LOGIC.

Logic is the study of correct reasoning. A host of philosophical themes have clustered around this central concern: the nature of truth and validity, of possibility and necessity; the semantics of words, sentences, and arguments; and even questions about substances and accidents, free will and determinism.

Aristotle

Aristotle (384–322 b.c.e.) was the first person to formulate an explicit theory of correct reasoning, as he himself claimed in Sophistical Refutations. He owed a good deal to the exploration into forms of argument carried out in the course of argument contests, such as those illustrated by Plato in some of his Socratic dialogues. Book 8 of his own Topics reads like a handbook for contestants, and the Topics as a whole is designed to teach its readers how to construct "dialectical" arguments: arguments that, in keeping with the idea of a real contest, use generally accepted premises that will be granted by the interlocutor. In an argument, says Aristotle, "when certain things have been laid down, something other than what has been laid down necessarily results from them." This definition captures the idea of logical consequence, and in his Prior Analytics Aristotle develops his "syllogistic," a formal theory of logical consequence, which he applies to "demonstrations," arguments in which the premises must not be merely accepted, but true.

Syllogisms (in the narrow sense considered in the Prior Analytics ) consist of three assertoric sentences, two of them premises, from which the third, the conclusion, follows. In an assertoric sentence, something is "predicated" of a subject, and a predicate can stand in one of just five relations to a subject: it may be its definition, its genus ("Man is an animal"), its differentia (the element of the definition that differentiates things of one species from another: "Man is rational"), an accident (a characteristic the particular thing happens to have: "Socrates is curly-haired"), or its characteristic property (a feature that all and only things of the subject's species have, but is not part of its definition: "Man is able to laugh").

The two premises of a syllogism share a common ("middle") term, and they have "quantity" (universal/particular) and "quality" (affirmative/negative). They may be, then, universal affirmative (A-sentences: "Y belongs to every X"), universal negative (E-sentences: "Y belongs to no X"), particular affirmative (I-sentences: "Y belongs to some X"), or particular negative (O-sentences: "Y does not belong to some X"). Depending on the position of the middle term—predicate of both premises ("third figure"), subject of both premises ("second figure"), or subject of the first, predicate of the second ("first figure")—from some combinations of two A, E, I, and O sentences as premises, there follows an A, E, I, or O sentence as a conclusion—and this conclusion follows purely in virtue of the form of the argument. (Although ancient logicians rarely used false premises as their examples, they too made their conclusions follow logically.) So, for example, in the first figure, the patterns AAA, EAE, AII, and EIO are valid arguments. First-figure syllogisms were held by Aristotle to be self-evident: for example, Mortal belongs to every man (every man is mortal); man belongs to every philosopher; thus mortal belongs to every philosopher. Aristotle also shows how second-and third-figure syllogisms can be reduced to first-figure ones, using a set of conversion rules.

Aristotle's other logical works both fill in the discussions in the Topics and the Prior Analytics and introduce new philosophical dimensions. On Interpretation discusses assertoric statements and their relations (such as contradiction and contrariety). It also proposes a basic semantics, in which sentences are signs for thoughts, and thoughts for things, and it ventures into difficult questions of possibility and necessity. If it is true that there will be a sea battle tomorrow, then how can we avoid the unpalatable conclusion that it is a matter of necessity that the battle will take place tomorrow? The Posterior Analytics uses the theory of demonstration as a basis for a theory of scientific knowledge. The Sophistical Refutations explore fallacious but apparently valid arguments. The Categories has, in part, the aspect of a preface to the Topics, but it is in part a work of metaphysics—a forerunner of Aristotle's treatise of that name.

The Stoics

The Stoics developed a logic different from Aristotle's, and to a large extent independently from him. Their greatest logician, Chrysippus, lived from about 280 to 206 b.c.e. and, as with most of the Stoics, his thought has mostly to be reconstructed from reports and fragments in later writers. Whereas Aristotelian syllogistic is a term-logic, Stoic logic was propositional: it explored the relations between what they called "assertibles"—that is to say, sentences that can be used to make assertions. Assertibles can be simple ("It is day") or complex ("If it is day, it is light"/ "It is day or it is not light"). The argument forms classified by the Stoics involve one complex and one simple assertible: for example, "If it is day, it is light. It is day. So, it is light." This is the first of five "indemonstrables"—basic argument forms—distinguished by Chrysippus. The Stoics had a schematic way of representing the indemonstrables—what they called their "modes"—using ordinal numbers. The four remaining modes of the five indemonstrables are (2) If the first, then the second; not the second; so not the first; (3) Not both the first and the second; the first; so not the second; (4) Either the first or the second; the first; so not the second; (5) Either the first or the second; not the first; so the second. Since the assertibles could be either negative or positive, and the complex assertible could itself include complex assertibles ("If both the first and the second, then the third"), there was quite a wide range of indemonstrable argument schemes. But Stoic logic was not limited to them. Nonindemonstrable forms of argument could be valid, and the Stoics had a theory of "analysis" in which, using certain basic rules (themata ) and, if wanted, additional theorems, the nonindemonstrable arguments were shown to be made up of demonstrable ones or of conversions of them.

Stoic logicians also explored modal concepts. One of their main starting points was provided by the fourth-century b.c.e. Megaric logician Diodorus Cronus, who formulated a "Master Argument," the subject of many attempts at reconstruction by modern historians, which attempts to show that, from the premises that true past propositions are necessary, and that an impossibility never follows from what is possible, it follows that nothing is possible except what is or will be true. The Stoic logicians rejected the argument by querying one or other of its premises, and Chrysippus developed his own understanding of possibility and necessity.

The Neoplatonists

Although Aristotle's pupil and successor, Theophrastus (c. 372–c. 287 b.c.e.), wrote extensively on logic, and Alexander of Aphrodisias (fl. c. 200 c.e.) cultivated Aristotelian logic, the most important enthusiasts of Aristotelian logic were, surprisingly, the Neoplatonists, from Porphyry (c. 234–c. 305) onward. Porphyry wrote an introduction (Eisagoge) to the Categories, which itself became for later students a part of the Aristotelian logical corpus (known as the Organon ), and he wrote extensive commentaries on Aristotle's logical works. Despite his Platonic metaphysics, Porphyry believed that logic, which is concerned with the world of appearances that is the subject of normal discourse, should be studied in strictly Aristotelian terms. Although later Neoplatonic commentators, following the lead of Iamblichus, tended more to introduce their characteristic metaphysical ideas into discussions of logic, Porphyry's approach was transmitted to the medieval Latin West by Boethius (c. 480–c. 524), who translated into Latin most of the Organon and wrote commentaries on some of it and logical textbooks on topical argument (the late ancient development of the Topics ), division, and on syllogisms.

The Medieval Latin West, 790–1200

The study of logic was revived in the Latin West at the court of Charlemagne; his adviser, Alcuin, wrote the first medieval logical textbook (On Dialectic ) in about 790. Logic was central to the intellectual life of medieval Europe in a way that it had not been in antiquity, and has not been since the Renaissance. Yet, until the 1130s, medieval logicians made do with what became known as the logica vetus ("old logic"): just Porphyry's Eisagoge, Aristotle's Categories and On Interpretation, and Boethius's commentaries and textbooks. They had only the most limited access to the Stoic tradition, through mentions by Boethius and the On Interpretation of Apuleius (second century c.e.). Ninth-century authors, such as John Scotus Erigena, were interested especially in the Categories —its metaphysical aspects and the question, raised by Augustine and by Boethius, of whether any of the ten categories distinguished by Aristotle apply to God. Anselm of Canterbury (1033 or 1034–1109) was a gifted logician who explored and questioned the Aristotelian doctrine of the Categories and made imaginative use of the ideas on possibility and necessity in On Interpretation.

In the twelfth century, the logica vetus was the central concern of the flourishing Paris schools. Peter Abelard (1079–1142), the greatest logician of the time, developed a nominalist metaphysics on its basis and elaborated a semantics to explain how sentences that use universal words (such as "Socrates is a man") are meaningful although there are no universal things, only particulars. Abelard also excelled in more purely logical matters. Starting from the hints and misunderstandings he found in Boethius, he rediscovered propositional logic and, in his Dialectica (c. 1116), he explored in great depth the truth conditions for conditional ("if … then …") sentences. Abelard was thus able to pioneer the analysis of sentences that are of ambiguous interpretation in terms of propositional logic, an aspect of logic that became especially popular from the 1130s onward, when On Sophistical Refutations and then the rest of Aristotle's logic (the logica nova —"new logic") became available. And he is one of the few logicians ever to have examined the logic of impersonal sentences, such as "It is good that he came today."

The Medieval Latin West, 1200–1500

From the middle of the twelfth century, logicians developed various branches of their subject, known as the logica modernorum ("contemporary logic"), that had not been treated specially, or at all, in antiquity. Peter of Spain's widely read Tractatus (often called Summulae logicales ) illustrates how parts of the logica modernorum had developed up until about the 1230s. There was a lull in interest and innovation in logic for nearly a century, but in the first half of the fourteenth century writers at Oxford, such as Walter Burley (d. 1344/45), William of Ockham (d. ?1349), Thomas Bradwardine (d. 1349), and William of Heytesbury (d. 1372/23), and, in Paris, John Buridan (d. after 1358) revived the branches of the logica modernorum and brought them to new levels of sophistication. The Logica magna (Great logic) of Paul of Venice (d. 1429) is a vast record of these achievements.

Some branches of the logica modernorum grew directly from the mid-twelfth-century interest in fallacies. For example, sophismata were a sort of disputation, involving a master and his pupils, built around sentences that either are apparently false but can be interpreted so as to make them true (e.g., "The whole Socrates is less than Socrates") or at least are open to different interpretations (e.g., "Every man is"). The ambiguities usually centered around the use of what were called "syncategorematic" words—words other than ordinary nouns, adjectives, and verbs with their own referential content: for instance, "only," "except," "all," "begins," "ceases"—and specialized treatises were devoted to studying these syncategoremata. Another type of disputation, "obligations," involved trying to force an opponent who has agreed to defend a particular statement into a self-contradiction, while following a very strict set of rules for what statements may be accepted or must be rejected. Liar paradoxes ("What I am now saying is false")—"insolubles"—formed another branch of study. In the theory of the "properties of terms" a highly elaborate theory was developed about the reference of words depending on their function within a sentence. Propositional logic was elaborated in treatises on what were called "consequences" (consequentiae ), although it remains questionable how far the approach was purely propositional.

Aristotle's texts and methods were not, however, forgotten in the later Middle Ages. Aristotelian syllogistic, studied earlier through Boethius's textbooks, could now be learned directly from the Prior Analytics. It was a basic tool for almost every medieval philosopher or theologian, and a set of mnemonics ("bArbArA," "cElArEnt," etc.) were devised to enable students to remember the valid patterns. On Interpretation continued to be central to discussions about possibility, necessity, and divine prescience, and the Posterior Analytics provided the criteria for organizing branches of knowledge as diverse as grammar and theology.

Medieval Islam

Logic was no less important for Islamic than for Christian philosophers and theologians. By the time of al-Farabi (c. 878–950), the first important Islamic logician, the whole of Aristotle's logical Organon was available in Arabic—far more material, then, than in the Latin West at this period, especially since the Arabic logicians tended to follow the habit of late antiquity in regarding the Rhetoric and the Poetics as parts of the Organon and assigning to them their own characteristic modes of argument, to contrast with demonstrative argument as taught in the Analytics and dialectical argument in the Topics.

Al-Farabi, who worked in Baghdad, Damascus, and elsewhere, saw his task as a logician to represent Aristotle faithfully, although this task involved him in a number of interpretations that went beyond the letter of the text. He was also concerned to vindicate logic in face of the grammarians, who doubted the need for this additional discipline; earlier in the tenth century, there had been a famous debate between the grammarian Abu Sa'id as-Sirafi and the logician Abu Bisr Matta, in which Abu Sa'id seems to have had the upper hand.

The great Persian philosopher Avicenna (Ibn Sina; 980–1037) respected al-Farabi and was also an ardent Aristotelian, but his approach to logic differed. In semantics, he rejected al-Farabi's theory that logic is concerned with expressions insofar as they signify meanings. Rather, he claimed, logic deals with meanings that classify meanings—so-called "second intentions." In his approach to syllogistic, Avicenna was far more inclined than al-Farabi had been to pursue his own train of analysis and accommodate Aristotle to it. For example, in modal logic he proposed a number of different readings of modal sentences: they could be taken absolutely (as in "God exists") or according to a condition—for example, "while something exists as a substance" (as in "man is necessarily a rational body") or "while something is described in the way it is" (as in "all mobile things are changing").

Logic continued to be studied in Islam because it was accepted by theologians as useful to their discipline rather than—as they often thought with regard to other areas of Aristotelian philosophy—a dangerous rival to it. Particularly important was the endorsement of al-Ghazali (1058–1111): whereas he wrote a work specifically designed to attack other areas of Aristotelian philosophy, he was himself the author of two short logical works, based on Avicenna. In the next century, Averroës (Ibn Rushd), who worked in Muslim Spain, followed al-Farabi (and the general direction of all his own work as the commentator par excellence on Aristotle) in seeking a greater fidelity to Aristotle, but it was Avicenna who remained the dominant influence on later Islamic logic.

See also Aristotelianism ; Neoplatonism ; Philosophy ; Philosophy, History of ; Scholasticism ; Stoicism .

bibliography

PRIMARY SOURCES

Aristotle. The Complete Works of Aristotle. Edited by Jonathan Barnes. Vol. 1. Princeton, N.J.: Princeton University Press, 1984.

——. Prior Analytics. Translated by Robin Smith. Indianapolis: Hackett, 1989.

Kretzmann, Norman, and Eleonore Stump, eds. Logic and the Philosophy of Language. Vol. 1 of The Cambridge Translations of Medieval Philosophical Texts. Cambridge, U.K.: Cambridge University Press, 1988.

Peter of Spain. Summulae logicales. Translated by Francis P. Dinneen. Amsterdam and Philadelphia: Benjamins, 1990.

SECONDARY SOURCES

Barnes, Jonathan, Suzanne Bobzien, Mario Mignucci, and Dink M. Schenkeveld. "Part 2: Logic and Language." In The Cambridge History of Hellenistic Philosophy, edited by Keimpe Algra et al. Cambridge, U.K.: Cambridge University Press, 1999. Excellent, up-to-date survey.

Kretzmann, Norman, Anthony Kenny, and Jan Pinborg, eds. Cambridge History of Later Medieval Philosophy. Cambridge, U.K.: Cambridge University Press, 1982. Pages 101–381 contain the most complete available account of medieval logic in the Latin West.

Marenbon, John. Boethius. New York: Oxford University Press, 2003. See pages 17–65. Includes bibliography for Greek Neoplatonic tradition. Martin, C. J. "Embarrassing Arguments and Surprising Conclusions in the Development of Theories of the Conditional in the Twelfth Century." In Gilbert de Poitiers et ses contemporains, edited by Jean Jolivet and Alain De Libera. Naples: Bibliopolis, 1987. Discusses Abelard and the twelfth-century rediscovery of propositional logic.

Smith, Robin. "Logic." In The Cambridge Companion to Aristotle, edited by Jonathan Barnes. Cambridge, U.K.: Cambridge University Press, 1995. Excellent bibliography on pp. 308–324.

John Marenbon