Logics

Classical logical theory is far from perfect ^{[ uncertainty ]} : its main content is formulated in a special language created for its own purposes , and uses objective thinking. It does not imply the use of control of pragmatic errors, errors, nonlinearities of the used reference systems, boundary description errors, scale relativism (relativity of objects and their spatial characteristics, for example: a person is large relative to an ant, but at the same time small relative to an elephant), etc. n. As a result, it is generally accepted that the fact of the presence of paradoxes and a priori statements, bush effects of a dictionary, etc., is normal in her language. Just as the ability to speak existed even before it arose veniya science of grammar and the art of right thinking existed long before the science of logic. Logical operations : definition , classification , proof , refutation , etc. - are often used by each person in his mental activity unconsciously and with errors. Some people tend to consider their own thinking as a natural process that does not require analysis and control more than, say, breathing or movement , but real thinking is not simply reduced to a logical sequence. In the process of solving emerging problems are also essential: intuition, emotions, imaginative vision of the world and much more ^[3] . However, the laxity of thinking does not mean that it is not subject to logic ^[4] . The main goal (function) of logic has always remained unchanged: the study of how one can derive others from some statements . It is assumed that the conclusion depends only on the method of communication of the statements included in it and their structure, and not on their specific content. Studying “what follows”, logic reveals the most general or, as they say, formal conditions for correct thinking. The sphere of specific interests of logic has changed significantly throughout its history.

Meaning of the word

The word “logic” is also used in the meanings “internal regularity inherent in one or another phenomenon” or “correct, reasonable course of reasoning” ^[5] . In particular, the following things can be called this word:

• the process of thinking - if we are talking about logical and illogical thinking , when the sequence of statements corresponds to the schemes studied in logic, in contrast to completely incoherent and reasoning by analogy with arbitrary images or stereotypes that the author liked.
• In electronics , a type of circuit designed for information processing and control. Unlike power schemes of transformation and distribution of energy. And low-power, but processing atomic signals - filtering, registration, generation.
• In arbitrary phenomena - attributed or detected in them certain functioning, repeating processes that can be described in logical categories - state, submission, reflection, dependence, etc.

Informal, formal, symbolic, and dialectical logic

Informal logic (the term is adopted primarily in English-language literature) is the study of argumentation in a natural language. One of its main tasks is the study of logical errors - see Logical semantics , philosophical logic , theory of argumentation , logical analysis of language . Any conclusion made in a natural language has a purely formal content (the meaning of reasoning can be divided into the form of thought and the content itself), if it can be shown that it is a particular application of an abstract universal rule that is distracted from any particular object, property or relation. It is this conclusion with purely formal content that is called the logical conclusion and the main subject of logic. An analysis of the conclusion that reveals this purely formal content is called formal logic . Symbolic logic studies symbolic abstractions that capture the formal structure of logical inference.

Dialectical logic is the science of thinking in Marxism . Here the concept of thinking is used in the sense of the Logos as an object of ancient philosophy, and dialectical logic - already in the sense of a separate science, as physics or formal logic. Dialectical reasoning takes into account the laws of formal logic. At the same time, while analyzing the dynamics of the transition of concepts into its opposite, it assumes that the opposites coincide, focuses on the laws of dialectics .

Within the framework of formal logic, there is a group of logics called nonclassical (sometimes the term “alternative logics” is also used). This group of logics differs significantly from classical logics by various variations of laws and rules (for example, logics that cancel the law of the excluded third , change the truth tables , etc.). Thanks to these variations, it is possible to construct various models of logical consequences and logical truth ^[6] .

Relation to other sciences

Historically, logic has been studied as part of philosophy and rhetoric . Now symbolic logic is also being studied as part of mathematics , computer science .

Metatheoretical Problems of Logic

• Consistency of formalized theories
• The completeness of formalized theories
• The decidability of formalized theories
• The independence of the axioms of formalized theories
• The correctness of the formal system
• Definability
• Comparative analysis of logical theories

Logic Concepts

The concepts of logic differ among themselves primarily in the ways of solving the metatheoretical problems of logic related to the foundations of mathematics :

• Psychologism
• Logicism
• Formalism (Mathematics)
• Intuitionism
• Constructive mathematics
• Conservatism (logic)

The problems of axiomatization of set theory

• Logical paradoxes
• Semantic paradoxes

Although many cultures have developed complex systems of reasoning, logic as an explicit analysis of the methods of reasoning was thoroughly developed initially in only three traditions: in Chinese , Indian and Greek . Although the exact dates are not very reliable (especially in the case of India), most likely, the logic arose in all three cultures in the IV century BC. e. . Modern logic, developed formally sophisticated, ultimately comes from the Greek tradition ( Aristotelian logic ), which, however, was not directly perceived, but through the mediation and commentary activities of Arab-Muslim philosophers and medieval European logicians. The following historical and regional forms of logic can be distinguished (their names are also given, which historically existed and are accepted in the literature on the history of formal logic):

• Ancient Chinese logic
• Indian logic
• European and Middle Eastern logic: traditional logic (in a broad sense)
  • Antique and Early Medieval Logic: Dialectics
  • Medieval logic
    • Arab and Jewish medieval logic
    • Eastern Christian Medieval Logic
    • Western European Medieval Logic: Scholastic Logic , Dialectics
  • The logic of the European Renaissance; dialectics
  • The logic of the New Age: traditional logic (in the narrow sense), formal logic
• Modern logic (global, from the second half of the 19th century ): mathematical logic , symbolic logic , logistics (the latter - as a rule, in Western literature).

Logic in its development has passed three thresholds:

• threshold for formalizing reasoning (in all three traditions)
• introduction of conditional (symbolic, alphabetic and numerical) designations (only European traditional logic)
• the scientific revolution with which modern logic began was mathematization (the introduction of mathematical methods into logic).

Logic in Ancient China

The logic in China appeared during the emergence of a large number of schools, competition and discussions between them. Confucius Mo-tzu’s contemporary (“Teacher of Mo”, “Sage of Mo”; V — IV centuries BC) was known as the founder of Moism ( Moji school), whose representatives were looking for sources of reliable reasoning and the conditions for its correctness. In the field of argumentation, they preferred the development of reasoning by analogy with the development of deduction. In the process of analyzing the semantics of the language, Moists developed a method for classifying names according to the degree of their commonality and dividing things by type (the “three rules”, “three fa” method).

One of the branches of Moism, logic ( Ming Chia , the school of names , V — III centuries BC), began to study the formal logic itself (its representatives approached the discovery of a categorical syllogism earlier or simultaneously with its formulation by Aristotle).

Later, during the Qin Dynasty , this line of research disappeared in China, since then the philosophy of legism brutally suppressed all other philosophical schools. The logic in China reappeared only with the penetration of the Indian logic of the Buddhists there and further lagged far behind the development of European and Middle Eastern logic.

Indian Logic

The origins of logic in India can be traced in the grammatical texts of the fifth century BC. e. . Two of the six orthodox Hindu (Vedic) schools of Indian philosophy - Nyaya and Vaisesika - were engaged in the methodology of cognition, and logic stood out from this problem field.

The very name of the school “nyaya” means “logic”. Its main achievement was the development of logic and methodology, which later became the common property (cf. Aristotelian logic in Europe). The main text of the school was the Nyaya Sutras of Akshapada Gautama ( II century A.D. ). Since the Nyaiks believed that the only way to free themselves from suffering was to achieve reliable knowledge, they developed subtle methods for distinguishing reliable sources of knowledge from false opinions. There are only four sources of knowledge (four pramanas ): perception, inference, comparison, and testimony. A strict five-term inference scheme included: an initial premise, a basis, an example, an application, and a conclusion.

Buddhist philosophy (not one of the six orthodox schools) was the main opponent of the Nyaik in logic. Nagarjuna , the founder of Madhyamika (the “middle way”), developed a reasoning known as “katuskoti,” or tetralemm. This four-sided argument systematically tested and rejected the statement of the statement, its negation, the combination of the statement and the negation and, finally, the rejection of its statement and its negation.

In Dignagi and his follower, Dharmakirti, Buddhist logic has reached the pinnacle. The central point of their analysis was the establishment (determination) of the necessary logical inherentness (inclusion in the definition), “vyapti”, also known as “constant adherence” or “conviction”. For this purpose, they developed the doctrine of "apocha" or distinction, about the rules for including signs in the definition or excluding them from it.

The Navya-Nyaya School (“New Nyaya”, “New Logic”) was founded in the 13th century by Ganesha Upadhyaya from Mytila, the author of “ Tattvachintamami ” (“Treasure of Thoughts on Reality”). However, he also relied on the work of his predecessors of the X century .

European and Middle Eastern Logic

In the history of European logic, the following stages can be distinguished:

• Aristotelian (traditional) lasted hundreds of years, during which logic developed very slowly;
• the scholastic stage of development, the peak of which falls on the XIV century ;
• modern stage.

The Logic of Antiquity

The founder of logic in ancient Greek philosophy is considered the ancient Greek philosopher Aristotle , as it is believed that he derived the first logical theory. Aristotle's predecessors in the development of logical science in ancient Greece were Parmenides , Zeno of Elea , Socrates, and Plato . Aristotle , for the first time, systematized the available knowledge of logic, substantiated the forms and rules of logical thinking. His series of works “ Organon ” consists of six works devoted to logic: “Categories”, “On Interpretation”, “Topic”, “First Analytics ” and “Second Analytics”, “Sophistic refutations”.

After Aristotle in Ancient Greece, logic was also developed by representatives of the Stoic school. A great contribution to the development of this science was made by the orator Cicero and the ancient Roman theorist of oratory Quintilian .

Logic in the Middle Ages

As you approach the Middle Ages, logic has become more widespread. It began to be developed by Arabic-speaking scholars, for example, Al-Farabi (c. 870 - 950 gg.). Medieval logic is called scholastic, and its heyday in the 14th century is associated with the names of scholars William Ockham , Albert of Saxony and Walter Burley .

Logic in the Renaissance and the New Age

This historical period in logic is marked by the appearance of many publications that are extremely significant for science.

Francis Bacon in 1620 publishes his “ New Organon ”, which contains the basics of inductive methods, later improved by John Stuart Mill and called methods for establishing causal relationships between Bacon-Mill phenomena. The essence of induction (generalization) is in ascent (in the process of cognition) from particular cases to general rules. It is also necessary to look for the causes of your mistakes.

In 1662, the textbook “ The Logic of Por Royal ” was published in Paris , authored by P. Nicole and A. Arnault , who created a logical teaching based on the methodological principles of Rene Descartes .

Modern Logic

In the late XIX - early XX centuries , the foundations of the so-called. mathematical, or symbolic, logic. Its essence lies in the fact that mathematical methods can be used to detect the truth value of natural language expressions. It is the use of symbolic logic that distinguishes modern logical science from traditional.

A huge contribution to the development of symbolic logic was made by such scientists as J. Bull , O. de Morgan , G. Frege , C. Pierce and others. In the XX century, mathematical logic took shape as an independent discipline within the framework of logical science.

The beginning of the 20th century was marked by the formation of ideas of non-classical logic, many important provisions of which were anticipated and / or laid down by N. A. Vasiliev and I. E. Orlov .

In the middle of the 20th century, the development of computer technology led to the appearance of logic elements, logical blocks, and devices of computer technology, which was associated with the additional development of such areas of logic as problems of logical synthesis, logical design, and problems of logical modeling of logical devices and computer technology.

In the 80s of the XX century, research began in the field of artificial intelligence based on languages ​​and logical programming systems. The creation of expert systems began using and developing automatic proof of theorems, as well as evidence-based programming methods for verifying computer algorithms and programs.

In the 80s, changes in education also began. The advent of personal computers in secondary schools led to the creation of computer science textbooks with the study of elements of mathematical logic to explain the logical principles of logical circuits and computing devices, as well as the principles of logical programming for fifth generation computers, and the development of computer science textbooks with the study of predicate calculus for database design knowledge.

The concepts of logic necessary for understanding the subject: ^[7]

Deductive and inductive reasoning in traditional logic

• Induction
• Deduction
• Transduction

Syllogistics

• Syllogism
• Syllogistic theories

Mathematical Logic Apparatus

Propositional Logic

• ( Propositional logic )

Predicate Logic

• Quantifier logic
• First order logic
• Second order logic

Calculus and logical methods

• Solvability
• Semantic tree
• Beta Tables
• Axiomatics
• Natural conclusion
• Sequence calculus

Logical Semantics

• Algebraic semantics
• Set-theoretic semantics
• Relational semantics of possible worlds
• The problem of content semantics of logical systems
• Categorical semantics
• Theory of Semantic Categories

Laws of Logic

• Law of identity
• Law of Excluded Third
• Law of contradiction
• Law of sufficient reason
• Laws of de Morgan
• Laws of deductive inferences
• Clavius ​​Law
• Laws of division
• Duns Scott Law

Model Theory

Evidence Theory

Inference Theories

• Inference Theories ( Inference Theory )
• Follow Theories ( Follow Theory )
• Implication Theories ( Implication Theory )
• Material implication

Logics with a non-classical understanding of following

• Relevant logic
• Paraconsistent logic
• Nonmonotonic Logic
  • Dynamic logic

Logics abrogating the law of the excluded third

• Intuitionistic logic
• Constructive logic
• The logic of quantum mechanics

Logics changing truth tables

• Multi-valued logic
• Two-digit logic
• Three digit logic

Logics Expanding the Sentence

• Logic of questions
• Grade Logic
• Logic of norms

Modal Logic

• Modality
• Aletic modalities ( aletic modality , aletic modal logic , aletic modal logic )
• Deontic modalities ( deontic modality , deontic modal logic , deontic modal logics )
• Epistemological modalities ( epistemological modality , epistemological modal logic , epistemological modal logics )
• Temporal modalities ( temporal modality , temporary modal logic , temporary modal logic )
• Strict implication
• Material implication

Non-deductive logical theories

• Inductive logic
• Probabilistic logic
• Decision logic
• The logic of fuzzy concepts (the logic of fuzzy sets , fuzzy logic )
• Analogy ( inference by analogy ).

Other Non-Classical Logics

• Categorical Logic
• Combinatorial logic is logic that replaces variables with functions in order to clarify intuitive operations with variables such as substitution. The arithmetic system built on the basis of combinatorial logic contains all partially recursive functions and avoids Gödel incompleteness.
• Conditional logic ( conditional logic ). Her subject is the truth of conditional sentences (in particular, the subjunctive mood). The logic of counterfeit claims.

Applied problems of logic (see Applied logic ) and logical semantics

• Applications of logic in the methodology of science
• Applications of logic in philosophy
• Applications of Logic in Theology
• Applications of logic in psychology
• Applications of logic in the legal sciences
• Logic Applications in Linguistics
• Logic applications in other disciplines
• Artificial Intelligence

Applications of logic in the analysis of cognitive procedures

Logical analysis of forms and methods of cognition

• Forms of thinking
• Definition
• Classification
• Abstraction
• Idealization
• Axiomatization
• Formalization
• Logical problems of argumentation
• Logic of evidence

Applications of logic in the methodology of science

• Science methodology
• Logic of science
• Logic and Empiricism

Applications of Logic in Philosophy

• Applications of logic in philosophy
• Applications of logic in ontology
• Applications of Logic in Epistemology
• Logic Applications in Ethics
• Logical problems of argumentation ( theory of argumentation )
• Analytic philosophy

Applications of logic in psychology

• Cognitive science
• Cognitive psychology
• The logic of discovery

Since logic establishes laws and patterns of thinking , there is a problem of correlating logic with creativity , which is based on intuition . Creativity without limits is an idealization : it is limited by the psychological laws of perception or, for example, the laws of composition in the visual arts. Creativity involves not only the ability to put forward an interesting idea, but also the ability to convincingly substantiate it and put it into practice according to certain rules, therefore, it must follow some rules of thinking.

Logic Applications in Linguistics

• Logical language analysis
• Analytic philosophy

Computer Science Logic Applications

• Dynamic Logic ( Dynamic Logic )
• Program Logics ( Program Logic )
• Logic of expert systems ( logic of expert systems )
• Logic in Computer Science
• Evidence-based programming
• Automatic proof of theorems
• Logic programming

• Logical symbol list
• Transcendental logic

• Logic / V. A. Bocharov // New Philosophical Encyclopedia : in 4 volumes / before. scientific ed. Council V. S. Styopin . - 2nd ed., Rev. and add. - M .: Thought , 2010 .-- 2816 p.

Research

• Husserl E. Logical studies. T. 1 // Philosophy as a rigorous science. - Novocherkassk: Saguna, 1994 .-- 357 p. - ISBN ISBN 5-7593-0138-1 .
• Vasiliev N.A. Imaginary logic. Selected Works. - Science, 1989 .-- 264 p. - 6200 copies. - ISBN 5-02-007946-4 .

Educational and reference books

• Getmanova A. D. A textbook on logic . - M .: Vlados, 1995 .-- 303 p. - ISBN 5-87065-009-7
• Kondakov N.I .: A logical dictionary-reference book . - M .: Nauka, 1975 .-- 720 p.
• Kondakov N.I. Introduction to logic. - M .: Nauka, 1967 on the website of Runivers
• Ivlev Yu. V. Textbook of logic: Semester course: Textbook. - M .: Business, 2003. - 208 s - ISBN 5-7749-0317-6
• Bocharov V.A., Markin V.I. Fundamentals of Logic: A Textbook. - M .: INFRA-M, 2001 .-- 296 p. - ISBN 5-16-000496-3
• Bocharov V. A. , Markin V. I. Chapter I. The subject and basic concepts of logic // Fundamentals of logic: textbook . - M .: INFRA-M, 1998 .-- S. 224. - 9 p. - ISBN 5-86225-595-8 . Archived March 7, 2016 on the Wayback Machine
• Ivin A. A. Logic: Textbook. - Ed. 2nd. - M .: Knowledge, 1998. - ( On the portal "Philosophy in Russia" ; on the site of the Glory of Yanko )
• Ivin A.A., Nikiforov A.L. Dictionary of Logic - M .: Tumanit, VLADOS, 1997 .-- 384 s - ISBN 5-691-00099-3 .
• Gorsky D.P. Logic: A manual for pedagogical schools. (unavailable link) - Ed. 3rd - M .: Uchpedgiz, 1961 .-- 160 p.
• Chelpanov G. I. Textbook of logic . - M., 1994.
• Formal Logic / Ed. I. Ya. Chupakhin, I.N. Brodsky . - L .: LSU, 1977 .-- 357 p.

Literature on the History of Logic

• Bazhanov V.A. History of logic in Russia and the USSR. - M.: Canon +, 2007 .-- 336 p. - ISBN 5-88373-032-9
• Makovelsky A.O. History of logic . - M., 1967. - 504 p.
• Popov P. S. History of the logic of the new time. - M., Publishing House of Moscow State University, 1960.
• Styazhkin N. I. Formation of mathematical logic. - M., 1967.
• Scholtz H. Geschichte der Logik, 1931. (Concise History of Logic. - New York, 1961).

Literature on Chinese Logic

• Spirin BC On the “third” and “fifth” concepts in the logic of ancient China // Far East. Collection of articles on philology, history, philosophy. - M., 1961.
• Krol Yu. L. Spore as a cultural phenomenon of ancient China // Peoples of Asia and Africa. - 1987. - No. 2.
• Krushinsky A. A. Names and Realities in Ancient Chinese Logic and Methodology (Review) // Modern Historical and Scientific Research: Science in Traditional China. - M., 1987.
• Pan Shimo (PRC). The Logic of Ancient China (short outline) // Philosophical Sciences. - 1991. - No. 12.
• Zhou Yunzhi. The main milestones of the development of ancient Chinese logic min bian, its main features and real achievements // Rationalist tradition and modernity. China. 1993. no. - S. 152-178.
• Krushinsky A. A. The Logic of I Ching. Deduction in ancient China. - M., 1999.
• Kvartalova N. P. Logical ideas of the treatise "Gunsun Lun-tzu" // Man and the spiritual culture of the East. Almanac. Vol. I. - M., 2003 .-- S. 167-172.
• Kobzev A.I. School of Names (Ming Chia): a Conflict of Logic and Dialectics // China in the Dialogue of Civilization: On the 70th Anniversary of Academician M. L. Titarenko. - M. 2004. - S. 550–557.

• Institute of Logic, Cognitology and Personality Development (ILKiRL)
• Federal educational portal "Social and humanitarian and political science education." Section "Philosophy". Subsection "Logic"
• Logic in the Electronic Library for Philosophy
• Philosophy in Russia - philosophical portal philosophy.ru
• History of Ancient Culture> History and Culture of Ancient Greece> Feat of Socrates