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The history of logic is the study of 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. Of these, only the treatment of logic descending from the Greek tradition, particularly Aristotelian logic, found wide application and acceptance in science and mathematics. The Greek tradition was further developed by Islamic logicians and then medieval European logicians. Not until the 19th century does the next great advance in logic arise, with the development of symbolic logic by George Boole and its subsequent development into formal calculable logical systems by Gottlob Frege and set theorists such as Georg Cantor and Giuseppe Peano, ushering in the Information Age.

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

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 ancient 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 absurdum used by Zeno of Elea, a pre-Socratic philosopher of the fifth century B.C. This is the technique of drawing an obviously false, absurd or impossible conclusion from an assumption, thus demonstrating that the assumption is false. Plato's Parmenides portrays Zeno as 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:

• The Categories, a study of the ten kinds of primitive term.
• Topics, with an appendix called On Sophistical Refutations, a discussion of dialectics.
• On Interpretation - an analysis of simple categorical propositions, into simple terms, nouns and verbs, negation, and signs of quantity.
• Prior Analytics a formal analysis of valid argument or 'syllogism'.
• Posterior Analytics a study of scientific demonstration, containing Aristotle's mature views on logic.

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 possible', 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 ancient Asia

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.

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 to some extent 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.

Al-Farabi's logic

Though Al-Farabi (Alfarabi) (873–950) was mainly an Aristotelian logician, he introduced a number of non-Aristotelian elements of logic. He discussed the topics of future contingents, the number and relation of the categories, the relation between logic and grammar, and non-Aristotelian forms of inference. He is credited for categorizing logic into two separate groups, the first being "idea" and the second being "proof".

Al-Farabi also introduced the theories of conditional syllogism and analogical inference, which were not part of the Aristotelian tradition. Another addition al-Farabi made to the Aristotelian tradition was his introduction of the concept of poetic syllogism in a commentary on Aristotle's Poetics.

Avicennian logic

Ibn Sina (Avicenna) (980–1037) developed his own system of logic known as "Avicennian logic" as an alternative to Aristotelian logic. After the Latin translations of the 12th century, Avicennian logic also influenced early medieval European logicians such as Albertus Magnus, though Aristotelian logic later became more popular in Europe due to the strong influence of Averroism.

Avicenna developed an early theory on hypothetical syllogism, which formed the basis of his early risk factor analysis. He also developed an early theory on propositional calculus, which was an area of logic not covered in the Aristotelian tradition. The first criticisms on Aristotelian logic were also written by Avicenna, who developed an original theory on temporal modal syllogism. He also contributed inventively to the development of inductive logic, being the first to describe the methods of agreement, difference and concomitant variation which are critical to inductive logic and the scientific method.

Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806-1873). Systematic refutations of Greek logic were written by the Illuminationist school, founded by Shahab al-Din Suhrawardi (1155-1191), who developed the idea of "decisive necessity", which refers to the reduction of all modalities (necessity, possibility, contingency and impossibility) to the single mode of necessity. Ibn al-Nafis (1213-1288) wrote a book on Avicennian logic, which was a commentary of Avicenna's Al-Isharat (The Signs) and Al-Hidayah (The Guidance). Another systematic refutation of Greek logic was written by Ibn Taymiyyah (1263-1328), who wrote the ar-Radd 'ala al-Mantiqiyyin (Refutation of Greek Logicians), in which he gave a proof for induction being the only true form of argument, which had an important influence on the development of the scientific method of observation and experimentation. The Sharh al-takmil fi'l-mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century was the last major Arabic work on logic.

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

"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. It is still reprinted today, but mostly for historical purposes.

The account of propositions that Locke gives in the Essay is essentially that of Port-Royal: "Verbal propositions, which are words, 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 rejected 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.

The rise of modern logic

The period which followed the important developments in logic in the thirteenth and early fourteenth century and the beginning of the nineteenth century was largely one of decline and neglect, and is generally regarded as barren by historians of logic. The revival of logic happened in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics. The development of the modern so-called 'symbolic' or 'mathematical' logic during this period is the most significant in the two thousand-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history .

A number of features distinguish modern from the old Aristotelian or traditional logic, the most important of which are as follows. Modern logic is fundamentally a calculus. whose rules of operation are determined only by the shape, and not by the meaning of the symbols it employs, as in mathematics. Many logicians were impressed by the 'success' of mathematics, in that there has been no prolonged dispute about any properly mathematical result. C.S. Peirce noted that even though a mistake in the evaluation of a definite integral by Laplace led to an error concerning the moon's orbit that persisted for nearly 50 years, the mistake, once spotted, was corrected without any serious dispute. Peirce contrasted this with the disputation and uncertainty surrounding traditional logic, and especially reasoning in metaphysics. He argued that a truly 'exact' logic will depends upon mathematical, i.e. 'diagrammatic' or 'iconic' thought. "Those who follow such methods will ... escape all error except such as will be speedily corrected after it is once suspected". Modern logic is also 'constructive' rather than 'abstractive' i.e. rather than abstracting and formalising theorems derived from ordinary language (or from psychological intuitions about validity) it constructs theorems by formal methods, then looks for an interpretation in ordinary language. It is entirely symbolic, meaning that even the logical constants (which the medevial logicians called 'syncategoremata') as well as the categoric terms are expressed in symbols. Finally, modern logic strictly avoids psychological, epistemological and metaphysical questions.

Periods of modern logic

The development of modern logic falls into roughly five periods, as follows.

1. The pre-historical or embryonic period from Leibniz to 1847, where the notion of a logical calculus is discussed and developed, particularly by Leibniz, but where no schools are formed, and where isolated periodic attempts were abandoned or went unnoticed.
2. The algebraic period from Boole's Analysis to Schroeder's Vorlesungen. In this period there are more practitioners, and a greater continuity of development.
3. The logicist period from the Begriffsschrift of Frege to the Principia Mathematica of Russell and Whitehead. This was dominated by the 'logicist school', whose aim was to incorporate the logic of all mathematical and scientific discourse in a single unified system, and which, taking as a fundamental principle that all mathematical truths are logical, did not accept any non-logical terminology. The major logicists were Frege, Russell, and the early Wittgenstein (Companion p.499). It culminates with the Principia, an important work which includes a thorough examination and attempted solution of the antimonies which had been an obstacle to earlier progress.
4. The metalogical period from 1910 to the 1930's which saw the development of metalogic, in the finitist system of Hilbert, and the non-finitist system of Lowenheim and Skolem, the combination of logic and metalogic in the work of Godel and Tarski. Godel's incompleteness theorem of 1931 was one of the greatest achievements in the history of logic. Later in the 1930's Godel developed the notion of set-theoretic constructibility.
5. The period after the World war II, and important proof discovered by Paul Cohen of the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory, the most widely accepted axiomatization of set theory, made possible by his invention of the powerful technique called forcing

Embryonic period

The idea that inference could be represented by a purely mechanical process is found as early as Raymond Lull, who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. Three hundred years later, the English philosopher and logician Thomas Hobbes suggested that all logic and reasoning could be reduced to the mathematical operations of addition and subtraction. The same idea is found in the work of Leibniz, who had read both Lull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like both Lull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was never published until long after his death. Leibniz says that ordinary language are subject to 'countless ambiguities' and are unsuited for a calculus, whose task is to expose mistakes in inference arising from the forms and structures of words. Gergonne (1816) says that reasoning does not have to be about objects which we have perfectly clear ideas about, since algebraic operations can be carried out without our having any idea of the meaning of the symbols involved. Bolzano anticipated a fundamental idea of modern proof theory when he defined logical consequence or 'deducibility' in terms of variables: a set of propositions n, o, p ... are deducible from propositions a, b, c ... in respect of the variables i, j, ... if any substitution for i, j that have the effect of making a, b, c ... true, simultaneously make the propositions n, o, p ... also . This is now known as semantic validity.

Algebraic period

Modern logic begins with the so-called 'algebraic school', originating with Boole and including Peirce, Jevons, Schroeder and Venn. Their objective was to develop a calculus to formalise reasoning in the area of classes, propositions and probabilities.

The school begins with Boole's seminal work Mathematical Analysis of Logic which appeared in 1847, although De Morgan (1847) is its immediate precursor. The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an usher in a private school in Lincoln. For example, let x and y stand for classes let the symbol = signify that the classes have the same members, xy stand for the class containing all and only the members of x and y and so on. Boole calls these elective symbols, i.e. symbols which select certain objects for consideration. An expression in which elective symbols are used is called an elective function, and an equation of which the members are elective functions, is an elective equation. The theory of elective functions and their 'development' is essentially the modern idea of truth-functions and their expression in disjunctive normal form.

Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between 'primary propositions' which are the subject of syllogistic theory, and 'secondary propositions', which are the subject of propositional logic, and showed how under different 'interpretations' the same algeberaic system could represent both. An example of a primary proposition is 'All inhabitants are either Europeans or Asiatics'. An example of a secondary proposition is 'Either all inhabitants are Europeans or they are all Asiatics'. These are easily distinguished in modern propositional calculus, where it is also possible to show that the first follows from the second, but it is a significant disadvantage that there is no way of representing this in the Boolean system.

In his Symbolic Logic (1881), John Venn used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's methods could be mechanised, and constructed a 'logical machine' which he showed to the Royal Society the following year. In 1885 Allan Marquand proposed an electrical version of the machine that is still extant (picture at the Firestone Library).

The defects in Boole's system (such as the use of the letter v for existential propositions) were all remedied by his followers. Jevons published Pure Logic, or the Logic of Quality apart from Quantity in 1864, where he suggested a symbol to signify exclusive or, which allowed Boole's system to be greatly simplified. This was usefully exploited by Schroeder when he set out theorems in parallel columns in his Vorlesungen (1890-1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive 'either ... or', however, like many of Peirce's innovations, this passed without notice until Sheffer rediscovered it in 1913. Boole's early work also lacks the idea of the logical sum which originates in Peirce (1867), Schroeder (1877) and Jevons (1890), and the concept of inclusion, first suggested by Gergonne (1816) and clearly articulated by Peirce (1870).

The success of Boole's algebraic system suggested that all logic must be capable of algebraic representation, and there were attempts to express a logic of relations in such form, of which the most ambitious was Schroder's monumental Vorlesungen über die Algebra der Logik (vol iii 1895), although the original idea was again anticipated by Peirce.

Logicist period

After Boole, the next great advances were made by the German mathematician Gottlob Frege. Frege's objective was the program of Logicism, i.e. demonstrating that arithmetic is identical with logic. Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or Begriffschrift is considered the greatest single achievement in the entire history of logic. Frege also tried to show that the concept of number can be defined by purely logical means, so that (if he was right) logic includes arithmetic and all branches of mathematics that are reducible to arithmetic. He was not the first writer to suggest this. In his pioneering work Die Grundlagen der Arithmetik (The Foundations of Arithmetic), sections 15-17, he acknowledges the efforts of Leibniz, J.S. Mill as well as Jevons, citing the latter's claim that 'algebra is a highly developed logic, and number but logical discrimination'.

Frege's first work, the Begriffschrift is a rigorously axiomatised system of propositional logic, relying on just two connectives (negational and conditional), two rules of inference (modus ponens and substitution), and six axioms. Frege referred to the 'completeness' of this system, but was unable to prove this. The most signification innovation, however, was his explanation of the quantifier in terms of the mathematical functions. Traditional logic regards the sentence 'Caesar is a man' as of fundamentally the same form as 'all men are mortal'. Sentences with a proper name subject were regarded as universal in character, interpretable as 'every Caesar is a man'. Frege argued that the quantifier expression 'all men' does not have the same logical or semantic form as 'all men', and that the universal proposition 'every A is B' is a complex proposition involving two functions, namely ' – is A' and ' – is B' such that whatever satisfies the first, also satisfies the second. In modern notation, this would be expressed as

(x) Ax -> Bx

In English, 'for all x, if Ax then Bx'. Thus only singular propositions are of subject-predicate form, and they are irreducibly singular, i.e. not reducible to a general proposition. Universal and particular propositions, by contrast, are not of simple subject-predicate form at all. If 'all mammals' were the logical subject of the sentence 'all mammals are land-dwellers', then to negate the whole sentence we would have to negate the predicate to give 'all mammals are not land-dwellers'. But this is not the case. This functional analysis of ordinary-language sentences later had a great impact on philosophy and linguistics.

This means that in Frege's calculus, Boole's 'primary' propositions can be represented in a different way from 'secondary' propositions. 'All inhabitants are either Europeans or Asiatics' is

(x)

whereas 'All the inhabitants are Europeans or all the inhabitants are Asiatics' is

(x) (I(x) -> E(x)) v (x) (I(x) -> A(x))

As Frege remarked in a critique of Boole's calculus:

"The real difference is that I avoid division into two parts … and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it'

As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient problem of multiple generality. The ambiguity of 'every girl kissed a boy' is difficult to express in traditional logic, but Frege's logic captures this through the different scope of the quantifiers. Thus

(x)

means that to every girl there corresponds some boy (any one will do) who the girl kissed. But

(x)

means that there is some particular boy whom every girl kissed. Without this device, the project of logicism would have been doubtful or impossible. Using it, Frege provided a definition of the ancestral relation, of the many-to-one relation, and hence of mathematical induction.

Extra bit

This period overlaps with the work of the so-called 'mathematical school' which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting whose objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory.

Inclusion

The modern symbol for inclusion first appears in Gergonne (1816), who defines it as one idea 'containing' or being 'contained' by another, using the upside-down letter 'C' to express this. Peirce articulated this clearly in 1870, arguing also that inclusion was a wider concept than equality, and hence a logically simpler one. Schroder (also Frege) calls the same concept 'subordination'.

References

• Beaney, Michael, The Frege Reader, London: Blackwell 1997).
• Bolzano., Wissenschaftslehre, 4 Bde Neudr., 2. verb, A. hrsg. W. Schultz, Leipzig I-II 1929, III 1930, IV 1931 (trans. as Theory of science, attempt at a detailed and in the main novel exposition of logic with constant attention to earlier authors. (Edited and translated by Rolf George University of California Press, Berkeley and Los Angeles 1972)
• Theory of science (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell - D. Reidel Publishing Company, Dordrecht and Boston 1973)
• Honderich, Ted (ed.). The Oxford Companion to Philosophy (New York: Oxford University Press, 1995) ISBN 0-19-866132-0
• Bochenski, I.M., A History of Formal Logic, Notre Dame press, 1961.
• Gergonne, (1816) "Essai de dialectique rationelle", in Annales de mathem, pures et appl. 7, 1816/7, 189-228
• Peirce, C.S., (1896), "The Regenerated Logic", The Monist, vol. VII, No. 1, pp. 19-40, The Open Court Publishing Co., Chicago, IL, 1896, for the Hegeler Institute. Reprinted (CP 3.425-455). Internet Archive The Monist 7.
• Boole, George (1847) The Mathematical Analysis of Logic (Cambridge and London); repr. in Studies in Logic and Probability, ed. R. Rhees (London 1952)
• Boole, George (1854) The Laws of Thought (London and Cambridge); repr. as Collected Logical Works. Vol. 2, (Chicago and London: Open Court, 1940).

Notes

1. Kneale & Kneale, p.2
2. Heath
3. Kneale & Kneale, p.16
4. Kneale & Kneale p. 21
5. Bochenski p. 63
6. Metaphysics Eta 3, 1046b 29
7. Boethius, Commentary on the Perihermenias, Meiser p. 234
8. Epictetus, Dissertationes ed. Schenkel ii. 19. I.
9. Alexander p. 177
10. Sextus, Adv. Math. viii 113
11. Sextus viii. 11, 12
12. Bochenski p.446
13. ^ History of logic: Arabic logic, Encyclopædia Britannica.
14. Seymour Feldman (1964), "Rescher on Arabic Logic", The Journal of Philosophy 61 (22), p. 724-734 .
15. Ludescher, Tanyss (February 1996), "The Islamic roots of the poetic syllogism", College Literature, retrieved 2008-02-29
16. Richard F. Washell (1973), "Logic, Language, and Albert the Great", Journal of the History of Ideas 34 (3), p. 445-450 .
17. ^ Lenn Evan Goodman (2003), Islamic Humanism, p. 155, Oxford University Press, ISBN 0195135806.
18. Lenn Evan Goodman (1992), Avicenna, p. 188, Routledge, ISBN 041501929X.
19. ^ Muhammad Iqbal, The Reconstruction of Religious Thought in Islam, "The Spirit of Muslim Culture" (cf. and )
20. Dr. Lotfollah Nabavi, Sohrevardi's Theory of Decisive Necessity and kripke's QSS System, Journal of Faculty of Literature and Human Sciences.
21. Dr. Abu Shadi Al-Roubi (1982), "Ibn Al-Nafis as a philosopher", Symposium on Ibn al-Nafis, Second International Conference on Islamic Medicine: Islamic Medical Organization, Kuwait (cf. Ibn al-Nafis As a Philosopher, Encyclopedia of Islamic World).
22. Nicholas Rescher and Arnold vander Nat, "The Arabic Theory of Temporal Modal Syllogistic", in George Fadlo Hourani (1975), Essays on Islamic Philosophy and Science, p. 189-221, State University of New York Press, ISBN 0873952243.
23. Richard F. Washell (1973), "Logic, Language, and Albert the Great", Journal of the History of Ideas 34 (3), p. 445-450 .
24. Oxford Companion p. 498, Bochenski, Part I Introduction, passim
25. Oxford Companion p. 500
26. Bochenski, p. 266
27. Peirce 1896
28. See Bochenski p.269
29. El. philos. sect. I de corp 1.1.2
30. (Bochenski p.274)
31. (Essai de dial. rat, 211n, quoted in Bochenski p.277)
32. Wissenschaftslehre II 198ff, quoted in Bochenski 280, see 'Oxford 'Companion p. 498
33. Before publishing, he wrote to De Morgan, who was just finishing his work Formal Logic. De Morgan suggested they should publish first, and thus the two books appeared at the same time, possibly even reaching the bookshops on the same day. cf. Kneale p. 404
34. Kneale p. 404
35. Kneale p. 407
36. Boole (1847) p. 16
37. Kneale p. 407
38. Boole 1847 pp.58-9
39. Beaney p.11
40. Kneale p. 407
41. Kneale p. 422
42. Trans. Amer. Math. Soc., xiv (1913), pp. 481-8. This is now known as the Sheffer stroke
43. Bochenski 296
44. See CP III
45. Kneale p.435
46. ibid p.435
47. Jevons, The Principles of Science, London 1879, p. 156, quoted in Grundlagen 15
48. Beaney p.10 – the completeness of Frege's system was eventually proved by Lukasiewicz in 1934
49. see e.g. the argument by the medieval logician William of Ockham that singular propositions are universal, in Summa Logicae III. 8 (??)
50. "On concept and object" p.198, Geach p. 48
51. BLC p. 14, quoted in Beaney p. 12
52. "Descr. of a notation", CP III 28
53. Vorlesungen I, 127

See also

• Term Logic
• Ernst Schröder
• Charles Sanders Peirce

Notes

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, Handbook of the History of Logic 2004. 1. Greek, Indian and Arabic logic; 2. Mediaeval and Renaissance logic; 3. The rise of modern logic: from Leibniz to Frege; 4. British logic in the Nineteenth century; 5. Logic from Russell to Church; 6. Sets and extensions in the Twentieth century (not yet published); 7. Logic and the modalities in the Twentieth century; 8. The many-valued and nonmonotonic turn in logic; 9. Logic and computation (not yet published); 10. Inductive logic (not yet published); 11. Logic: A history of its central concepts (not yet published) Elsevier, ISBN 0-444-51611-5.
• Leila Haaparanta (ed.) 2009. The Development of Modern Logic Oxford University Press.
• Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton University Press.
• Heath, T.L., 1949. Mathematics in Aristotle Oxford University Press.
• 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.

External links

• Ontology and History of Logic in Western Thought. An Introduction Annotated bibliography
• Overview of Indian and Greek Development of Logic and Language
• Petrus Hispanus (Stanford Encyclopedia of Philosophy)
• Paul Spade's "Thoughts Words and Things"
• John of St Thomas
• Joyce's Principles of Logic (Traditional Logic Primer)
• Article on Logic in Britannica 1911 - a good summary of developments in logic before Frege-Russell

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