History of logic

The history of logic deals with the origin and development of logic and all of its sub-disciplines.

Probably due to different languages ​​and cultures, different traditions of logic have emerged in different parts of the world:

• The EU- western logic has its beginning in ancient Greece and can be divided into two streams: in the tradition of the Aristotelian- scholastic logic and
• In the modern or mathematical logic from 1847.

• 3.1 The " traditional logic "
• 3.2 The early modern period and non-formal logics
• 3.3 Leibniz
• 3.4 The heyday of classical logic
• 3.5 Modern

• 4.1 India and Tibet
• 4.2 China
• 4.3 Japan
• 4.4 Arabic- space

Ancient logic

Precursor

Among the precursors of the ancient logic include the pre-Socratics ( 6th and 5th centuries BC), the Sophist (from the 5th century BC) and Plato ( 4th century BC). The Sophists taught, usually against payment, for example, how to make speeches before public meetings or in court and was able to persuade the interlocutor or audience. To this end, they taught rhetoric, single logical artifices and sometimes how you could apply fallacies.

Plato has indeed left a logical system and not even a logical font, in his dialogues can be found but already many sites that deal with topics of logic and have had a strong influence on the founder of logic, his student Aristotle. The employment of Plato with the order and the laws of thought is likely seen as a response to the rejected by him, arbitrary, often deliberately -misleading conceptual and reasoning acrobatics of the Sophists. The most important logical discovery Plato was probably the concept of division ( diaeresis ). It is a method that makes it possible to define a given phrase by lower subsumed under higher terms. Besides the Dihairesis the constant exercise discussions of teachers and students of the Platonic Academy have influenced the subsequent history of logic. In the first logical font of the young Aristotle, the Topics, it is, in fact, the formulation of rules of correct reasoning.

Apart from the logical vocabulary that Plato the method of Dihairesis ( then as definition, difference ( in Aristotle specific difference ), genus, species ) used is worth noting that already contains a provision of the statement and the true or false statement. Plato distinguishes ( in dialogue Sophist 261c ) a true term connection of a main and a verb, " Theaetetus sits ", " Theaetetus flies " from the incorrect.

• Tool ( altgri.: Organon) ( German ) Categories ( altgri.: Kategoriai, Latin: Categoriae ) ( altgri. )
• Doctrine of the sentence ( altgri.: Peri hermeneias, Latin: De interpretatione ) ( altgri. )
• First analysis ( altgri.: Analytikon proteron, Latin: Analytica priora ) ( altgri. )
• Second analysis ( altgri.: Analytikon hysteron, Latin: Analytica posteriora ) ( altgri. )
• Topik ( altgri.: topoi, Latin: Topica ) ( altgri. )
• Sophistical refutations ( altgri.: Peri sophistikon elenchon, Latin: De sophisticis elenchis ) ( altgri. )

The Aristotelian term logic

A first system of logic one then finds in Aristotle ( 384-322 BC ), considered not only as the founder of logic, but also of incomparable importance for the further history of logic and its development was. His logical works Organon consists of six individual writings in which all essential parts of logic are discussed: the term ( categories), which consists of concepts statement ( De Interpretatione ) and consists of statements circuit ( Analytica Priora Analytica and posteriora ). Further, the practice of reasoning is treated ( Topik and sophistical refutations ), except in the Organon come in the fourth book of the Metaphysics logical problems discussed.

The Aristotelian logic is a logical system in which terms are related to each other. So this is not a propositional logic, but rather a term or term logic. In the immediate post-history came Aristotle's logic quickly forgotten until Late Antiquity dominated the Stoic propositional logic. Only in the Middle Ages, it begins to dominate and influence the development of critical logic.

Categories

In Scripture the words Categories in ten species are divided ( the categories). These ten parts of speech (Socrates refers to a particular person, human being is a general term that white is a property, etc.) differ so that each part of speech with some of the other parts of speech can be connected to a set. A set consists of at least two words ( Man running; Socrates is a man ). In contrast to words sentences are either affirmatively or negatively. Each affirmation and negation, each is either true or false, ie sentences have a truth value. Next, there are four different kinds of things, one of which always kind can act only as a subject of a sentence, never as a predicate of a sentence ( things that are certain individuals, such as " Socrates "). Other both as a subject, as well as the predicate of a sentence (eg "man" as a subject: "Man is an animal " and as a predicate: " Socrates is a man ").

De Interpretatione

Gist of De Interpretatione is an analysis of the logical statement. As part of this analysis, Aristotle refers to the affirmation (S = P) and the negation of (S ≠ P) the same terms as a contradiction. The principle of contradiction ( S can not be P and at the same time not P) is now considered a fundamental law of logic. Next is introduced what is known today as a quantifier: before general concepts can set the following quantifiers " every man is a sense of nature ", " no man is a turtle ", " not everyone is called Socrates " or " some people called Socrates ". Between the statements "every man is white 'and' not every man is white ' is - just one of the two is correct - a contradictory contrast; between the statements "every man is white 'and' no man is white " is - both are wrong - a second type of contrast, the contradictory opposition. A second law of logic, the law of excluded middle also first appears in Aristotle. So must be true one of two contradictory opposite statements S = P and S ≠ P. However, this law does not apply in the following case. Neither of the two contradictory opposite statements " morning this house will collapse " and " Tomorrow will not collapse this house " can be described as true or false. For statements testify the things to come, you could - in addition to true and false - introduce a third truth value. Aristotle has thus anticipated the Multi-valued logic. Also, the verb is already addressed by Aristotle in his dual function: first, it is attributed to subjects, to testify their existence: " Socrates is ", secondly, it serves as a link (now copula called ) between subject and predicate of a statement: " Socrates is a man. " Further, private ions treated ( Not man, unjust, odd) distinguished and different types of predicates: predicates such as white and well come to the subject person to akzidentiell; Predicates such as bipedal creatures and come the subject person, however, much too, they can be combined into a definition of the subject. With the introduction of today's so-called Modalbegriffe Aristotle founded the modal logic. Modalbegriffe refer to statements: possible ( problematic statement: it is possible that SP is ) and necessary ( apodictic statement: it is necessary so that SP is ).

The Analytics

As a logical major work, the two extensive analytics apply. Here Aristotle developed the syllogism, its proof and final lesson that is a formal logical system in the modern sense. In a conclusion ( premise ) is closed on a third statement ( conclusion ) of two statements. These three statements are in turn made ​​up of three terms ( subject - Medium term - predicate ) composed. For example, from the premises Socrates (subject) is a human ( middle term ) and all men (M) are living organisms ( predicate ) follows the conclusion Socrates (S ) is a living being (P). Aristotle distinguishes ( he calls them the three figures ), which are now called deduction, induction and abduction of three types of circuits.

The Megarian - Stoic propositional logic

Away from the Aristotelian term logic first developed in the Megarian, then in the influential Stoic school of philosophy, the bivalent propositional logic ( 4th and 3rd centuries BC). Where should firstly be noted that quite a term logic of these schools have existed, but has been lost and secondly, that Aristotle pupil Theophrastus extended the syllogistic to propositional logic circuits. Powerful action was at first only the Stoic logic, which spread its logic in manuals. In the Middle Ages it was almost completely replaced by the Aristotelian- scholastic logic to be only in 1934 rediscovered by Łukasiewicz quasi. Benson Mates and Michael Frede have written monographs for the Megarian - Stoic logic. The source location is bad, one is V.A. Sextus Empiricus rely on, Diogenes Laertius and Galen.

• Topik (Latin: Topica ) (Latin ) ( eng.)

• About the statement ( gri.: Peri hermēneías, Latin: Peri hermeniae ) (Latin )

• Introduction ( altgri.: Isagoge ) ( eng.)
• Among the categories of Aristotle in question and answer ( altgri.: Ice tas ARISTOTELOUS katēgorías kata peúsin kai apókrisin )

• By defining (Latin: De diffinitione )

• Ten categories (Latin: Categoriae decem ) (Latin )

• From the classification (Latin: De Divisione )
• About the categorical syllogism (Latin: De syllogismo categorico )
• Introduction to categorical syllogisms (Latin: Introductio ad syllogismos categoricos )
• About hypothetical syllogisms (Latin: De hypotheticis syllogismis )
• About the topical differences (Latin: De topicis differentiis )

• Martianus Capella ( 5th or 6th century ): The marriage of Philology with Mercury (Latin: De nuptiis Philologiae et Mercurii )
• Cassiodorus († 580): Institutions (Latin: Institutiones ) (Latin ) and
• Isidore of Seville ( † 636): etymology (Latin: Etymologiae ) (Latin ).

Eubulides the first to formulate the Liar Paradox, Philon the oldest truth table. Also of Philo 's statements linking comes through the words if and then, the so-called material implication ( if A, then B, and in words, if Stefan comes to the party, then he takes with Luke ). More links statements come from Chrysippus: the conjunction (A and B; in words: Stefan comes and Luke comes ), the exclusive disjunction (either A or B; in words: Either I will marry you or I marry Judith ). For the Stoics, the enclosing alternative ( at least A or B) is preserved. Diodorus Cronus, Philo and Chrysippus also provided contributions to modal logic. The Stoics developed an axiomatization of its propositional logic.

Comments and material collections

The Latin tradition of logic begins with Cicero ( 1st century BC) and his translations into Latin. On Apuleius (2nd century) also go numerous Latin terms and graphical schema of the logical square back.

For the transition period from antiquity to the early Middle Ages no significant logical texts have survived, but one was concerned with material collections and comments by the logic of Plato, Aristotle and the Stoics. Mention may be made Galen ( 2nd century ), Alexander of Aphrodisias (2nd / 3rd century ) and Porphyry ( 3rd century ) with its porphyrianischen tree. From Diogenes Laertius an extensive work on the history of philosophy, and thus the logic is preserved.

Influential was Boethius (5th / 6th century ), who translated not only older texts, but also busy with its own logic. Other factors include Isidor (5th / 6th century ) and Cassiodorus ( sixth century).

Middle Ages

• Logic (Latin: Dialectica )

• King Charles works against the Synod (Latin: Opus Caroli Regis contra Synodum ) here: Chapter IV, 23

• Latin: Dicta Albini de imagine Dei
• Latin: Dicta Candidi de imagine Dei

• About the divine predestination (Latin: De divina praedestinatione )
• About natures ( gri.: Periphyseon, lat: De Divisione Naturae ) (Latin )

• About sensible and use of reason (Latin: De ratione et rational uti )

• About hypothetical syllogisms (Latin: De syllogismis hypotheticis )
• About categorical syllogisms (Latin: De syllogismis cathegoricis )

• About syllogisms (Latin: Quid sit syllogism )
• Smaller fonts (all Latin ): Incipit de partibus Logice, Quis sit dialecticus, De difinitione philosophy

• About the grammarians (Latin: De Grammatico )

• Logic (Latin: Dialectica )

• Logic " ingredientibus " (Latin: Logica " ingredientibus " )
• Logic (Latin: Dialectica )
• Smaller fonts: Introductory Logic ( Latin: Introductiones parvulorum ), Latin: Logica " nostrorum petitioni Sociorum ", Latin: Tractatus de intellectibus, Latin: Sententiae secundum Magistrum Petrum

• Latin: Ars Meliduna

• Latin: Summa Lamberti

• Latin: Parva logicalia

• Latin: Ars magna

• Latin: Introductiones in Logicam

• Latin: Summulae Logicales

• Latin: Summa Logicae

• Latin: Summula de Dialectica
• Latin: consequentiae
• Latin: Sophismata

• Latin: De Puritate Artis Logicae

• Latin: Summa Logicae
• Latin: Perutilis Logica

• Latin: Logica Magna

• Latin: Animadversiones Aristotelicae

The Middle Ages is an important epoch in the history of logic. She was strongly influenced by - among others known about teaching the Arabic logic - logic of Aristotle. In the medieval university operation, the logic had as one of the septem artes liberales their place in the so-called " Faculty of Arts " ( facultas artium ). The study of the artes was a prerequisite for the study of all other faculties. In the Early Middle Ages ( as before 1100 ) initially oriented themselves to the encyclopaedic works of Late Antiquity (of Cassiodorus, Isidore, Martianus Capella ). Since the 12th century, the subject matter of logic then comprised three separate corpora:

• Logica vetus: As the " old logic " the collection of ancient works on logic is called, who used the medieval logicians until about 1150. For the corpus logica vetus included at least the Latin translations of the following three headings: the Isagoge of Porphyry and the Categories and De Interpretatione of Aristotle. During the 11th century, three works by Boethius came this: About the categorical syllogism, hypothetical syllogisms and About About the topical differences. Rather loosely belonged to the corpus of the logica vetus, De diffinitione of Marius Victorinus and topical agents of Cicero.
• Logica nova: The "new logic " was also based on the Aristotelian writings now available Analytica priora, Analytica posteriora, the Topics the Sophistic refutations.
• Logica moderna: In the course of medieval logic, there was also original medieval logics. In these self- creations away from the ancient models a whole range of new problems in the fields of logic and semantics have been developed and discussed in independent treatises.

Some of the specific medieval logical topics:

• The distinction between syncategorematic and kate gore matic expressions: suction. syncategorematic expressions ( each ) mean nothing in themselves, but can be kate gore matic expressions ( humans ) are added and thus exert their function ( each person ). The kate gore matic expressions are usually nouns and verbs.
• Neither the ancient nor modern logic is the doctrine of supposition known: termini (general terms how living things ) can be used in different ways in sentences. Some types of supposition: suppositio materialis: In person has 6 letters is the word man for man.
• Suppositio personaliz: In The ball has been kicked into the goal is for a particular ball ball for a single thing.
• Suppositio simplex: In The tree is a plant tree stands for the concept tree, which falls like plant under other terms.

The medieval logic was mainly borne by the theologically influenced scholastic philosophy. One can therefore speak of a " scholastic logic ", by the way - as scholasticism itself - continues even in modern times.

Modern Times

As a result of the invention of the printing press, appeared in the 16th century on first logic books that were not written in Latin. The first known today logic book on German dates from 1534, the first in Italian in 1547, the first English from 1551 and the first French out in 1555. At universities dominated in Europe until about 1700, however, continue Latin, although there is no Latin native speakers gave.

The " traditional logic "

In the 17th century, developed a type of formal logic that is still familiar today known under the name " traditional logic ". As a deputy of the early writings of this flow, the influential handbook logic of Port -Royal and the Logica Hamburgensis can be mentioned. In this early classical logic is also a ( non-formal logical ) strand, which reached its climax in Kant developed: after One began to wonder how the knowing subject ever comes to terms, statements and conclusions - that is, after the epistemological presuppositions and implications of logic.

The early modern period and non-formal logics

Generally it can be for the earlier modern philosophy a certain lack of interest in formal logic diagnose ( Descartes, Spinoza, Locke, Hume, Kant, Hegel, etc.). It was limited to the relaying of textbook knowledge; and so it is not surprising that major philosophers such as Kant and Hegel, the term "logic" in today misleading manner for the intended non-formal parts of their systems - the transcendental (Kant) and the dialectical logic ( Hegel) - used. Despite otherwise Views Kant, however, had nothing against formal ( he says "general" ) objected logic; he goes out with his transcendental logic only about this. The traditional - and he also taught - formal logic of his time also flowed in many places in his Critique of Pure Reason.

Leibniz

Significant achievements in the field of formal logic in the earlier modern period provided Gottfried Wilhelm Leibniz. He had successors (including James I Bernoulli, Gottfried Ploucquet, Lambert, Bolzano), but since most of his logical writings were not published until long after his death, he was at first no great influence on the history of logic. To mention is V.A. his early attempt to further advance the logic by means of a specially constructed logical language in which instead of real terms and propositional variables are used.

The heyday of classical logic

Only in the middle of the nineteenth century is the formal logic again wider attention, at first mainly in England. Leading the field is George Boole with the shorter treatise " The Mathematical Analysis of Logic " (1847 ) and his later masterpiece " Laws of Thought" (1854 ). Boole's idea is to regard logic as a mathematical calculus, which is restricted to the values ​​1 and 0 ( true and false ). In class symbols as algebraic operations such as addition, multiplication, etc. can be performed. In this way, Boole developed a complete system of single-digit predicate logic, which contains the syllogistic as a subsystem. At the same time Augustus De Morgan, Boole published his work " Formal Logic", 1847. De Morgan here is interested in, among other things for a generalization of syllogistic to statements of the form "Most A are B". Another logician John Venn in England, who published his book " Symbolic Logic " with the famous Venn diagrams 1881. At the logical research Ernst Schröder are also involved in America Charles Sanders Peirce and Germany.

The real breakthrough for modern logic succeeds, however, Gottlob Frege, who must be regarded as the most significant addition to Aristotle, logicians at all well. In his Begriffsschrift (1879 ) he presents for the first time in front of a full second-order logic predicate. He also developed here is the idea of a formal language and, based on the idea of ​​formal proof in which nothing " left to the divining " according to Frege's words remain. It is these ideas form an essential theoretical basis for the development of modern computer technology and computer science. Frege's work is, however, at first hardly noticed by his contemporaries; this may, inter alia, be due to its very hard to read logical notation. In both 1893 and 1903 published volumes of " basic laws of arithmetic " Frege tried to axiomatize all of mathematics in a kind of set theory. However, this system contains a contradiction ( the so-called Russell's antinomy ), as Frege must learn in a famous letter from Bertrand Russell in 1902.

Russell himself, it reserves the right to submit together with Alfred North Whitehead in Principia Mathematica (1910 ), the first consistent set-theoretic foundations of mathematics. The authors acknowledge Frege in the preface, they owed him most in " logical- analytical issues." In contrast to Frege's work, the Principia Mathematica is a resounding success. One reason for this may be, inter alia, seen in the notation used by Russell / Whitehead, which is to a large extent still common. Impetus for this notation delivered Giuseppe Peano, another important logicians of the 19th century, the Russell met in 1900 at a congress. In addition to his thoughts on the logical notation Peano is mainly for his axiomatization of number theory ( the so-called Peano axioms ) are known.

Modern

The propositional fragment of the " Principia Mathematica " is used as a starting point for the development of a whole series metalogischer terms. In his habilitation thesis in 1918 Paul Bernays shows ( building on the work of David Hilbert ) consistency, syntactic and semantic completeness and decidability and assesses the independence of the axioms (where he notes that one of the axioms actually dependent, so unnecessary, is ).

In addition to the axiomatic method of the " Principia " more calculus types are developed. 1934 Gerhard Gentzen presented his system of natural deduction and the sequent calculus. Based on this developed Evert Willem Beth 1959 tableau calculus. Again, this is Paul Lorenzen guide its dialogical logic.

The modern logic also brings the development of a semantics of predicate logic with it. An important preliminary to this is the famous Löwenheim - Skolem represents ( first proved by Leopold Löwenheim in 1915, a more general result shows Albert Thoralf Skolem 1920). Kurt Gödel proved in 1929 the completeness of first-order predicate logic ( Gödel's completeness theorem ), 1930, the incompleteness of Peano arithmetic ( Gödel's incompleteness theorem ). Alfred Tarski in 1933 formulated a theory of truth for predicate logic.

Other important events in the history of modern logic are the development of intuitionistic logic, modal logic, lambda - calculus, type theory and the logic levels ( logic higher level ). An important trend in modern logic is the development of theorem provers (see also artificial intelligence) and the application of logic in computer science through formal methods.

Logic in the non-European philosophies

Contrary to the widely spread views, there are also outside the Western philosophical traditions logical thinking based on the same fundamental laws and basic idea ( principle of contradiction, law of excluded middle, logic as the science of force close, etc ) are based and independent of the European tradition, a very have reached high levels.

India and Tibet

After a few until the 7th century BC, reaching back the precursors Nyaya Sutra form, in the 2nd century. AD was present in final form, and his comments the real beginning of Indian logic. Between 500 and 1300 the logic especially by monks of the Mahayana Buddhism was maintained, the developed their own scholasticism. The most significant logicians are Vasubandhu (4th cent.), Dignaga - (. 7th century AD) ( 480 540 AD) and Dharmakirti, and the modern period ( from 900 AD) dominated Gangeśa ( 13th cent. AD) and the Navya Nyaya - (New Nyaya, the " new logical school ").

China

The Chinese tradition of logic begins in the 5th century BC with Mozi who founded the mohistische logic. From the ancient Chinese philosophical schools of the Nine currents then dealt especially which emerged from the school of Mohism names, such as the philosopher Hui Shi, with logical questions. After penetration of Buddhism in China AD were also writings of Dignaga of Xuanzang and his colleagues translated into Chinese in the 7th century. Overall, the logic in Chinese philosophy has but despite the suggestions from India not as well developed as in Europe, India and Japan.

Japan

From the Japanese reception of Buddhism on Chinese and Indian sources AD even exceeding tradition of thought developed in the 8th century, especially in Buddhist scholasticism ( especially in the Sanron shū ), a highly differentiated and high requirements on the Indian logic.

Arabic- space

The logic in the Arab region has its classical phase in the Middle Ages. It was heavily influenced by Aristotelian logic and seemed even turn on the medieval European logic back. During the heyday of Islam built Al -Kindi ( ca.800 -873 ), Latinized Alkindus, his philosophy to mathematics at first. Al -Kindi had numerous works of Aristotle and other Greek philosophers by employees who were part of the Greek- Christian origin, translate. He is considered the first great philosopher and logician of Islam and was one of the founders of a mathematical way of thinking in philosophy. Other main representatives were Abu Nasr al -Farabi ( ca.870 -950 ), Avicenna ( 980-1037 ) and Averroes ( 1126-1198 ).