History of Logic

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The History of Logic documents the development of logic in various cultures and traditions throughout history. Although many cultures have employed intricate systems of reasoning, and even logical thinking was already implicit in Babylon in some sense, logic as an explicit analysis of reasoning methods has received substantial treatment only originally in three traditions: Ancient Chinese, Ancient India and Ancient Greece.

Although the exact dates are uncertain, particularly in the case of India, it is likely that the logic emerged in all three societies around the fourth century BC. C. The formally sophisticated treatment of logic comes from the Greek tradition, especially the Aristotelian Organon, whose achievements would be developed by Islamic logicians and, later, by the logicians of the European Middle Ages. The discovery of Indian logic among British scholars in the eighteenth century influenced modern logic as well.

The history of logic is the product of the confluence of four lines of thought, which appear at different historical moments:Aristotelian logic, followed by the contributions of the Megarians and the Stoics. Centuries later, Ramon Llull and Leibniz studied the possibility of a single, complete and exact language for reasoning. At the beginning of the 19th century, investigations into the foundations of algebra and geometry, followed by the development of the first complete calculus by Frege. Already in the 20th century, Bertrand Russell and Whitehead completed the process of creating mathematical logic. From this moment on, new developments and new schools and trends will not cease. Another interesting perspective on how to approach the study of logical history is offered by Alberto Moretti and is synthesized by Diego Letzen.

Old age

Logic, as an explicit analysis of reasoning methods, originally developed in three civilizations in ancient history: China, India, and Greece, between the 5th and 1st centuries BC. c.

Mesopotamia

In Mesopotamia, Esagil-kin-apli's Medical Diagnostic Manual, written in the 11th century B.C. C., was based on a logical set of axioms and assumptions, including the modern view that, through the examination and inspection of a patient's symptoms, it is possible to determine the problem of the same, its etiology and its origin. future development, and chances of recovery.

During the 7th and 8th centuries, Babylonian astronomers began to use an internal logic in their planetary prediction systems which was an important contribution to logic and the philosophy of science. Babylonian thought had a considerable influence on the thought of the archaic Greece.

Ancient Greece

In Ancient Greece, two opposing logical traditions emerged. Stoic logic was rooted in Euclid of Megara, a pupil of Socrates, and with its concentration on propositional logic is perhaps closest to modern logic. However, the tradition that survived the influences of later cultures was the Peripatetic, which had its origin in the body of works by Aristotle known as the Organon (instrument), the first systematic Greek work on logic. Aristotle's discussion of the syllogism allows for interesting comparisons with the Indian scheme of inference and the less rigid Chinese discussion.

The philosopher Parmenides who formulated the logical principle of identity, where he states "what is is and what is not is not" and from this principle the principle of non-contradiction is deduced, where "what is cannot not be". He competes with Aristotle for the title of "father of logic". Heraclitus was said to have denied such by asserting the constant flow of things (panta rei). However, Allan Bloom considers that the first known explicit statement of the principle of non-contradiction occurs in Plato 's Republic where the character Socrates says, "it is clear that the same thing will not be willing to do or suffer things at the same time." contrary with respect to the same and in relation to the same object".It is clear that Plato and Aristotle had bases in the Eleatic pre-Socratics to formulate this principle.

Aristotle is considered the founder of logic as a propaedeutic or basic tool for all sciences. Aristotle was the first to formalize reasoning, using letters to represent terms. This precision must be taken into account since logic is prior to Aristotle in its aspect of informal logic, as he himself acknowledges.He was also the first to use the term "logic" to refer to the study of arguments within the "apophantic language" as manifesting truth in science. He held that the truth manifests itself in the true judgment and the valid argument in the syllogism: "Syllogism is an argument in which, certain things established, necessarily results from them, because they are what they are, something else." He addressed in various writings of his Órganon such questions as concept, proposition, definition, proof, and fallacy. In his main logical work, the Primitive Analytics, he developed the syllogism, a rigidly structured logical system. Aristotle also formalized the opposition table of judgments and categorized the valid forms of the syllogism. Furthermore, Aristotle recognized and studied inductive arguments, basis of what constitutes experimental science, whose logic is closely linked to the scientific method. The influence of Aristotle's achievements was so great that in the eighteenth century Immanuel Kant went so far as to say that Aristotle had practically completed the science of logic.

In Europe, Aristotle was the first to develop logic. Aristotelian logic was widely accepted in science and mathematics and remained in wide use in the West until the early 19th century. Aristotle's system of logic was responsible for the introduction of the hypothetical syllogism, temporal modal logic, inductive logic, as well as influential terms such as terms, predicables, syllogisms, and propositions. In Europe during the late medieval period, great efforts were made to show that Aristotle's ideas were compatible with the Christian faith. During the High Middle Ages, logic became the main focus of philosophers, who would engage in critical logical analyzes of philosophical arguments, often using variations of the methodology of Scholasticism. In 1323, William of Ockham's influential Summa Logicae was published. By the 18th century, the structured approach to storylines had degenerated and fallen out of favour, as shown in Holberg's satirical play Erasmus Montanus.

The Stoic philosophers introduced the hypothetical syllogism and announced propositional logic, but it did not get much development. The term "logic" is found in the ancient Peripatetics and Stoics as a theory of argumentation or closed argument. In this way, the argumentative form responds to the principle of knowledge that supposes that it adequately represents reality. Therefore, without losing its condition of formality, they are not formalists and they have not just detached themselves from the structures of language.On the other hand, informal logic was cultivated by rhetoric, oratory, and philosophy, among other branches of knowledge. These studies focused mainly on the identification of fallacies and paradoxes, as well as on the correct construction of discourses.

In the Roman period logic had little development, rather summaries and comments were made to the works received, the most notable being: Cicero, Porphyry and Boethius. In the Byzantine period, Philopon.

Until the nineteenth century, Aristotelian and Stoic logic always maintained a relationship with arguments formulated in natural language. That is why although they were formal, they were not formalists. Today that relationship is treated from a completely different point of view. The strict formalization has shown the limitations of traditional or Aristotelian logic, which today is interpreted as a small part of class logic.

Through Latin in Western Europe and different Eastern languages ​​such as Arabic, Armenian and Georgian, the Aristotelian tradition was considered in a special way for the codification of the laws of reasoning. Only from the 19th century did this approach change.

Ancient indian

Two of the six Indian schools of thought are related to logic: Nyāya and Vaisheshika. The Nyaya Sutras of Aksapada Gautama constitute the core texts of the Nyaya school, one of the six orthodox schools of Hindu philosophy. This realist school worked with a rigid five-member inference scheme that encompasses an initial premise, a reason, an example, an application, and a conclusion. The idealistic Buddhist philosophy became the main opponent of the Naiyayikas. Nāgārjuna, the founder of the Madhyamika middle path, developed an analysis known as "catuskoti" or tetralemma. This four-pronged argumentation systematically examined and rejected the affirmation of a proposition, its negation, the joint affirmation and negation, and finally, the rejection of its affirmation and negation. But it was with Dignāga and his successor Dharmakirti that Buddhist logic reached its greatest height. The analysis of it, centered on the definition of the necessarily logical implication, "vyapti", also known as concomitance or invariable penetration. To this end, a doctrine known as "apoha" or differentiation was developed. It comprises what might be called the inclusion and exclusion of defining properties. The difficulties concerning this enterprise, in part, stimulated the neo-scholastic school of Navya-Nyāya, which introduced a formal analysis of inference in the sixteenth century. Also known as concomitance or invariant penetration. To this end, a doctrine known as "apoha" or differentiation was developed. It comprises what might be called the inclusion and exclusion of defining properties. The difficulties concerning this enterprise, in part, stimulated the neo-scholastic school of Navya-Nyāya, which introduced a formal analysis of inference in the sixteenth century. Also known as concomitance or invariant penetration. To this end, a doctrine known as "apoha" or differentiation was developed. It comprises what might be called the inclusion and exclusion of defining properties. The difficulties concerning this enterprise, in part, stimulated the neo-scholastic school of Navya-Nyāya, which introduced a formal analysis of inference in the sixteenth century.

In India, innovations in the scholastic school, called the Nyaya, continued from ancient times to the early 18th century with the Navya-Nyaya school. Around the 16th century, theories similar to modern logic were developed, such as Gottlob Frege's "distinction between sense and reference of proper names" and his "definition of number", as well as the theory of "restrictive conditions for universals" anticipating some of Developments in modern set theory. Since 1824, Indian logic has attracted the attention of many Western students and has influenced important 19th-century logicians such as Charles Babbage, Augustus De Morgan, and George Boole. In the 20th century, Western philosophers such as Stanislaw Schayer and Klaus Glashoff have investigated Indian logic more extensively.

Ancient china

In China, a contemporary of Confucius, Mozi, "Master Mo", is considered to be the founder of the Mohist school (Mohism), whose principles are related to issues such as valid inference and the conditions of correct conclusions. In particular, one of the schools that followed Mohism, the logicians, is considered by many experts to be the first to investigate formal logic. Unfortunately, due to rigid legal regulations during the Qin dynasty, that line of research disappeared from China until the introduction of Indian philosophy by Buddhism. Translation and school research in logic was suppressed by the Qin dynasty, in accordance with legal philosophy. In India, the logic lasted considerably longer: it developed (for example with the nyāya) until the Asharite school appeared in the Islamic world, which suppressed some of the original work in logic. Despite the above, there were Indian scholastic innovations until the early 19th century, but not much survived within colonial India. The sophisticated and formal treatment of modern logic apparently comes from the Greek tradition.

The Chinese logical philosopher Gong Sunlong (325-250 BC) proposed the paradox "One and one cannot be two, since neither becomes two." [24] In China, the tradition of academic research in logic, however, was suppressed by the Qin dynasty following Han Feizi's legalistic philosophy.

Middle Ages

In the High Middle Ages, logic maintains the condition of propaedeutic science under the name of dialectics. It continues to be studied as one of the liberal arts but without great contributions.

In its evolution towards the Late Middle Ages, the Arab contributions of Al-Farabi, Avicenna and Averroes are important, since it was the Arabs who reintroduced the writings of Aristotle in Europe. In the Late Middle Ages, his study was a requirement to enter any university. From the middle of the thirteenth century three separate bodies of the text are included in the logic. In the logica vetus and logica nova is traditional logical writings, especially Aristotle's Organon and the comments of Boethius and Porphyry. The parva logicalia can be considered as representative of medieval logic.

The critical evolution that is developing from the contributions of Abelardo dynamized the logical and epistemological problem from the thirteenth century (Pedro Hispano, Raimundo Lulio Lambert de Auxerre, Guillermo de Sherwood) that culminated in all the problems of the fourteenth century, with William of Ockham, Jean Buridan, John Wyclif and Pedro of Spain, Richard Kilvington and Albert of Saxony.

Here are treated a number of new problems on the frontier of logic and semantics that were not treated by the ancient thinkers. Of special relevance is the problem regarding the assessment of the terms of language in relation to universal concepts, as well as their epistemological and ontological status and the problem of individuation.

Islamic world

For a time after Muhammad's death, Islamic law considered it important to formulate standards for arguments, which gave rise to a new approach to logic in Kalam, but this approach was later displaced by ideas borrowed from Greek and Hellenistic philosophy. with the rise of the philosophers of the Mu'tazili school, who highly valued Aristotle's Organon. The works of Islamic philosophers with Hellenistic influences were crucial to the reception of Aristotelian logic in medieval Europe, along with commentaries on the Organon.made by Averroes. The works of al-Farabi, Avicenna, al-Ghazali, and other Muslim logicians, who at times criticized and corrected Aristotelian logic and introduced their own forms of logic, also played a central role in the subsequent development of medieval European logic.

Islamic logic includes not only the study of formal models of inference and their validation, but also elements of the philosophy of language and elements of epistemology and metaphysics. Due to disputes with Arabic grammarians, Islamic philosophers were very interested in working on the study of the relations between logic and language, and devoted much discussion to the question of the object of interest and goals of logic in relation to reasoning and speech.. In the area of ​​logical-formal analysis, they developed the theory of terms, propositions, and syllogisms. They regarded the syllogism as the form to which all rational argument could be reduced, and they regarded the syllogistic theory as the central point of logic. Even poetics was considered, in certain aspects,

Among the most important developments made by Muslim logicians is that of Avicenna's logic as a substitute for Aristotelian logic. Avicenna's logical system was responsible for the introduction of the hypothetical syllogism, temporal-mode logic, and inductive logic.Another important development in Islamic philosophy is that of a strict science of citation, the isnad or "review", and the development of a scientific method of open inquiry to question certain claims, ijtihad, which could normally be applied to many kinds of issues. From the twelfth century, despite the logical sophistication of al-Ghazali, the rise of the Asharite school at the end of the Middle Ages gradually limited the original work on logic in the Islamic world, although it continued later in the fifteenth century.

Medieval Europe

By 'medieval logic' (also known as 'scholastic logic') is usually understood the form of Aristotelian logic developed in medieval Europe in the period c 1200–1600. This task began after the Latin translations of the 12th century, when Arabic texts on Aristotelian logic and the logic of Avicenna were translated into the language of Rome. Although Avicenna's logic influenced early medieval European logicians such as Albert the Great, the Aristotelian tradition became dominant due to the significant influence of Averroism.

The application of Aristotelian logic proceeded by having the student memorize a long set of syllogisms. Memorization consisted of diagrams, or learning a key sentence, with the first letter of each word reminding the student of the names of the syllogisms. Each syllogism had a name, for example Modus Ponens had the form 'If A is true, then B is true. A is true, therefore B is true. Most college logic students memorized Aristotle's 19 two-subject syllogisms, allowing them to correctly connect a subject and an object. A few geniuses developed three-subject systems, or described a way of making three-subject rules.

A developmental feature of Aristotelian logic is known as the theory of supposition, a study of the semantics of the terms of the proposition.

After the initial phase of translations, the tradition of medieval logic was developed in manuals such as that of Petrus Hispanus in the thirteenth century, of unknown identity, who authored a standard manual on logic, the Tractatus, which was well known in Europe. for several centuries.

The tradition reached its height in the 14th century, with the works of William of Ockham (c. 1287–1347) and Jean Buridan. Great works of this tradition are Metaphysical Disputations by Francisco Suárez (1548–1617), and Lógica by Juan Poinsot (1589–1644, known as Juan de Santo Tomás). This tradition is continued to the present day by Neo-Aristotelians and Neo-Thomists.

Modern age

In the seventeenth century, logic acquires a new focus in the rationalist interpretations of Port Royal (Antoine Arnauld, Pierre Nicole) but they did not represent a radical change in the concept of logic as a science.

Rationalist philosophers, by placing the origin of philosophical reflection in consciousness, contributed, through the development of analysis as a scientific method of thinking, the issues that will mark the development of formal logic. Of special importance are Descartes' idea of ​​a Mathesis Universalis and Leibniz who, with his Characteristica universalis, supposes the possibility of a universal language, specified with mathematical precision on the basis that the syntax of words should be in correspondence with the entities designated as individuals or metaphysical elements, which would make possible an algorithmic calculation or computation in the discovery of truth.The first attempts and realizations of calculating machines appear (Pascal and Leibniz) and although their development was not effective, nevertheless the idea of ​​a Mathesis Universalis or Characteristica universalis is the immediate antecedent of the development of symbolic logic from the 20th century.. Leibniz and Descartes followed the Jesuit school very closely, especially Francisco Suárez, who in turn used the Mexican Logic, by Fray Antonio de Rubio, a Mexican philosopher (Novohispano). In addition, it is considered that modernizing logics never achieved precision. of these studies. Sander Pierce, Gottlob Frege, Saussure and Wittgenstein followed neo-scholastic criteria to formulate their more complete logical theories.

In the 18th century, Kant considered that logic, being an a priori science, had practically found its full development with Aristotelian logic, so it had hardly been modified since then. But he makes a new use of the word "logic" as transcendental logic, in the sense of investigating the pure concepts of understanding or transcendental categories.

The logic of transcendental thinking ends up situating itself in a dialectical process as subjective idealism in Fichte; objective idealism in Schelling and finally an absolute idealism in Hegel, who considers logic within the Absolute as a dialectical process of the Absolute Spirit that produces its determinations as a concept and its reality as a result in the becoming of the Idea of ​​the Absolute as Subject whose truth is manifested in the result of the movement through the contradiction in three successive moments, thesis-antithesis-synthesis. Epistemology and ontology are united and exposed in Philosophy, understood as the Absolute System.

Contemporary age

Historically, Descartes may have been the first philosopher to have had the idea of ​​using algebra, especially its techniques for solving unknown quantities in equations, as a vehicle for scientific exploration. The idea of ​​a reasoning calculus was also cultivated by Gottfried Wilhelm Leibniz. Leibniz was the first to formulate the notion of a generally applicable system of mathematical logic. However, the relevant papers were not published until 1901 and many of them remain unpublished, and the current understanding of the power of Leibniz's discoveries did not begin to develop until the 1980s.

Gottlob Frege in his Begriffsschrift (1879) extended formal logic beyond propositional logic to include constructs like "all" and "some". He showed how to introduce variables and quantifiers to reveal the logical structure of sentences, which might be hidden behind their grammatical structure. For example, "All humans are mortal" becomes "Everything x is such that if x is a human then x is mortal." Frege's peculiar double dimensional notation caused his work to be ignored for many years.

In a masterful 1885 paper read by Peano, Ernst Schröder, and others, Charles Peirce introduced the term "Second Order Logic" providing most of the modern logical notation, including the prefixed symbols for universal and existential quantification. Logicians of the late nineteenth and early twentieth centuries were most familiar with the Peirce-Schröder system of logic, although Frege is generally recognized as the Father of modern logic.

In 1889 Giuseppe Peano published the first version of the logical axiomatization of arithmetic. Five of the nine axioms are known as the Peano axioms. One of these axioms was a formalization of the principle of mathematical induction.

XIX century

From the second half of the 19th century, logic would be profoundly revolutionized. In 1847 George Boole published a short treatise entitled The Mathematical Analysis of Logic, and in 1854 another more important one entitled The Laws of Thought. Boole's idea was to construct logic as a calculation in which the truth values ​​are represented by 0 (false) and 1 (true), and to which mathematical operations such as addition and multiplication are applied.

At the same time, Augustus De Morgan published his work Formal Logic in 1847, where he introduced De Morgan's laws and tried to generalize the notion of syllogism. Another important English contributor was John Venn, who in 1881 published his book Symbolic Logic, where he introduced the famous Venn diagrams.

Charles Sanders Peirce and Ernst Schröder also made important contributions.

However, the real revolution in logic came at the hands of Gottlob Frege, who is often considered the greatest logician in history, along with Aristotle. In his 1879 work, the Conceptography, Frege first offers a complete system of predicate logic. He also develops the idea of ​​a formal language and defines the notion of proof. These ideas formed a fundamental theoretical basis for the development of computers and computer science, among other things. Despite this, his contributions were overlooked by Frege's contemporaries, probably because of the complicated notation he developed. In 1893 and 1903, Frege published The Laws of Arithmetic in two volumes., where he tries to deduce all mathematics from logic, in what is known as the logicist project. His system, however, contained a contradiction (Russell's paradox).

Twentieth century

The 20th century would be one of enormous developments in logic. As of the 20th century, logic began to be studied for its intrinsic interest, and not only for its virtues as a propaedeutic, for which reason it was studied at much more abstract levels.

In 1910, Bertrand Russell and Alfred North Whitehead published Principia mathematica, a monumental work in which they achieved a large part of mathematics from logic, avoiding the paradoxes into which Frege fell. The authors acknowledge Frege's credit in the preface. In contrast to Frege's work, Principia mathematica was a resounding success, coming to be considered one of the most important and influential works of non-fiction of the entire 20th century. Principia mathematica uses a notation inspired by that of Giuseppe Peano, some of which is still widely used today.

Although in the light of contemporary systems Aristotelian logic may seem wrong and incomplete, Jan Łukasiewicz showed that, despite its great difficulties, Aristotelian logic was consistent, although it had to be interpreted as class logic, which is not the case. small modification. For this reason, the syllogistic has practically no use today.

In addition to propositional logic and predicate logic, the 20th century saw the development of many other logical systems; among which the many modal logics stand out.

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