History of logic

The history of logic documents the development of logic as it occurs in various cultures and traditions in history. While many cultures have employed intricate systems of reasoning, logic as an explicit analysis of the methods of reasoning received sustained development originally only in three traditions: China, India and Greece. Although exact dates are uncertain, especially in the case of India, it is possible that logic emerged in all three societies in the fourth century B.C.E. The notions of systems of reasoning and logic, however, are sufficiently imprecise that various answers to the questions of what they are and how they are to be understood have been given. The formally sophisticated treatment of modern logic descends from the Greek tradition, but comes not wholly through Europe, but instead comes from the transmission of Aristotelian logic and commentary upon it by Islamic philosophers to logicians in Medieval Europe.

Logic in China

In China, a contemporary of Confucius, Mozi, "Master Mo," is credited with founding the Mohist school, whose canons dealt with issues relating to valid inference and the conditions of correct conclusions. In particular, one of the schools that grew out of Mohism, the Logicians, are credited by some scholars for their early investigation of formal logic. Unfortunately, due to the harsh rule of Legalism in the subsequent Qin Dynasty, this line of investigation disappeared in China until the introduction of Indian philosophy by Buddhists.

At least one commentator has noted that Chinese logic seems to be based on coherence and analogy, usually consisting of a series of picturesque metaphors, parables, and anecdotes strung together to illustrate certain main ideas. This results in making Chinese philosophy more poetic than logical, at least as logic is understood in Western thought. "Chinese thought tries to bring emotional rather than intellectual conviction and its main appeal is to the heart rather than to the mind." (Hansen, "Language and Logic in Ancient China")

Logic in India

Two of the six Indian schools of thought deal with logic: Nyaya 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-materialist, school worked out a rigid five-member schema of inference involving an initial premise, a reason, an example, an application and a conclusion.

The idealist Buddhist philosophy became the chief opponent to the Naiyayikas. Nagarjuna, the founder of the Madhyamika "Middle Way" developed an analysis known as the "catuskoti" or tetralemma. This four-cornered argumentation systematically examined and rejected the affirmation of a proposition, its denial, the joint affirmation and denial, and finally, the rejection of its affirmation and denial. But it was with Dignaga and his successor Dharmakirti that Buddhist logic reached its height. Their analysis centered on the definition of necessary logical entailment, "vyapti," also known as invariable concomitance or pervasion. To this end a doctrine known as "apoha" or differentiation was developed. This involved what might be called inclusion and exclusion of defining properties. The difficulties involved in this enterprise, in part, stimulated the neo-scholastic school of Navya-Nyaya, which introduced a formal analysis of inference in the sixteenth century.

Logic in Greece

In Greece, two main competing logical traditions emerged. Stoic logic traced its roots back to Euclid of Megara (c. 430 - c. 360 B.C.E.), a pupil of Socrates, and with its concentration on propositional logic was perhaps closer to modern logic. The Megarians were interested in puzzles, and studied modality and conditionals. The Stoics used numbers as variables for replacing whole propositions. The most important Stoic logician was Chrysippus (c. 279 - 206 B.C.E.), who discussed five basic or valid inference schemata, and from them derived or proved many other valid inference schemata.

There was also a Medieval tradition that held that the Greek philosopher Parmenides, in fifth century B.C.E., invented logic while living on a rock in Egypt. In any case, his disciple, Zeno of Elea did produce many supposedly logical arguments, known as Zeno's paradoxes. These were given in support of Parmenides' philosophy—a philosophy that denied motion and multiplicity—and purported to show that a non-Parmenidean view leads to absurdity. This method of proving something by assuming its alternative and showing that this assumption leads to absurdity is known as reduction ad absurdum and Zeno's use of it suggests that he knew of the general pattern of such argument. Zeno's paradoxes do, however, all contain fatal mistakes, but showing what the mistakes are often required waiting until much later developments in logic and mathematical logic.

The Greek tradition that survived to influence later cultures, however, was the Peripatetic tradition which originated in Aristotle's collection of works known as the "Organon" or instrument, the first systematic Greek work on logic. In fact, Aristotle is often called the first great logician. Although he did not use these terms himself, Aristotle introduced the formal study of what is now known as formal logic, that is; logic that is concerned with the form, not the content, of statements or propositions, and the relationships that exist between different statements on the basis of their form—some statements being accepted (as premises), other statement(s) follow (as conclusion(s)) from those accepted statements because of their form.

Aristotle held that a proposition involves two terms, a subject and a predicate. Propositions can be universal ("all," "no") or particular ("some"), and affirmative or negative. Aristotle's formal logic was confined to examination of syllogisms, which consist of three propositions. The first two are the premises, and must share only one term. The third proposition is the conclusion, which contains the two terms that are not shared by the premises. Aristotle also investigated how the common term (shared by the two premises) can occur and the effects of its different ways of occurring. Aristotle's work on syllogisms bears interesting comparison with the Indian schema of inference and the less rigid Chinese discussion.

Aristotle also formulated certain theses about logic (sometimes called metalogical principles): The Law of Noncontradiction, the Principle of the Excluded Middle, and the Law of Bivalence. In addition he investigated some of what are now known as informal fallacies, fallacies that occur for some reason other than the form of the argument, such as argumetum ad hominem, and appeal to the crowd.

Aristotle's successor as head of his school, Theophrastus of Eresus (c. 371 - c. 286 B.C.E.), carried on Aristotle's investigations of logic and added to them.

Through Latin in Western Europe, and disparate languages more to the East, such as Arabic, Armenian and Georgian, the Aristotelian tradition was considered to codify pre-eminently the laws of reasoning. It was only in the nineteenth century that this viewpoint changed; it has been suggested by a few commentators that this change may have been facilitated by an acquaintance with the classical literature of India and deeper knowledge of China.

Except for what was done in the Arabic world, there was little work in logic between that of Boethius (480 - 524 or 525 C.E.) and Peter Abelard (1079-1142) in the twelfth century.

Logic in Islamic philosophy

After Muhammed's death, Islamic law placed importance on formulating standards of argument, which gave rise to a novel approach to argumentation in Kalam, but this approach was displaced by ideas from Greek philosophy with the rise of the Mutazilite philosophers, who valued highly Aristotle's Organon. The work of Greek-influenced Islamic philosophers was crucial in the reception of Greek logic in medieval Europe, and the commentaries on the Organon by Averroes played a central role in the subsequent flowering of medieval European logic.

Despite the logical sophistication of Al-Ghazali, the rise of the Asharite school slowly suffocated original work on logic in the Islamic world.

Medieval Logic

Medieval Logic (also known as Scholastic Logic) generally means the form of Aristotelian logic developed in medieval Occident throughout the period c. 1200-1600. The first great medieval logician was Peter Abelard, who wrote commentaries on Aristotle's work on logic. Among other things, Abelard wrote on the role of the copula in categorical propositions ("all" or "none"), the effects of placing the negation sign in different positions, modal notions such as "possible," and conditional propositions (if___ then … ).

During the medieval period mnemonic names were created for the valid moods of the syllogism that had been discussed in Aristotle's Prior Analytics. Two of those moods were BARBARA, in which the three propositions of the syllogism consist entirely of universal affirmatives, and CELARENT, in which one premise is a universal negative, the other a universal affirmative, and the conclusion is a universal negative. Medieval logicians also investigated modal logic.

Logic in the medieval period was developed through textbooks such as that by Peter of Spain in the thirteenth century, but whose exact identity is unknown, who was the author of a standard textbook on logic, the Tractatus which was well known in Europe for many centuries. This tradition of medieval logic reached a high point in the fourteenth century, with the works of William of Ockham (c. 1287-1347) and Jean Buridan.

One feature was the development of Aristotelian logic through what is known as Supposition Theory, a study of the semantics of the terms of the proposition, or theory of reference (in general) and theory or personal reference. The last great works in this tradition are the Logic of John Poinsot (1589-1644, known as John of St Thomas), and the Metaphysical Disputations of Francisco Suarez (1548-1617).

In the sixteenth century, however, what we now know as logic was largely displaced by interest in and study of dialectic. Thus the three works of Philip Melanchthon (1497-1560), Compendiaria dialectics ratio (1520), Dialectics libri quattuor (1528), and Erotemata dialectics (1547) each carried the term dialectics in its title, instead of logic, and the same was true of works by Petrus Ramus (1515-1572) and the scholar known as the Portugese Aristotle, Petrus Fonseca, S.J., whose Institutionum dialecticarum libri octo first appeared in 1564.

In the eighteenth century, there was a return to the use of the term "logic." Christoph Scheibler (1589-1653), known as the Protestant Suarez, published an encyclopedic book Opus Logicum in Marburg, Germany, in 1633. Other books with the term "logic" in their titles appeared, such as Logica Hamburgensis in 1638 from Joachim Jungius (1587-1657), Logica vetus et nova (1654) by the German Cartesian Johannes Clauberg (1622-1655), and some others. The most notable and important work of this era was the Port Royal Logic.

Traditional Logic

What has become known as traditional logic generally means the textbook tradition that begins with Antoine Arnauld and Pierre Nicole's Logic, or the Art of Thinking, better known as the Port-Royal Logic. Published in 1662, it was the most influential work on logic in England until John Stuart Mill's System of Logic in 1825. The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather than terms, for example) within a framework that is broadly derived from Aristotelian and medieval term logic. Between 1664 and 1700 there were eight editions, and the book had considerable influence after that. It was frequently reprinted in English up to the end of the nineteenth century.

The account of propositions that John Locke gave in the Essay is essentially that of Port-Royal. "Verbal propositions, which are words, [are] the signs of our ideas, put together or separated in affirmative or negative sentences. So that proposition consists in the putting together or separating these signs, according as the things which they stand for agree or disagree." (Locke, An Essay Concerning Human Understanding, IV. 5. 6)

Works in this tradition include Isaac Watts' Logick: Or, the Right Use of Reason (1725), Richard Whately's Logic (1826), and John Stuart Mill's A System of Logic (1843), which was one of the last great works in that tradition.

The Transition to Modern Logic

The idea of a calculus of reasoning was cultivated by Gottfried Wilhelm Leibniz, who was the first to formulate the notion of a broadly applicable system of mathematical logic. However, the relevant documents were not published until 1901 or remain unpublished to the present day, and the current understanding of the power of Leibniz's discoveries did not emerge until the 1980s. [See Lenzen's chapter in Gabbay and Woods (2004)].

John Venn 1834-1923, was a Cambridge logician who published three standard texts in logic, The Logic of Chance 1866, Symbolic Logic 1881, and The Principles of Empirical Logic 1889. Today he is remembered mostly for his logical diagrams, known as Venn diagrams, used for representing syllogisms. He was not the originator of using geometrical representations to illustrate syllogistic logic; Leibniz had often used such methods. Venn became critical of the diagrams used in the nineteenth century, especially those of logicians George Boole 1815-1864, and Augustus de Morgan 1806-1871. Boole was the inventor of what is now known as Boolean algebra, which is the basis of all modern computer arithmetic; he is regarded as being one of the founders of the field of computer science, although computers did not exist in his day. De Morgan was an Indian-born British mathematician and logician who formulated what are now known as De Morgan's laws and was the first to introduce the term mathematical induction and make rigorous the idea. Venn wrote the book Symbolic Logic to interpret and make his corrections on Boole's work. Prior to publishing this book, Venn wrote a paper entitled "On the Diagrammatic and Mechanical Representation of Prepositions and Reasonings" introducing Venn diagrams. This paper was published in the Philosophical Magazine and Journal of Science in July, 1880. In Symbollic Logic, Venn further elaborated on these diagrams, and they became the most important part of his work.

In an 1885 article read by Giuseppe Peano, Ernst Schröder, and others, Charles Sanders Peirce introduced the term second-order logic and provided us with much of our modern logical notation, including prefixed symbols for universal and existential quantification. Logicians in the late nineteenth and early twentieth centuries were thus more familiar with the Peirce-Schröder system of logic, although Frege is generally recognized today as being the "Father of modern logic."

Modern Logic

Near the end of the nineteenth century three overlapping traditions in the development of logic emerged. One originated with the work of Boole and includes the work of Peirce, Jevons, Schröoder, and Venn. This can be called the algebraic school; its work focused on regularities in correct reasoning and on operations such as addition and subtraction. This work begins with a group of related operations and then finds within them a common abstract structure. It then formulated a set of axioms that is satisfied by each of those systems.

A second tradition could be called the logicist school. It attempted to codify the underlying logic of all scientific discourse into one single system. In this view logic is concerned with the most general or abstract features of precise discourse, apart from the subject-matter of that discourse. The major members of this school were Bertrand Russell and Alfred North Whitehead (in their monumental work Principia Mathematica), the early Ludwig Wittgenstein, and Gottlob Frege (1848-1925).

Frege has been called the greatest logician since Aristotle. His work was the foundation or beginning point for an enormous outpouring of work in formal logic, beginning at the end of the nineteenth and continuing into the twentieth century. Frege's 1879 Begriffsschrift developed a formal language with mathematical rigor. He extended formal logic beyond propositional logic to include constructors such as "all," and "some." He showed how to introduce variables and quantifiers to reveal the logical structure of sentences, which may have been obscured by their grammatical structure. For instance, "All humans are mortal" becomes "All things x are such that, if x is a human then x is mortal." Frege's peculiar two dimensional notation led to his work being ignored for many years. Frege held that arithmetic and analysis are parts of logic; this was at least partly a negative reply to Kant's claim that arithmetic is synthetic a priori. Ultimately, however, Frege's system was found to be inconsistent (because Russell's Paradox could be derived within Frege's system), and various responses were made in an attempt to recapture the logicist program and avoid the inconsistency. The first of those being Russell and Whitehead's Principia Mathematica, which used a theory of types (membership in any set was restricted to only certain types of things).

The third tradition can be called the mathematical school. [See the article Mathematical Logic.] This tradition or school includes the work of Richard Dedekind (1831-1916), Giuseppe Peano (1858-1932), David Hilbert (1862-1943), Ernst Zermelo (1871-1953), and many others since then. Its goal was the axiomatization of particular branches of mathematics, including geometry, arithmetic, analysis, and set theory. This school continues to this day, with considerable activity still occurring in it.

In 1889 Peano published the first version of the logical axiomatization of arithmetic. Five of the nine axioms he came up with are now known as the Peano axioms. One of these axioms was a formalized statement of the principle of mathematical induction.

Ernst Zermelo's axiomatic set theory was another attempt to escape Russell's Paradox. Its axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory, with the axiom of choice, commonly abbreviated ZFC.

In Poland, under Jan Łukasiewicz (1878-1956) there was a variation on the mathematical school: logic became the branch of mathematics that was to be brought within the axiomatic methodology. Łukasiewicz worked on multi-valued logics; his three-valued propositional calculus, introduced in 1917, was the first explicitly axiomatized non-classical logical calculus. He is responsible for one of the most elegant axiomatizations of classical propositional logic; it has just three axioms and is one of the most used axiomatizations today.

The discovery of non-Euclidian geometry spurred mathematicians to consider alternative interpretations of their mathematical languages and to consider metalogical questions about their systems. Those metalogical or metamethematical questions included those of the independence, consistency, categoricity, and completeness of axiomatic systems.

This intensive work on metamathematical issues culminated in the work of Kurt Gödel (1906-1978), a logician of the caliber of Aristotle and Frege. He proved a number of important metamathematical statements, including his most famous, the incompleteness theorem which shows that for axiomatizations of sufficient richness for arithmetic, there is a sentence which is neither provable nor refutable within that axiomatic system.

Gödel was also one of the central figures in the study of computability. Others included Alonzo Church (1903-1995), Alan Turing (1912-1954), and others. Church proved that Peano arithmetic and first-order logic are undecidable. The latter result is known as Church's theorem. Turing is often considered to be the father of modern computer science. He provided an influential formalization of the concept of the algorithm and computation with the Turing machine, formulating the now widely accepted "Turing" version of the Church–Turing thesis, namely that any practical computing model has either the equivalent or a subset of the capabilities of a Turing machine.

Mathematical logic has come to be a central part of contemporary analytic philosophy, especially with the work of Willard Van Orman Quine, Saul Kripke, Donald Davidson, and Michael Dummet. Some of the topics covered have been modal logic, tense logic, many-valued logic, deontic logic, relevance logic, and non-standard logic.

Logic and Philosophy

The history of logic cannot be separated from general philosophy and the philosophy of logic because the philosophical point of view that is adopted and the conclusions reached will determine, at least to a large extent, what is comprehended under or taken to count as logic.

Throughout the history of western philosophy what is called logic has included, in addition to the formal logic discussed above, the Transcendental Logic of Immanuel Kant (1724-1804), and the dialectical logic of Johann Gottlieb Fichte (1762-1814), Friedrich Wilhelm Joseph Schelling (1775-1854), and especially G.W.F. Hegel (1770-1831). There has also been the materialist dialectic logic of Karl Marx (1818-1883), and psychologistic logic of such figures as Wilhelm Wundt (1832-1920) and others. There has also been the phenomenology of Edmund Husserl (1859-1938) and his followers, including Martin Heidegger (1889-1976), and Jean-Paul Sartre (1905-1980), the deconstructionism of Jacques Derrida (1930-2004) and others, and other outgrowths of Continental philosophy.

Another major topic of enormous discussion and disagreement in Western philosophy, at least since the time of David Hume (1711-1776) and his devastating critiques of it, is the existence and status of supposed "inductive logic." The problem of induction arises because all inductive inferences are, technically speaking, invalid because the premises of an inductive argument can all be true and the conclusion nevertheless be false. Yet the sciences seem to require or rely on inductive logic and methods. There has been a great deal of work on supposed methods of inductive logic, including John Stuart Mill's Mill's Methods, Charles Sanders Peirce's account of inductive logic, and the work of Rudolf Carnap and many others, especially the proponents of logical positivism, who seemed to need an inductive procedure in order to work out their program. Karl Popper, however, claimed that he had solved the problem of induction by discarding it in favor of his method of falsification. This controversy about whether there is any inductive logic, and if so how it is to be understood and accounted for, continues.

In addition to those, there is today what is often known as fuzzy logic, or deviant logic, advocated by Susan Haack and others. This movement prizes vagueness, among other things, and is based, at least partly, on quantum mechanics, which seems to defy classical logic. This movement also owes a great deal to Quine and his famous paper "Two Dogmas of Empiricism," in which he suggested, by implication if not directly, that even the supposed laws of logic are subject to pragmatic considerations, and change if necessary.

References
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Church, Alonzo, "A Bibliography of Symbolic Logic," Journal of Symbolic Logic 1: 121-218; 3:178-212, 1936-38.
Dumitriu, Anton, History of Logic, Volume III, Revised and updated and enlarged translation of the Roumanian work, Tunbridge Wells, Kent, UK: Abacus Press, 1977. ISBN 0856261424
Gabbay, Dov, and John Woods, (eds) Handbook of the History of Logic. Vol. 1: Greek, Indian and Arabic logic; Vol. 3: The Rise of Modern Logic I: Leibniz to Frege. Amsterdam & Boston: Elsevier, 2004. ISBN 0444515968 (set)
Grattan-Guinness, Ivor. The Search for Mathematical Roots 1870-1940. Princeton, NJ: Princeton University Press, 2000. ISBN 9780691058580
Haack, Susan. Deviant Logic, Fuzzy Logic. Chicago & London: University of Chicago Press, 1996. ISBN 0226311333
Hansen, Chad. "Language and Logic in Ancient China". Retrieved December 16, 2007.
Kneale, William and Martha Kneale. The Development of Logic. New York: Oxford University Press, [1962] 1985. ISBN 0198247737
Putnam, Hilary, Philosophy of Logic. New York, Harper & Row, 1971. ISBN 0061360422
Quine, W.V., Philosophy of Logic. Cambridge, Mass.: Harvard University Press, 1986. ISBN 0674665635
Van Heijenoort, Jean. Selected Essays: History of Logic 3. Napoli: Bibliopolis, 1985. ISBN 8870881229

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