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The history of logic deals with the study of the development of the science of valid
inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infer, infer'' means to "carry forward". Inference is theoretically traditionally divided into deductive reasoning, deduction and in ...
(logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
). Formal logics developed in ancient times in India
India, officially the Republic of India (Hindi: ), is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area, the List of countries and dependencies by population, second-most populous ...
, China
China, officially the People's Republic of China (PRC), is a country in East Asia. It is the world's List of countries and dependencies by population, most populous country, with a Population of China, population exceeding 1.4 billion, slig ...
, and Greece
Greece,, or , romanized: ', officially the Hellenic Republic, is a country in Southeast Europe. It is situated on the southern tip of the Balkans, and is located at the crossroads of Europe, Asia, and Africa. Greece shares land borders wi ...
. Greek methods, particularly Aristotelian logic
In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...
(or term logic) as found in the '' Organon'', found wide application and acceptance in Western science and mathematics for millennia.Boehner p. xiv The Stoics
Stoicism is a school of Hellenistic philosophy founded by Zeno of Citium in Athens in the early 3rd century BCE. It is a philosophy of personal virtue ethics informed by its system of logic and its views on the natural world, asserting tha ...
, especially Chrysippus
Chrysippus of Soli (; grc-gre, Χρύσιππος ὁ Σολεύς, ; ) 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 the Stoic philosopher Cleanthes. When C ...
, began the development of predicate logic.
Christian
Christians () are people who follow or adhere to Christianity, a monotheistic Abrahamic religion based on the life and teachings of Jesus Christ. The words ''Christ'' and ''Christian'' derive from the Koine Greek title ''Christós'' (Χρι� ...
and Islamic
Islam (; ar, ۘالِإسلَام, , ) is an Abrahamic monotheistic religion centred primarily around the Quran, a religious text considered by Muslims to be the direct word of God (or '' Allah'') as it was revealed to Muhammad, the ma ...
philosophers such as Boethius
Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the t ...
(died 524), Ibn Sina
Ibn Sina ( fa, ابن سینا; 980 – June 1037 CE), commonly known in the West as Avicenna (), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, philosophers, and writers of the Islami ...
(Avicenna, died 1037) and William of Ockham
William of Ockham, OFM (; also Occam, from la, Gulielmus Occamus; 1287 – 10 April 1347) was an English Franciscan friar, scholastic philosopher, apologist, and Catholic theologian, who is believed to have been born in Ockham, a small vil ...
(died 1347) further developed Plato's logic in the Middle Ages
In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire ...
, reaching a high point in the mid-fourteenth century, with Jean Buridan. The period between the fourteenth century and the beginning of the nineteenth century saw largely decline and neglect, and at least one historian of logic regards this time as barren.Oxford Companion p. 498; Bochenski, Part I Introduction, ''passim'' Empirical methods
Empirical research is research using empirical evidence. It is also a way of gaining knowledge by means of direct and indirect observation or experience. Empiricism values some research more than other kinds. Empirical evidence (the record of ...
ruled the day, as evidenced by Sir Francis Bacon
Francis Bacon, 1st Viscount St Alban (; 22 January 1561 – 9 April 1626), also known as Lord Verulam, was an English philosopher and statesman who served as Attorney General and Lord Chancellor of England. Bacon led the advancement of both ...
's ''Novum Organon
The ''Novum Organum'', fully ''Novum Organum, sive Indicia Vera de Interpretatione Naturae'' ("New organon, or true directions concerning the interpretation of nature") or ''Instaurationis Magnae, Pars II'' ("Part II of The Great Instauration ...
'' of 1620.
Logic revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a rigorous and formal discipline which took as its exemplar the exact method of proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
used in mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a hearkening back to the Greek tradition. The development of the modern "symbolic" or "mathematical" logic during this period by the likes of Boole
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Irel ...
, Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
, Russell, and Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. 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 sta ...
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
Intellectual history (also the history of ideas) is the study of the history of human thought and of intellectuals, people who conceptualize, discuss, write about, and concern themselves with ideas. The investigative premise of intellectual hist ...
.Oxford Companion p. 500
Progress in mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
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
Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United ...
and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...
, temporal logic In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I ...
, deontic logic, and relevance logic.
Logic in the East
Logic in India
Hindu logic
Origin
TheNasadiya Sukta
The Nāsadīya Sūkta (after the incipit ', or "not the non-existent"), also known as the Hymn of Creation, is the 129th hymn of the 10th mandala of the Rigveda (10:129). It is concerned with cosmology and the origin of the universe.
Nasadiya Su ...
of the ''Rigveda
The ''Rigveda'' or ''Rig Veda'' ( ', from ' "praise" and ' "knowledge") is an ancient Indian collection of Vedic Sanskrit hymns (''sūktas''). It is one of the four sacred canonical Hindu texts ('' śruti'') known as the Vedas. Only on ...
'' ( RV 10.129) contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of '' catuskoti'': "A", "not A", "A and 'not A'", and "not A and not not A".
Logic began independently in ancient India
According to consensus in modern genetics, anatomically modern humans first arrived on the Indian subcontinent from Africa between 73,000 and 55,000 years ago. Quote: "Y-Chromosome and Mt-DNA data support the colonization of South Asia by ...
and continued to develop to early modern times without any known influence from Greek logic.
Before Gautama
Though the origins in India of public debate (''pariṣad''), one form of rational inquiry, are not clear, we know that public debates were common in preclassical India, for they are frequently alluded to in various '' Upaniṣads'' and in the early Buddhist literature. Public debate is not the only form of public deliberations in preclassical India. Assemblies (''pariṣad'' or ''sabhā
A sabhā in Ancient India was an assembly, congregation, or council. Personified as a deity, Sabhā is a daughter of Prajapati in the Atharvaveda. The term has also given rise to modern terms of Parliament of India, such as Lok Sabha (Lower Hou ...
'') of various sorts, comprising relevant experts, were regularly convened to deliberate on a variety of matters, including administrative, legal and religious matters.
Dattatreya
A philosopher named Dattatreya is stated in theBhagavata purana
The ''Bhagavata Purana'' ( sa, भागवतपुराण; ), also known as the ''Srimad Bhagavatam'', ''Srimad Bhagavata Mahapurana'' or simply ''Bhagavata'', is one of Hinduism's eighteen great Puranas (''Mahapuranas''). Composed in S ...
to have taught Anvlksikl to Aiarka, Prahlada and others. It appears from the Markandeya purana
The ''Markandeya Purana'' ( sa, मार्कण्डेय पुराण; IAST: ) is a Sanskrit text of Hinduism, and one of the eighteen major Puranas. The text's title Markandeya refers to a sage in Hindu History, who is the central c ...
that the Anvlksikl-vidya expounded by him consisted of a mere disquisition on soul in accordance with the yoga philosophy. Dattatreya expounded the philosophical side of Anvlksiki and not its logical aspect.
Medhatithi Gautama
While the teachers mentioned before dealt with some particular topics of Anviksiki, the credit of founding the Anviksiki in its special sense of a science is to be attributed to Medhatithi Gautama (c. 6th century BC). Guatama founded the ''anviksiki
Ānvīkṣikī is a term in Sanskrit denoting roughly the "science of inquiry" and it should have been recognized in India as a distinct branch of learning as early as 650 BCE. However, over the centuries its meaning and import have undergone co ...
'' school of logic. The ''Mahabharata
The ''Mahābhārata'' ( ; sa, महाभारतम्, ', ) is one of the two major Sanskrit epics of ancient India in Hinduism, the other being the '' Rāmāyaṇa''. It narrates the struggle between two groups of cousins in the K ...
'' (12.173.45), around the 5th century BC, refers to the ''anviksiki'' and ''tarka'' schools of logic.
Panini
(c. 5th century BC) developed a form of logic (to whichBoolean logic
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas ...
has some similarities) for his formulation of Sanskrit grammar
The grammar of the Sanskrit language has a complex verbal system, rich nominal declension, and extensive use of compound nouns. It was studied and codified by Sanskrit grammarians from the later Vedic period (roughly 8th century BCE), culminati ...
. Logic is described by Chanakya
Chanakya ( Sanskrit: चाणक्य; IAST: ', ; 375–283 BCE) was an ancient Indian polymath who was active as a teacher, author, strategist, philosopher, economist, jurist, and royal advisor. He is traditionally identified as Kauṭi ...
(c. 350-283 BC) in his ''Arthashastra
The ''Arthashastra'' ( sa, अर्थशास्त्रम्, ) is an Ancient Indian Sanskrit treatise on statecraft, political science, economic policy and military strategy. Kautilya, also identified as Vishnugupta and Chanakya, is ...
'' as an independent field of inquiry.
Nyaya-Vaisheshika
Two of the six Indian schools of thought deal with logic:Nyaya
(Sanskrit: न्याय, ''nyā-yá''), literally meaning "justice", "rules", "method" or "judgment",Vaisheshika. The Nyāya Sūtras of Aksapada Gautama (c. 2nd century AD) constitute the core texts of the Nyaya school, one of the six orthodox schools of
''P'' and not ''P''; that is, being and not being.
# not (''P'' or not ''P''); that is, neither being nor not being.Under
Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c. 54 years older Thales. The systematic study of proof seems to have begun with the school of Pythagoras (i. e. the Pythagoreans) in the late sixth century BC. Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize ''form'' rather than ''matter''.
In contrast to Heraclitus,
None of the surviving works of the great fourth-century philosopher
The logic of 
These works are of outstanding importance in the history of logic. In the ''Categories'', he attempts to discern all the possible things to which a term can refer; this idea underpins his philosophical work ''
The Stoics adopted the Megarian logic and systemized it. The most important member of the school was
The works of
''
an
Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a Logic in Islamic philosophy#Post-Avicennian logic, 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
, ''Encyclopedia of Islamic World'').
"Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logic developed in
''Traditional logic'' generally means the textbook tradition that begins with
The idea that inference could be represented by a purely mechanical process is found as early as Raymond Llull, who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. The work of logicians such as the
Modern logic begins with what is known as the "algebraic school", originating with Boole and including Peirce, Jevons, Schröder, and
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 
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.
Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to ''Laws of Thought'' Corcoran also wrote a point-by-point comparison of ''Prior Analytics'' and ''Laws of Thought''. According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by 1) providing it with mathematical foundations involving equations, 2) extending the class of problems it could treat — from assessing validity to solving equations — and 3) expanding the range of applications it could handle — e.g. from propositions having only two terms to those having arbitrarily many.
More specifically, Boole agreed with what
After Boole, the next great advances were made by the German mathematician 
:
whereas "All the inhabitants are men or all the inhabitants are women" is
:
As Frege remarked in a critique of Boole's calculus:
: "The real difference is that I avoid he Booleandivision into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it'
As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient problem of multiple generality. The ambiguity of "every girl kissed a boy" is difficult to express in traditional logic, but Frege's logic resolves 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 
This period overlaps with the work of what is known as the "mathematical school", which included Dedekind, Pasch,
The names of Gödel and Tarski dominate the 1930s, a crucial period in the development of metamathematics – the study of mathematics using mathematical methods to produce metatheory, metatheories, or mathematical theories about other mathematical theories. Early investigations into metamathematics had been driven by Hilbert's program. Work on metamathematics culminated in the work of Gödel, who in 1929 showed that a given first-order logic, first-order sentence is Provability logic, deducible if and only if it is logically valid – i.e. it is true in every structure (mathematical logic), 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 method, effective procedure such as an algorithm or computer program is capable of proving all facts about the natural numbers. 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 
Alfred Tarski, a pupil of Jan Łukasiewicz, Łukasiewicz, is best known for his definition of truth and logical consequence, and the semantic concept of Open sentence, 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 (logic), interpretation. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of
After World War II,
vol. VII
No. 1,
pp. 19
40, The Open Court Publishing Co., Chicago, IL, 1896, for the Hegeler Institute. Reprinted (CP 3.425–455). ''Internet Archive'
''The Monist'' 7
*
Set Theory and its Philosophy
', Oxford University Press.
The History of Logic from Aristotle to Gödel
with annotated bibliographies on the history of logic * * *
Paul Spade's "Thoughts Words and Things"
An Introduction to Late Mediaeval Logic and Semantic Theory
Open Access pdf download; Insights, Images, Bios, and links for 178 logicians
by David Marans {{bots, deny=Yobot History of logic, Logic History of science by discipline, Logic
Hindu
Hindus (; ) are people who religiously adhere to Hinduism. Jeffery D. Long (2007), A Vision for Hinduism, IB Tauris, , pages 35–37 Historically, the term has also been used as a geographical, cultural, and later religious identifier for ...
philosophy. This realist school developed a rigid five-member schema of inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infer, infer'' means to "carry forward". Inference is theoretically traditionally divided into deductive reasoning, deduction and in ...
involving an initial premise, a reason, an example, an application, and a conclusion. The idealist
In philosophy, the term idealism identifies and describes metaphysical perspectives which assert that reality is indistinguishable and inseparable from perception and understanding; that reality is a mental construct closely connected to ...
Buddhist philosophy
Buddhist philosophy refers to the philosophical investigations and systems of inquiry that developed among various schools of Buddhism in India following the parinirvana of The Buddha and later spread throughout Asia. The Buddhist path combin ...
became the chief opponent to the Naiyayikas.
Jain Logic
Jains
Jainism ( ), also known as Jain Dharma, is an Indian religion. Jainism traces its spiritual ideas and history through the succession of twenty-four tirthankaras (supreme preachers of ''Dharma''), with the first in the current time cycle being ...
made its own unique contribution to this mainstream development of logic by also occupying itself with the basic epistemological issues, namely, with those concerning the nature of knowledge, how knowledge is derived, and in what way knowledge can be said to be reliable.
The Jains have doctrines of relativity used for logic and reasoning:
* Anekāntavāda – the theory of relative pluralism or manifoldness;
* Syādvāda – the theory of conditioned predication and;
* Nayavāda – The theory of partial standpoints.
These Jain philosophical concepts made most important contributions to the ancient Indian philosophy
Indian philosophy refers to philosophical traditions of the Indian subcontinent. A traditional Hindu classification divides āstika and nāstika schools of philosophy, depending on one of three alternate criteria: whether it believes the Veda ...
, especially in the areas of skepticism and relativityBuddhist logic
Nagarjuna
Nagarjuna
Nāgārjuna . 150 – c. 250 CE (disputed)was an Indian Mahāyāna Buddhist thinker, scholar-saint and philosopher. He is widely considered one of the most important Buddhist philosophers.Garfield, Jay L. (1995), ''The Fundamental Wisdom of ...
(c. 150-250 AD), the founder of the Madhyamaka
Mādhyamaka ("middle way" or "centrism"; ; Tibetan: དབུ་མ་པ ; ''dbu ma pa''), otherwise known as Śūnyavāda ("the emptiness doctrine") and Niḥsvabhāvavāda ("the no ''svabhāva'' doctrine"), refers to a tradition of Buddhis ...
("Middle Way") developed an analysis known as the catuṣkoṭi (Sanskrit), a "four-cornered" system of argumentation that involves the systematic examination and rejection of each of the 4 possibilities of a proposition, ''P'':
# ''P''; that is, being.
# not ''P''; that is, not being.
# ''P'' and not ''P''; that is, being and not being.
# not (''P'' or not ''P''); that is, neither being nor not being.Under propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
, De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British math ...
imply that this is equivalent to the third case (''P'' and not ''P''), and is therefore superfluous; there are actually only 3 cases to consider.
Dignaga
However, Dignāga (c 480-540 AD) is sometimes said to have developed a formal syllogism, and it was through him and his successor, Dharmakirti, that Buddhist logic reached its height; it is contested whether their analysis actually constitutes a formal syllogistic system. In particular, their analysis centered on the definition of an inference-warranting relation, "vyapti
a Sanskrit expression, in Hindu philosophy refers to the state of pervasion. It is considered as the logical ground of inference which is one of the means to knowledge. No conclusion can be inferred without the knowledge of vyapti. Vyapti guara ...
", 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.
Dignāga's famous "wheel of reason" (''Hetucakra ''Hetucakra'' or ''Wheel of Reasons'' is a Sanskrit text on logic written by Dignaga (c 480–540 CE). It concerns the application of his 'three modes’ ( trairūpya), conditions or aspects of the middle term called ''hetu'' ("reason" for a conc ...
'') is a method of indicating when one thing (such as smoke) can be taken as an invariable sign of another thing (like fire), but the inference is often inductive and based on past observation. Matilal remarks that Dignāga's analysis is much like John Stuart Mill's Joint Method of Agreement and Difference, which is inductive.
Syllogism and influence
In addition, the traditional five-member Indian syllogism, though deductively valid, has repetitions that are unnecessary to its logical validity. As a result, some commentators see the traditional Indian syllogism as a rhetorical form that is entirely natural in many cultures of the world, and yet not as a logical form—not in the sense that all logically unnecessary elements have been omitted for the sake of analysis.Logic in China
In China, a contemporary ofConfucius
Confucius ( ; zh, s=, p=Kǒng Fūzǐ, "Master Kǒng"; or commonly zh, s=, p=Kǒngzǐ, labels=no; – ) was a Chinese philosopher and politician of the Spring and Autumn period who is traditionally considered the paragon of Chinese sages. C ...
, Mozi
Mozi (; ; Latinized as Micius ; – ), original name Mo Di (), was a Chinese philosopher who founded the school of Mohism during the Hundred Schools of Thought period (the early portion of the Warring States period, –221 BCE). The ancie ...
, "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. Due to the harsh rule of Legalism in the subsequent Qin Dynasty
The Qin dynasty ( ; zh, c=秦朝, p=Qín cháo, w=), or Ch'in dynasty in Wade–Giles romanization ( zh, c=, p=, w=Ch'in ch'ao), was the first dynasty of Imperial China. Named for its heartland in Qin state (modern Gansu and Shaanxi), ...
, this line of investigation disappeared in China until the introduction of Indian philosophy by Buddhists
Buddhism ( , ), also known as Buddha Dharma and Dharmavinaya (), is an Indian religion or philosophical tradition based on teachings attributed to the Buddha. It originated in northern India as a -movement in the 5th century BCE, and ...
.
Logic in the West
Prehistory of 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 withgeometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, which originally meant the same as "land measurement". The ancient Egyptians discovered geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, including the formula for the volume of a truncated pyramid.Kneale p. 3 Ancient Babylon was also skilled in mathematics. Esagil-kin-apli's medical ''Diagnostic Handbook'' in the 11th century BC was based on a logical set of axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s and assumptions,H. F. J. Horstmanshoff, Marten Stol, Cornelis Tilburg (2004), ''Magic and Rationality in Ancient Near Eastern and Graeco-Roman Medicine'', p. 99, Brill Publishers
Brill Academic Publishers (known as E. J. Brill, Koninklijke Brill, Brill ()) is a Dutch international academic publisher founded in 1683 in Leiden, Netherlands. With offices in Leiden, Boston, Paderborn and Singapore, Brill today publishes 2 ...
, . while Babylonian astronomers
Babylonian astronomy was the study or recording of celestial objects during the early history of Mesopotamia.
Babylonian astronomy seemed to have focused on a select group of stars and constellations known as Ziqpu stars. These constellations m ...
in the 8th and 7th centuries BC employed an internal logic
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent ...
within their predictive planetary systems, an important contribution to the philosophy of science
Philosophy of science is a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ultim ...
.D. Brown (2000), ''Mesopotamian Planetary Astronomy-Astrology '', Styx Publications, .
Ancient Greece before Aristotle
While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrativeproof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
. Both Thales
Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded ...
and Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His poli ...
of the Pre-Socratic philosophers
Pre-Socratic philosophy, also known as early Greek philosophy, is ancient Greek philosophy before Socrates. Pre-Socratic philosophers were mostly interested in cosmology, the beginning and the substance of the universe, but the inquiries of th ...
seemed aware of geometric methods.
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
The Academy ( Ancient Greek: Ἀκαδημία) was founded by Plato in c. 387 BC in Athens. Aristotle studied there for twenty years (367–347 BC) before founding his own school, the Lyceum. The Academy persisted throughout the Hellenisti ...
. The proofs of Euclid of Alexandria are a paradigm of Greek geometry. The three basic principles of geometry are as follows:
* Certain propositions must be accepted as true without demonstration; such a proposition is known as an axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
of geometry.
* Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry; such a demonstration is known as a proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a c ...
or a "derivation" of the proposition.
* The proof must be ''formal''; that is, the derivation of the proposition must be independent of the particular subject matter in question.
Further evidence that early Greek 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. In the case of the classical Greek city-states, interest in argumentation was also stimulated by the activities of the Rhetoric
Rhetoric () is the art of persuasion, which along with grammar and logic (or dialectic), is one of the three ancient arts of discourse. Rhetoric aims to study the techniques writers or speakers utilize to inform, persuade, or motivate par ...
ians or Orators and the Sophists
A sophist ( el, σοφιστής, sophistes) was a teacher in ancient Greece in the fifth and fourth centuries BC. Sophists specialized in one or more subject areas, such as philosophy, rhetoric, music, athletics, and mathematics. They taught ' ...
, who used arguments to defend or attack a thesis, both in legal and political contexts.
Thales
It is said Thales, most widely regarded as the first philosopher in the Greek tradition, measured the height of thepyramids
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
by their shadows at the moment when his own shadow was equal to his height. Thales was said to have had a sacrifice in celebration of discovering Thales' theorem just as Pythagoras had the Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposit ...
.
Thales is the first known individual to use deductive reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...
applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has been attributed. Indian
Indian or Indians may refer to:
Peoples South Asia
* Indian people, people of Indian nationality, or people who have an Indian ancestor
** Non-resident Indian, a citizen of India who has temporarily emigrated to another country
* South Asia ...
and Babylonian mathematicians knew his theorem for special cases before he proved it. It is believed that Thales learned that an angle inscribed in a semicircle
In mathematics (and more specifically geometry), a semicircle is a one-dimensional locus of points that forms half of a circle. The full arc of a semicircle always measures 180° (equivalently, radians, or a half-turn). It has only one line o ...
is a right angle during his travels to Babylon
''Bābili(m)''
* sux, 𒆍𒀭𒊏𒆠
* arc, 𐡁𐡁𐡋 ''Bāḇel''
* syc, ܒܒܠ ''Bāḇel''
* grc-gre, Βαβυλών ''Babylṓn''
* he, בָּבֶל ''Bāvel''
* peo, 𐎲𐎠𐎲𐎡𐎽𐎢 ''Bābiru''
* elx, 𒀸𒁀𒉿𒇷 ''Babi ...
.
Pythagoras
Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c. 54 years older Thales. The systematic study of proof seems to have begun with the school of Pythagoras (i. e. the Pythagoreans) in the late sixth century BC. Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize ''form'' rather than ''matter''.
Heraclitus and Parmenides
The writing ofHeraclitus
Heraclitus of Ephesus (; grc-gre, Ἡράκλειτος , "Glory of Hera"; ) was an ancient Greek pre-Socratic philosopher from the city of Ephesus, which was then part of the Persian Empire.
Little is known of Heraclitus's life. He wrot ...
(c. 535 – c. 475 BC) was the first place where the word ''logos
''Logos'' (, ; grc, λόγος, lógos, lit=word, discourse, or reason) is a term used in Western philosophy, psychology and rhetoric and refers to the appeal to reason that relies on logic or reason, inductive and deductive reasoning. Aris ...
'' was given special attention in ancient Greek philosophy, Heraclitus held that everything changes and all was fire and conflicting opposites, seemingly unified only by this ''Logos''. He is known for his obscure sayings.
In contrast to Heraclitus, Parmenides
Parmenides of Elea (; grc-gre, Παρμενίδης ὁ Ἐλεάτης; ) was a pre-Socratic Greek philosopher from Elea in Magna Graecia.
Parmenides was born in the Greek colony of Elea, from a wealthy and illustrious family. His date ...
held that all is one and nothing changes. He may have been a dissident Pythagorean, disagreeing that One (a number) produced the many. "X is not" must always be false or meaningless. What exists can in no way not exist. Our sense perceptions with its noticing of generation and destruction are in grievous error. Instead of sense perception, Parmenides advocated ''logos'' as the means to Truth. He has been called the discoverer of logic,
Zeno of Elea
Zeno of Elea (; grc, Ζήνων ὁ Ἐλεᾱ́της; ) was a pre-Socratic Greek philosopher of Magna Graecia and a member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic. He is best known ...
, a pupil of Parmenides, had the idea of a standard argument pattern found in the method of proof known as '' reductio ad absurdum''. This is the technique of drawing an obviously false (that is, "absurd") conclusion from an assumption, thus demonstrating that the assumption is false. Therefore, Zeno and his teacher are seen as the first to apply the art of logic. Plato's dialogue Parmenides
Parmenides of Elea (; grc-gre, Παρμενίδης ὁ Ἐλεάτης; ) was a pre-Socratic Greek philosopher from Elea in Magna Graecia.
Parmenides was born in the Greek colony of Elea, from a wealthy and illustrious family. His date ...
portrays Zeno as claiming to have written a book defending the monism
Monism attributes oneness or singleness (Greek: μόνος) to a concept e.g., existence. Various kinds of monism can be distinguished:
* Priority monism states that all existing things go back to a source that is distinct from them; e.g., i ...
of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. Zeno famously used this method to develop his paradoxes in his arguments against motion. Such ''dialectic'' reasoning later became popular. The members of this school were called "dialecticians" (from a Greek word meaning "to discuss").
Plato
None of the surviving works of the great fourth-century philosopher Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
(428–347 BC) include any formal logic, but they include important contributions to the field of philosophical logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
. 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 (Θεαίτητος) is a Greek name which could refer to:
* Theaetetus (mathematician) (c. 417 BC – 369 BC), Greek geometer
* ''Theaetetus'' (dialogue), a dialogue by Plato, named after the geometer
* Theaetetus (crater)
Theaetetus ...
'', where Plato identifies thought or opinion with talk or discourse (''logos''). The second question is a result of Plato's theory of Forms
The theory of Forms or theory of Ideas is a philosophical theory, fuzzy concept, or world-view, attributed to Plato, that the physical world is not as real or true as timeless, absolute, unchangeable ideas. According to this theory, ideas in th ...
. Forms are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called universals
In metaphysics, a universal is what particular things have in common, namely characteristics or qualities. In other words, universals are repeatable or recurrent entities that can be instantiated or exemplified by many particular things. For exa ...
, namely an abstract entity common to each set of things that have the same name. In both the ''Republic
A republic () is a " state in which power rests with the people or their representatives; specifically a state without a monarchy" and also a "government, or system of government, of such a state." Previously, especially in the 17th and 18th ...
'' and the ''Sophist
A sophist ( el, σοφιστής, sophistes) was a teacher in ancient Greece in the fifth and fourth centuries BC. Sophists specialized in one or more subject areas, such as philosophy, rhetoric, music, athletics, and mathematics. They taught ' ...
'', Plato suggests that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary connection between "forms". The third question is about definition
A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories: intensional definitions (which try to give the sense of a term), and extensional definiti ...
. 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 understanding, and is the foundation of all valid inference. This had a great influence on Plato's student Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
, in particular Aristotle's notion of the essence
Essence ( la, essentia) is a polysemic term, used in philosophy and theology as a designation for the property or set of properties that make an entity or substance what it fundamentally is, and which it has by necessity, and without which it ...
of a thing.
Aristotle
The logic of Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
, and particularly his theory of the syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be tru ...
, has had an enormous influence in Western thought. Aristotle was the first logician to attempt a systematic analysis of logical syntax, of noun (or '' term''), and of verb. He was the first ''formal logician'', in that he demonstrated the principles of reasoning by employing variables to show the underlying logical form of an argument. He sought relations of dependence which characterize necessary inference, and distinguished the validity of these relations, from the truth of the premises. He was the first to deal with the principles of contradiction and excluded middle in a systematic way.Bochenski p. 63
The Organon
His logical works, called the '' Organon'', are the earliest formal study of logic that have come down to modern times. Though it is difficult to determine the dates, the probable order of writing of Aristotle's logical works is: * '' The Categories'', a study of the ten kinds of primitive term. * '' The Topics'' (with an appendix called '' On Sophistical Refutations''), a discussion of dialectics. * ''On Interpretation
''De Interpretatione'' or ''On Interpretation'' ( Greek: Περὶ Ἑρμηνείας, ''Peri Hermeneias'') is the second text from Aristotle's '' Organon'' and is among the earliest surviving philosophical works in the Western tradition to dea ...
'', an analysis of simple categorical proposition
In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments ...
s into simple terms, negation, and signs of quantity.
* '' The Prior Analytics'', a formal analysis of what makes a syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be tru ...
(a valid argument, according to Aristotle).
* '' The Posterior Analytics'', a study of scientific demonstration, containing Aristotle's mature views on logic.
These works are of outstanding importance in the history of logic. In the ''Categories'', he attempts to discern all the possible things to which a term can refer; this idea underpins his philosophical work ''Metaphysics
Metaphysics is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
'', which itself had a profound influence on Western thought.
He also developed a theory of non-formal logic (''i.e.,'' the theory of fallacies
A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was intr ...
), which is presented in ''Topics'' and ''Sophistical Refutations''.
''On Interpretation'' contains a comprehensive treatment of the notions of opposition
Opposition may refer to:
Arts and media
* ''Opposition'' (Altars EP), 2011 EP by Christian metalcore band Altars
* The Opposition (band), a London post-punk band
* '' The Opposition with Jordan Klepper'', a late-night television series on Com ...
and conversion; chapter 7 is at the origin of the square of opposition
In term logic (a branch of philosophical logic), the square of opposition is a diagram representing the relations between the four basic categorical propositions.
The origin of the square can be traced back to Aristotle's tractate '' On Interp ...
(or logical square); chapter 9 contains the beginning of modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...
.
The ''Prior Analytics'' contains his exposition of the "syllogism", 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.
Stoics
The other great school of Greek logic is that of theStoics
Stoicism is a school of Hellenistic philosophy founded by Zeno of Citium in Athens in the early 3rd century BCE. It is a philosophy of personal virtue ethics informed by its system of logic and its views on the natural world, asserting tha ...
. Stoic logic traces its roots back to the late 5th century BC philosopher Euclid of Megara
Euclid of Megara (; grc-gre, Εὐκλείδης ; c. 435 – c. 365 BC) was a Greek Socratic philosopher who founded the Megarian school of philosophy. He was a pupil of Socrates in the late 5th century BC, and was present at his death. H ...
, a pupil of Socrates
Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no t ...
and slightly older contemporary of Plato, probably following in the tradition of Parmenides and Zeno. His pupils and successors were called "Megarians
Megara (; el, Μέγαρα, ) is a historic town and a municipality in West Attica, Greece. It lies in the northern section of the Isthmus of Corinth opposite the island of Salamis, which belonged to Megara in archaic times, before being taken ...
", or "Eristics", and later the "Dialecticians". The two most important dialecticians of the Megarian school were Diodorus Cronus
Diodorus Cronus ( el, Διόδωρος Κρόνος; died c. 284 BC) was a Greek philosopher and dialectician connected to the Megarian school. He was most notable for logic innovations, including his master argument formulated in response to A ...
and Philo
Philo of Alexandria (; grc, Φίλων, Phílōn; he, יְדִידְיָה, Yəḏīḏyāh (Jedediah); ), also called Philo Judaeus, was a Hellenistic Jewish philosopher who lived in Alexandria, in the Roman province of Egypt.
Philo's de ...
, 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 of Soli (; grc-gre, Χρύσιππος ὁ Σολεύς, ; ) 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 the Stoic philosopher Cleanthes. When C ...
(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 Laërtius
Diogenes Laërtius ( ; grc-gre, Διογένης Λαέρτιος, ; ) was a biographer of the Greek philosophers. Nothing is definitively known about his life, but his surviving ''Lives and Opinions of Eminent Philosophers'' is a principal sour ...
, Sextus Empiricus
Sextus Empiricus ( grc-gre, Σέξτος Ἐμπειρικός, ; ) was a Greek Pyrrhonist philosopher and Empiric school physician. His philosophical works are the most complete surviving account of ancient Greek and Roman Pyrrhonism, and ...
, Galen
Aelius Galenus or Claudius Galenus ( el, Κλαύδιος Γαληνός; September 129 – c. AD 216), often Anglicized as Galen () or Galen of Pergamon, was a Greek physician, surgeon and philosopher in the Roman Empire. Considered to be o ...
, Aulus Gellius
Aulus Gellius (c. 125after 180 AD) was a Roman author and grammarian, who was probably born and certainly brought up in Rome. He was educated in Athens, after which he returned to Rome. He is famous for his ''Attic Nights'', a commonplace book, ...
, Alexander of Aphrodisias, and Cicero
Marcus Tullius Cicero ( ; ; 3 January 106 BC – 7 December 43 BC) was a Roman statesman, lawyer, scholar, philosopher, and academic skeptic, who tried to uphold optimate principles during the political crises that led to the esta ...
.
Three significant contributions of the Stoic school were (i) their account of modality
Modality may refer to:
Humanities
* Modality (theology), the organization and structure of the church, as distinct from sodality or parachurch organizations
* Modality (music), in music, the subject concerning certain diatonic scales
* Modaliti ...
, (ii) their theory of the Material conditional
The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q i ...
, and (iii) their account of meaning and truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as belief ...
.
* ''Modality''. According to Aristotle, the Megarians of his day claimed there was no distinction between potentiality and actuality
In philosophy, potentiality and actuality are a pair of closely connected principles which Aristotle used to analyze motion, causality, ethics, and physiology in his ''Physics'', ''Metaphysics'', ''Nicomachean Ethics'', and '' De Anima''.
The ...
. 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 what is known as his Master argument, which states that each pair of the following 3 propositions contradicts the third proposition:
:* Everything that is past is true and necessary.
:* The impossible does not follow from the possible.
:* What neither is nor will be is possible.
: 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 premise and said that the impossible could follow from the possible.
* ''Conditional statements''. The first logicians to debate conditional statements were Diodorus and his pupil Philo of Megara. Sextus Empiricus refers three times to a debate between Diodorus and Philo. Philo regarded a conditional as true unless it has both a true antecedent and a false consequent
A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...
. Precisely, let ''T0'' and ''T1'' be true statements, and let ''F0'' and ''F1'' be false statements; then, according to Philo, each of the following conditionals is a true statement, because it is not the case that the consequent is false while the antecedent is true (it is not the case that a false statement is asserted to follow from a true statement):
:* If ''T0'', then ''T1''
:* If ''F0'', then ''T0''
:* If ''F0'', then ''F1''
: The following conditional does not meet this requirement, and is therefore a false statement according to Philo:
:* If ''T0, then ''F0''
: Indeed, Sextus says "According to [Philo], there are three ways in which a conditional may be true, and one in which it may be false."Sextus Empiricus, ''Adv. Math.'' viii, Section 113 Philo's criterion of truth is what would now be called a truth-functional definition of "if ... then"; it is the definition used in modern logic.
:In contrast, Diodorus allowed the validity of conditionals only when the antecedent clause could never lead to an untrue conclusion. A century later, the Stoic philosopher Chrysippus
Chrysippus of Soli (; grc-gre, Χρύσιππος ὁ Σολεύς, ; ) 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 the Stoic philosopher Cleanthes. When C ...
attacked the assumptions of both Philo and Diodorus.
* ''Meaning and truth''. The most important and striking difference between Megarian-Stoic logic and Aristotelian logic is that Megarian-Stoic logic concerns propositions, not terms, and is thus closer to modern propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
. 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 signifies, that which is signified, and the object; for example, that which signifies is the word ''Dion'', and that which is signified is what Greeks understand but barbarians do not, and the object is Dion himself.
Medieval logic
Logic in the Middle East
The works of Al-Kindi
Abū Yūsuf Yaʻqūb ibn ʼIsḥāq aṣ-Ṣabbāḥ al-Kindī (; ar, أبو يوسف يعقوب بن إسحاق الصبّاح الكندي; la, Alkindus; c. 801–873 AD) was an Arab Muslim philosopher, polymath, mathematician, physician ...
, Al-Farabi
Abu Nasr Muhammad Al-Farabi ( fa, ابونصر محمد فارابی), ( ar, أبو نصر محمد الفارابي), known in the West as Alpharabius; (c. 872 – between 14 December, 950 and 12 January, 951)PDF version was a renowned early Isl ...
, Avicenna
Ibn Sina ( fa, ابن سینا; 980 – June 1037 CE), commonly known in the West as Avicenna (), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, philosophers, and writers of the Islamic ...
, Al-Ghazali
Al-Ghazali ( – 19 December 1111; ), full name (), and known in Persian-speaking countries as Imam Muhammad-i Ghazali (Persian: امام محمد غزالی) or in Medieval Europe by the Latinized as Algazelus or Algazel, was a Persian poly ...
, Averroes
Ibn Rushd ( ar, ; full name in ; 14 April 112611 December 1198), often Latinized as Averroes ( ), was an
Andalusian polymath and jurist who wrote about many subjects, including philosophy, theology, medicine, astronomy, physics, psy ...
and other Muslim logicians were based on Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West. Al-Farabi
Abu Nasr Muhammad Al-Farabi ( fa, ابونصر محمد فارابی), ( ar, أبو نصر محمد الفارابي), known in the West as Alpharabius; (c. 872 – between 14 December, 950 and 12 January, 951)PDF version was a renowned early Isl ...
(Alfarabi) (873–950) was an Aristotelian logician who discussed the topics of future contingent
Future contingent propositions (or simply, future contingents) are statements about states of affairs in the future that are ''contingent:'' neither necessarily true nor necessarily false.
The problem of future contingents seems to have been firs ...
s, the number and relation of the categories, the relation between logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
and grammar
In linguistics, the grammar of a natural language is its set of structural constraints on speakers' or writers' composition of clauses, phrases, and words. The term can also refer to the study of such constraints, a field that includes doma ...
, and non-Aristotelian forms of inference
Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''wikt:infer, infer'' means to "carry forward". Inference is theoretically traditionally divided into deductive reasoning, deduction and in ...
. 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.
Maimonides
Musa ibn Maimon (1138–1204), commonly known as Maimonides (); la, Moses Maimonides and also referred to by the acronym Rambam ( he, רמב״ם), was a Sephardic Jewish philosopher who became one of the most prolific and influential Torah ...
(1138-1204) wrote a ''Treatise on Logic'' (Arabic: ''Maqala Fi-Sinat Al-Mantiq''), referring to Al-Farabi as the "second master", the first being Aristotle.
Ibn Sina
Ibn Sina ( fa, ابن سینا; 980 – June 1037 CE), commonly known in the West as Avicenna (), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, philosophers, and writers of the Islami ...
(Avicenna) (980–1037) was the founder of Avicennian logic
Ibn Sina ( fa, ابن سینا; 980 – June 1037 CE), commonly known in the West as Avicenna (), was a Persian polymath who is regarded as one of the most significant physicians, astronomers, philosophers, and writers of the Islamic G ...
, which replaced Aristotelian logic as the dominant system of logic in the Islamic world, and also had an important influence on Western medieval writers such as Albertus Magnus
Albertus Magnus (c. 1200 – 15 November 1280), also known as Saint Albert the Great or Albert of Cologne, was a German Dominican friar, philosopher, scientist, and bishop. Later canonised as a Catholic saint, he was known during his li ...
. Avicenna wrote on the hypothetical syllogism and on the propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
, which were both part of the Stoic logical tradition. He developed an original "temporally modalized" syllogistic theory, involving temporal logic In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I ...
and modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...
.History of logic: Arabic logic''
Encyclopædia Britannica
The (Latin for "British Encyclopædia") is a general knowledge English-language encyclopaedia. It is published by Encyclopædia Britannica, Inc.; the company has existed since the 18th century, although it has changed ownership various t ...
''. He also made use of inductive logic
Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' rea ...
, such as the methods of agreement, difference, and concomitant variation which are critical to the scientific method
The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century (with notable practitioners in previous centuries; see the article history of scientifi ...
.Goodman, Lenn Evan (2003), ''Islamic Humanism'', p. 155, Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print book ...
, . One of Avicenna's ideas had a particularly important influence on Western logicians such as William of Ockham
William of Ockham, OFM (; also Occam, from la, Gulielmus Occamus; 1287 – 10 April 1347) was an English Franciscan friar, scholastic philosopher, apologist, and Catholic theologian, who is believed to have been born in Ockham, a small vil ...
: 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
Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics.
Epi ...
, 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
John Stuart Mill (20 May 1806 – 7 May 1873) was an English philosopher, political economist, Member of Parliament (MP) and civil servant. One of the most influential thinkers in the history of classical liberalism, he contributed widely to ...
(1806–1873).Muhammad Iqbal
Sir Muhammad Iqbal ( ur, ; 9 November 187721 April 1938), was a South Asian Muslim writer, philosopher, Quote: "In Persian, ... he published six volumes of mainly long poems between 1915 and 1936, ... more or less complete works on philos ...
, '' The Reconstruction of Religious Thought in Islam'', "The Spirit of Muslim Culture" ( cf.br>an
Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a Logic in Islamic philosophy#Post-Avicennian logic, 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
Concepts are defined as abstract ideas. They are understood to be the fundamental building blocks of the concept behind principles, thoughts and beliefs.
They play an important role in all aspects of cognition. As such, concepts are studied by ...
ions and assents. In response to this tradition, Nasir al-Din al-Tusi
Muhammad ibn Muhammad ibn al-Hasan al-Tūsī ( fa, محمد ابن محمد ابن حسن طوسی 18 February 1201 – 26 June 1274), better known as Nasir al-Din al-Tusi ( fa, نصیر الدین طوسی, links=no; or simply Tusi in the West ...
(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.
The Illuminationist school was 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
Possibility is the condition or fact of being possible. Latin origins of the word hint at ability.
Possibility may refer to:
* Probability, the measure of the likelihood that an event will occur
* Epistemic possibility, a topic in philosophy an ...
, 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'').Dr. Abu Shadi Al-Roubi (1982), "Ibn Al-Nafis as a philosopher", ''Symposium on Ibn al-Nafis'', Second International Conference on Islamic Medicine: Islamic Medical Organization, Kuwait ( cf.br>Ibn al-Nafis As a Philosopher, ''Encyclopedia of Islamic World'').
Ibn Taymiyyah
Ibn Taymiyyah (January 22, 1263 – September 26, 1328; ar, ابن تيمية), birth name Taqī ad-Dīn ʾAḥmad ibn ʿAbd al-Ḥalīm ibn ʿAbd al-Salām al-Numayrī al-Ḥarrānī ( ar, تقي الدين أحمد بن عبد الحليم � ...
(1263–1328), wrote the ''Ar-Radd 'ala al-Mantiqiyyin'', where he argued against the usefulness, though not the validity, of the syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be tru ...
and in favour of inductive reasoning
Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' re ...
. Ibn Taymiyyah also argued against the certainty of syllogistic arguments and in favour of analogy
Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ...
; 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., pp. 16-36 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.
Logic in medieval Europe
"Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logic developed in medieval Europe
In the history of Europe, the Middle Ages or medieval period lasted approximately from the late 5th to the late 15th centuries, similar to the post-classical period of global history. It began with the fall of the Western Roman Empire a ...
throughout roughly the period 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
Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the t ...
, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics.Kneale p. 198 Until the twelfth century, the only works of Aristotle available in the West were the ''Categories'', ''On Interpretation'', and Boethius's translation of the Isagoge
The ''Isagoge'' ( el, Εἰσαγωγή, ''Eisagōgḗ''; ) 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 ...
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
Peter Abelard (; french: link=no, Pierre Abélard; la, Petrus Abaelardus or ''Abailardus''; 21 April 1142) was a medieval French scholastic philosopher, leading logician, theologian, poet, composer and musician. This source has a detailed des ...
(1079–1142). His direct influence was small, but his influence through pupils such as John of Salisbury
John of Salisbury (late 1110s – 25 October 1180), who described himself as Johannes Parvus ("John the Little"), was an English author, philosopher, educationalist, diplomat and bishop of Chartres.
Early life and education
Born at Salisbury, E ...
was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed. The proof for the principle of explosion
In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (, 'from falsehood, anything ollows; or ), or the principle of Pseudo-Scotus, is the law according to which any statement can be proven from a ...
, also known as the principle of seudo-Scotus, the law according to which any proposition can be proven from a contradiction (including its negation), was first given by the 12th century French logician William of Soissons.
By the early thirteenth century, the remaining works of Aristotle's ''Organon'', including the ''Prior Analytics
The ''Prior Analytics'' ( grc-gre, Ἀναλυτικὰ Πρότερα; la, Analytica Priora) is a work by Aristotle on reasoning, known as his syllogistic, composed around 350 BCE. Being one of the six extant Aristotelian writings on logic a ...
'', ''Posterior Analytics
The ''Posterior Analytics'' ( grc-gre, Ἀναλυτικὰ Ὕστερα; la, Analytica Posteriora) is a text from Aristotle's '' Organon'' that deals with demonstration, definition, and scientific knowledge. The demonstration is distinguis ...
'', and the ''Sophistical Refutations
''Sophistical Refutations'' ( el, Σοφιστικοὶ Ἔλεγχοι, Sophistikoi Elenchoi; la, De Sophisticis Elenchis) is a text in Aristotle's ''Organon'' in which he identified thirteen fallacies.Sometimes listed as twelve. According to ...
'' (collectively known as the ''Logica Nova
In the history of logic, the term ''logica nova'' (Latin, meaning "new logic") refers to a subdivision of the logical tradition of Western Europe, as it existed around the middle of the twelfth century. The ''Logica vetus'' ("old logic") referred ...
'' or "New Logic"), 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 supposition
Supposition 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, within an Aristotelian context. Philosophers such as John Burid ...
. 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 just in the present, or does the range include past and future men? Can a term supposit for a non-existing individual? Some medievalists have argued that this idea is a precursor of modern first-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. "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 syncategoremata. 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 Ockham
William of Ockham, OFM (; also Occam, from la, Gulielmus Occamus; 1287 – 10 April 1347) was an English Franciscan friar, scholastic philosopher, apologist, and Catholic theologian, who is believed to have been born in Ockham, a small vil ...
's work Summa Logicae. There, Ockham distinguishes between 'material' and 'formal' consequences, which are roughly equivalent to the modern material implication and logical implication respectively. Similar accounts are given by Jean Buridan and Albert of Saxony
en, Frederick Augustus Albert Anthony Ferdinand Joseph Charles Maria Baptist Nepomuk William Xavier George Fidelis
, image = Albert of Saxony by Nicola Perscheid c1900.jpg
, image_size =
, caption = Photograph by Nicola Persch ...
.
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).
Traditional logic
The textbook tradition
''Traditional logic'' generally means the textbook tradition that begins with Antoine Arnauld
Antoine Arnauld (6 February 16128 August 1694) was a French Catholic theologian, philosopher and mathematician. He was one of the leading intellectuals of the Jansenist group of Port-Royal and had a very thorough knowledge of patristics. C ...
's 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 after Aristotle until the nineteenth century.Buroker xxiii 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
In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, ...
. Between 1664 and 1700, there were eight editions, and the book had considerable influence after that. The Port-Royal introduces the concepts of extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* Ext ...
and intension
In any of several fields of study that treat the use of signs — for example, in linguistics, logic, mathematics, semantics, semiotics, and philosophy of language — an intension is any property or quality connoted by a word, phrase, or ano ...
. The account of proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s that Locke gives in the ''Essay'' is essentially that of the Port-Royal: "Verbal propositions, which are words, rethe 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."
Dudley Fenner
Dudley Fenner (1587) was an English puritan divine. He helped popularise Ramist logic in the English language. Fenner was also one of the first theologians to use the term "covenant of works" to describe God's relationship with Adam in the ''Bo ...
helped popularize Ramist logic, a reaction against Aristotle. Another influential work was the '' Novum Organum'' by Francis Bacon
Francis Bacon, 1st Viscount St Alban (; 22 January 1561 – 9 April 1626), also known as Lord Verulam, was an English philosopher and statesman who served as Attorney General and Lord Chancellor of England. Bacon led the advancement of both ...
, published in 1620. The title translates as "new instrument". This is a reference to Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
's work known as the '' Organon''. In this work, Bacon rejects the syllogistic method of Aristotle in favor 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 is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' re ...
, a method which starts from empirical observation and proceeds to lower axioms or propositions; from these lower axioms, more general ones can be induced. For example, in finding the cause of a ''phenomenal nature'' such as heat, 3 lists should be constructed:
* The presence list: a list of every situation where heat is found.
* The absence list: a list of every situation that is similar to at least one of those of the presence list, except for the lack of heat.
* The variability list: a list of every situation where heat can vary.
Then, the ''form nature'' (or cause) of heat may be defined as that which is common to every situation of the presence list, and which is lacking from every situation of the absence list, and which varies by degree in every situation of the variability list.
Other works in the textbook tradition include Isaac Watts's ''Logick: Or, the Right Use of Reason'' (1725), Richard Whately
Richard Whately (1 February 1787 – 8 October 1863) was an English academic, rhetorician, logician, philosopher, economist, and theologian who also served as a reforming Church of Ireland Archbishop of Dublin. He was a leading Broad Churchman, ...
's ''Logic'' (1826), and John Stuart Mill
John Stuart Mill (20 May 1806 – 7 May 1873) was an English philosopher, political economist, Member of Parliament (MP) and civil servant. One of the most influential thinkers in the history of classical liberalism, he contributed widely to ...
'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 lie in introspection influenced the view that logic is best understood as a branch of psychology, a view which dominated the next fifty years of its development, especially in Germany.
Logic in Hegel's philosophy
G.W.F. Hegel
Georg Wilhelm Friedrich Hegel (; ; 27 August 1770 – 14 November 1831) was a German philosopher. He is one of the most important figures in German idealism and one of the founding figures of modern Western philosophy. His influence extends a ...
indicated the importance of logic to his philosophical system when he condensed his extensive ''Science of Logic
''Science of Logic'' (''SL''; german: Wissenschaft der Logik, ''WdL''), first published between 1812 and 1816, is the work in which Georg Wilhelm Friedrich Hegel outlined his vision of logic. Hegel's logic is a system of '' dialectics'', i.e., ...
'' 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 may refer to:
Companies
* Absolute Entertainment, a video game publisher
* Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK
* Absolute Software Corporation, specializes in security and data risk manag ...
", 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"); this compulsion is, supposedly, not a matter of individual psychology, because it 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 ( grc-gre, διαλεκτική, ''dialektikḗ''; related to dialogue; german: Dialektik), also known as the dialectical method, is a discourse between two or more people holding different points of view about a subject but wishing ...
.
Although Hegel's ''Logic'' has had little impact on mainstream logical studies, its influence can be seen elsewhere:
* Carl von Prantl's ''Geschichte der Logik im Abendland'' (1855–1867).
* The work of the British Idealists, such as F.H. Bradley's ''Principles of Logic'' (1883).
* The economic, political, and philosophical studies of Karl Marx
Karl Heinrich Marx (; 5 May 1818 – 14 March 1883) was a German philosopher, economist, historian, sociologist, political theorist, journalist, critic of political economy, and socialist revolutionary. His best-known titles are the 1848 ...
, and in the various schools of Marxism
Marxism is a Left-wing politics, left-wing to Far-left politics, far-left method of socioeconomic analysis that uses a Materialism, materialist interpretation of historical development, better known as historical materialism, to understand S ...
.
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 ofpsychology
Psychology is the science, scientific study of mind and behavior. Psychology includes the study of consciousness, conscious and Unconscious mind, unconscious phenomena, including feelings and thoughts. It is an academic discipline of immens ...
. The German psychologist Wilhelm Wundt
Wilhelm Maximilian Wundt (; ; 16 August 1832 – 31 August 1920) was a German physiologist, philosopher, and professor, known today as one of the fathers of modern psychology. Wundt, who distinguished psychology as a science from philosophy and ...
, 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 (; 28 July 1851 – 17 October 1914) was a German philosopher, famed for his theory regarding aesthetics, creating the framework for the concept of ''Einfühlung'' ( empathy)'','' defined as, "projecting oneself onto the object of ...
described logic as "a specific discipline of psychology".
* Christoph von Sigwart
Christoph von Sigwart (28 March 1830 – 4 August 1904) was a German philosopher and logician. He was the son of philosopher Heinrich Christoph Wilhelm Sigwart (31 August 1789 – 16 November 1844).
Life
After a course of philosophy a ...
understood logical necessity as grounded in the individual's compulsion to think in a certain way.
* Benno Erdmann
Benno Erdmann (30 May 1851, Guhrau – 7 January 1921, Berlin) was a German neo-Kantian philosopher, logician, psychologist and scholar of Immanuel Kant.
Biography
Erdmann received his Ph.D. in 1873 from the University of Berlin with a disse ...
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
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
. It was also subjected to an extended and destructive critique by Edmund Husserl
, thesis1_title = Beiträge zur Variationsrechnung (Contributions to the Calculus of Variations)
, thesis1_url = https://fedora.phaidra.univie.ac.at/fedora/get/o:58535/bdef:Book/view
, thesis1_year = 1883
, thesis2_title ...
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, also spelled scepticism, is a questioning attitude or doubt toward knowledge claims that are seen as mere belief or dogma. For example, if a person is skeptical about claims made by their government about an ongoing war then the p ...
and relativism
Relativism is a family of philosophical views which deny claims to objectivity within a particular domain and assert that valuations in that domain are relative to the perspective of an observer or the context in which they are assessed. Ther ...
were unavoidable consequences.
Such criticisms did not immediately extirpate what is called " psychologism". For example, the American philosopher Josiah Royce
Josiah Royce (; November 20, 1855 – September 14, 1916) was an American objective idealist philosopher and the founder of American idealism. His philosophical ideas included his version of personalism, defense of absolutism, idealism and his ...
, 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 inmathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
. The development of the modern "symbolic" or "mathematical" logic during this period is the most significant in the 2000-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 had been no prolonged dispute about any truly mathematical result. C.S. Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for t ...
noted that even though a mistake in the evaluation of a definite integral by Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
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 is the branch of philosophy that studies the fundamental nature of reality, the first principles of being, identity and change, space and time, causality, necessity, and possibility. It includes questions about the nature of conscio ...
. 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.
Modern logic
The development of modern logic falls into roughly five periods: * The embryonic period fromLeibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
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
George Boole (; 2 November 1815 – 8 December 1864) was a largely self-taught English mathematician, philosopher, and logician, most of whose short career was spent as the first professor of mathematics at Queen's College, Cork in Irel ...
's Analysis to Schröder's ''Vorlesungen''. In this period, there were more practitioners, and a greater continuity of development.
* The logicist
In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all ...
period from the Begriffsschrift
''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.
''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept nota ...
of Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
to the ''Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
'' of Russell and Whitehead. The aim of the "logicist school" was to incorporate the logic of all mathematical and scientific discourse in a single unified system which, taking as a fundamental principle that all mathematical truths are logical, did not accept any non-logical terminology. The major logicists were Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
, Russell, and the early Wittgenstein
Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is consi ...
. It culminates with the ''Principia'', an important work which includes a thorough examination and attempted solution of the antinomies
Antinomy (Greek ἀντί, ''antí'', "against, in opposition to", and νόμος, ''nómos'', "law") refers to a real or apparent mutual incompatibility of two laws. It is a term used in logic and epistemology, particularly in the philosophy of I ...
which had been an obstacle to earlier progress.
* The metamathematical period from 1910 to the 1930s, which saw the development of metalogic
Metalogic is the study of the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems.Harry GenslerIntroduction to Logic Routledge, ...
, in the finitist system of Hilbert, and the non-finitist system of Löwenheim and Skolem, the combination of logic and metalogic in the work of Gödel and Tarski. Gödel's incompleteness theorem
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies ...
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 logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
branched into four inter-related but separate areas of research: model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
, proof theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
, computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has sinc ...
, and set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, and its ideas and methods began to influence philosophy
Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. ...
.
Embryonic period
The idea that inference could be represented by a purely mechanical process is found as early as Raymond Llull, who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. The work of logicians such as the Oxford Calculators
The Oxford Calculators were a group of 14th-century thinkers, almost all associated with Merton College, Oxford; for this reason they were dubbed "The Merton School". These men took a strikingly logical and mathematical approach to philosoph ...
led to a method of using letters instead of writing out logical calculations (''calculationes'') in words, a method used, for instance, in the ''Logica magna'' by Paul of Venice. Three hundred years after Llull, the English philosopher and logician Thomas Hobbes
Thomas Hobbes ( ; 5/15 April 1588 – 4/14 December 1679) was an English philosopher, considered to be one of the founders of modern political philosophy. Hobbes is best known for his 1651 book '' Leviathan'', in which he expounds an influ ...
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
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
, who had read both Llull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like Llull 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'' that would make all arguments "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) said that reasoning does not have to be about objects about which one has perfectly clear ideas, because algebraic operations can be carried out without having any idea of the meaning of the symbols involved. Bolzano
Bolzano ( or ; german: Bozen, (formerly ); bar, Bozn; lld, Balsan or ) is the capital city of the province of South Tyrol in northern Italy. With a population of 108,245, Bolzano is also by far the largest city in South Tyrol and the third ...
anticipated a fundamental idea of modern proof theory when he defined logical consequence or "deducibility" in terms of variables:Hence I say that propositions , , ,… are ''deducible'' from propositions , , , ,… with respect to variable parts , ,…, if every class of ideas whose substitution for , ,… makes all of , , , ,… true, also makes all of , , ,… true. Occasionally, since it is customary, I shall say that propositions , , ,… ''follow'', or can be ''inferred'' or ''derived'', from , , , ,…. Propositions , , , ,… I shall call the ''premises'', , , ,… the ''conclusions.''This is now known as semantic validity.
Algebraic period
Modern logic begins with what is known as the "algebraic school", originating with Boole and including Peirce, Jevons, Schröder, and Venn
Venn is a surname and a given name. It may refer to:
Given name
* Venn Eyre (died 1777), Archdeacon of Carlisle, Cumbria, England
* Venn Pilcher (1879–1961), Anglican bishop, writer, and translator of hymns
* Venn Young (1929–1993), New Ze ...
. 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
Lincoln () is a cathedral city, a non-metropolitan district, and the county town of Lincolnshire, England. In the 2021 Census, the Lincoln district had a population of 103,813. The 2011 census gave the urban area of Lincoln, including North H ...
. 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.Kneale p. 407 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-function
In logic, a truth function is a function that accepts truth values as input and produces a unique truth value as output. In other words: The input and output of a truth function are all truth values; a truth function will always output exactly on ...
s 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 predicate logic, 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 Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
the following year. In 1885 Allan Marquand proposed an electrical version of the machine that is still extantpicture 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
Exclusive or or exclusive disjunction is a logical operation that is true if and only if its arguments differ (one is true, the other is false).
It is symbolized by the prefix operator J and by the infix operators XOR ( or ), EOR, EXOR, , ...
, 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.
Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic John Corcoran in an accessible introduction to ''Laws of Thought'' Corcoran also wrote a point-by-point comparison of ''Prior Analytics'' and ''Laws of Thought''. According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by 1) providing it with mathematical foundations involving equations, 2) extending the class of problems it could treat — from assessing validity to solving equations — and 3) expanding the range of applications it could handle — e.g. from propositions having only two terms to those having arbitrarily many.
More specifically, Boole agreed with what Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say.
First, in the realm of foundations, Boole reduced the four propositional forms of Aristotelian logic to formulas in the form of equations — by itself a revolutionary idea.
Second, in the realm of logic's problems, Boole's addition of equation solving to logic — another revolutionary idea — involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving.
Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle".
Logicist period
After Boole, the next great advances were made by the German mathematician Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
. Frege's objective was the program of Logicism, i.e. demonstrating that arithmetic is identical with logic.Kneale p. 435 Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or Begriffsschrift
''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.
''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept nota ...
is important. Frege also tried to show that the concept of number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual 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". At the outset Frege abandons the traditional "concepts ''subject'' and ''predicate''", replacing them with ''argument'' and ''function'' respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words ''if, and, not, or, there is, some, all,'' and so forth, deserves attention". 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
Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Ling ...
.
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 men or women" is
:
whereas "All the inhabitants are men or all the inhabitants are women" is
:
As Frege remarked in a critique of Boole's calculus:
: "The real difference is that I avoid he Booleandivision into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it'
As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient problem of multiple generality. The ambiguity of "every girl kissed a boy" is difficult to express in traditional logic, but Frege's logic resolves 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
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help ...
.
This period overlaps with the work of what is known as the "mathematical school", which included Dedekind, Pasch, Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. 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 sta ...
, Hilbert, Zermelo, Huntington, Veblen and Heyting. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was Hilbert's Program, which sought to ground 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. The standard axiomatization
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contai ...
of the natural numbers is named the Peano axioms eponymously. Peano maintained a clear distinction between mathematical and logical symbols. While unaware of Frege's work, he independently recreated his logical apparatus based on the work of Boole and Schröder.
The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by Bertrand Russell. This proved Frege's naive set theory led to a contradiction. Frege's theory contained the axiom 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). Russell's paradox symbolically is as follows:
:
The monumental Principia Mathematica
The ''Principia Mathematica'' (often abbreviated ''PM'') is a three-volume work on the foundations of mathematics written by mathematician–philosophers Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913. ...
, 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
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...
s and inference rules in Mathematical logic, symbolic logic.
Metamathematical period
The names of Gödel and Tarski dominate the 1930s, a crucial period in the development of metamathematics – the study of mathematics using mathematical methods to produce metatheory, metatheories, or mathematical theories about other mathematical theories. Early investigations into metamathematics had been driven by Hilbert's program. Work on metamathematics culminated in the work of Gödel, who in 1929 showed that a given first-order logic, first-order sentence is Provability logic, deducible if and only if it is logically valid – i.e. it is true in every structure (mathematical logic), 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 method, effective procedure such as an algorithm or computer program is capable of proving all facts about the natural numbers. 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
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
, 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 Jan Łukasiewicz, Łukasiewicz, is best known for his definition of truth and logical consequence, and the semantic concept of Open sentence, 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 (logic), 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
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
. Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as completeness (logic), completeness, decidability (logic), decidability, consistency and Structure (mathematical logic), 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
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
are Undecidable problem, 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
Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United ...
and philosophical logic
Understood in a narrow sense, philosophical logic is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical ...
, particularly from the 1950s onwards, in subjects such as modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend ot ...
, temporal logic In logic, temporal logic is any system of rules and symbolism for representing, and reasoning about, propositions qualified in terms of time (for example, "I am ''always'' hungry", "I will ''eventually'' be hungry", or "I will be hungry ''until'' I ...
, deontic logic, and relevance logic.
Logic after WWII
After World War II, mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
branched into four inter-related but separate areas of research: model theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (math ...
, proof theory
Proof theory is a major branchAccording to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding part ...
, computability theory
Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has sinc ...
, and set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
.
In set theory, the method of Forcing (mathematics), forcing revolutionized the field by providing a robust method for constructing models and obtaining independence results. Paul Cohen introduced this method in 1963 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 Turing degree, 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 non-standard analysis, 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, ''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 calculus, 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 logics extend the scope of formal logic to include the elements of Linguistic modality, modality (for example, possibility
Possibility is the condition or fact of being possible. Latin origins of the word hint at ability.
Possibility may refer to:
* Probability, the measure of the likelihood that an event will occur
* Epistemic possibility, a topic in philosophy an ...
and Necessary and sufficient conditions#Necessary conditions, necessity). The ideas of Saul Kripke, particularly about possible worlds, and the formal system now called Kripke semantics have had a profound impact on analytic philosophy
Analytic philosophy is a branch and tradition of philosophy using analysis, popular in the Western world and particularly the Anglosphere, which began around the turn of the 20th century in the contemporary era in the United Kingdom, United ...
. His best known and most influential work is ''Naming and Necessity'' (1980).See ''Philosophical Analysis in the Twentieth Century: Volume 2: The Age of Meaning'', Scott Soames: "''Naming and Necessity'' is among the most important works ever, ranking with the classical work of Frege in the late nineteenth century, and of Russell, Tarski and Wittgenstein in the first half of the twentieth century". Cited in Byrne, Alex and Hall, Ned. 2004. 'Necessary Truths'. ''Boston Review'' October/November 2004 Deontic logics are closely related to modal logics: they attempt to capture the logical features of obligation, Permission (philosophy), permission and related concepts. Although some basic novelties syncretism, syncretizing mathematical and philosophical logic were shown by Bernard Bolzano#Metaphysics, Bolzano in the early 1800s, it was Ernst Mally, a pupil of Alexius Meinong, who was to propose the first formal deontic system in his ''Grundgesetze des Sollens'', based on the syntax of Whitehead's and Russell's propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
.
Another logical system founded after World War II was fuzzy logic by Azerbaijani mathematician Lotfi Asker Zadeh in 1965.
See also
* History of deductive reasoning * History of inductive reasoning * History of abductive reasoning * History of the function concept * History of Mathematics * History of Philosophy * Plato's beard * Timeline of mathematical logicNotes
References
; Primary Sources * Alexander of Aphrodisias, ''In Aristotelis An. Pr. Lib. I Commentarium'', ed. Wallies, Berlin, C.I.A.G. vol. II/1, 1882. * Avicenna, ''Avicennae Opera'' Venice 1508. *Boethius
Anicius Manlius Severinus Boethius, commonly known as Boethius (; Latin: ''Boetius''; 480 – 524 AD), was a Roman senator, consul, ''magister officiorum'', historian, and philosopher of the Early Middle Ages. He was a central figure in the t ...
''Commentary on the Perihermenias'', Secunda Editio, ed. Meiser, Leipzig, Teubner, 1880.
* Bernard Bolzano, Bolzano, Bernard ''Wissenschaftslehre'', (1837) 4 Bde, Neudr., hrsg. W. Schultz, Leipzig I-II 1929, III 1930, IV 1931 (''Theory of Science'', four volumes, translated by Rolf George and Paul Rusnock, New York: Oxford University Press, 2014).
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External links
The History of Logic from Aristotle to Gödel
with annotated bibliographies on the history of logic * * *
Paul Spade's "Thoughts Words and Things"
An Introduction to Late Mediaeval Logic and Semantic Theory
Open Access pdf download; Insights, Images, Bios, and links for 178 logicians
by David Marans {{bots, deny=Yobot History of logic, Logic History of science by discipline, Logic