LOGIC, HISTORY OF

In 1900 logic was poised on the brink of the most active period in its history. The late 19th-century work of Frege, Peano, and Cantor, as well as Peirce's and Schrder's extensions of Boole's insights, had broken new ground, raised considerable interest, established international lines of communication, and formed a new alliance between logic and mathematics. Five projects internal to late 19th-century logic coalesced in the early 20th century, especially in works such as Russell and Whitehead's Principia Mathematica. These were the development of a consistent set or property theory (originating in the work of Cantor and Frege), the application of the axiomatic method (including non-symbolically), the development of quantificational logic, and the use of logic to understand mathematical objects and the nature of mathematical proof. The five projects were unified by a general effort to use symbolic techniques, sometimes called mathematical, or formal, techniques. Logic became increasingly mathematical, then, in two senses. First, it attempted to use symbolic methods like those that had come to dominate mathematics. Second, an often dominant purpose of logic came to be its use as a tool for understanding the nature of mathematicssuch as in defining mathematical concepts, precisely characterizing mathematical systems, or describing the nature of ideal mathematical proof. (See mathematics, history of: Mathematics in the 19th and 20th centuries, and mathematics, foundations of.) Russell and Whitehead's Principia Mathematica The three-volume Principia Mathematica (191013) was optimistically named after the Philosophiae naturalis principia mathematica of another hugely important Cambridge thinker, Isaac Newton. Like Newton's Principia, it was imbued with an optimism about the application of mathematical techniques, this time not to physics but to logic and to mathematics itselfwhat the first sentence of their preface calls the mathematical treatment of the principles of mathematics. It was intended by Russell and Whitehead both as a summary of then-recent work in logic (especially by Frege, Cantor and Peano) and as a ground-breaking, large-scale treatise systematically developing mathematical logic and deriving basic mathematical principles from the principles of logic alone. The Principia was the natural outcome of Russell's earlier polemical book, The Principles of Mathematics (published in 1903 but largely written in 1900), and his views were later summarized in Introduction to Mathematical Philosophy (1919). Whitehead's A Treatise on Universal Algebra (1898) was more in the algebraic tradition of Boole, Peirce, and Schrder, but there is a sense in which Principia Mathematica became the second volume both of it and of Russell's Principles. The main idea in the Principia is the view, taken from Frege, that all of mathematics could be derived from the principles of logic alone. This view later came to be known as logicism and was one of the principal philosophies of mathematics in the early 20th century. Number theory, the core of mathematics, was organized around the Peano postulates, stated in works by Peano of 1889 and 1895 (and anticipated by similar but less influential theories of Peirce and Dedekind). These postulates state and organize the fundamental laws of natural (integral, positive) numbers, and thus of all of mathematics: 0 is a number. The successor of any number is also a number. No two distinct numbers have the same successor. 0 is not the successor of any number. If any property is possessed by 0 and also by the successor of any number having the property, then all numbers have that property. If some entities satisfying these conditions could be derived or constructed in logic, it would have been shown that mathematics was (or at least could be) founded in pure logic, requiring no additional assumptions. Although his language actually used the intensional and second-order language of functions and properties, Frege had claimed to have accomplished precisely this, identifying 0 with the empty set, 1 with the set of all single-membered sets (singletons), 2 with the set of all dual-membered sets (doubletons), and so on. These sets of equinumerous sets were then what numbers really were. Unfortunately, Russell showed through his famous paradox that the theory is inconsistent and, hence, that any statement at all can be derived in Frege's system, not merely desired logical truths, the Peano postulates, and what follows from them. Russell, in a famous letter to Frege, asked him to consider the set of all those sets not members of themselves. Paradox follows if one assumes such a set is empty, or is not empty. After meditating on this paradox and a great many other paradoxes devised by Burali-Forti, George Godfrey Berry, and others, Russell and Whitehead concluded that the main difficulty lies in allowing the construction of entities that contain a vicious circlei.e., entities that are used in the construction or definition of themselves. Russell and Whitehead sought to rule out this possibility while at the same time allowing a great many of the operations that Frege had deemed desirable. The result was the theory of types: all sets and other entities have a logical type, and sets are always constructed from specifying members with lower types. (F.P. Ramsay offered a criticism that was subsequently accommodated in later editions of Principia Mathematica; as modified, the theory came to be known as the ramified theory of types.) Consequently, to speak of sets that are, or are not, members of themselves is simply to violate this rule governing the specification of sets. There is some evidence that Cantor had been aware of the difficulties created when there is no such restriction (he permitted large collective entities that do not obey the usual rules for sets), and a parallel intuition concerning the pitfalls of certain operations was independently followed by Ernst Zermelo in the development of his set theory. In addition to its notation (much of it borrowed from Peano), its masterful development of logical systems for propositional and predicate logic, and its overcoming of difficulties that had beset earlier logical theories and logistic conceptions, the Principia offered discussions of functions, definite descriptions, truth, and logical laws that had a deep influence on discussions in analytical philosophy and logic throughout the 20th century. What is perhaps missing is any hesitation or perplexity about the limits of logic: whether this logic is, for example, provably consistent, complete, or decidable, or whether there are concepts expressible in natural languages but not in this logical notation. This is somewhat odd, given the well-known list of problems posed by Hilbert in 1900 that came to animate 20th-century logic, especially German logic. The Principia is a work of confidence and mastery and not of open problems and possible difficulties and shortcomings; it is a work closer to the naive progressive elements of the Jahrhundertwende than to the agonizing fin de sicle. the history of the discipline from its origins among the ancient Greeks to the present time. Additional reading A broad survey of the history of logic is found in William Kneale and Martha Kneale, The Development of Logic (1962, reprinted 1984), covering ancient, medieval, modern, and contemporary periods. Articles on particular authors and topics are found in The Encyclopedia of Philosophy, ed. by Paul Edwards, 8 vol. (1967); and New Catholic Encyclopedia, 18 vol. (196789). I.M. Bochenski, Ancient Formal Logic (1951, reprinted 1968), is an overview of early Greek developments. On Aristotle, see Jan Lukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, 2nd ed., enlarged (1957, reprinted 1987); Gnther Patzig, Aristotle's Theory of the Syllogism (1968; originally published in German, 2nd ed., 1959); Otto A. Bird, Syllogistic and Its Extensions (1964); and Storrs McCall, Aristotle's Modal Syllogisms (1963). I.M. Bochenski, La Logique de Thophraste (1947, reprinted 1987), is the definitive study of Theophrastus' logic. On Stoic logic, see Benson Mates, Stoic Logic (1953, reprinted 1973); and Michael Frede, Die stoische Logik (1974).Detailed treatment of medieval logic is found in Norman Kretzmann, Anthony Kenny, and Jan Pinborg (eds.), The Cambridge History of Later Medieval Philosophy: From the Rediscovery of Aristotle to the Disintegration of Scholasticism, 11001600 (1982); and translations of important texts of the period are presented in Norman Kretzmann and Eleonore Stump (eds.), Logic and the Philosophy of Language (1988). For Boethius, see Margaret Gibson (ed.), Boethius, His Life, Thought, and Influence (1981); and, for Arabic logic, Nicholas Rescher, The Development of Arabic Logic (1964). L.M. de Rijk, Logica Modernorum: A Contribution to the History of Early Terminist Logic, 2 vol. in 3 (19621967), is a classic study of 12th- and early 13th-century logic, with full texts of many important works. Norman Kretzmann (ed.), Meaning and Inference in Medieval Philosophy (1988), is a collection of topical studies.A broad survey of modern logic is found in Wilhelm Risse, Die Logik der Neuzeit, 2 vol. (196470). See also Robert Adamson, A Short History of Logic (1911, reprinted 1965); C.I. Lewis, A Survey of Symbolic Logic (1918, reissued 1960); Jrgen Jrgensen, A Treatise of Formal Logic: Its Evolution and Main Branches with Its Relations to Mathematics and Philosophy, 3 vol. (1931, reissued 1962); Alonzo Church, Introduction to Mathematical Logic (1956); I.M. Bochenski, A History of Formal Logic, 2nd ed. (1970; originally published in German, 1962); Heinrich Scholz, Concise History of Logic (1961; originally published in German, 1959); Alice M. Hilton, Logic, Computing Machines, and Automation (1963); N.I. Styazhkin, History of Mathematical Logic from Leibniz to Peano (1969; originally published in Russian, 1964); Carl B. Boyer, A History of Mathematics, 2nd ed., rev. by Uta C. Merzbach (1991); E.M. Barth, The Logic of the Articles in Traditional Philosophy: A Contribution to the Study of Conceptual Structures (1974; originally published in Dutch, 1971); Martin Gardner, Logic Machines and Diagrams, 2nd ed. (1982); and E.J. Ashworth, Studies in Post-Medieval Semantics (1985).Developments in the science of logic in the 20th century are reflected mostly in periodical literature. See Warren D. Goldfarb, Logic in the Twenties: The Nature of the Quantifier, The Journal of Symbolic Logic 44:351368 (September 1979); R.L. Vaught, Model Theory Before 1945, and C.C. Chang, Model Theory 19451971, both in Leon Henkin et al. (eds.), Proceedings of the Tarski Symposium (1974), pp. 153172 and 173186, respectively; and Ian Hacking, What is Logic? The Journal of Philosophy 76:285319 (June 1979). Other journals devoted to the subject include History and Philosophy of Logic (biannual); Notre Dame Journal of Formal Logic (quarterly); and Modern Logic (quarterly). Modern logic It is customary to speak of logic since the Renaissance as modern logic. This is not to suggest that there was a smooth development of a unified conception of reasoning, or that the logic of this period is modern in the usual sense. Logic in the modern era has exhibited an extreme diversity, and its chaotic development has reflected all too clearly the surrounding political and intellectual turmoil. These upheavals include the Renaissance itself, the diminishing role of the Roman Catholic church and of Latin, the Reformation and subsequent religious wars, the scientific revolution and the growth of modern mathematics, the rise and fall of empires and nation-states, and the waxing influence of the New World and the former Soviet Union. The 16th century Renaissance writers sometimes denounced all of scholastic logic. The humanism of the Renaissance is often seen as promoting the study of Greek and Roman classics, but Aristotle's logic was frequently regarded as being so hopelessly bound together with sterile medieval logic as to constitute an exception to this spirit of rebirth. Some, such as Martin Luther (14831546), were repelled by any hint of Aristotelianism. Others, such as the great humanist essayist Desiderius Erasmus (14661536), occasionally praised Aristotle but never his logical theory; like many writers in the Renaissance, Erasmus found in the theory of the syllogism only subtlety and arid ingenuity (Johan Huizinga, Erasmus ). The German Lutheran humanist Philipp Melanchthon (14971560) had a more balanced appreciation of Aristotle's logic. Melanchthon's Compendaria dialectices ratio (Brief Outline of Dialects) of 1520, built upon his Institutiones Rhetoricae of the previous year, became a popular Lutheran text. There he described his purpose as presenting a true, pure and uncomplicated logic, just as we have received it from Aristotle and some of his judicious commentators. Elsewhere, influential writers such as Rabalais, Petrarch, and Montaigne had few kind words for logic as they knew it. The French reformer and pamphleteer Petrus Ramus (Pierre de la Rame) was also the author of extremely influential Reform logical texts. His Dialectique (Dialectics) of 1555 (translated into English in 1574) was the first major logical work in a modern language. In this work and in his Dialecticae libri duo (Two Books of Dialectics) of 1556 he combined attacks on scholastic logic, an emphasis on the use of logic in actual arguments (dialectics), and a presentation of a much simplified approach to categorical syllogism (without an attempt to follow Aristotle). Elsewhere, he proposed that reasoning should be taught by using Euclid's Elements rather than by the study of the syllogism. He devoted special attention to valid syllogisms with singular premises, such as Octavius is the heir of Caesar. I am Octavius. Therefore, I am the heir of Caesar. Singular terms (such as proper names) had been treated by earlier logicians: Pseudo-Scotus, among others, had proposed assimilating them to universal propositions by understanding Julius Caesar is mortal as All Julius Caesars are mortal. Although Ramus' proposals for singular terms were not widely accepted, his concern for explicitly addressing them and his refusal to use artificial techniques to convert them to standard forms prefigured more recent interests. Although it had its precursors in medieval semantic thought, Ramus' division of thought into a hierarchy composed of concepts, judgments, arguments, and method was influential in the 17th and 18th centuries. Representations of the universal affirmative, All A's are B's in modern logic. Scholastic logic remained alive, especially in predominantly Roman Catholic universities and countries, such as Italy and Spain. Some of this work had considerable value, even though it was outside of the mainstream logical tradition, from which it diverged in the 16th century. If the Reform tradition of Melanchthon and Ramus represents one major tradition in modern logic, and the neo-scholastic tradition another, then (here following the historian of logic Nicholai Ivanovich Styazhkin) a third tradition is found in the followers of the Spanish (Majorcan) soldier, priest, missionary, and mystic Ramn Lull (12351315). His Ars magna, generalis et ultima (1501; Great, General and Ultimate Art) represents an attempt to symbolize concepts and derive propositions that form various combinations of possibilities. These notions, associated with lore of the Kabbala, later influenced Pascal and Leibniz and the rise of probability theory. Lull's influence can be seen more directly in the work of his fellow Spaniard Juan Luis Vives (14921540), who used a V-shaped symbol to indicate the inclusion of one term in another (see illustration). Other work inspired by Lull includes the logic and notational system of the German logician Johann Heinrich Alsted (15881638). The work of Vives and Alsted represents perhaps the first systematic effort at a logical symbolism. With the 17th century came increasing interest in symbolizing logic. These symbolizations sometimes took graphic or pictorial forms but more often used letters in the manner of algebra to stand for propositions, concepts, classes, properties, and relations, as well as special symbols for logical notions. Inspired by the triumphs achieved in mathematics after it had turned to the systematic use of special symbols, logicians hoped to imitate this success. The systematic application of symbols and abbreviations and the conscious hope that through this application great progress could be made have been a distinguishing characteristic of modern logic into the 20th century. The modern era saw major changes not only in the external appearance of logical writings but also in the purposes of logic. Logic for Aristotle was a theory of ideal human reasoning and inference that also had clear pedagogical value. Early modern logicians stressed what they called dialectics (or rhetoric), because logic had come to mean an elaborate scholastic theory of reasoning that was not always directed toward improving reasoning. A related goal was to extend the scope of human reasoning beyond textbook syllogistic theory and to acknowledge that there were important kinds of valid inference that could not be formulated in traditional Aristotelian syllogistic. But another part of the rejection of Aristotelian logic (broadly conceived to include scholastic logic) is best explained by the changing and quite new goals that logic took on in the modern era. One such goal was the development of an ideal logical language that naturally expressed ideal thought and was more precise than natural languages. Another goal was to develop methods of thinking and discovery that would accelerate or improve human thought or would allow its replacement by mechanical devices. Whereas Aristotelian logic had seen itself as a tool for training natural abilities at reasoning, later logics proposed vastly improving meagre and wavering human tendencies and abilities. The linking of logic with mathematics was an especially characteristic theme in the modern era. Finally, in the modern era came an intense consciousness of the importance of logical form (forms of sentences, as well as forms or patterns of arguments). Although the medievals made many distinctions among patterns of sentences and arguments, the modern logical notion of form perhaps first crystallized in the work of Sir William Rowan Hamilton and the English mathematician and logician Augustus De Morgan (De Morgan's Formal Logic of 1847). The now standard discussions of validity, invalidity, and the self-conscious separation of formal from nonformal aspects of sentences and arguments all trace their roots to this work.