Logic: Laws and Forms of Correct Thinking

Judgment

Judgment (proposition) is a form of thought that affirms or denies the existence of objects, the relationships among them, or the presence of certain properties and relations in those objects. For instance: "The constitution is the fundamental law of the state." An important cognitive feature of judgment is that it can be evaluated as true or false.

The logical structure of a judgment consists of the subject (S) — the concept of the object of judgment; the predicate (P) — the concept that expresses what is said about the subject; the copula — the assertion of the presence or absence of the property expressed by the predicate in the subject, typically articulated by words such as "is," "is not," "are," etc.; and the quantifier — which indicates whether the characteristic expressed by the predicate pertains to the entirety of the subject's extension or only to a part of it, expressed through terms like "all," "none," "each," "not one," "some," "individual," "many."

The subject and predicate are referred to as the terms of the judgment. The schema of a judgment appears as follows: S is (is not) P.

Similar to concepts, judgments are inseparable from the linguistic form in which they manifest, primarily declarative sentences.

Judgments can be classified as simple or complex. Simple judgments express the relationship between two concepts: "It is raining." A complex judgment consists of several simple ones: "It is raining, and there is no one outside."

Simple judgments are further divided into various types based on two criteria: by quality — into affirmative (S is P) and negative (S is not P) — and by quantity — into singular, particular (Some S are (are not) P), and universal (All S are P, No S is P).

An integrated classification of judgments based on quantity and quality encompasses four types of judgments denoted by corresponding Latin letters: A - universal affirmative: All S are P; E - universal negative: No S is P; I - particular affirmative: Some S are P; O - particular negative: Some S are not P.

Since judgments relate the extensions of two concepts, they can be interpreted using Euler circles, which are useful for analyzing syllogisms.

The relationship between the extensions of the judgment terms — the subject and the predicate — is termed the distribution of terms. A term is considered distributed if its extension is wholly included in or wholly excluded from the extension of another term, while it is undistributed if its extension is only partially included or partially excluded from another term. Distribution is denoted by the symbol “+,” while non-distribution is indicated by “-.”

In a universal affirmative, the subject is always distributed (S+), while the predicate is typically undistributed (P-). In a universal negative, both the subject and the predicate are always distributed (S+, P+). In a particular affirmative, both the subject and the predicate are generally undistributed (S-, P-). In a particular negative, the subject is undistributed (S-), while the predicate is distributed (P+). Understanding the distribution of terms in judgments helps prevent errors in logical operations involving judgments and in reasoning processes related to proof and refutation.

If judgments share the same subjects and predicates but differ only in quantity and quality, they are called judgments of the same matter, meaning they are comparable and possess relationships of compatibility and incompatibility. Compatible judgments can be simultaneously true (A and I, E and O, I and O), whereas incompatible judgments cannot be true at the same time (A and E, A and O, E and I).

The relationships among simple judgments of the same matter (determining truth) are examined through a particular mnemonic scheme known as the logical square. The vertices of the square represent simple categorical judgments A, E, I, O, while the sides and diagonals illustrate the relationships among the judgments.

Compatibility can manifest as subordination and subcontrariety. The relationship of subordination exists between the judgments A and I, E and O. The truth of a universal judgment determines the truth of the corresponding particular one. Subcontrariety (partial compatibility) pertains to the relationship between the judgments I and O: they can both be true simultaneously, but they cannot both be false at the same time. If one is false, the other is true; however, the truth of one leads to the indeterminacy of the other, which may be either true or false.

Incompatibility can take the form of opposition (contrariety) and contradiction (contradictoriness). The relationship of opposition (contrariety) exists between the judgments A and E: they cannot both be true simultaneously, but they may both be false at the same time. If one is true, the other must be false. However, if one is false, the status of the other remains indeterminate. The relationship of contradiction (contradictoriness) exists between the judgments A and O, E and I. They cannot both be true or both be false simultaneously. Between them, there exists alternative incompatibility, which is expressed by the law of excluded middle: the truth of one necessarily entails the falsehood of the other, and vice versa — the falsehood of one leads to the truth of the other.

Complex judgments are formed from simple judgments through logical connections: conjunction — “and” (denoted as “l”), disjunction — “or” — weak (denoted as “ν”) and strong (denoted as “^”), implication — “if, then” (denoted as “<θ”), equivalence — “if and only if” (denoted as “·θ·”).

A conjunctive judgment is true only when all its components are true. It is false if at least one of its components is false. A weak disjunctive judgment is true when at least one component is true and is false when all components are false. A strong disjunctive judgment is true only when one component is true and the others are false. An implicative (conditional) judgment is true in all cases except one: when the premise (p) is true, and the conclusion (q) is false. An equivalent judgment is true when both the premise and the conclusion share the same truth value: both true or both false.