Definitions

History of logic

The history of logic traces the development of the science of valid inference (logic). While many cultures have employed intricate systems of reasoning, and logical methods are evident in all human thought, an explicit analysis of the principles of reasoning was developed only in three traditions: those of China, India, and Greece. Although exact dates are uncertain, particularly in the case of India, it is possible that logic emerged in all three societies by the fourth century BC. The formally sophisticated treatment of modern logic descends from the Greek tradition, particularly Aristotelian logic, which was further developed by Islamic logicians and then medieval European logicians.

Logic was known as 'dialectic' or 'analytic' in Ancient Greece. The word 'logic' (from the Greek logos, meaning discourse or sentence) does not appear in the modern sense until the commentaries of Alexander of Aphrodisias, writing in the third century A.D.

Logic in Ancient Greece

Logic before Plato

People have employed valid reasoning in all periods of human history. However, logic studies the principles of valid reasoning, inference and demonstration, and there is almost no historic evidence of such study before the time of Plato and Aristotle. It is probable that the idea of demonstrating a conclusion first developed in connection with Geometry, which originally meant the same as 'land measurement'. The Egyptians discovered some truths of geometry (such as the formula for a truncated pyramid) empirically, but the great achievement of the Ancient Greeks was to replace empirical methods by demonstrative science. The systematic study of this seems to have begun with the school of Pythagoras in the late sixth century B.C. The three basic principles of geometry are that certain propositions must be accepted as true without demonstration, that all other propositions of the system are derived from these, and that the derivation must be formal, i.e. independent of the special subject matter in question. Fragments of early proofs are preserved in the works of Plato and Aristotle, and it is probable that the idea of a deductive system was known in the Pythagorean school, and in the Platonic Academy.

Separately from geometry, the idea of a standard argument pattern is found in the reductio ad impossibile used by Zeno of Elea, a pre-Socratic philosopher of the fifth century B.C. This is the technique of drawing on obviously false or absurd or impossible conclusion from an assumption, thus demonstrating that the assumption is false. In his book Parmenides, Plato has Zeno claiming to have written a book defending the monism of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. Other philosophers who practised such dialectic reasoning were the so-called Minor Socratics, including Euclid of Megara, who were probably followers of Parmenides and Zeno. The members of this school were called 'dialecticians' (from a Greek word meaning 'to discuss').

Further evidence that pre-Aristotelian thinkers were concerned with the principles of reasoning is found in the fragment called Dissoi Logoi, probably written at the beginning of the fourth century B.C.. This is part of a protracted debate about truth and falsity.

Plato's Logic

None of the surviving works of the great fourth century philosopher Plato (428 – 347) include any formal logic, but he is certainly the first major thinker in the field of philosophical logic. Plato raises three important logical questions:

• What is it that can properly be called true or false?
• What is the nature of the connection between the assumptions of a valid argument and its conclusion?
• What is the nature of definition?

The first question arises in the dialogue Theaetetus in the attempt to define knowledge. Plato identifies thought or opinion with talk or discourse (logos): 'forming an opinion is talking, and opinion is speech that is held not with someone else or aloud but in silence with oneself' (Theaetetus 189E-190A). The second question arises with Plato's theory of Forms. Forms are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called universals, namely an abstract entity common to each set of things that have the same name. Both in The Republic and The Sophist it is strongly suggested by Plato that correct thinking is following out the connection between forms. The necessary connection between the premisses and the conclusion of an argument is a relation between thoughts determined by the 'forms' which underlie the thoughts. The third question involves the nature of definition. Many of Plato's dialogues concern the search for a definition of some important concept (justice, truth, the Good), and it is likely that Plato was impressed by the importance of definition in mathematics. What underlies every definition is a Platonic Form, the common nature present in different particular things. Thus a definition reflects the ultimate object of our understanding, and is the foundation of all valid inference. Plato's conception of definition had a great influence on Aristotle, in particular Aristotle's notion of the essence of a thing, the 'what it is to be' a particular thing of a certain kind.

Aristotle's logic

The logic of Aristotle, and particularly his theory of the syllogism, has had an enormous influence in Western thought. His logical works, called the Organon, are the earliest formal study of logic that have come down to modern times. Though it is difficult to determine the dates, the probable order of writing of Aristotle's logical works is:

These works are of outstanding importance in the history of logic. Aristotle is the first logician to attempt a systematic analysis of logical syntax, into noun or term, and verb. In the Categories, he attempts to classify all the possible things that a term can refer to. This idea underpins his philosophical work, the Metaphysics, which later had a great influence on Western thought. Aristotle was the first formal logician (i.e. he gives the principles of reasoning using variables to show the underlying logical form of arguments). He is looking for relations of dependence which characterise necessary inference, and distinguishes the validity of these relations, from the truth of the premisses (the soundness of the argument). The Prior Analytics contains his exposition of the 'syllogistic', where three important principles are applied for the first time in history: the use of variables, a purely formal treatment, and the use of an axiomatic system.

Stoic Logic

The other great school of Greek logic is that of the Stoics. Stoic logic traces its roots back to the late fifth century philosopher, Euclid of Megara, a pupil of Socrates and slightly older contemporary of Plato. He was probably a disciple of Parmenides. His pupils and successors were called 'Megarians', or 'Eristics', and later the 'Dialecticians'. Among his pupils were Eubulides (according to tradition), and Stilpo. Unlike with Aristotle, we have no complete works by writers of this school, and have to rely on accounts (sometimes hostile) of Sextus Empiricus, writing in the third century A.D. The three most important contributions of the Stoic school were (i) their account of modality, (ii) their theory of the Material conditional, and (iii) their account of meaning and truth.

(1) Modality. According to Aristotle, the Megarians of his day claimed there was no distinction between potentiality and actuality. Diodorus Cronus (2nd half 4th century BC) defined the possible as that which either is or will be, the impossible as what will not be true, and the contingent as that which either is already, or will be false. Diodorus is also famous for his so-called Master argument, that the three propositions 'everything that is past is true and necessary', 'The impossible does not follow from the impossible', and 'What neither is nor will be is possible' are inconsistent. Diodorus used the plausibility of the first two to prove that nothing is possible if it neither is nor will be true. Chrysippus (c.280–c.207 BC), by contrast, denied the second premiss and said that the impossible could follow from the possible.

(2) Conditional statements. The first logicians to debate conditional statements were Diodorus and his pupil Philo of Megara (fl. 300 BC). Sextus Empiricus refers three times to a debate between Diodorus and Philo. Philo argued that a true conditional is one that does not begin with a truth and end with a falsehood. such as 'if it is day, then I am talking'. But Diodorus argued that a true conditional is what could not possibly begin with a truth and end with falsehood - thus the conditional quoted above could be false if it were day and I became silent. Philo's criterion of truth is what would now be called a truth-functional definition of 'if ... then'. In a second reference, Sextus says 'According to him there are three ways in which a conditional may be true, and one in which it may be false'.

(3) Meaning and truth. The most important and striking difference between Megarian-Stoic logic and Aristotelian logic is that it concerns propositions, not terms, and is thus closer to modern propositional logic. The Stoics distinguished between utterance (phone) , which may be noise, speech (lexis), which is articulate but which may be meaningless, and discourse (logos), which is meaningful utterance. The most original part of their theory is the idea that what is expressed by a sentence, called a lekton, is something real. This corresponds to what is now called a proposition. Sextus says that according to the Stoics, three things are linked together, that which is signified, that which signifies, and the object. For example, what signifies is the word 'Dion', what is signified is what Greeks understand but barbarians do not, and the object is Dion himself .

Logic in Islamic philosophy

For a time after the Prophet Muhammad's death, Islamic law placed importance on formulating standards of argument, which gave rise to a novel approach to logic in Kalam, but this approach was later displaced by ideas from Greek philosophy and Hellenistic philosophy with the rise of the Mu'tazili theologians, who highly valued Aristotle's Organon. The works of Hellenistic-influenced Islamic philosophers were crucial in the reception of Aristotelian logic in medieval Europe, along with the commentaries on the Organon by Averroes. The works of al-Farabi, Avicenna, al-Ghazali and other Muslim logicians who often 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 not only included the study of formal patterns of inference and their validity but also elements of the philosophy of language and elements of epistemology and metaphysics. Due to disputes with Arabic grammarians, Islamic philosophers were interested in working out the relationship between logic and language, and they devoted much discussion to the question of the subject matter and aims of logic in relation to reasoning and speech. In the area of formal logical analysis, they elaborated upon the theory of terms, propositions and syllogisms. They considered the syllogism to be the form to which all rational argumentation could be reduced, and they regarded syllogistic theory as the focal point of logic. Even poetics was considered as a syllogistic art in some fashion by many major Islamic logicians.

Logic in medieval Europe

"Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logic developed in medieval Europe throughout the period c 1200–1600. For centuries after Stoic logic had been formulated, it was the dominant system of logic in the classical world. When the study of logic resumed after the Dark Ages, the main source was the work of the Christian philosopher Boethius. Alcuin, who taught at York in the eighth century AD, mentions that the library there contained Aristotle, Marius Victorinus, and Boethius (although we do not know how much of Aristotle was included there). Until the twelfth century the only works of Aristotle available in the West were the Categories, On Interpretation and Boethius' translation of the Isagoge of Porphyry (a commentary on the Categories). These works were known as the 'Old Logic' (Logica Vetus or Ars Vetus). A significant and original work on the old logic was the Logica Ingredientibus of Peter Abelard.

The 'rediscovery' of the works of antiquity began in the Latin West in the late twelfth century, when Arabic texts on Aristotelian logic and works by Islamic logicians were translated into Latin. While the logic of Arabic writers such as Avicenna had an influence on early medieval European logicians such as Albertus Magnus, the Aristotelian tradition became more dominant due to the strong influence of Averroism.

After the initial translation phase, the tradition of Medieval logic was developed through textbooks such as that by Peter of Spain (fl. thirteenth century), 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. The tradition reached its high point in the fourteenth century, with the works of William of Ockham (c. 1287–1347) and Jean Buridan.

One feature of the development of Aristotelian logic through what is known as supposition theory, a study of the semantics of the terms of the proposition.

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).

"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 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 Locke gives 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 the tradition.

Another influential work was the Novum Organum by Francis Bacon, published in 1620. The title translates as "new instrument". This is a reference to Aristotle's work Organon. In this work, Bacon repudiated the syllogistic method of Aristotle in favour of an alternative procedure 'which by slow and faithful toil gathers information from things and brings it into understanding' (Farrington, 1964, 89). This method is known as Induction. The inductive method starts from empirical observation and proceeds to lower axioms or propositions. From the lower axioms more general ones can be derived (by induction). In finding the cause of a phenomenal nature such as heat, one must list all of the situations where heat is found. Then another list should be drawn up, listing situations that are similar to those of the first list except for the lack of heat. A third table lists situations where heat can vary. The form nature, or cause, of heat must be that which is common to all instances in the first table, is lacking from all instances of the second table and varies by degree in instances of the third table.

Descartes may have been the first philosopher to have had the idea of using algebra, especially techniques for solving for unknown quantities in equations, as a vehicle for scientific exploration. The idea of a calculus of reasoning was also developed by Gottfried Wilhelm Leibniz. Leibniz 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).

Gottlob Frege in his 1879 Begriffsschrift extended formal logic beyond propositional logic to include constructors such as "all", "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.

In a masterly 1885 article read by Peano, Ernst Schröder, and others, Charles 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 19th and early 20th 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".

In 1889 Giuseppe 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.

Logic in India

Formal logic also developed in India, without the influence, so far as is known, of Greek logic. 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 school developed 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-Nyāya, which developed a formal analysis of inference in the sixteenth century.

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.

References

• Alexander of Aphrodisias, In Aristotelis An. Pr. Lib. I Commentarium, ed. Wallies, C.I.A.G.
• Boethius Commentary on the Perihermenias, Secunda Editio, ed. Meiser.
• Bochenski, I.M., A History of Formal Logic, Notre Dame press, 1961.
• Alonzo Church, 1936-8. "A bibliography of symbolic logic". Journal of Symbolic Logic 1: 121-218; 3:178-212.
• Epictetus, Dissertationes ed. Schenkl.
• Farrington, B., The Philosophy of Francis Bacon, Liverpool 1964.
• Dov Gabbay and John Woods, eds, 2004. Handbook of the History of Logic. Vol. 1: Greek, Indian and Arabic logic; Vol. 3: The Rise of Modern Logic I: Leibniz to Frege. Elsevier, ISBN 0-444-51611-5.
• Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton University Press.
• Heath, T.L. Mathematics in Aristotle.
• Kneale, William and Martha, 1962. The development of logic. Oxford University Press, ISBN 0-19-824773-7.
• Sextus Empiricus, Against the Grammarians (Adversos Mathematicos I). David Blank (trans.) (Oxford: Clarendon Press, 1998). ISBN 0-19-824470-3.