Self-evident concepts are those concepts that understood with certainty. But considering that these are concepts rather than propositions, we need a better understanding of self-evident concepts. A self-evident concept is a concept that is known (or knowable) apart from a definition.
Most concepts are known by their definitions. This does not mean that we are consciously aware of these definitions, or that we consciously apply these definitions to particular cases. It also does not exclude vagueness. These misconceptions aside, definitions are simply short descriptions that explain what something is. So human beings are a kind of animal that is rational, while rocks are any form of mineral that is sufficiently small. Each definition has two parts: the genus and the difference. The genus is what category of thing the concept applies to while the difference is what makes it different from other things in the same genus.
It is impossible that all concepts have definitions. Suppose that this was false. To know a concept is to know its definition. But a definition is only known if we know the definition of the genus and the difference. But they are also composed of a genus and difference of their own. So unless those are known, the concept will not be known. But human understanding is sequential. And such a series of definitions would be an actual infinite. Therefore, it is impossible for any human to understand them because it is impossible to create an actual infinite by addition. Therefore, some concepts lack definitions.
But it is not necessary to believe that all self-evident concepts are ones that lack definitions. It is simply enough to know these concepts through some other means than their definition. Remember that concepts are intellectual understandings of something – not imaginative pictures. Therefore, to know a concept with certainty is to understand it. The concept red could be understood partially by someone who is blind and had never seen red. Someone could know certain facts related to the concept such as which things are red, what wavelengths of light are red and how human beings see red. But all of these things – even together – do not amount to an understanding of red. The concept is only known by those who have seen red. The extra understanding do not improve our understanding of redness, but of things related to red and of the effects of redness. So red is a self-evident concept.
To know a concept apart from a definition is to know the concept with the same kind of certainty that comes from knowing the definition of a concept. For that reason, no definition – even if it did exist – would improve our knowledge of a self-evident concept. This also explains why some concepts may be self-evident to some and not to others. If one person knows a concept apart from a definition while another knows it by a definition, then that concept is self-evident to one person but not the other.
After understanding what self-evident concepts are, it is necessary to ask how they are known.