Logic: Laws and Forms of Correct Thinking

Inference as a Form of Thought

A vast array of knowledge about the world that humans acquire is of a deductive nature, obtained through logical reasoning. This encompasses understanding the fundamental properties of natural phenomena—such as the laws of biological evolution, the structure of DNA, and the composition of the microcosm. The form through which this knowledge is attained is inference.

Inference is a mode of thinking wherein one or more propositions yield a new proposition that embodies fresh knowledge. For instance: “Every accused individual has the right to defense,” “Polishchuk is an accused individual,” therefore, “Polishchuk has the right to defense.”

The structure of an inference consists of premises—propositions that contain the foundational knowledge; a conclusion—a proposition that conveys new knowledge derived from the premises; and the derivation—a logical transition from the premises to the conclusion.

Depending on the direction of logical progression and the reliability of the resulting outcome, three types of inferences are distinguished: deductive inferences involve a transition from general knowledge to specific, with the conclusion necessarily following from the premises; inductive inferences entail a transition from specific knowledge to general, with the conclusion being probabilistic in nature; and abductive inferences (by analogy) consist of a transition from one specific piece of knowledge to another, also yielding a probabilistic conclusion.

Deductive Inferences

Deductive inferences are classified into two categories: direct—in which the conclusion arises from a single premise; and indirect—in which the conclusion is drawn from two or more premises.

The primary forms of direct deductive inferences include:

• Transformation—the reconfiguration of a proposition from affirmative to negative and vice versa (a change in quality while the quantity remains constant). All four types of propositions can be transformed according to the following schemas: A (All S are P) ↔ E (No S are non-P); E (No S are P) ↔ A (All S are non-P); I (Some S are P) ↔ O (Some S are non-non-P); O (Some S are not P) ↔ I (Some S are non-P).
• Conversion—an operation in which the subject and predicate are swapped. The schemas of conversion are as follows: A (All S are P) generally converts to I (Some P are S), while if the predicate in the premise is distributed (definition or judgment with an emphasized subject), A converts to A (All P are S); E (No S are P) invariably converts to E (No P are S); I (Some S are P) typically (when S and P are undistributed) converts to I (Some P are S), while if the predicate in the premise is distributed (Some S and only S are P), I converts to A (All P are S); the proposition O does not undergo conversion.
• Contradiction—in this operation, the predicate becomes the concept that opposes the original predicate, while the subject becomes the original predicate. The schemas for deriving contradictions include: A (All S are P) ↔ E (No non-P are S); E (No S are P) ↔ I (Some non-P are S); O (Some S are not P) ↔ I (Some non-P are S); I cannot be subjected to contradiction.
• Inferences based on the Logical Square are constructed on the logical relations between the propositions A, E, I, O, as previously mentioned. The truth or falsity of one type of proposition will determine the truth, falsity, or uncertainty of another type. Since there exist contradictions between A and O, and between E and I, the truth of one implies the falsity of the other and vice versa. Given the relationship of opposition between A and E, the truth of one leads to the falsity of the other, while the falsity of one yields uncertainty regarding the other. The relationship of partial compatibility between I and O indicates that the falsity of one confirms the truth of the other, yet the truth of one dictates the uncertainty of the other. The hierarchical relationship between A and I, as well as between E and O, means that the truth of the subordinate proposition entails the truth of the subservient proposition, while the truth of the subordinate proposition results in uncertainty regarding the subservient proposition. The falsity of the subordinate proposition leads to the falsity of the subservient proposition, yet the falsity of the subservient proposition results in uncertainty regarding the subordinate proposition.

Types of Indirect Deductive Inferences

• Simple Categorical Syllogism
• Inferences from Complex Propositions (conditional, disjunctive).

A simple categorical syllogism is a deductive inference where both premises and the conclusion are categorical propositions. For instance: “Every crime is an act that poses a danger to society. Theft is a crime. Therefore, theft is a socially dangerous act.” In a syllogism, there are three terms: two extremities and one middle term. The extreme terms are those that serve as the subject and predicate of the conclusion, with the subject being termed the minor term (denoted as S), and the predicate as the major term (denoted as P). Accordingly, the premises containing these terms are labeled the major and minor premises. The middle term is the concept that appears in both premises but is absent from the conclusion. It is denoted as M (from the Latin medius meaning middle). Thus, a simple categorical syllogism is a conclusion about the relationship between the two extreme terms based on their relationship to the middle term. The middle term acts as a connecting link between the major and minor terms.

The structure of the syllogism appears as follows:

M — P
S — M
S — P

The legitimacy of the logical transition from the premises to the conclusion in a categorical syllogism is founded on the axiom of the syllogism: everything asserted or denied regarding all items of a certain class is asserted or denied concerning each item and any part of the items in that class.

Syllogisms vary according to the positioning of the middle term in the premises. These variations are termed figures of syllogism. There are four figures of syllogism: in the first, the middle term occupies the position of the subject in the major premise and the predicate in the minor; in the second, it occupies the position of the predicate in both premises; in the third, it occupies the position of the subject in both premises; and in the fourth, it occupies the position of the predicate in the major premise and the subject in the minor.

Depending on the types of propositions—A, E, I, O—that comprise the syllogism and how they are combined, each figure has certain moods. The moods are denoted by three letters corresponding to the propositions that construct the syllogism: the major premise, the minor premise, and the conclusion. The correct moods for the first figure are: AAA, AII, EAE, EIO; for the second: EAE, AEE, EIO, AOO; for the third: AAI, EAO, IAI, OAO, AII, EIO; and for the fourth: AAI, AEE, IAI, EAO, EIO.

The construction of syllogisms is governed by a series of general and specific rules (pertaining to each figure). Among the general rules are three rules regarding terms and four rules pertaining to premises:

• A syllogism must consist of exactly three terms. Violating this rule involves conflating different concepts that are treated as the middle term. This error is termed the "quaternization of terms."
• The middle term must be distributed at least once in one of the premises; otherwise, the connection between the extreme terms remains indeterminate.
• A term that is not distributed in a premise cannot be distributed in the conclusion, as this would result in the fallacy of illicit extension of the term.
• From two negative premises, a conclusion cannot necessarily follow. At least one of the premises must be an affirmative proposition.
• If one of the premises is a negative proposition, then the conclusion (if possible) must also be negative.
• From two particular premises, a conclusion cannot necessarily follow. In this case, the middle term is not distributed in either premise.
• If one of the premises is a particular proposition, then the conclusion must also be particular.

A purely conditional inference is one in which both premises and the conclusion are conditional propositions. It has the following structure: If A, then B; If B, then C; Therefore, if A, then C.

In a conditional-categorical syllogism, one of the premises is a conditional judgment, while the other premise and the conclusion are categorical judgments. Its structure is: If A, then B; A; Therefore, B.

The logical basis for the conclusions of a conditional-categorical syllogism is the axiom that affirming the antecedent necessarily leads to affirming the consequent, while denying the consequent leads to denying the antecedent.

The conditional-categorical syllogism has two valid modes:

• In the affirmative — modus ponens — the categorical premise affirms the antecedent of the conditional premise, and the conclusion affirms the consequent of the conditional premise: If A, then B; A; Therefore, B.
• In the negative — modus tollens — the categorical premise denies the consequent, and the conclusion denies the antecedent of the conditional premise: If A, then B; not-B; Therefore, not-A.

In principle, there are two more modes of conditional-categorical syllogism: If A, then B; not-A; Therefore, not-B. If A, then B; B; Therefore, A.

However, these modes do not yield valid conclusions. This is because the relationship between cause and effect is usually not unambiguous. The same effect can arise from multiple causes. This distinction changes when the premise of the conditional-categorical proposition is an equivalent conditional judgment: If and only if A, then B. Here, the relationship between cause and effect is unambiguous. The antecedent is both a necessary and sufficient condition for the existence of the consequent, and vice versa. The conditional-categorical syllogism with an equivalent judgment has not two but four valid modes.

A disjunctive-categorical inference is defined as an inference in which one of the premises is a disjunctive proposition, while the other premise and the conclusion are categorical judgments. This inference has two modes:

• Affirmative-negative — in which the categorical premise affirms one member of the disjunction, and the conclusion negates the other members of the disjunction: A is B, or C, or D; A is B; Therefore, A is neither C nor D.
• Negative-affirmative — in which the categorical premise denies one or more members of the disjunction, while the conclusion affirms the other members: A is B, or C, or D; A is neither B nor C; Therefore, A is D.

The conditions for the validity of conclusions in this inference are that the disjunction must be complete and strict.

Inductive inferences are characterized as those in which knowledge about specific items of a class or about certain parts of a class leads to knowledge about the class as a whole. Induction represents both an inference from the particular to the general and a method of scientific inquiry.

Induction can be complete or incomplete. In complete induction, the premises exhaust the entire class of objects subject to generalization, thereby ensuring the validity of its conclusions. In incomplete induction, a conclusion about the entire class of objects is drawn based on knowledge of only some of the objects (a part) of that class. The conclusion in incomplete inductive inferences does not necessarily follow; instead, it bears a merely plausible, probabilistic character.

The degree of probability of the conclusion in incomplete induction is determined by how methodically the selection of initial facts is conducted. Accordingly, three types of incomplete induction are distinguished: 1) popular induction; 2) selective induction through fact selection; 3) scientific induction. Sometimes the second and third types are regarded as variations of scientific induction.

In popular induction, or induction through simple enumeration, the samples for investigation are taken arbitrarily (those that happen to occur), and a generalizing conclusion about the class of objects is made based on the recurrence of a certain characteristic in all investigated items of this class, with no counterexamples among the observed cases. Typical errors encountered in the use of popular induction include the "hasty generalization" and the post hoc ergo propter hoc fallacy (Latin for "after this, therefore because of this"), which underlie the emergence of various superstitions, biases, and misconceptions. The degree of probability of conclusions drawn from popular induction depends solely on the quantity of facts being investigated. However, no matter how many phenomena are observed, there always remains the possibility of encountering something that contradicts the generalization. In such cases, it is revealed that the conclusion is erroneous.

Selective induction, akin to popular induction, is based on the recurrence of facts in the absence of counterexamples. However, in this case, not just any facts are taken, but rather an analysis and selection of facts are conducted according to a specific system that has been previously developed and validated in practice. As a result, we obtain a group of selected objects known as a sample. All elements of the sample are investigated, and the results are extrapolated to the entire class (the population). Scientifically developed selection systems allow for conclusions that are close to being valid.

In scientific induction, the general conclusion about a specific class of phenomena is drawn based on knowledge of the necessary characteristics or causal relationships of a portion of the phenomena within that class. This is the most refined form of induction. Scientific induction yields conclusions that are not only probable but also valid. In this case, the number of investigated facts is of no importance; a conclusion can be drawn based on even a single fact. For instance, having established that the ability to conduct electricity is a necessary and essential property of metals, one can confidently conclude that all metals are electrically conductive. The application of scientific induction has allowed for the formulation of scientific laws (e.g., Archimedes', Ohm's, etc.).

A special place in scientific induction is occupied by inductive conclusions regarding causal relationships between phenomena (the methods of Bacon and Mill). Contemporary logic describes five methods for establishing causal connections: the method of single similarity or the method of finding commonality among the diverse: if a certain circumstance consistently precedes the emergence of phenomenon a, while other circumstances change, then it is probable that this circumstance is the cause of a; the method of single difference or the method of finding distinctions within similarities: if a certain circumstance A is present when phenomenon a arises and absent when a does not occur, while all other circumstances remain unchanged, then A is likely the cause of a; the combined method of similarity and difference constitutes a synthesis of the first two methods, where through analysis of multiple cases, both similarities and distinctions are identified; the method of concomitant variation: if the occurrence or change of a preceding phenomenon consistently triggers the occurrence or change of another accompanying phenomenon, then the former is likely the cause of the latter; the method of residues: if complex circumstances give rise to a complex phenomenon, and it is known that a portion of the circumstances causes a certain portion of this phenomenon, then the residual portion of circumstances likely produces the remaining part of the investigated phenomenon.

Inferences by analogy are defined as abductive inferences, where knowledge about one subject is transferred to another based on their similarity in certain respects. For example, upon discovering that Mars bears many properties similar to Earth and knowing that life exists on Earth, one may surmise that life could also exist on Mars.

Through analogical inference, knowledge about one object is transposed to another. Like incomplete induction, analogy yields plausible knowledge. Strict analogy and simple (non-strict) analogy are differentiated. In strict analogy, it is known that there exists a substantial connection between the characteristic being transferred and the characteristics of similarity. Typically, scientific analogies are of a strict nature.

The degree of probability of an analogy increases as follows: the more characteristics of similarity established between the two compared phenomena; the more significant the nature of the characteristics of similarity; the closer the connection between the characteristics of similarity; the more substantial their relation to the characteristic being transferred to the investigated object; and the more thoroughly the differences between the compared subjects are accounted for.

The conclusion drawn by analogy serves as the foundation for the method of modeling.