Logic: Laws and Forms of Correct Thinking

Concept as a Form of Thought

A concept is defined as an elementary form of thought that reflects individual objects or groups of similar objects in their essential characteristics, as well as certain properties and relationships of things and phenomena.

For instance, the concept "Mars" reflects a specific object, while the concept "planet" denotes a group (class) of celestial bodies sharing similar characteristics; the concept "hardness" conveys certain physical properties of objects, and the concept "equality" expresses specific relationships between entities.

The logical structure of a concept is characterized by its content and extension. The content of a concept encompasses the aggregate of essential features of the object as conceived within that concept, representing the meaningful aspect of the concept. For example, within the concept of "tree," the attributes such as "being a plant" and "having a crown, trunk, and roots" are considered. The extension of a concept is the set of objects it encompasses; thus, the extension of the concept "house" includes all houses, as they share common essential characteristics.

The content and extension of a concept are closely intertwined, expressed through the law of inverse relation between the extension and content of a concept, according to which an increase in the content of a concept results in the formation of a concept with a smaller extension, and vice versa. For example, by enhancing the content of the concept "state" by adding a new attribute—“modern”—we transition to the concept of "modern state," which has a smaller extension.

Concepts can vary in their extension; they may be singular, general, or null. Depending on the nature of the elements forming the extension, concepts can be collective, where the content cannot be applied to each individual element of the collection, but only to the entire set (constellation), and non-collective (distributive), in which attributes are relevant to both the entire class of elements and each individual element within that class (star).

Logical Relations Between Concepts

If the content of concepts shares common characteristics indicating a common genus, they are comparable (such as “share” and “bond”), whereas if their content lacks shared generic attributes (like “circle” and “society”), they are non-comparable.

Comparative concepts can further be classified into compatible and incompatible. Compatible concepts have overlapping extensions, either entirely or partially, while incompatible concepts have extensions that do not overlap at all.

There are three types of compatibility relations: equivalence, when the same object is represented in both concepts (e.g., "isosceles triangle" and "equilateral triangle"); intersection, when the extension of one concept overlaps with that of another (e.g., "lawyer" and "lecturer"); and subordination (or hierarchical relation), when the extension of one concept is wholly contained within the extension of another as a part of it (e.g., "social relations" and "property relations").

The relationships between concepts are often illustrated using Euler circles, where each circle denotes the extension of a concept, and each point within the circle represents an object that falls within that extension.

Three types of non-compatibility relations exist: coordination, where concepts do not intersect but are subordinated to the same overarching concept; opposition, where one concept includes certain attributes that are incompatible with those of another (between oppositional concepts, a third possibility may exist; for instance, between "black" and "white" (the generic concept being "color"), the concept "gray" is conceivable); and contradiction, where one concept includes certain features while another excludes those very features. Here, the extensions of two contradictory concepts encompass the entire range of the generic concept, with no third possibility existing (e.g., "white" and "non-white").

Primary Operations with Concepts

The fundamental operations involving concepts include: generalization, the transition from a concept with a smaller extension to a concept with a larger extension, albeit with a "poorer" content (e.g., "government of Ukraine" to "government"); restriction, the transition from a concept with a larger extension to one with a smaller extension, yet with "richer" content (e.g., "production" to "commodity production"); definition—discussed further below; and division—also discussed below.

It is important to distinguish the operations of generalization and restriction from the illusory transition from part to whole and the identification of a part within a whole.

Definition is a logical operation that reveals the content of a concept. For instance: "Capital is value advanced in production with the aim of obtaining additional value through the exploitation of labor." A definition answers the question of what a given object is and delineates a specific object from all others. It comprises two components: the defined concept, whose content is to be revealed, and the defining concept, which elucidates the content of the defined.

The most common (classical) form of definition is through the nearest genus and specific difference. The symbolic representation of such a definition is: A = Bs, where A is the defined concept, Bs is the defining concept (B being the genus, and s being the specific difference).

The main rules of definition are: 1) the rule of commensurability— the extension of the defined concept must be equal (neither broader nor narrower) than that of the defining concept: A = Bs; 2) the rule against circularity or tautology; 3) the rule of clarity and comprehensibility—indicating attributes that do not require further definition and do not contain ambiguity; 4) the definition should not be merely negative.

Division of Concepts is a logical operation that uncovers the extension of a generic concept through the enumeration of its species. For example: "Agreements are made orally or in writing." In the structure of division, we distinguish the divisible, the members of the division, and the basis (or criterion) of division— the attribute by which the division is made. The operation of division underlies classification.

There are two types of division: by the alteration of attributes (e.g., "People are divided by gender into men and women"); and dichotomy (from Greek "dicha" and "tome," meaning division into two parts)—the division into two contradictory concepts (e.g., "Wars are classified as just and unjust").

The rules of division are as follows: 1) the division must be commensurate and relevant, meaning the sum of the extensions of the members of the division must equal the extension of the divisible concept; 2) the division must be conducted solely on one basis; 3) the members of the division must mutually exclude each other in terms of extension; 4) the division must be continuous and sequential, meaning that during the division of the generic concept, one should progress to the nearest species without skipping them; 5) the basis (or criterion) of division must be clearly defined and unambiguous.