The connotation of a concept refers to its essence or unique attributes. This is the standard statement in textbooks, but please don't ask me what is essence and what is attribute. That's not the content of logical investigation. Their exact meaning and the disputes they cause need to be solved by relevant specific disciplines.
Here we only need to know that the concept is different from other concepts by its own connotation, such as the aforementioned "milk". It doesn't matter logically whether "cow's" is essential, to what extent, or whether it is the attribute of milk. It is important to distinguish milk from other milk.
In the sense of textbooks, the connotation of concepts is constantly developing and changing with the deepening of our understanding. Therefore, when we use the connotation of the concept to define the concept and divide or classify the concept, there will be different results in different stages of understanding. Therefore, as we said before, the connotation of the concept is not static, it has only relative stability, but no absolute stability.
Therefore, the conceptual extension used to clarify the relationship between concepts will change accordingly. For example, when we define whales as mammals, we exclude them from the extension of fish and add them to the extension of mammals.
The extension of the concept is the object represented by the concept. For example, milk, its extension includes yellow milk, buffalo milk, flower milk, yak milk, African wild milk and so on; Another example is milk, its extension includes milk, goat milk, horse milk, dog milk and so on.
Therefore, for a concept with more than one extension, each extension is also a concept of this concept, which is itself a concept of all these extensions. The so-called "genus" and "species" mean that milk is a kind of milk.
This shows that there is a relationship between concepts: milk really contains milk and milk really contains milk. The concept of genus always really includes the concept of its species, and the concept of species is always really included in the concept of its genus.
There are more than one relationship between concepts, in addition to the same relationship (also called the same relationship), cross relationship and total difference relationship, and total difference relationship is divided into two kinds: contradictory relationship and antagonistic relationship. This is divided according to whether the extension of the concept coincides or not or how much it coincides.
The same relationship means that the extensions of the two concepts coincide completely, neither more nor less. For example: milk and milk, pen and pen are just another way of saying it; Another example: Lu Xun and the author of The True Story of Ah Q refer to the same person.
Cross-relationship refers to the extension of the two concepts overlap, and other parts do not overlap, such as women and teachers. Some women are teachers, but some are not; Some teachers are women, while others are not. Only part of the extensions of these two concepts are consistent.
A completely different relationship, as the name implies, is that the extensions of the two concepts are completely different, without any overlap, such as white and black. However, it should be noted that white and black are not contradictory concepts, but opposing concepts.
Why? It's simple, because the definition. Logical definition: contradiction means that a generic concept has only two extensions, and the two extensions do not overlap; If a generic concept has three or more non-overlapping extensions, then these extensions are opposite.
The extension of color concept includes not only white and black, but also red, orange, yellow, green, blue, blue and purple. They don't overlap with each other, so they just oppose each other, not contradict each other. White and black are the same. Therefore, according to the definition, white is not a contradictory concept of black, but a contradictory concept of "non-white".
According to different classification methods, concepts can be divided into many categories: single concept (Peking University), universal concept (university) and virtual concept (university on the moon); Positive concept (white) and negative concept (non-white); The concepts of entity (milk), nature (nutrition) and relationship (greater than) are literally easy to understand, so I won't repeat them here.