The philosophical study of reality is called metaphysics. The philosophical study of knowledge is called epistemology. One cannot have an epistemology—or systematic view of knowledge—without a metaphysics—or systematic view of what reality is like. Epistemology and metaphysics therefore cannot be separated. Together they form the foundation for every other belief we may hold. Metaphysics and epistemology form a foundation for ethics—(how people should act). Ethics in turn forms a foundation for politics (how people should act with the context of society) and aesthetics (the study of art and what constitutes good or bad art). A complete philosophical system will integrate all of these branches of philosophy.
Aristotle said, "The least initial deviation from the truth is multiplied later a thousandfold.” Since metaphysics and epistemology are the foundation of any philosophical system, if one makes a metaphysical or epistemological mistake, they will also make mistakes in the areas of ethics, politics, and aesthetics. In other words, mistaken premises lead to mistaken conclusions.
To validate a conclusion we must identify and validate the premises. How do we validate a premise? By identifying and validating the premises that support that premise. Here we can see a potential problem. Either,
each premise will have its own premise going down an infinite chain of premises, or
eventually we will find a stopping point—a foundation. That stopping point would be an irreducible premise that can stand alone.
If option (1) is true, then knowledge would not be possible because there would be no grounding on which to base our beliefs. If option (2) is true then knowledge is possible.
Knowledge is possible because irreducible premises do exist. I am going to call these premises "Axioms". An axiom is a foundational premise that is self evident. A belief is self evident when one must assume that belief even when they try to reject that belief. For example, if I try to argue against X, and while I am arguing against X, I am forced to assume X, then I will know that X is an axiom.
Knowledge and the ability to think rationally are dependent on the following axioms:
Reality exists. This is a self-evident truth because if a person tries to deny it through argumentation, he/she must first appeal to reality in order to defend their position that there is no reality to appeal to. It is clearly a self-contradictory statement. Reality is the widest of all concepts. It includes all that is known and unknown.
Consciousness exists. It is the awareness of reality. It could have been the case that a universe could have existed without consciousness, but we know that it exists because anyone that tries to deny consciousness through argumentation must appeal to reality to defend their position. And the very process of "appealing to reality" assumes that one is aware of reality. Axiom 1 and 2 combined show us that reality exists independently of our representations of it.
Axioms cannot be proven. The concept of proof is not irreducible. It relies on axioms. For example, the concept of "proof" assumes that there is an existence or reality that exists independently of consciousness. The concept of proof is meaningless without the concept of reality/existence by which beliefs can be compared. If we compare a given belief to reality and the belief does not match reality, then that belief is false. A belief can be proven when it is shown to match reality. This is why the existence of reality itself cannot be proven, because the concept of proof requires a belief in reality. In other words trying to prove that reality exists would result in a circular argument. Even though Axioms cannot be proven, they can be validated. Validation is a larger concept than proof. Axioms are validated because they are self-evident.
If one rejects these axioms, one necessarily undermines their own ability to think correctly. That is why the axioms are so important. If you come across any idea that tries to argue against these axioms, then you know it is false, because that idea is necessarily self-refuting. The axioms are sentries that guard our mind against confusion and ignorance.