History of logic
Encyclopedia
The history of logic is the study of the development of the science of valid inference (logic
). Formal logic was developed in ancient times in China
, India
, and Greece
. Greek logic, particularly Aristotelian logic, found wide application and acceptance in science and mathematics.
Aristotle's logic was further developed by Islamic
and Christian
philosophers in the Middle Ages, reaching a high point in the mid-fourteenth century. The period between the fourteenth century and the beginning of the nineteenth century was largely one of decline and neglect, and is regarded as barren by at least one historian of logic.
Logic was revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a 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.
Progress in mathematical logic
in the first few decades of the twentieth century, particularly arising from the work of Gödel
and Tarski
, had a significant impact on analytic philosophy
and philosophical logic
, particularly from the 1950s onwards, in subjects such as modal logic
, temporal logic
, deontic logic
, and relevance logic
.
Valid reasoning has been employed in all periods of human history. However, logic studies the principles of valid reasoning, inference and demonstration. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry
, which originally meant the same as "land measurement". In particular, the ancient Egypt
ians had empirical
ly discovered some truths of geometry
, such as the formula for the volume of a truncated pyramid
.
Another origin can be seen in Babylonia
. Esagil-kin-apli's medical Diagnostic Handbook in the 11th century BC was based on a logical set of axiom
s and assumptions, while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems, an important contribution to the philosophy of science
.
in the late sixth century BC. 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, that is, independent of the particular subject matter in question. Fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and 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 BC. 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 BC. This is part of a protracted debate about truth and falsity.
(428–347) include any formal logic, but they include important contributions to the field of philosophical logic
. Plato raises three questions:
The first question arises in the dialogue Theaetetus
, where Plato identifies thought or opinion with talk or discourse (logos). The second question is a result of 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. In both The Republic and The Sophist
, Plato suggests that the necessary connection between the premisses and the conclusion of an argument corresponds to a necessary connection between "forms". The third question is about 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. This 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.
, 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 was the first logician to attempt a systematic analysis of logical syntax, into noun (or term
), and verb. In the Categories, he attempted to classify all the possible things that a term can refer to. This idea underpins his philosophical work, the Metaphysics
, which also had a profound influence on Western thought. He was the first to deal with the principles of contradiction and excluded middle
in a systematic way. He was the first formal logician (i.e. he gave the principles of reasoning using variables to show the underlying logical form
of arguments). He was looking for relations of dependence which characterise necessary inference, and distinguished the validity
of these relations, from the truth of the premises (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. In the Topics and Sophistical Refutations he also developed a theory of non-formal logic (e.g. the theory of fallacies).
s. Stoic logic traces its roots back to the late 5th century BC philosopher, Euclid of Megara
, a pupil of Socrates
and slightly older contemporary of Plato. His pupils and successors were called "Megarians", or "Eristics", and later the "Dialecticians". The two most important dialecticians of the Megarian school were Diodorus Cronus
and Philo
who were active in the late 4th century BC. The Stoics adopted the Megarian logic and systemized it. The most important member of the school was Chrysippus
(c. 278–c. 206 BC), who was its third head, and who formalized much of Stoic doctrine. He is supposed to have written over 700 works, including at least 300 on logic, almost none of which survive. Unlike with Aristotle, we have no complete works by the Megarians or the early Stoics, and have to rely mostly on accounts (sometimes hostile) by later sources, including prominently Diogenes Laertius
, Sextus Empiricus
, Galen
, Aulus Gellius
, Alexander of Aphrodisias
and Cicero
.
Three significant 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
.
and continued to develop through to early modern times, without any known influence from Greek logic. Medhatithi Gautama (c. 6th century BCE) founded the anviksiki school of logic. The Mahabharata
(12.173.45), around the 5th century BCE, refers to the anviksiki and tarka schools of logic. (c. 5th century BCE) developed a form of logic (to which Boolean logic
has some similarities) for his formulation of Sanskrit grammar
. Logic is described by Chanakya
(c. 350-283 BCE) in his Arthashastra
as an independent field of inquiry anviksiki.
Two of the six Indian schools of thought deal with logic: Nyaya
and Vaisheshika
. The Nyaya Sutras
of Aksapada Gautama (c. 2nd century CE) 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
(c. 150-250 CE), the founder of the Madhyamika ("Middle Way") developed an analysis known as the catuskoti
(Sanskrit). 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
(c 480-540 CE), who developed a formal syllogistic, 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. This later school began around eastern India
and Bengal
, and developed theories resembling modern logic, such as Gottlob Frege
's "distinction between sense and reference of proper names" and his "definition of number," as well as the Navya-Nyaya theory of "restrictive conditions for universals" anticipating some of the developments in modern set theory
. Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as Charles Babbage
, Augustus De Morgan
, and particularly George Boole
, as confirmed by his wife Mary Everest Boole
who wrote in an "open letter to Dr Bose" titled "Indian Thought and Western Science in the Nineteenth Century" written in 1901: "Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan and George Boole on the mathematical atmosphere of 1830-1865"
, 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
.
, Avicenna
, Al-Ghazali
, Averroes
and other Muslim logicians both criticized and developed Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West. Al-Farabi
(Alfarabi) (873–950) was an Aristotelian logician who 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
. Al-Farabi also considered the theories of conditional syllogisms and analogical inference
, which were part of the Stoic
tradition of logic rather than the Aristotelian.
Ibn Sina
(Avicenna) (980–1037) was the founder of Avicennian logic, which replaced Aristotelian logic as the domininant system of logic in the Islamic world, and also had an important influence on Western medieval writers such as Albertus Magnus
. Avicenna wrote on the hypothetical syllogism
and on the propositional calculus
, which were both part of the Stoic logical tradition. He developed an original theory of “temporally
modalized
” syllogistic and made use of inductive logic
, such as the methods of agreement, difference and concomitant variation
which are critical to the scientific method
. One of Avicenna's ideas had a particularly important influence on Western logicians such as William of Ockham
. Avicenna's word for a meaning or notion (ma'na), was translated by the scholastic logicians as the Latin intentio. In medieval logic and epistemology, this is a sign in the mind that naturally represents a thing. This was crucial to the development of Ockham's conceptualism
. A universal term (e.g. "man") does not signify a thing existing in reality, but rather a sign in the mind (intentio in intellectu) which represents many things in reality. Ockham cites Avicenna's commentary on Metaphysics V in support of this view.
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). Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a Post-Avicennian logic. This was further elaborated by his student Afdaladdîn al-Khûnajî (d. 1249), who developed a form of logic revolving around the subject matter of concept
ions and assents
. In response to this tradition, Nasir al-Din al-Tusi
(1201–1274) began a tradition of Neo-Avicennian logic which remained faithful to Avicenna's work and existed as an alternative to the more dominant Post-Avicennian school over the following centuries.
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), the Ar-Radd 'ala al-Mantiqiyyin (Refutation of Greek Logicians), where he argued against the usefulness, though not the validity, of the syllogism
and in favour of inductive reasoning
. Ibn Taymiyyah also argued against the certainty of syllogistic arguments
and in favour of analogy
. His argument is that concepts founded on induction
are themselves not certain but only probable, and thus a syllogism based on such concepts is no more certain than an argument based on analogy. He further claimed that induction itself is founded on a process of analogy. His model of analogical reasoning was based on that of juridical arguments. This model of analogy has been used in the recent work of John F. Sowa
.
The Sharh al-takmil fi'l-mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century is the last major Arabic work on logic that has been studied. However, "thousands upon thousands of pages" on logic were written between the 14th and 19th centuries, though only a fraction of the texts written during this period have been studied by historians, hence little is known about the original work on Islamic logic produced during this later period.
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, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics. 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). An important work in this tradition was the Logica Ingredientibus of Peter Abelard
(1079–1142). His direct influence was small, but his influence through pupils such as John of Salisbury
was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed.
By the early thirteenth century the remaining works of Aristotle's Organon (including the Prior Analytics
, Posterior Analytics
and the Sophistical Refutations) had been recovered in the West. Logical work until then was mostly paraphrasis or commentary on the work of Aristotle. The period from the middle of the thirteenth to the middle of the fourteenth century was one of significant developments in logic, particularly in three areas which were original, with little foundation in the Aristotelian tradition that came before. These were:
The last great works in this tradition are the Logic of John Poinsot (1589–1644, known as John of St Thomas), the Metaphysical Disputations of Francisco Suarez
(1548–1617), and the Logica Demonstrativa of Giovanni Girolamo Saccheri
(1667–1733).
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 the nineteenth century. 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. The account of proposition
s 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)
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". This method is known as inductive reasoning
. 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.
Other works in the textbook 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). Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lay in introspection influenced the view that logic is best understood as a branch of psychology, an approach to the subject which dominated the next fifty years of its development, especially in Germany.
G.W.F. Hegel indicated the importance of logic to his philosophical system when he condensed his extensive Science of Logic
into a shorter work published in 1817 as the first volume of his Encyclopaedia of the Philosophical Sciences. The "Shorter" or "Encyclopaedia" Logic, as it is often known, lays out a series of transitions which leads from the most empty and abstract of categories – Hegel begins with "Pure Being" and "Pure Nothing" – to the "Absolute
" – the category which contains and resolves all the categories which preceded it. Despite the title, Hegel's Logic is not really a contribution to the science of valid inference. Rather than deriving conclusions about concepts through valid inference from premises, Hegel seeks to show that thinking about one concept compels thinking about another concept (one cannot, he argues, possess the concept of "Quality" without the concept of "Quantity"); and the compulsion here is not a matter of individual psychology, but arises almost organically from the content of the concepts themselves. His purpose is to show the rational structure of the "Absolute" – indeed of rationality itself. The method by which thought is driven from one concept to its contrary, and then to further concepts, is known as the Hegelian dialectic
.
Although Hegel's Logic has had little impact on mainstream logical studies, its influence can be seen in the work of the British Idealists
– for example in F.H. Bradley's Principles of Logic (1883) – and in the economic, political and philosophical studies of Karl Marx
and the various schools of Marxism
.
. The German psychologist Wilhelm Wundt
, for example, discussed deriving "the logical from the psychological laws of thought", emphasizing that "psychological thinking is always the more comprehensive form of thinking." This view was widespread among German philosophers of the period: Theodor Lipps
described logic as "a specific discipline of psychology"; Christoph von Sigwart
understood logical necessity as grounded in the individual's compulsion to think in a certain way; and Benno Erdmann argued that "logical laws only hold within the limits of our thinking" Such was the dominant view of logic in the years following Mill's work. This psychological approach to logic was rejected by Gottlob Frege
. It was also subjected to an extended and destructive critique by Edmund Husserl
in the first volume of his Logical Investigations (1900), an assault which has been described as "overwhelming". Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven, and that skepticism
and relativism
were unavoidable consequences.
Such criticisms did not immediately extirpate so-called "psychologism
". For example, the American philosopher Josiah Royce
, while acknowledging the force of Husserl's critique, remained "unable to doubt" that progress in psychology would be accompanied by progress in logic, and vice versa.
. The development of the modern so-called "symbolic" or "mathematical" logic during this period is the most significant in the 2,000-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 logic 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 would depend 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 medieval logicians called "syncategoremata
") and the categoric terms are expressed in symbols. Finally, modern logic strictly avoids psychological, epistemological and metaphysical questions.
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 Lull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death. Leibniz says that ordinary languages 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; hence, he proposed to identify an alphabet of human thought
comprising fundamental concepts which could be composed to express complex ideas, and create a calculus ratiocinator which would make reasoning "as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate."
Gergonne
(1816) says that reasoning does not have to be about objects about which we have perfectly clear ideas, 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.
Modern logic begins with the so-called "algebraic school", originating with Boole and including Peirce, Jevons
, Schröder
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, Lincolnshire
. 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 algebraic 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 Schröder 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 binary operation, "neither ... nor ...
" and equally well "not both ... and ...
", however, like many of Peirce's innovations, this remained unknown or unnoticed until Sheffer
rediscovered it in 1913. Boole's early work also lacks the idea of the logical sum which originates in Peirce (1867), Schröder
(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 Schröder's monumental Vorlesungen über die Algebra der Logik ("Lectures on the Algebra of Logic", vol iii 1895), although the original idea was again anticipated by Peirce.
. 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 Begriffsschrift
is important. 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 Begriffsschrift ("concept script") 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 significant innovation, however, was his explanation of the quantifier in terms of 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
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
whereas "All the inhabitants are Europeans or all the inhabitants are Asiatics" is
As Frege remarked in a critique of Boole's calculus:
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
means that to every girl there corresponds some boy (any one will do) who the girl kissed. But
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 of mathematical induction
.
This period overlaps with the work of the so-called "mathematical school", which included Dedekind
, Pasch
, Peano
, Hilbert
, Zermelo
, Huntington
, Veblen
and Heyting
. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory.
The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by Bertrand Russell
. This proved that the Frege's naive set theory
led to a contradiction. Frege's theory is that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not). This contradiction is now known as Russell's paradox
. One important method of resolving this paradox was proposed by Ernst Zermelo
. Zermelo set theory
was the first axiomatic set theory. It was developed into the now-canonical Zermelo–Fraenkel set theory
(ZF).
The monumental Principia Mathematica
, a three-volume work on the foundations of mathematics
, written by Russell and Alfred North Whitehead
and published 1910–13 also included an attempt to resolve the paradox, by means of an elaborate system of types: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak of the "set of all sets". The Principia was an attempt to derive all mathematical truths from a well-defined set of axiom
s and inference rules in symbolic logic
.
and Tarski
dominate the 1930s, a crucial period in the development of metamathematics
– the study of mathematics using mathematical methods to produce metatheories
, or mathematical theories about other mathematical theories. Early investigations into metamathematics had been driven by Hilbert's program
. which sought to resolve the ongoing crisis in the foundations of mathematics by grounding all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. Work on metamathematics culminated in the work of Gödel, who in 1929 showed that a given first-order sentence
is deducible
if and only if is logically valid – i.e. it is true in every structure
for its language. This is known as Gödel's completeness theorem
. A year later, he proved two important theorems, which showed Hibert's program to be unattainable in its original form. The first is that no consistent system of axioms whose theorems can be listed by an effective procedure
such as an algorithm
or computer program is capable of proving all facts about the natural number
s. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second is that if such a system is also capable of proving certain basic facts about the natural numbers, then the system cannot prove the consistency of the system itself. These two results are known as Gödel's incompleteness theorems
, or simply Gödel's Theorem. Later in the decade, Gödel developed the concept of set-theoretic constructibility, as part of his proof that the axiom of choice and the continuum hypothesis
are consistent with Zermelo-Fraenkel set theory.
In proof theory
, Gerhard Gentzen
developed natural deduction
and the sequent calculus
. The former attempts to model logical reasoning as it 'naturally' occurs in practice and is most easily applied to intuitionistic logic, while the latter was devised to clarify the derivation of logical proofs in any formal system. Since Gentzen's work, natural deduction and sequent calculi have been widely applied in the fields of proof theory, mathematical logic and computer science. Gentzen also proved normalization and cut-elimination theorems for intuitionistic and classical logic which could be used to reduce logical proofs to a normal form.
Alfred Tarski
, a pupil of Łukasiewicz, is best known for his definition of truth and logical consequence, and the semantic concept of logical satisfaction
. In 1933, he published (in Polish) The concept of truth in formalized languages, in which he proposed his semantic theory of truth
: a sentence such as "snow is white" is true if and only if snow is white. Tarski's theory separated the metalanguage
, which makes the statement about truth, from the object language
, which contains the sentence whose truth is being asserted, and gave a correspondence (the T-schema
) between phrases in the object language and elements of an interpretation
. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of model theory
. Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as completeness
, decidability
, consistency
and definability
. According to Anita Feferman, Tarski "changed the face of logic in the twentieth century".
Alonzo Church
and Alan Turing
proposed formal models of computability, giving independent negative solutions to Hilbert's Entscheidungsproblem
in 1936 and 1937, respectively. The Entscheidungsproblem asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the halting problem
as a key example of a mathematical problem without an algorithmic solution.
Church's system for computation developed into the modern λ-calculus, while the Turing machine
became a standard model for a general-purpose computing device. It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the Church–Turing thesis
that any deterministic algorithm
that can be carried out by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both Peano arithmetic and first-order logic
are undecidable
. Later work by Emil Post and Stephen Cole Kleene
in the 1940s extended the scope of computability theory and introduced the concept of degrees of unsolvability.
The results of the first few decades of the twentieth century also had an impact upon analytic philosophy
and philosophical logic
, particularly from the 1950s onwards, in subjects such as modal logic
, temporal logic
, deontic logic
, and relevance logic
.
After World War II, mathematical logic
branched into four inter-related but separate areas of research: model theory
, proof theory
, computability theory
, and set theory
.
In set theory, the method of forcing
revolutionized the field by providing a robust method for constructing models and obtaining independence results. Paul Cohen
introduced this method in 1962 to prove the independence of the continuum hypothesis
and the axiom of choice from Zermelo–Fraenkel set theory
. His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic.
Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. It developed into a study of abstract computability, which became known as recursion theory
. The priority method
, discovered independently by Albert Muchnik and Richard Friedberg in the 1950s, led to major advances in the understanding of the degrees of unsolvability and related structures. Research into higher-order computability theory demonstrated its connections to set theory. The fields of constructive analysis and computable analysis
were developed to study the effective content of classical mathematical theorems; these in turn inspired the program of reverse mathematics
. A separate branch of computability theory, computational complexity theory
, was also characterized in logical terms as a result of investigations into descriptive complexity
.
Model theory applies the methods of mathematical logic to study models of particular mathematical theories. Alfred Tarski published much pioneering work in the field, which is named after a series of papers he published under the title Contributions to the theory of models. In the 1960s, Abraham Robinson
used model-theoretic techniques to develop calculus and analysis based on infinitesimals
, a problem that first had been proposed by Leibniz.
In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability
method invented by Georg Kreisel
and Gödel's Dialectica interpretation
. This work inspired the contemporary area of proof mining
. The Curry-Howard correspondence emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and typed lambda calculi
used in computer science. As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. Advances were also made in ordinal analysis
and the study of independence results in arithmetic such as the Paris–Harrington theorem
.
This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, tense logic is a formalised system for representing, and reasoning about, propositions qualified in terms of time. The philosopher Arthur Prior
played a significant role in its development in the 1960s. Modal logic
s extend the scope of formal logic to include the elements of modality
(for example, possibility
and necessity). The ideas of Saul Kripke
, particularly about possible world
s, and the formal system now called Kripke semantics
have had a profound impact on analytic philosophy
. His best known and most influential work is Naming and Necessity
(1980). Deontic logic
s are closely related to modal logics: they attempt to capture the logical features of obligation
, permission and related concepts. Ernst Mally
, a pupil of Alexius Meinong
, was the first to propose a formal deontic system in his Grundgesetze des Sollens, based on the syntax of Whitehead's and Russell's propositional calculus
. Another logical system founded after World War II was fuzzy logic
by Iranian mathematician Lotfi Asker Zadeh
in 1965.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
). Formal logic was developed in ancient times in China
Logic in China
In the history of logic, logic in China plays a particularly interesting role due to its length and relative isolation from the strong current of development of the study of logic in Europe and the Islamic world, though it may have some influence from Indian logic due to the spread of...
, India
Indian logic
The development of Indian logic dates back to the anviksiki of Medhatithi Gautama the Sanskrit grammar rules of Pāṇini ; the Vaisheshika school's analysis of atomism ; the analysis of inference by Gotama , founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna...
, and Greece
Greek philosophy
Ancient Greek philosophy arose in the 6th century BCE and continued through the Hellenistic period, at which point Ancient Greece was incorporated in the Roman Empire...
. Greek logic, particularly Aristotelian logic, found wide application and acceptance in science and mathematics.
Aristotle's logic was further developed by Islamic
Logic in Islamic philosophy
Logic played an important role in Islamic philosophy .Islamic Logic or mantiq is similar science to what is called Traditional Logic in Western Sciences.- External links :*Routledge Encyclopedia of Philosophy: , Routledge, 1998...
and Christian
Christian philosophy
Christian philosophy may refer to any development in philosophy that is characterised by coming from a Christian tradition.- Origins of Christian philosophy :...
philosophers in the Middle Ages, reaching a high point in the mid-fourteenth century. The period between the fourteenth century and the beginning of the nineteenth century was largely one of decline and neglect, and is regarded as barren by at least one historian of logic.
Logic was revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. 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.
Progress in mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
in the first few decades of the twentieth century, particularly arising from the work of Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
and Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
, had a significant impact on analytic philosophy
Analytic philosophy
Analytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...
and philosophical logic
Philosophical logic
Philosophical logic is a term introduced by Bertrand Russell to represent his idea that the workings of natural language and thought can only be adequately represented by an artificial language; essentially it was his formalization program for the natural language...
, particularly from the 1950s onwards, in subjects such as modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
, temporal logic
Temporal logic
In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry...
, deontic logic
Deontic logic
Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts...
, and relevance logic
Relevance logic
Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
.
Prehistory of logic
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, which originally meant the same as "land measurement". In particular, the ancient Egypt
Ancient Egypt
Ancient Egypt was an ancient civilization of Northeastern Africa, concentrated along the lower reaches of the Nile River in what is now the modern country of Egypt. Egyptian civilization coalesced around 3150 BC with the political unification of Upper and Lower Egypt under the first pharaoh...
ians had empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
ly discovered some truths of geometry
Egyptian mathematics
Egyptian mathematics is the mathematics that was developed and used in Ancient Egypt from ca. 3000 BC to ca. 300 BC.-Overview:Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found at Tomb Uj at Abydos. These labels appear to have been used as tags for...
, such as the formula for the volume of a truncated pyramid
Frustum
In geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
.
Another origin can be seen in Babylonia
Babylonia
Babylonia was an ancient cultural region in central-southern Mesopotamia , with Babylon as its capital. Babylonia emerged as a major power when Hammurabi Babylonia was an ancient cultural region in central-southern Mesopotamia (present-day Iraq), with Babylon as its capital. Babylonia emerged as...
. Esagil-kin-apli's medical Diagnostic Handbook in the 11th century BC was based on a logical set of axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s and assumptions, while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems, an important contribution to the philosophy of science
Philosophy of science
The philosophy of science is concerned with the assumptions, foundations, methods and implications of science. It is also concerned with the use and merit of science and sometimes overlaps metaphysics and epistemology by exploring whether scientific results are actually a study of truth...
.
Before Plato
While the ancient Egyptians empirically discovered some truths of geometry, 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 PythagorasPythagoras
Pythagoras of Samos was an Ionian Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him...
in the late sixth century BC. 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, that is, independent of the particular subject matter in question. Fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy
Platonic Academy
The Academy was founded by Plato in ca. 387 BC in Athens. Aristotle studied there for twenty years before founding his own school, the Lyceum. The Academy persisted throughout the Hellenistic period as a skeptical school, until coming to an end after the death of Philo of Larissa in 83 BC...
.
Separately from geometry, the idea of a standard argument pattern is found in the Reductio ad absurdum
Reductio ad absurdum
In logic, proof by contradiction is a form of proof that establishes the truth or validity of a proposition by showing that the proposition's being false would imply a contradiction...
used by Zeno of Elea
Zeno of Elea
Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known for his paradoxes, which Bertrand Russell has described as "immeasurably subtle and profound".- Life...
, a pre-Socratic philosopher of the fifth century BC. 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
Parmenides
Parmenides of Elea was an ancient Greek philosopher born in Elea, a Greek city on the southern coast of Italy. He was the founder of the Eleatic school of philosophy. The single known work of Parmenides is a poem, On Nature, which has survived only in fragmentary form. In this poem, Parmenides...
portrays Zeno as claiming to have written a book defending the monism
Monism
Monism is any philosophical view which holds that there is unity in a given field of inquiry. Accordingly, some philosophers may hold that the universe is one rather than dualistic or pluralistic...
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
Euclid of Megara
Euclid of Megara was a Greek Socratic philosopher who founded the Megarian school of philosophy. He was a pupil of Socrates in the late 5th century BCE, and was present at his death. He held the supreme good to be one, eternal and unchangeable, and denied the existence of anything contrary to the...
, 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
Dissoi Logoi
Dissoi Logoi is a rhetorical exercise dating back at least to the 3rd century AD of arguing a topic from both sides...
, probably written at the beginning of the fourth century BC. 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 PlatoPlato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
(428–347) include any formal logic, but they include important contributions to the field of philosophical logic
Philosophical logic
Philosophical logic is a term introduced by Bertrand Russell to represent his idea that the workings of natural language and thought can only be adequately represented by an artificial language; essentially it was his formalization program for the natural language...
. Plato raises three 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
Theaetetus (dialogue)
The Theaetetus is one of Plato's dialogues concerning the nature of knowledge. The framing of the dialogue begins when Euclides tells his friend Terpsion that he had written a book many years ago based on what Socrates had told him of a conversation he'd had with Theaetetus when Theaetetus was...
, where Plato identifies thought or opinion with talk or discourse (logos). The second question is a result of Plato's theory of Forms
Theory of Forms
Plato's theory of Forms or theory of Ideas asserts that non-material abstract forms , and not the material world of change known to us through sensation, possess the highest and most fundamental kind of reality. When used in this sense, the word form is often capitalized...
. 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. In both The Republic and The Sophist
Sophist (dialogue)
The Sophist is a Platonic dialogue from the philosopher's late period, most likely written in 360 BCE. Having criticized his Theory of Forms in the Parmenides, Plato presents a new conception of the forms in the Sophist, more mundane and down-to-earth than its predecessor...
, Plato suggests that the necessary connection between the premisses and the conclusion of an argument corresponds to a necessary connection between "forms". The third question is about definition
Definition
A definition is a passage that explains the meaning of a term , or a type of thing. The term to be defined is the definiendum. A term may have many different senses or meanings...
. 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. This had a great influence on Aristotle, in particular Aristotle's notion of the essence
Essence
In philosophy, essence is the attribute or set of attributes that make an object or substance what it fundamentally is, and which it has by necessity, and without which it loses its identity. Essence is contrasted with accident: a property that the object or substance has contingently, without...
of a thing, the "what it is to be" a particular thing of a certain kind.
Aristotle's logic
The logic of AristotleAristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, and particularly his theory of the syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
, has had an enormous influence in Western thought
Western thought
The term Western thought is usually associated with the cultural tradition that traces its origins to Greek thought and the Abrahamic religions...
. His logical works, called the Organon
Organon
The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...
, 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 CategoriesCategories (Aristotle)The Categories is a text from Aristotle's Organon that enumerates all the possible kinds of thing that can be the subject or the predicate of a proposition...
, a study of the ten kinds of primitive term. - The TopicsTopics (Aristotle)The Topics is the name given to one of Aristotle's six works on logic collectively known as the Organon. The other five are:*Categories*De Interpretatione*Prior Analytics*Posterior Analytics*On Sophistical Refutations...
(with an appendix called On Sophistical RefutationsOn Sophistical RefutationsSophistical Refutations is a text in Aristotle's Organon.Aristotle identified thirteen fallacies, as follows:Verbal fallacies* Accent or emphasis* Amphibology* Equivocation* Composition* Division...
), a discussion of dialectics. - On Interpretation, an analysis of simple categorical propositionCategorical propositionA categorical proposition contains two categorical terms, the subject and the predicate, and affirms or denies the latter of the former. Categorical propositions occur in categorical syllogisms and both are discussed in Aristotle's Prior Analytics....
s, into simple terms, negation, and signs of quantity; and a comprehensive treatment of the notions of oppositionSquare of oppositionIn the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four propositions of the system are logically related to each of the others...
and conversion. - The Prior AnalyticsPrior AnalyticsThe Prior Analytics is Aristotle's work on deductive reasoning, specifically the syllogism. It is also part of his Organon, which is the instrument or manual of logical and scientific methods....
, a formal analysis of valid argument or "syllogism". - The Posterior AnalyticsPosterior AnalyticsThe Posterior Analytics is a text from Aristotle's Organon that deals with demonstration, definition, and scientific knowledge. The demonstration is distinguished as a syllogism productive of scientific knowledge, while the definition marked as the statement of a thing's nature, .....
, a study of scientific demonstration, containing Aristotle's mature views on logic.
These works are of outstanding importance in the history of logic. Aristotle was the first logician to attempt a systematic analysis of logical syntax, into noun (or term
Terminology
Terminology is the study of terms and their use. Terms are words and compound words that in specific contexts are given specific meanings, meanings that may deviate from the meaning the same words have in other contexts and in everyday language. The discipline Terminology studies among other...
), and verb. In the Categories, he attempted to classify all the possible things that a term can refer to. This idea underpins his philosophical work, the Metaphysics
Metaphysics (Aristotle)
Metaphysics is one of the principal works of Aristotle and the first major work of the branch of philosophy with the same name. The principal subject is "being qua being", or being understood as being. It examines what can be asserted about anything that exists just because of its existence and...
, which also had a profound influence on Western thought. He was the first to deal with the principles of contradiction and excluded middle
Law of excluded middle
In logic, the law of excluded middle is the third of the so-called three classic laws of thought. It states that for any proposition, either that proposition is true, or its negation is....
in a systematic way. He was the first formal logician (i.e. he gave the principles of reasoning using variables to show the underlying logical form
Logical form
In logic, the logical form of a sentence or set of sentences is the form obtained by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks on a form...
of arguments). He was looking for relations of dependence which characterise necessary inference, and distinguished the validity
Validity
In logic, argument is valid if and only if its conclusion is entailed by its premises, a formula is valid if and only if it is true under every interpretation, and an argument form is valid if and only if every argument of that logical form is valid....
of these relations, from the truth of the premises (the soundness
Soundness
In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word...
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. In the Topics and Sophistical Refutations he also developed a theory of non-formal logic (e.g. the theory of fallacies).
Stoic logic
The other great school of Greek logic is that of the StoicSTOIC
STOIC was a variant of Forth.It started out at the MIT and Harvard Biomedical Engineering Centre in Boston, and was written in the mid 1970s by Jonathan Sachs...
s. Stoic logic traces its roots back to the late 5th century BC philosopher, Euclid of Megara
Euclid of Megara
Euclid of Megara was a Greek Socratic philosopher who founded the Megarian school of philosophy. He was a pupil of Socrates in the late 5th century BCE, and was present at his death. He held the supreme good to be one, eternal and unchangeable, and denied the existence of anything contrary to the...
, a pupil of Socrates
Socrates
Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
and slightly older contemporary of Plato. His pupils and successors were called "Megarians", or "Eristics", and later the "Dialecticians". The two most important dialecticians of the Megarian school were Diodorus Cronus
Diodorus Cronus
Diodorus Cronus was a Greek philosopher and dialectician connected to the Megarian school. He was most notable for logic innovations, including his master argument fomulated in response to Aristotle's discussion of future contingents.-Life:...
and Philo
Philo the Dialectician
Philo the Dialectician was a dialectic philosopher of the Megarian school. He is often called Philo of Megara although the city of his birth is unknown...
who were active in the late 4th century BC. The Stoics adopted the Megarian logic and systemized it. The most important member of the school was Chrysippus
Chrysippus
Chrysippus of Soli was a Greek Stoic philosopher. He was a native of Soli, Cilicia, but moved to Athens as a young man, where he became a pupil of Cleanthes in the Stoic school. When Cleanthes died, around 230 BC, Chrysippus became the third head of the school...
(c. 278–c. 206 BC), who was its third head, and who formalized much of Stoic doctrine. He is supposed to have written over 700 works, including at least 300 on logic, almost none of which survive. Unlike with Aristotle, we have no complete works by the Megarians or the early Stoics, and have to rely mostly on accounts (sometimes hostile) by later sources, including prominently Diogenes Laertius
Diogenes Laertius
Diogenes Laertius was a biographer of the Greek philosophers. Nothing is known about his life, but his surviving Lives and Opinions of Eminent Philosophers is one of the principal surviving sources for the history of Greek philosophy.-Life:Nothing is definitively known about his life...
, Sextus Empiricus
Sextus Empiricus
Sextus Empiricus , was a physician and philosopher, and has been variously reported to have lived in Alexandria, Rome, or Athens. His philosophical work is the most complete surviving account of ancient Greek and Roman skepticism....
, Galen
Galen
Aelius Galenus or Claudius Galenus , better known as Galen of Pergamon , was a prominent Roman physician, surgeon and philosopher...
, Aulus Gellius
Aulus Gellius
Aulus Gellius , was a Latin author and grammarian, who was probably born and certainly brought up in Rome. He was educated in Athens, after which he returned to Rome, where he held a judicial office...
, Alexander of Aphrodisias
Alexander of Aphrodisias
Alexander of Aphrodisias was a Peripatetic philosopher and the most celebrated of the Ancient Greek commentators on the writings of Aristotle. He was a native of Aphrodisias in Caria, and lived and taught in Athens at the beginning of the 3rd century, where he held a position as head of the...
and Cicero
Cicero
Marcus Tullius Cicero , was a Roman philosopher, statesman, lawyer, political theorist, and Roman constitutionalist. He came from a wealthy municipal family of the equestrian order, and is widely considered one of Rome's greatest orators and prose stylists.He introduced the Romans to the chief...
.
Three significant contributions of the Stoic school were (i) their account of modality
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
, (ii) their theory of the Material conditional
Material conditional
The material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q is logically equivalent to the negative compound: not . A material conditional compound itself is often simply called a conditional...
, and (iii) their account of meaning and truth
Truth
Truth has a variety of meanings, such as the state of being in accord with fact or reality. It can also mean having fidelity to an original or to a standard or ideal. In a common usage, it also means constancy or sincerity in action or character...
.
- Modality. According to Aristotle, the Megarians of his day claimed there was no distinction between potentiality and actuality. Diodorus Cronus 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, by contrast, denied the second premiss and said that the impossible could follow from the possible.
- Conditional statements. The first logicians to debate conditional statementsMaterial conditionalThe material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q is logically equivalent to the negative compound: not . A material conditional compound itself is often simply called a conditional...
were Diodorus and his pupil Philo of Megara. 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."
- 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 India
Formal logic began independently in ancient IndiaHistory of India
The history of India begins with evidence of human activity of Homo sapiens as long as 75,000 years ago, or with earlier hominids including Homo erectus from about 500,000 years ago. The Indus Valley Civilization, which spread and flourished in the northwestern part of the Indian subcontinent from...
and continued to develop through to early modern times, without any known influence from Greek logic. Medhatithi Gautama (c. 6th century BCE) founded the anviksiki school of logic. The Mahabharata
Mahabharata
The Mahabharata is one of the two major Sanskrit epics of ancient India and Nepal, the other being the Ramayana. The epic is part of itihasa....
(12.173.45), around the 5th century BCE, refers to the anviksiki and tarka schools of logic. (c. 5th century BCE) developed a form of logic (to which Boolean logic
Boolean logic
Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
has some similarities) for his formulation of Sanskrit grammar
Vyakarana
The Sanskrit grammatical tradition of ' is one of the six Vedanga disciplines. It has its roots in late Vedic India, and includes the famous work, The Sanskrit grammatical tradition of ' is one of the six Vedanga disciplines. It has its roots in late Vedic India, and includes the famous work, ...
. Logic is described by Chanakya
Chanakya
Chānakya was a teacher to the first Maurya Emperor Chandragupta , and the first Indian emperor generally considered to be the architect of his rise to power. Traditionally, Chanakya is also identified by the names Kautilya and VishnuGupta, who authored the ancient Indian political treatise...
(c. 350-283 BCE) in his Arthashastra
Arthashastra
The Arthashastra is an ancient Indian treatise on statecraft, economic policy and military strategy which identifies its author by the names Kautilya and , who are traditionally identified with The Arthashastra (IAST: Arthaśāstra) is an ancient Indian treatise on statecraft, economic policy and...
as an independent field of inquiry anviksiki.
Two of the six Indian schools of thought deal with logic: Nyaya
Nyaya
' is the name given to one of the six orthodox or astika schools of Hindu philosophy—specifically the school of logic...
and Vaisheshika
Vaisheshika
Vaisheshika or ' is one of the six Hindu schools of philosophy of India. Historically, it has been closely associated with the Hindu school of logic, Nyaya....
. The Nyaya Sutras
Nyaya Sutras
The Nyāya Sūtras are an ancient Indian text on of philosophy composed by ' . The sutras contain five chapters, each with two sections...
of Aksapada Gautama (c. 2nd century CE) constitute the core texts of the Nyaya school, one of the six orthodox schools of Hindu
Hindu
Hindu refers to an identity associated with the philosophical, religious and cultural systems that are indigenous to the Indian subcontinent. As used in the Constitution of India, the word "Hindu" is also attributed to all persons professing any Indian religion...
philosophy. This realist
Philosophical realism
Contemporary philosophical realism is the belief that our reality, or some aspect of it, is ontologically independent of our conceptual schemes, linguistic practices, beliefs, etc....
school developed a rigid five-member schema of inference
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
involving an initial premise, a reason, an example, an application and a conclusion. The idealist
Idealism
In philosophy, idealism is the family of views which assert that reality, or reality as we can know it, is fundamentally mental, mentally constructed, or otherwise immaterial. Epistemologically, idealism manifests as a skepticism about the possibility of knowing any mind-independent thing...
Buddhist philosophy
Buddhist philosophy
Buddhist philosophy deals extensively with problems in metaphysics, phenomenology, ethics, and epistemology.Some scholars assert that early Buddhist philosophy did not engage in ontological or metaphysical speculation, but was based instead on empirical evidence gained by the sense organs...
became the chief opponent to the Naiyayikas. Nagarjuna
Nagarjuna
Nāgārjuna was an important Buddhist teacher and philosopher. Along with his disciple Āryadeva, he is credited with founding the Mādhyamaka school of Mahāyāna Buddhism...
(c. 150-250 CE), the founder of the Madhyamika ("Middle Way") developed an analysis known as the catuskoti
Catuṣkoṭi
Catuṣkoṭi is a logical argument of a 'suite of four discrete functions' or 'an indivisible quaternity' that has multiple applications and has been important in the Dharmic traditions of Indian logic and the Buddhadharma logico-epistemological traditions, particularly those of the Madhyamaka...
(Sanskrit). 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
Dignaga
Dignāga was an Indian scholar and one of the Buddhist founders of Indian logic.He was born into a Brahmin family in Simhavakta near Kanchi Kanchipuram), and very little is known of his early years, except that he took as his spiritual preceptor Nagadatta of the Vatsiputriya school, before being...
(c 480-540 CE), who developed a formal syllogistic, and his successor Dharmakirti
Dharmakirti
Dharmakīrti , was an Indian scholar and one of the Buddhist founders of Indian philosophical logic. He was one of the primary theorists of Buddhist atomism, according to which the only items considered to exist are momentary states of consciousness.-History:Born around the turn of the 7th century,...
that Buddhist logic
Buddhist logic
Buddhist Logic, the categorical nomenclature modern Western discourse has extended to Buddhadharma traditions of 'Hetuvidya' and 'Pramanavada' , which arose circa 500CE, is a particular development, application and lineage of continuity of 'Indian Logic', from which it seceded...
reached its height. Their analysis centered on the definition of necessary logical entailment
Entailment
In logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
, "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
Navya-Nyaya
The Navya-Nyāya or Neo-Logical darśana of Indian logic and Indian philosophy was founded in the 13th century CE by the philosopher Gangeśa Upādhyāya of Mithila. It was a development of the classical Nyāya darśana. Other influences on Navya-Nyāya were the work of earlier philosophers Vācaspati...
, which developed a formal analysis of inference in the sixteenth century. This later school began around eastern India
East India
East India is a region of India consisting of the states of West Bengal, Bihar, Jharkhand, and Orissa. The states of Orissa and West Bengal share some cultural and linguistic characteristics with Bangladesh and with the state of Assam. Together with Bangladesh, West Bengal formed the...
and Bengal
Bengal
Bengal is a historical and geographical region in the northeast region of the Indian Subcontinent at the apex of the Bay of Bengal. Today, it is mainly divided between the sovereign land of People's Republic of Bangladesh and the Indian state of West Bengal, although some regions of the previous...
, and developed theories resembling modern logic, such as Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
's "distinction between sense and reference of proper names" and his "definition of number," as well as the Navya-Nyaya theory of "restrictive conditions for universals" anticipating some of the developments in modern set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
. Since 1824, Indian logic attracted the attention of many Western scholars, and has had an influence on important 19th-century logicians such as Charles Babbage
Charles Babbage
Charles Babbage, FRS was an English mathematician, philosopher, inventor and mechanical engineer who originated the concept of a programmable computer...
, Augustus De Morgan
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....
, and particularly George Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
, as confirmed by his wife Mary Everest Boole
Mary Everest Boole
Mary Everest Boole was a self-taught mathematician who is best known as an author of didactic works on mathematics, such as Philosophy and Fun of Algebra, and as the wife of fellow mathematician George Boole...
who wrote in an "open letter to Dr Bose" titled "Indian Thought and Western Science in the Nineteenth Century" written in 1901: "Think what must have been the effect of the intense Hinduizing of three such men as Babbage, De Morgan and George Boole on the mathematical atmosphere of 1830-1865"
Logic in China
In China, a contemporary of ConfuciusConfucius
Confucius , literally "Master Kong", was a Chinese thinker and social philosopher of the Spring and Autumn Period....
, Mozi
Mozi
Mozi |Lat.]] as Micius, ca. 470 BC – ca. 391 BC), original name Mo Di , was a Chinese philosopher during the Hundred Schools of Thought period . Born in Tengzhou, Shandong Province, China, he founded the school of Mohism, and argued strongly against Confucianism and Daoism...
, "Master Mo", is credited with founding the Mohist school
Mohism
Mohism or Moism was a Chinese philosophy developed by the followers of Mozi , 470 BC–c.391 BC...
, 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
Logicians
The Logicians or School of Names was a Chinese philosophical school that grew out of Mohism in the Warring States Period 479–221 BCE....
, are credited by some scholars for their early investigation of formal logic
Formal logic
Classical or traditional system of determining the validity or invalidity of a conclusion deduced from two or more statements...
. Unfortunately, due to the harsh rule of Legalism in the subsequent Qin Dynasty
Qin Dynasty
The Qin Dynasty was the first imperial dynasty of China, lasting from 221 to 207 BC. The Qin state derived its name from its heartland of Qin, in modern-day Shaanxi. The strength of the Qin state was greatly increased by the legalist reforms of Shang Yang in the 4th century BC, during the Warring...
, this line of investigation disappeared in China until the introduction of Indian philosophy by Buddhists
Buddhism
Buddhism is a religion and philosophy encompassing a variety of traditions, beliefs and practices, largely based on teachings attributed to Siddhartha Gautama, commonly known as the Buddha . The Buddha lived and taught in the northeastern Indian subcontinent some time between the 6th and 4th...
.
Logic in Islamic philosophy
The works of Al-FarabiAl-Farabi
' known in the West as Alpharabius , was a scientist and philosopher of the Islamic world...
, Avicenna
Avicenna
Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Sīnā , commonly known as Ibn Sīnā or by his Latinized name Avicenna, was a Persian polymath, who wrote almost 450 treatises on a wide range of subjects, of which around 240 have survived...
, Al-Ghazali
Al-Ghazali
Abu Hāmed Mohammad ibn Mohammad al-Ghazzālī , known as Algazel to the western medieval world, born and died in Tus, in the Khorasan province of Persia was a Persian Muslim theologian, jurist, philosopher, and mystic....
, Averroes
Averroes
' , better known just as Ibn Rushd , and in European literature as Averroes , was a Muslim polymath; a master of Aristotelian philosophy, Islamic philosophy, Islamic theology, Maliki law and jurisprudence, logic, psychology, politics, Arabic music theory, and the sciences of medicine, astronomy,...
and other Muslim logicians both criticized and developed Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West. Al-Farabi
Al-Farabi
' known in the West as Alpharabius , was a scientist and philosopher of the Islamic world...
(Alfarabi) (873–950) was an Aristotelian logician who discussed the topics of future contingents, the number and relation of the categories, the relation between logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and grammar
Grammar
In linguistics, grammar is the set of structural rules that govern the composition of clauses, phrases, and words in any given natural language. The term refers also to the study of such rules, and this field includes morphology, syntax, and phonology, often complemented by phonetics, semantics,...
, and non-Aristotelian forms of inference
Inference
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.Human inference Inference is the act or process of deriving logical conclusions...
. Al-Farabi also considered the theories of conditional syllogisms and analogical inference
Analogy
Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...
, which were part of the Stoic
STOIC
STOIC was a variant of Forth.It started out at the MIT and Harvard Biomedical Engineering Centre in Boston, and was written in the mid 1970s by Jonathan Sachs...
tradition of logic rather than the Aristotelian.
Ibn Sina
Avicenna
Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Sīnā , commonly known as Ibn Sīnā or by his Latinized name Avicenna, was a Persian polymath, who wrote almost 450 treatises on a wide range of subjects, of which around 240 have survived...
(Avicenna) (980–1037) was the founder of Avicennian logic, which replaced Aristotelian logic as the domininant system of logic in the Islamic world, and also had an important influence on Western medieval writers such as Albertus Magnus
Albertus Magnus
Albertus Magnus, O.P. , also known as Albert the Great and Albert of Cologne, is a Catholic saint. He was a German Dominican friar and a bishop, who achieved fame for his comprehensive knowledge of and advocacy for the peaceful coexistence of science and religion. Those such as James A. Weisheipl...
. Avicenna wrote on the hypothetical syllogism
Hypothetical syllogism
In logic, a hypothetical syllogism has two uses. In propositional logic it expresses one of the rules of inference, while in the history of logic, it is a short-hand for the theory of consequence.-Propositional logic:...
and on the propositional calculus
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
, which were both part of the Stoic logical tradition. He developed an original theory of “temporally
Temporal logic
In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry...
modalized
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
” syllogistic and made use of inductive logic
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
, such as the methods of agreement, difference and concomitant variation
Mill's Methods
Mill's Methods are five methods of induction described by philosopher John Stuart Mill in his 1843 book A System of Logic. They are intended to illuminate issues of causation.-Direct method of agreement:...
which are critical to the scientific method
Scientific method
Scientific method refers to a body of techniques for investigating phenomena, acquiring new knowledge, or correcting and integrating previous knowledge. To be termed scientific, a method of inquiry must be based on gathering empirical and measurable evidence subject to specific principles of...
. One of Avicenna's ideas had a particularly important influence on Western logicians such as William of Ockham
William of Ockham
William of Ockham was an English Franciscan friar and scholastic philosopher, who is believed to have been born in Ockham, a small village in Surrey. He is considered to be one of the major figures of medieval thought and was at the centre of the major intellectual and political controversies of...
. Avicenna's word for a meaning or notion (ma'na), was translated by the scholastic logicians as the Latin intentio. In medieval logic and epistemology, this is a sign in the mind that naturally represents a thing. This was crucial to the development of Ockham's conceptualism
Conceptualism
Conceptualism is a philosophical theory that explains universality of particulars as conceptualized frameworks situated within the thinking mind. Intermediate between Nominalism and Realism, the conceptualist view approaches the metaphysical concept of universals from a perspective that denies...
. A universal term (e.g. "man") does not signify a thing existing in reality, but rather a sign in the mind (intentio in intellectu) which represents many things in reality. Ockham cites Avicenna's commentary on Metaphysics V in support of this view.
Fakhr al-Din al-Razi
Fakhr al-Din al-Razi
Abu Abdullah Muhammad ibn Umar ibn al-Husayn al-Taymi al-Bakri al-Tabaristani Fakhr al-Din al-Razi , most commonly known as Fakhruddin Razi was a well-known Persian Sunni Muslim theologian and philosopher....
(b. 1149) criticised Aristotle's "first figure
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill
John Stuart Mill
John Stuart Mill was a British philosopher, economist and civil servant. An influential contributor to social theory, political theory, and political economy, his conception of liberty justified the freedom of the individual in opposition to unlimited state control. He was a proponent of...
(1806–1873). Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a Post-Avicennian logic. This was further elaborated by his student Afdaladdîn al-Khûnajî (d. 1249), who developed a form of logic revolving around the subject matter of concept
Concept
The word concept is used in ordinary language as well as in almost all academic disciplines. Particularly in philosophy, psychology and cognitive sciences the term is much used and much discussed. WordNet defines concept: "conception, construct ". However, the meaning of the term concept is much...
ions and assents
Grammar of Assent
An Essay in Aid of a Grammar of Assent is John Henry Newman's seminal work. While it was completed in 1870, Newman revealed to friends that it took him 20 years to write the book....
. In response to this tradition, Nasir al-Din al-Tusi
Nasir al-Din al-Tusi
Khawaja Muḥammad ibn Muḥammad ibn Ḥasan Ṭūsī , better known as Naṣīr al-Dīn al-Ṭūsī , was a Persian polymath and prolific writer: an astronomer, biologist, chemist, mathematician, philosopher, physician, physicist, scientist, theologian and Marja Taqleed...
(1201–1274) began a tradition of Neo-Avicennian logic which remained faithful to Avicenna's work and existed as an alternative to the more dominant Post-Avicennian school over the following centuries.
Systematic refutations of Greek logic were written by the Illuminationist school, founded by Shahab al-Din Suhrawardi
Shahab al-Din Suhrawardi
Other important Muslim mystics carry the name Suhrawardi, particularly Abu 'l-Najib al-Suhrawardi and his paternal nephew Abu Hafs Umar al-Suhrawardi."Shahāb ad-Dīn" Yahya ibn Habash as-Suhrawardī was a Persian...
(1155–1191), who developed the idea of "decisive necessity", which refers to the reduction of all modalities (necessity
Necessity
In U.S. criminal law, necessity may be either a possible justification or an exculpation for breaking the law. Defendants seeking to rely on this defense argue that they should not be held liable for their actions as a crime because their conduct was necessary to prevent some greater harm and when...
, possibility
Logical possibility
A logically possible proposition is one that can be asserted without implying a logical contradiction. This is to say that a proposition is logically possible if there is some coherent way for the world to be, under which the proposition would be true...
, contingency and impossibility
Impossibility
In contract law, impossibility is an excuse for the nonperformance of duties under a contract, based on a change in circumstances , the nonoccurrence of which was an underlying assumption of the contract, that makes performance of the contract literally impossible...
) 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), the Ar-Radd 'ala al-Mantiqiyyin (Refutation of Greek Logicians), where he argued against the usefulness, though not the validity, of the syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
and in favour of inductive reasoning
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
. Ibn Taymiyyah also argued against the certainty of syllogistic arguments
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
and in favour of analogy
Analogy
Analogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...
. His argument is that concepts founded on induction
Induction
-General use:* Induction , induction of childbirth* Rite of passage** Introduction of an individual into a body such as the armed forces** Formal introduction of a priest into possession of the position to which she or he has been presented and instituted...
are themselves not certain but only probable, and thus a syllogism based on such concepts is no more certain than an argument based on analogy. He further claimed that induction itself is founded on a process of analogy. His model of analogical reasoning was based on that of juridical arguments. This model of analogy has been used in the recent work of John F. Sowa
John F. Sowa
John Florian Sowa is the computer scientist who invented conceptual graphs, a graphic notation for logic and natural language, based on the structures in semantic networks and on the existential graphs of Charles S. Peirce. He is currently developing high-level "ontologies" for artificial...
.
The Sharh al-takmil fi'l-mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century is the last major Arabic work on logic that has been studied. However, "thousands upon thousands of pages" on logic were written between the 14th and 19th centuries, though only a fraction of the texts written during this period have been studied by historians, hence little is known about the original work on Islamic logic produced during this later period.
Logic in medieval Europe
"Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logicOrganon
The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...
developed in medieval Europe
Middle Ages
The Middle Ages is a periodization of European history from the 5th century to the 15th century. The Middle Ages follows the fall of the Western Roman Empire in 476 and precedes the Early Modern Era. It is the middle period of a three-period division of Western history: Classic, Medieval and Modern...
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, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics. Until the twelfth century the only works of Aristotle available in the West were the Categories, On Interpretation and Boethius' translation of the Isagoge
Isagoge
The Isagoge or "Introduction" to Aristotle's "Categories", written by Porphyry in Greek and translated into Latin by Boethius, was the standard textbook on logic for at least a millennium after his death. It was composed by Porphyry in Sicily during the years 268-270, and sent to Chrysaorium,...
of Porphyry
Porphyry (philosopher)
Porphyry of Tyre , Porphyrios, AD 234–c. 305) was a Neoplatonic philosopher who was born in Tyre. He edited and published the Enneads, the only collection of the work of his teacher Plotinus. He also wrote many works himself on a wide variety of topics...
(a commentary on the Categories). These works were known as the "Old Logic" (Logica Vetus or Ars Vetus). An important work in this tradition was the Logica Ingredientibus of Peter Abelard
Peter Abelard
Peter Abelard was a medieval French scholastic philosopher, theologian and preeminent logician. The story of his affair with and love for Héloïse has become legendary...
(1079–1142). His direct influence was small, but his influence through pupils such as John of Salisbury
John of Salisbury
John of Salisbury , who described himself as Johannes Parvus , was an English author, educationalist, diplomat and bishop of Chartres, and was born at Salisbury.-Early life and education:...
was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed.
By the early thirteenth century the remaining works of Aristotle's Organon (including the Prior Analytics
Prior Analytics
The Prior Analytics is Aristotle's work on deductive reasoning, specifically the syllogism. It is also part of his Organon, which is the instrument or manual of logical and scientific methods....
, Posterior Analytics
Posterior Analytics
The Posterior Analytics is a text from Aristotle's Organon that deals with demonstration, definition, and scientific knowledge. The demonstration is distinguished as a syllogism productive of scientific knowledge, while the definition marked as the statement of a thing's nature, .....
and the Sophistical Refutations) had been recovered in the West. Logical work until then was mostly paraphrasis or commentary on the work of Aristotle. The period from the middle of the thirteenth to the middle of the fourteenth century was one of significant developments in logic, particularly in three areas which were original, with little foundation in the Aristotelian tradition that came before. These were:
- The theory of suppositionSupposition theorySupposition theory was a branch of medieval logic that was probably aimed at giving accounts of issues similar to modern accounts of reference, plurality, tense, and modality, from within an Aristotelian context. Philosophers such as John Buridan, William of Ockham, William of Sherwood, Walter...
. Supposition theory deals with the way that predicates (e.g. 'man' range over a domain of individuals (e.g. all men). In the proposition 'every man is an animal', does the term 'man' range over or 'supposit for' men existing in the present? Or does the range include past and future men? Can a term supposit for non-existing individuals? Some medievalists have argued that this idea was a precursor of modern first order logic. "The theory of supposition with the associated theories of copulatio (sign-capacity of adjectival terms), ampliatio (widening of referential domain), and distributio constitute one of the most original achievements of Western medieval logic".
- The theory of syncategoremataSyncategorematic termIn scholastic logic, a syncategorematic term is a word that cannot serve as the subject or the predicate of a proposition, and thus cannot stand for any of Aristotle's categories, but can be used with other terms to form a proposition...
. Syncategoremata are terms which are necessary for logic, but which, unlike categorematic terms, do not signify on their own behalf, but 'co-signify' with other words. Examples of syncategoremata are 'and', 'not', 'every', 'if', and so on.
- The theory of consequences. A consequence is a hypothetical, conditional proposition: two propositions joined by the terms 'if ... then'. For example 'if a man runs, then God exists' (Si homo currit, Deus est). A fully developed theory of consequences is given in Book III of William of OckhamWilliam of OckhamWilliam of Ockham was an English Franciscan friar and scholastic philosopher, who is believed to have been born in Ockham, a small village in Surrey. He is considered to be one of the major figures of medieval thought and was at the centre of the major intellectual and political controversies of...
's work Summa Logicae. There, Ockham distinguishes between 'material' and 'formal' consequences, which are roughly equivalent to the modern material implicationMaterial conditionalThe material conditional, also known as material implication, is a binary truth function, such that the compound sentence p→q is logically equivalent to the negative compound: not . A material conditional compound itself is often simply called a conditional...
and logical implication respectively. Similar accounts are given by Jean BuridanJean BuridanJean Buridan was a French priest who sowed the seeds of the Copernican revolution in Europe. Although he was one of the most famous and influential philosophers of the late Middle Ages, he is today among the least well known...
and Albert of SaxonyAlbert of SaxonyAlbert of Saxony may refer to:* Albert of Saxony * Albert I, Duke of Saxony * Albert, Duke of Saxony * Prince Albert of Saxony, Duke of Teschen * Albert of Saxony...
.
The last great works in this tradition are the Logic of John Poinsot (1589–1644, known as John of St Thomas), the Metaphysical Disputations of Francisco Suarez
Francisco Suárez
Francisco Suárez was a Spanish Jesuit priest, philosopher and theologian, one of the leading figures of the School of Salamanca movement, and generally regarded among the greatest scholastics after Thomas Aquinas....
(1548–1617), and the Logica Demonstrativa of Giovanni Girolamo Saccheri
Giovanni Girolamo Saccheri
Giovanni Girolamo Saccheri was an Italian Jesuit priest, scholastic philosopher, and mathematician....
(1667–1733).
The Textbook Tradition
"Traditional Logic" generally means the textbook tradition that begins with Antoine ArnauldAntoine Arnauld
Antoine Arnauld — le Grand as contemporaries called him, to distinguish him from his father — was a French Roman Catholic theologian, philosopher, and mathematician...
and Pierre Nicole
Pierre Nicole
Pierre Nicole was one of the most distinguished of the French Jansenists.Born in Chartres, he was the son of a provincial barrister, who took in charge his education...
's Logic, or the Art of Thinking, better known as the Port-Royal Logic
Port-Royal Logic
Port-Royal Logic, or Logique de Port-Royal, is the common name of La logique, ou l'art de penser, an important textbook on logic first published anonymously in 1662 by Antoine Arnauld and Pierre Nicole, two prominent members of the Jansenist movement, centered around Port-Royal. Blaise Pascal...
. Published in 1662, it was the most influential work on logic in England until the nineteenth century. 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
Term logic
In philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...
. Between 1664 and 1700 there were eight editions, and the book had considerable influence after that. The account of proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
s that Locke
John Locke
John Locke FRS , widely known as the Father of Liberalism, was an English philosopher and physician regarded as one of the most influential of Enlightenment thinkers. Considered one of the first of the British empiricists, following the tradition of Francis Bacon, he is equally important to social...
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)
Another influential work was the Novum Organum by Francis Bacon
Francis Bacon
Francis Bacon, 1st Viscount St Albans, KC was an English philosopher, statesman, scientist, lawyer, jurist, author and pioneer of the scientific method. He served both as Attorney General and Lord Chancellor of England...
, published in 1620. The title translates as "new instrument". This is a reference to Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's work Organon
Organon
The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic:* Categories* On Interpretation* Prior Analytics* Posterior Analytics...
. 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". This method is known as inductive reasoning
Inductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
. 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.
Other works in the textbook tradition include Isaac Watts
Isaac Watts
Isaac Watts was an English hymnwriter, theologian and logician. A prolific and popular hymnwriter, he was recognised as the "Father of English Hymnody", credited with some 750 hymns...
' Logick: Or, the Right Use of Reason (1725), Richard Whately
Richard Whately
Richard Whately was an English rhetorician, logician, economist, and theologian who also served as the Church of Ireland Archbishop of Dublin.-Life and times:...
's Logic (1826), and John Stuart Mill
John Stuart Mill
John Stuart Mill was a British philosopher, economist and civil servant. An influential contributor to social theory, political theory, and political economy, his conception of liberty justified the freedom of the individual in opposition to unlimited state control. He was a proponent of...
's A System of Logic (1843). Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lay in introspection influenced the view that logic is best understood as a branch of psychology, an approach to the subject which dominated the next fifty years of its development, especially in Germany.
Logic in Hegel's philosophy
Science of Logic
Hegel's work The Science of Logic outlined his vision of logic, which is an ontology that incorporates the traditional Aristotelian syllogism as a sub-component rather than a basis...
into a shorter work published in 1817 as the first volume of his Encyclopaedia of the Philosophical Sciences. The "Shorter" or "Encyclopaedia" Logic, as it is often known, lays out a series of transitions which leads from the most empty and abstract of categories – Hegel begins with "Pure Being" and "Pure Nothing" – to the "Absolute
Absolute (philosophy)
The Absolute is the concept of an unconditional reality which transcends limited, conditional, everyday existence. It is sometimes used as an alternate term for "God" or "the Divine", especially, but by no means exclusively, by those who feel that the term "God" lends itself too easily to...
" – the category which contains and resolves all the categories which preceded it. Despite the title, Hegel's Logic is not really a contribution to the science of valid inference. Rather than deriving conclusions about concepts through valid inference from premises, Hegel seeks to show that thinking about one concept compels thinking about another concept (one cannot, he argues, possess the concept of "Quality" without the concept of "Quantity"); and the compulsion here is not a matter of individual psychology, but arises almost organically from the content of the concepts themselves. His purpose is to show the rational structure of the "Absolute" – indeed of rationality itself. The method by which thought is driven from one concept to its contrary, and then to further concepts, is known as the Hegelian dialectic
Dialectic
Dialectic is a method of argument for resolving disagreement that has been central to Indic and European philosophy since antiquity. The word dialectic originated in Ancient Greece, and was made popular by Plato in the Socratic dialogues...
.
Although Hegel's Logic has had little impact on mainstream logical studies, its influence can be seen in the work of the British Idealists
British idealism
A species of absolute idealism, British idealism was a philosophical movement that was influential in Britain from the mid-nineteenth century to the early twentieth century. The leading figures in the movement were T.H. Green , F. H. Bradley , and Bernard Bosanquet . They were succeeded by the...
– for example in F.H. Bradley's Principles of Logic (1883) – and in the economic, political and philosophical studies of Karl Marx
Karl Marx
Karl Heinrich Marx was a German philosopher, economist, sociologist, historian, journalist, and revolutionary socialist. His ideas played a significant role in the development of social science and the socialist political movement...
and the various schools of Marxism
Marxism
Marxism is an economic and sociopolitical worldview and method of socioeconomic inquiry that centers upon a materialist interpretation of history, a dialectical view of social change, and an analysis and critique of the development of capitalism. Marxism was pioneered in the early to mid 19th...
.
Logic and psychology
Between the work of Mill and Frege stretched half a century during which logic was widely treated as a descriptive science, an empirical study of the structure of reasoning, and thus essentially as a branch of psychologyPsychology
Psychology is the study of the mind and behavior. Its immediate goal is to understand individuals and groups by both establishing general principles and researching specific cases. For many, the ultimate goal of psychology is to benefit society...
. The German psychologist Wilhelm Wundt
Wilhelm Wundt
Wilhelm Maximilian Wundt was a German physician, psychologist, physiologist, philosopher, and professor, known today as one of the founding figures of modern psychology. He is widely regarded as the "father of experimental psychology"...
, for example, discussed deriving "the logical from the psychological laws of thought", emphasizing that "psychological thinking is always the more comprehensive form of thinking." This view was widespread among German philosophers of the period: Theodor Lipps
Theodor Lipps
Theodor Lipps was a German philosopher. Lipps was one of the most influential German university professors of his time, attracting many students from other countries. Lipps was very concerned with conceptions of art and the aesthetic, focusing much of his philosophy around such issues...
described logic as "a specific discipline of psychology"; Christoph von Sigwart
Christoph von Sigwart
Christoph von Sigwart was a German philosopher and logician. He was the son of philosopher Heinrich Christoph Wilhelm Sigwart .-Life:...
understood logical necessity as grounded in the individual's compulsion to think in a certain way; and Benno Erdmann argued that "logical laws only hold within the limits of our thinking" Such was the dominant view of logic in the years following Mill's work. This psychological approach to logic was rejected by Gottlob Frege
Gottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
. It was also subjected to an extended and destructive critique by Edmund Husserl
Edmund Husserl
Edmund Gustav Albrecht Husserl was a philosopher and mathematician and the founder of the 20th century philosophical school of phenomenology. He broke with the positivist orientation of the science and philosophy of his day, yet he elaborated critiques of historicism and of psychologism in logic...
in the first volume of his Logical Investigations (1900), an assault which has been described as "overwhelming". Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven, and that skepticism
Skepticism
Skepticism has many definitions, but generally refers to any questioning attitude towards knowledge, facts, or opinions/beliefs stated as facts, or doubt regarding claims that are taken for granted elsewhere...
and relativism
Relativism
Relativism is the concept that points of view have no absolute truth or validity, having only relative, subjective value according to differences in perception and consideration....
were unavoidable consequences.
Such criticisms did not immediately extirpate so-called "psychologism
Psychologism
Psychologism is a generic type of position in philosophy according to which psychology plays a central role in grounding or explaining some other, non-psychological type of fact or law...
". For example, the American philosopher Josiah Royce
Josiah Royce
Josiah Royce was an American objective idealist philosopher.-Life:Royce, born in Grass Valley, California, grew up in pioneer California very soon after the California Gold Rush. He received the B.A...
, while acknowledging the force of Husserl's critique, remained "unable to doubt" that progress in psychology would be accompanied by progress in logic, and vice versa.
Rise of modern logic
The period between the fourteenth century and the beginning of the nineteenth century had been largely one of decline and neglect, and is generally regarded as barren by historians of logic. The revival of logic occurred in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in mathematicsMathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
. The development of the modern so-called "symbolic" or "mathematical" logic during this period is the most significant in the 2,000-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 logic 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
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
. He argued that a truly "exact" logic would depend 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 medieval logicians called "syncategoremata
Syncategorematic term
In scholastic logic, a syncategorematic term is a word that cannot serve as the subject or the predicate of a proposition, and thus cannot stand for any of Aristotle's categories, but can be used with other terms to form a proposition...
") and 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:- The embryonic period from Leibniz to 1847, when the notion of a logical calculus was discussed and developed, particularly by Leibniz, but no schools were formed, and isolated periodic attempts were abandoned or went unnoticed.
- The algebraic period from Boole's Analysis to SchröderErnst SchröderErnst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
's Vorlesungen. In this period there were more practitioners, and a greater continuity of development. - The logicist period from the BegriffsschriftBegriffsschriftBegriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book...
of Frege to the Principia MathematicaPrincipia MathematicaThe Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
of RussellBertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
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 FregeGottlob FregeFriedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
, RussellBertrand RussellBertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
, and the early WittgensteinLudwig WittgensteinLudwig Josef Johann Wittgenstein was an Austrian philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He was professor in philosophy at the University of Cambridge from 1939 until 1947...
. It culminates with the Principia, an important work which includes a thorough examination and attempted solution of the antinomiesAntinomyAntinomy literally means the mutual incompatibility, real or apparent, of two laws. It is a term used in logic and epistemology....
which had been an obstacle to earlier progress. - The metamathematical period from 1910 to the 1930s, which saw the development of metalogicMetalogicMetalogic is the study of the metatheory of logic. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves...
, in the finitist system of HilbertDavid HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, and the non-finitist system of LöwenheimLeopold LöwenheimLeopold Löwenheim was a German mathematician, known for his work in mathematical logic. The Nazi regime forced him to retire because under the Nuremberg Laws he was considered only three quarters Aryan. In 1943 much of his work was destroyed during a bombing raid on Berlin...
and Skolem, the combination of logic and metalogic in the work of GödelGodelGodel or similar can mean:*Kurt Gödel , an Austrian logician, mathematician and philosopher*Gödel...
and Tarski. Gödel's incompleteness theorem of 1931 was one of the greatest achievements in the history of logic. Later in the 1930s Gödel developed the notion of set-theoretic constructibility. - The period after World War II, when mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
branched into four inter-related but separate areas of research: model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, proof theoryProof theoryProof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
, computability theoryComputability theoryComputability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
, and set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, and its ideas and methods began to influence philosophyPhilosophyPhilosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
.
Embryonic period
Ramon Llull
Ramon Llull was a Majorcan writer and philosopher, logician and tertiary Franciscan. He wrote the first major work of Catalan literature. Recently-surfaced manuscripts show him to have anticipated by several centuries prominent work on elections theory...
, 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
Thomas Hobbes
Thomas Hobbes of Malmesbury , in some older texts Thomas Hobbs of Malmsbury, was an English philosopher, best known today for his work on political philosophy...
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 Lull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death. Leibniz says that ordinary languages 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; hence, he proposed to identify an alphabet of human thought
Alphabet of human thought
The alphabet of human thought is a concept originally proposed by Gottfried Leibniz that provides a universal way to represent and analyze ideas and relationships, no matter how complicated, by breaking down their component pieces...
comprising fundamental concepts which could be composed to express complex ideas, and create a calculus ratiocinator which would make reasoning "as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate."
Gergonne
Joseph Diaz Gergonne
Joseph Diaz Gergonne was a French mathematician and logician.-Life:In 1791, Gergonne enlisted in the French army as a captain. That army was undergoing rapid expansion because the French government feared a foreign invasion intended to undo the French Revolution and restore Louis XVI to full power...
(1816) says that reasoning does not have to be about objects about which we have perfectly clear ideas, since algebraic operations can be carried out without our having any idea of the meaning of the symbols involved. Bolzano
Bernard Bolzano
Bernhard Placidus Johann Nepomuk Bolzano , Bernard Bolzano in English, was a Bohemian mathematician, logician, philosopher, theologian, Catholic priest and antimilitarist of German mother tongue.-Family:Bolzano was the son of two pious Catholics...
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
William Stanley Jevons
William Stanley Jevons was a British economist and logician.Irving Fisher described his book The Theory of Political Economy as beginning the mathematical method in economics. It made the case that economics as a science concerned with quantities is necessarily mathematical...
, Schröder
Ernst Schröder
Ernst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
and Venn
John Venn
Donald A. Venn FRS , was a British logician and philosopher. He is famous for introducing the Venn diagram, which is used in many fields, including set theory, probability, logic, statistics, and computer science....
. 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
Augustus De Morgan
Augustus De Morgan was a British mathematician and logician. He formulated De Morgan's laws and introduced the term mathematical induction, making its idea rigorous. The crater De Morgan on the Moon is named after him....
(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, Lincolnshire
Lincoln, Lincolnshire
Lincoln is a cathedral city and county town of Lincolnshire, England.The non-metropolitan district of Lincoln has a population of 85,595; the 2001 census gave the entire area of Lincoln a population of 120,779....
. 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
Disjunctive normal form
In boolean logic, a disjunctive normal form is a standardization of a logical formula which is a disjunction of conjunctive clauses. As a normal form, it is useful in automated theorem proving. A logical formula is considered to be in DNF if and only if it is a disjunction of one or more...
.
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 algebraic 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
John Venn
Donald A. Venn FRS , was a British logician and philosopher. He is famous for introducing the Venn diagram, which is used in many fields, including set theory, probability, logic, statistics, and computer science....
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
Royal Society
The Royal Society of London for Improving Natural Knowledge, known simply as the Royal Society, is a learned society for science, and is possibly the oldest such society in existence. Founded in November 1660, it was granted a Royal Charter by King Charles II as the "Royal Society of London"...
the following year. In 1885 Allan Marquand
Allan Marquand
Allan Marquand was an art historian at Princeton University and a curator of the Princeton University Art Museum.Marquand was the son of Henry Gurdon Marquand, a prominent philanthropist and art collector. After graduating from Princeton in 1874, Allan obtained his Ph.D. in Philosophy in 1880, at...
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 Schröder 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 binary operation, "neither ... nor ...
Logical NOR
In boolean logic, logical nor or joint denial is a truth-functional operator which produces a result that is the negation of logical or. That is, a sentence of the form is true precisely when neither p nor q is true—i.e. when both of p and q are false...
" and equally well "not both ... and ...
Sheffer stroke
In Boolean functions and propositional calculus, the Sheffer stroke, named after Henry M. Sheffer, written "|" , "Dpq", or "↑", denotes a logical operation that is equivalent to the negation of the conjunction operation, expressed in ordinary language as "not both"...
", however, like many of Peirce's innovations, this remained unknown or unnoticed until Sheffer
Henry M. Sheffer
Henry Maurice Sheffer was an American logician.Sheffer was a Polish Jew born in the western Ukraine, who immigrated to the USA in 1892 with his parents and six siblings. He studied at the Boston Latin School before entering Harvard University, learning logic from Josiah Royce, and completing his...
rediscovered it in 1913. Boole's early work also lacks the idea of the logical sum which originates in Peirce (1867), Schröder
Ernst Schröder
Ernst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
(1877) and Jevons (1890), and the concept of inclusion
Inclusion (logic)
In logic and mathematics inclusion is the concept that all the contents of one object are also contained within a second object. The modern symbol for inclusion first appears in Gergonne , who defines it as one idea 'containing' or being 'contained' by another, using the backward letter 'C' to...
, 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 Schröder's monumental Vorlesungen über die Algebra der Logik ("Lectures on the Algebra of Logic", 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 FregeGottlob Frege
Friedrich Ludwig Gottlob Frege was a German mathematician, logician and philosopher. He is considered to be one of the founders of modern logic, and made major contributions to the foundations of mathematics. He is generally considered to be the father of analytic philosophy, for his writings on...
. Frege's objective was the program of Logicism
Logicism
Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic. Bertrand Russell and Alfred North Whitehead championed this theory fathered by Richard Dedekind...
, 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 Begriffsschrift
Begriffsschrift
Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book...
is important. Frege also tried to show that the concept of number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
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 Begriffsschrift ("concept script") 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 significant innovation, however, was his explanation of the quantifier in terms of 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
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....
.
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) [ I(x) -> (E(x) v A(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 [the Boolean] 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
Problem of multiple generality
The problem of multiple generality names a failure in traditional logic to describe certain intuitively valid inferences. For example, it is intuitively clear that if:then it follows logically that:The syntax of traditional logic permits exactly four sentence types: "All As are Bs", "No As are...
. 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
Scope
The word scope may refer to many different devices or viewing instruments, constructed for many different purposes. It may refer to a telescopic sight, an optical device commonly used on firearms. Other uses of scope or Scopes may refer to:...
of the quantifiers. Thus
- (x) [ girl(x) -> E(y) (boy(y) & kissed(x,y)) ]
means that to every girl there corresponds some boy (any one will do) who the girl kissed. But
- E(x) [ boy(x) & (y) (girl(y) -> kissed(y,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
Ancestral relation
In mathematical logic, the ancestral relation of an arbitrary binary relation R is defined below.The ancestral makes its first appearance in Frege's Begriffsschrift. Frege later employed it in his Grundgesetze as part of his definition of the natural numbers...
, of the many-to-one relation
Injective function
In mathematics, an injective function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain. In other words, every element of the function's codomain is mapped to by at most one element of its domain...
, and of mathematical induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
.
This period overlaps with the work of the so-called "mathematical school", which included Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
, Pasch
Moritz Pasch
Moritz Pasch was a German mathematician specializing in the foundations of geometry. He completed his Ph.D. at the University of Breslau at only 22 years of age...
, Peano
Giuseppe Peano
Giuseppe Peano was an Italian mathematician, whose work was of philosophical value. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The standard axiomatization of the natural numbers is named the Peano axioms in...
, Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
, Zermelo
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...
, Huntington
Edward Vermilye Huntington
Edward Vermilye Huntington was an American mathematician....
, Veblen
Oswald Veblen
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...
and Heyting
Arend Heyting
Arend Heyting was a Dutch mathematician and logician. He was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic...
. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory.
The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
. This proved that the Frege's naive set theory
Naive set theory
Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics , and the everyday usage of set theory concepts in most...
led to a contradiction. Frege's theory is that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not). This contradiction is now known as Russell's paradox
Russell's paradox
In the foundations of mathematics, Russell's paradox , discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction...
. One important method of resolving this paradox was proposed by Ernst Zermelo
Ernst Zermelo
Ernst Friedrich Ferdinand Zermelo was a German mathematician, whose work has major implications for the foundations of mathematics and hence on philosophy. He is known for his role in developing Zermelo–Fraenkel axiomatic set theory and his proof of the well-ordering theorem.-Life:He graduated...
. Zermelo set theory
Zermelo set theory
Zermelo set theory, as set out in an important paper in 1908 by Ernst Zermelo, is the ancestor of modern set theory. It bears certain differences from its descendants, which are not always understood, and are frequently misquoted...
was the first axiomatic set theory. It was developed into the now-canonical Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
(ZF).
The monumental Principia Mathematica
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
, a three-volume work on the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
, written by Russell and Alfred North Whitehead
Alfred North Whitehead
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
and published 1910–13 also included an attempt to resolve the paradox, by means of an elaborate system of types: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak of the "set of all sets". The Principia was an attempt to derive all mathematical truths from a well-defined set of axiom
Axiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s and inference rules in symbolic logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
.
Metamathematical period
The names of GödelKurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
and Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
dominate the 1930s, a crucial period in the development of metamathematics
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
– the study of mathematics using mathematical methods to produce metatheories
Metatheory
A metatheory or meta-theory is a theory whose subject matter is some other theory. In other words it is a theory about a theory. Statements made in the metatheory about the theory are called metatheorems....
, or mathematical theories about other mathematical theories. Early investigations into metamathematics had been driven by Hilbert's program
Hilbert's program
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert, was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies...
. which sought to resolve the ongoing crisis in the foundations of mathematics by grounding all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. Work on metamathematics culminated in the work of Gödel, who in 1929 showed that a given first-order sentence
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
is deducible
Provability logic
Provability logic is a modal logic, in which the box operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory, such as Peano arithmetic....
if and only if is logically valid – i.e. it is true in every structure
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
for its language. This is known as Gödel's completeness theorem
Gödel's completeness theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It was first proved by Kurt Gödel in 1929....
. A year later, he proved two important theorems, which showed Hibert's program to be unattainable in its original form. The first is that no consistent system of axioms whose theorems can be listed by an effective procedure
Effective method
In computability theory, an effective method is a procedure that reduces the solution of some class of problems to a series of rote steps which, if followed to the letter, and as far as may be necessary, is bound to:...
such as an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
or computer program is capable of proving all facts about the natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second is that if such a system is also capable of proving certain basic facts about the natural numbers, then the system cannot prove the consistency of the system itself. These two results are known as Gödel's incompleteness theorems
Gödel's incompleteness theorems
Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of...
, or simply Gödel's Theorem. Later in the decade, Gödel developed the concept of set-theoretic constructibility, as part of his proof that the axiom of choice and the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
are consistent with Zermelo-Fraenkel set theory.
In proof theory
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
, Gerhard Gentzen
Gerhard Gentzen
Gerhard Karl Erich Gentzen was a German mathematician and logician. He had his major contributions in the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus...
developed natural deduction
Natural deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning...
and the sequent calculus
Sequent calculus
In proof theory and mathematical logic, sequent calculus is a family of formal systems sharing a certain style of inference and certain formal properties. The first sequent calculi, systems LK and LJ, were introduced by Gerhard Gentzen in 1934 as a tool for studying natural deduction in...
. The former attempts to model logical reasoning as it 'naturally' occurs in practice and is most easily applied to intuitionistic logic, while the latter was devised to clarify the derivation of logical proofs in any formal system. Since Gentzen's work, natural deduction and sequent calculi have been widely applied in the fields of proof theory, mathematical logic and computer science. Gentzen also proved normalization and cut-elimination theorems for intuitionistic and classical logic which could be used to reduce logical proofs to a normal form.
Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
, a pupil of Łukasiewicz, is best known for his definition of truth and logical consequence, and the semantic concept of logical satisfaction
Open sentence
In mathematics, an open sentence is described as "open" in the sense that its truth value is meaningless until its variables are replaced with specific numbers, at which point the truth value can usually be determined...
. In 1933, he published (in Polish) The concept of truth in formalized languages, in which he proposed his semantic theory of truth
Semantic theory of truth
A semantic theory of truth is a theory of truth in the philosophy of language which holds that truth is a property of sentences.-Origin:The semantic conception of truth, which is related in different ways to both the correspondence and deflationary conceptions, is due to work published by Polish...
: a sentence such as "snow is white" is true if and only if snow is white. Tarski's theory separated the metalanguage
Metalanguage
Broadly, any metalanguage is language or symbols used when language itself is being discussed or examined. In logic and linguistics, a metalanguage is a language used to make statements about statements in another language...
, which makes the statement about truth, from the object language
Object language
An object language is a language which is the "object" of study in various fields including logic, linguistics, mathematics and theoretical computer science. The language being used to talk about an object language is called a metalanguage...
, which contains the sentence whose truth is being asserted, and gave a correspondence (the T-schema
T-schema
The T-schema or truth schema is used to give an inductive definition of truth which lies at the heart of any realisation of Alfred Tarski's semantic theory of truth...
) between phrases in the object language and elements of an interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
. Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as completeness
Completeness
In general, an object is complete if nothing needs to be added to it. This notion is made more specific in various fields.-Logical completeness:In logic, semantic completeness is the converse of soundness for formal systems...
, decidability
Decidability (logic)
In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas. Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas can be effectively...
, consistency
Consistency
Consistency can refer to:* Consistency , the psychological need to be consistent with prior acts and statements* "Consistency", an 1887 speech by Mark Twain...
and definability
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
. According to Anita Feferman, Tarski "changed the face of logic in the twentieth century".
Alonzo Church
Alonzo Church
Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...
and Alan Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
proposed formal models of computability, giving independent negative solutions to Hilbert's Entscheidungsproblem
Entscheidungsproblem
In mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...
in 1936 and 1937, respectively. The Entscheidungsproblem asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the halting problem
Halting problem
In computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...
as a key example of a mathematical problem without an algorithmic solution.
Church's system for computation developed into the modern λ-calculus, while the Turing machine
Turing machine
A Turing machine is a theoretical device that manipulates symbols on a strip of tape according to a table of rules. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a...
became a standard model for a general-purpose computing device. It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the Church–Turing thesis
Church–Turing thesis
In computability theory, the Church–Turing thesis is a combined hypothesis about the nature of functions whose values are effectively calculable; in more modern terms, algorithmically computable...
that any deterministic algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
that can be carried out by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both Peano arithmetic and first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
are undecidable
Undecidable problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is impossible to construct a single algorithm that always leads to a correct yes-or-no answer....
. Later work by Emil Post and Stephen Cole Kleene
Stephen Cole Kleene
Stephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science...
in the 1940s extended the scope of computability theory and introduced the concept of degrees of unsolvability.
The results of the first few decades of the twentieth century also had an impact upon analytic philosophy
Analytic philosophy
Analytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...
and philosophical logic
Philosophical logic
Philosophical logic is a term introduced by Bertrand Russell to represent his idea that the workings of natural language and thought can only be adequately represented by an artificial language; essentially it was his formalization program for the natural language...
, particularly from the 1950s onwards, in subjects such as modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
, temporal logic
Temporal logic
In logic, the term temporal logic is used to describe any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time. In a temporal logic we can then express statements like "I am always hungry", "I will eventually be hungry", or "I will be hungry...
, deontic logic
Deontic logic
Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts...
, and relevance logic
Relevance logic
Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related. They may be viewed as a family of substructural or modal logics...
.
Logic after WWII
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
branched into four inter-related but separate areas of research: model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, proof theory
Proof theory
Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as plain lists, boxed lists, or trees, which are constructed...
, computability theory
Computability theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
, and set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
.
In set theory, the method of forcing
Forcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
revolutionized the field by providing a robust method for constructing models and obtaining independence results. Paul Cohen
Paul Cohen (mathematician)
Paul Joseph Cohen was an American mathematician best known for his proof 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.-Early years:Cohen was born in Long Branch, New Jersey, into a...
introduced this method in 1962 to prove the independence of the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
and the axiom of choice from Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory
In mathematics, Zermelo–Fraenkel set theory with the axiom of choice, named after mathematicians Ernst Zermelo and Abraham Fraenkel and commonly abbreviated ZFC, is one of several axiomatic systems that were proposed in the early twentieth century to formulate a theory of sets without the paradoxes...
. His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic.
Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. It developed into a study of abstract computability, which became known as recursion theory
Recursion theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
. The priority method
Turing degree
In computer science and mathematical logic the Turing degree or degree of unsolvability of a set of natural numbers measures the level of algorithmic unsolvability of the set...
, discovered independently by Albert Muchnik and Richard Friedberg in the 1950s, led to major advances in the understanding of the degrees of unsolvability and related structures. Research into higher-order computability theory demonstrated its connections to set theory. The fields of constructive analysis and computable analysis
Computable analysis
In mathematics, computable analysis is the study of which parts of real analysis and functional analysis can be carried out in a computable manner. It is closely related to constructive analysis.- Basic results :...
were developed to study the effective content of classical mathematical theorems; these in turn inspired the program of reverse mathematics
Reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of...
. A separate branch of computability theory, computational complexity theory
Computational complexity theory
Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other...
, was also characterized in logical terms as a result of investigations into descriptive complexity
Descriptive complexity
Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the...
.
Abraham Robinson
Abraham Robinson was a mathematician who is most widely known for development of non-standard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics....
used model-theoretic techniques to develop calculus and analysis based on infinitesimals
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
, a problem that first had been proposed by Leibniz.
In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability
Realizability
Realizability is a part of proof theory which can be used to handle information about formulas instead of about the proofs of formulas. A natural number n is said to realize a statement in the language of arithmetic of natural numbers...
method invented by Georg Kreisel
Georg Kreisel
Georg Kreisel FRS is an Austrian-born mathematical logician who has studied and worked in Great Britain and America. Kreisel came from a Jewish background; his family sent him to England before the Anschluss, where he studied mathematics at Trinity College, Cambridge and then, during World War...
and Gödel's Dialectica interpretation
Dialectica interpretation
In proof theory, the Dialectica interpretation is a proof interpretation of intuitionistic arithmetic into a finite type extension of primitive recursive arithmetic, the so-called System T. It was developed by Kurt Gödel to provide a consistency proof of arithmetic...
. This work inspired the contemporary area of proof mining
Proof mining
In proof theory, a branch of mathematical logic, proof mining is a research program that analyzes formalized proofs, especially in analysis, to obtain explicit bounds or rates of convergence from proofs that, when expressed in natural language, appear to be nonconstructive.This research has led to...
. The Curry-Howard correspondence emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and typed lambda calculi
Typed lambda calculus
A typed lambda calculus is a typed formalism that uses the lambda-symbol to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered...
used in computer science. As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. Advances were also made in ordinal analysis
Ordinal analysis
In proof theory, ordinal analysis assigns ordinals to mathematical theories as a measure of their strength. The field was formed when Gerhard Gentzen in 1934 used cut elimination to prove, in modern terms, that the proof theoretic ordinal of Peano arithmetic is ε0.-Definition:Ordinal...
and the study of independence results in arithmetic such as the Paris–Harrington theorem
Paris–Harrington theorem
In mathematical logic, the Paris–Harrington theorem states that a certain combinatorial principle in Ramsey theory is true, but not provable in Peano arithmetic...
.
This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, tense logic is a formalised system for representing, and reasoning about, propositions qualified in terms of time. The philosopher Arthur Prior
Arthur Prior
Arthur Norman Prior was a noted logician and philosopher. Prior founded tense logic, now also known as temporal logic, and made important contributions to intensional logic, particularly in Prior .-Biography:Prior was entirely educated in New Zealand, where he was fortunate to have come under the...
played a significant role in its development in the 1960s. Modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
s extend the scope of formal logic to include the elements of modality
Linguistic modality
In linguistics, modality is what allows speakers to evaluate a proposition relative to a set of other propositions.In standard formal approaches to modality, an utterance expressing modality can always roughly be paraphrased to fit the following template:...
(for example, possibility
Logical possibility
A logically possible proposition is one that can be asserted without implying a logical contradiction. This is to say that a proposition is logically possible if there is some coherent way for the world to be, under which the proposition would be true...
and necessity). The ideas of Saul Kripke
Saul Kripke
Saul Aaron Kripke is an American philosopher and logician. He is a professor emeritus at Princeton and teaches as a Distinguished Professor of Philosophy at the CUNY Graduate Center...
, particularly about possible world
Possible world
In philosophy and logic, the concept of a possible world is used to express modal claims. The concept of possible worlds is common in contemporary philosophical discourse and has also been disputed.- Possibility, necessity, and contingency :...
s, and the formal system now called Kripke semantics
Kripke semantics
Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
have had a profound impact on analytic philosophy
Analytic philosophy
Analytic philosophy is a generic term for a style of philosophy that came to dominate English-speaking countries in the 20th century...
. His best known and most influential work is Naming and Necessity
Naming and Necessity
Naming and Necessity is a book by the philosopher Saul Kripke that was first published in 1980 and deals with the debates of proper nouns in the philosophy of language. The book is based on a transcript of three lectures given at Princeton University in 1970...
(1980). Deontic logic
Deontic logic
Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts...
s are closely related to modal logics: they attempt to capture the logical features of obligation
Obligation
An obligation is a requirement to take some course of action, whether legal or moral. There are also obligations in other normative contexts, such as obligations of etiquette, social obligations, and possibly...
, permission and related concepts. Ernst Mally
Ernst Mally
Ernst Mally was an Austrian philosopher affiliated with the so-called Graz School of phenomenology. A pupil of Alexius Meinong, he was one of the founders of deontic logic and is mainly known for his contributions in that field of research.- Life :Mally was born in the town of Kranj in the Duchy...
, a pupil of Alexius Meinong
Alexius Meinong
Alexius Meinong was an Austrian philosopher, a realist known for his unique ontology...
, was the first to propose a formal deontic system in his Grundgesetze des Sollens, based on the syntax of Whitehead's and Russell's propositional calculus
Propositional calculus
In mathematical logic, a propositional calculus or logic is a formal system in which formulas of a formal language may be interpreted as representing propositions. A system of inference rules and axioms allows certain formulas to be derived, called theorems; which may be interpreted as true...
. Another logical system founded after World War II was fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
by Iranian mathematician Lotfi Asker Zadeh
Lotfi Asker Zadeh
Lotfali Askar Zadeh , better known as Lotfi A. Zadeh, is a mathematician, electrical engineer, computer scientist, artifical intelligence researcher and professor emeritus of computer science at the University of California, Berkeley...
in 1965.
External links
- History of Logic in Relationship to Ontology Annotated bibliography on the history of logic
- Paul Spade's "Thoughts Words and Things"
- John of St Thomas