. I am ignorant of the true origins of this knowledge.
Second doubt is about the a priori-ness of our knowledge of our feelings. Whether I love somebody etc. are in a sense physical facts. Introspection can also be looked at as a kind of keen observation of the internal world. The internal world is very like the external world. It contains facts like what I feel, what my opinions are etc.
What's the actual distinction between a priori and posteriori? The distinction between the internal world on the one hand, and the external on the other? Or is it the distinction between observed fact on the one hand, and deduction, induction and prediction on the other?
Currently, I feel like looking at it this way: All knowledge of facts, whether pertaining to the external world or the internal (emotional, psychological, intellectual) are all posteriori. Everything derived from these facts are a priori. In fact, the nature of the knowledge is closely intertwined with the nature of the fact itself. Certain facts just exist out of nowhere. Something else in their place would have resulted in a different, but probably equally valid, world. They contribute to posteriori knowledge. On the other hand, facts that can be deduced, or predicted, from the set of posteriori knowledge are all a priori. Their induction to the set of knowledge that a person may have (for example, whether a person learns a formula by heart or derives it from first principles) doesn't affect the nature of that knowledge.