History of logic

It is said Thales, most widely regarded as the first philosopher in the

Thales is the first known individual to use deductive reasoning applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has been attributed.^[29] Indian and Babylonian mathematicians knew his theorem for special cases before he proved it.^[30] It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon.^[31]

Before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c. 54 years older Thales.^[32] The systematic study of proof seems to have begun with the school of Pythagoras (i. e. the Pythagoreans) in the late sixth century BC.^[20] Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize form rather than matter.^[33]

The writing of Heraclitus (c. 535 – c. 475 BC) was the first place where the word logos was given special attention in ancient Greek philosophy,^[34] 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.

This logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. For though all things come to be in accordance with this logos, humans are like the inexperienced when they experience such words and deeds as I set out, distinguishing each in accordance with its nature and saying how it is. But other people fail to notice what they do when awake, just as they forget what they do while asleep.

— 

Diels-Kranz

, 22B1

In contrast to Heraclitus, Parmenides held that all is one and nothing changes. He may have been a dissident Pythagorean, disagreeing that One (a number) produced the many.^[35] "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,^[36][37]

For this view, that That Which Is Not exists, can never predominate. You must debar your thought from this way of search, nor let ordinary experience in its variety force you along this way, (namely, that of allowing) the eye, sightless as it is, and the ear, full of sound, and the tongue, to rule; but (you must) judge by means of the Reason (Logos) the much-contested proof which is expounded by me.

— B 7.1–8.2

Let no one ignorant of geometry enter here.

— Inscribed over the entrance to Plato's Academy.

None of the surviving works of the great fourth-century philosopher Plato (428–347 BC) include any formal logic,^[40] but they include important contributions to the field of philosophical logic. 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

The logic of

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.
• On Sophistical Refutations
  ), a discussion of dialectics.
• On Interpretation, an analysis of simple categorical propositions
  into simple terms, negation, and signs of quantity.
• The Prior Analytics, a formal analysis of what makes a syllogism (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, which itself had a profound influence on Western thought.

He also developed a theory of non-formal logic (i.e., the theory of

On Interpretation contains a comprehensive treatment of the notions of opposition and conversion; chapter 7 is at the origin of the square of opposition (or logical square); chapter 9 contains the beginning of modal logic.

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.

The other great school of Greek logic is that of the Stoics.^[48] Stoic logic traces its roots back to the late 5th century BC philosopher Euclid of Megara, a pupil of Socrates and slightly older contemporary of Plato, probably following in the tradition of Parmenides and Zeno. His pupils and successors were called "Megarians", or "Eristics", and later the "Dialecticians". The two most important dialecticians of the Megarian school were Diodorus Cronus and Philo, who were active in the late 4th century BC.

The Stoics adopted the Megarian logic and systemized it. The most important member of the school was

• 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.[55] Chrysippus, by contrast, denied the second premise and said that the impossible could follow from the possible.^[56]

• 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. 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."

modern logic

.

In contrast, Diodorus allowed the validity of conditionals only when the antecedent clause could never lead to an untrue conclusion.^[57][58][59]

• 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.^[60] 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.^[61]

The works of

Maimonides (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.

Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873).^[71] Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a Post-Avicennian logic. This was further elaborated by his student Afdaladdîn al-Khûnajî (d. 1249), who developed a form of logic revolving around the subject matter of conceptions and assents. In response to this tradition, Nasir al-Din al-Tusi (1201–1274) began a tradition of Neo-Avicennian logic which remained faithful to Avicenna's work and existed as an alternative to the more dominant Post-Avicennian school over the following centuries.^[72]

The

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.[78] 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.^[72]

"Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logic developed in

• The theory of supposition. Supposition theory deals with the way that predicates (e.g., 'man') range over a domain of individuals (e.g., all men).^[85] 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.^[86] "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".^[87]
• 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
  logical implication respectively. Similar accounts are given by Jean Buridan and Albert of Saxony
  .

Traditional logic generally means the textbook tradition that begins with Antoine Arnauld's and Pierre Nicole's Logic, or the Art of Thinking, better known as the Port-Royal Logic.^[89] Published in 1662, it was the most influential work on logic after Aristotle until the nineteenth century.^[90] The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather than terms, for example) within a framework that is broadly derived from Aristotelian and medieval term logic. Between 1664 and 1700, there were eight editions, and the book had considerable influence after that.^[90] The Port-Royal introduces the concepts of extension and intension. The account of propositions that Locke gives in the Essay is essentially that of the Port-Royal: "Verbal propositions, which are words, [are] the signs of our ideas, put together or separated in affirmative or negative sentences. So that proposition consists in the putting together or separating these signs, according as the things which they stand for agree or disagree."^[91]

Other works in the textbook tradition include Isaac Watts's Logick: Or, the Right Use of Reason (1725), Richard Whately's Logic (1826), and John Stuart Mill's A System of Logic (1843). Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lie in introspection^[93] 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.^[94]

• Carl von Prantl's Geschichte der Logik im Abendland (1855–1867).^[95]
• 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, and in the various schools of Marxism.

Between the work of Mill and Frege stretched half a century during which logic was widely treated as a descriptive science, an empirical study of the structure of reasoning, and thus essentially as a branch of psychology.^[96] The German psychologist Wilhelm Wundt, for example, discussed deriving "the logical from the psychological laws of thought", emphasizing that "psychological thinking is always the more comprehensive form of thinking."^[97] This view was widespread among German philosophers of the period:

• Theodor Lipps described logic as "a specific discipline of psychology".^[98]
• Christoph von Sigwart understood logical necessity as grounded in the individual's compulsion to think in a certain way.^[99]
• Benno Erdmann argued that "logical laws only hold within the limits of our thinking".^[100]

Such was the dominant view of logic in the years following Mill's work.^[101] This psychological approach to logic was rejected by Gottlob Frege. It was also subjected to an extended and destructive critique by Edmund Husserl in the first volume of his Logical Investigations (1900), an assault which has been described as "overwhelming".^[102] Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven, and that skepticism and relativism were unavoidable consequences.

Such criticisms did not immediately extirpate what is called "psychologism". For example, the American philosopher Josiah Royce, while acknowledging the force of Husserl's critique, remained "unable to doubt" that progress in psychology would be accompanied by progress in logic, and vice versa.^[103]

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.^[2] The revival of logic occurred in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in 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.^[4]

A number of features distinguish modern logic from the old Aristotelian or traditional logic, the most important of which are as follows:

The development of modern logic falls into roughly five periods:[106]

• The embryonic period from Leibniz to 1847, when the notion of a logical calculus was discussed and developed, particularly by Leibniz, but no schools were formed, and isolated periodic attempts were abandoned or went unnoticed.
• The algebraic period from
  Boole's Analysis to Schröder
  's Vorlesungen. In this period, there were more practitioners, and a greater continuity of development.
• The
  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, Russell, and the early Wittgenstein.^[107] It culminates with the Principia, an important work which includes a thorough examination and attempted solution of the antinomies
  which had been an obstacle to earlier progress.
• The metamathematical period from 1910 to the 1930s, which saw the development of
  set-theoretic constructibility
  .
• The period after World War II, when mathematical logic branched into four inter-related but separate areas of research: model theory, proof theory, computability theory, and set theory, and its ideas and methods began to influence philosophy.

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^[108] 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 suggested that all logic and reasoning could be reduced to the mathematical operations of addition and subtraction.^[109] The same idea is found in the work of Leibniz, 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;^[110] hence, he proposed to identify an alphabet of human thought comprising fundamental concepts which could be composed to express complex ideas,^[111] 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."^[112]

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.^[113] Bolzano anticipated a fundamental idea of modern proof theory when he defined logical consequence or "deducibility" in terms of variables:^[114]

Hence I say that propositions ${\displaystyle M}$, ${\displaystyle N}$, ${\displaystyle O}$,... are deducible from propositions ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$,... with respect to variable parts ${\displaystyle i}$, ${\displaystyle j}$,..., if every class of ideas whose substitution for ${\displaystyle i}$, ${\displaystyle j}$,... makes all of ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$,... true, also makes all of ${\displaystyle M}$, ${\displaystyle N}$, ${\displaystyle O}$,... true. Occasionally, since it is customary, I shall say that propositions ${\displaystyle M}$, ${\displaystyle N}$, ${\displaystyle O}$,... follow, or can be inferred or derived, from ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$,.... Propositions ${\displaystyle A}$, ${\displaystyle B}$, ${\displaystyle C}$, ${\displaystyle D}$,... I shall call the premises, ${\displaystyle M}$, ${\displaystyle N}$, ${\displaystyle O}$,... 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

truth-functions and their expression in disjunctive normal form.^[118]

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."[120] 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.^[121]

In his Symbolic Logic (1881), John Venn used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's methods could be mechanised, and constructed a "logical machine" which he showed to the Royal Society the following year.^[118] In 1885 Allan Marquand proposed an electrical version of the machine that is still extant (picture at the Firestone Library).

The defects in Boole's system (such as the use of the letter v for existential propositions) were all remedied by his followers. Jevons published Pure Logic, or the Logic of Quality apart from Quantity in 1864, where he suggested a symbol to signify

logical sum which originates in Peirce (1867), Schröder (1877) and Jevons (1890),^[125] 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.^[126]

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.^[127] Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought.^[128] 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 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

J. S. Mill as well as Jevons, citing the latter's claim that "algebra is a highly developed logic, and number but logical discrimination."^[130]

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.[131] 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".^[132] 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".^[133] 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

${\displaystyle \forall \;x{\big (}A(x)\rightarrow B(x){\big )}}$

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.^[134] This functional analysis of ordinary-language sentences later had a great impact on philosophy and linguistics.

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

Frege

's "Concept Script"

${\displaystyle \forall \;x{\Big (}I(x)\rightarrow {\big (}M(x)\lor W(x){\big )}{\Big )}}$

whereas "All the inhabitants are men or all the inhabitants are women" is

${\displaystyle \forall \;x{\big (}I(x)\rightarrow M(x){\big )}\lor \forall \;x{\big (}I(x)\rightarrow W(x){\big )}}$

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

"The real difference is that I avoid [the Boolean] division into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it."^[135]

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

${\displaystyle \forall \;x{\Big (}G(x)\rightarrow \exists \;y{\big (}B(y)\land K(x,y){\big )}{\Big )}}$

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

${\displaystyle \exists \;x{\Big (}B(x)\land \forall \;y{\big (}G(y)\rightarrow K(y,x){\big )}{\Big )}}$

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.^[136]

This period overlaps with the work of what is known as the "mathematical school", which included

axiomatization 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.^[137]

The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by

axiomatic set theory. It was developed into the now-canonical Zermelo–Fraenkel set theory

(ZF). Russell's paradox symbolically is as follows:

${\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}$

The monumental

inference rules in symbolic logic

.

Metamathematical period

The names of

set-theoretic constructibility, as part of his proof that the axiom of choice and the continuum hypothesis are consistent with Zermelo–Fraenkel set theory

. In proof theory, Gerhard Gentzen developed natural deduction and the sequent calculus. The former attempts to model logical reasoning as it 'naturally' occurs in practice and is most easily applied to intuitionistic logic, while the latter was devised to clarify the derivation of logical proofs in any formal system. Since Gentzen's work, natural deduction and sequent calculi have been widely applied in the fields of proof theory, mathematical logic and computer science. Gentzen also proved normalization and cut-elimination theorems for intuitionistic and classical logic which could be used to reduce logical proofs to a normal form.^[141]

logical satisfaction. In 1933, he published (in Polish) The concept of truth in formalized languages, in which he proposed his semantic theory of truth: a sentence such as "snow is white" is true if and only if snow is white. Tarski's theory separated the metalanguage, which makes the statement about truth, from the object language, which contains the sentence whose truth is being asserted, and gave a correspondence (the T-schema) between phrases in the object language and elements of an interpretation. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of model theory.^[142] Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as completeness, decidability, consistency and definability. According to Anita Feferman, Tarski "changed the face of logic in the twentieth century".^[143]

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

degrees of unsolvability

.

The results of the first few decades of the twentieth century also had an impact upon analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic.

Logic after WWII

After World War II, mathematical logic branched into four inter-related but separate areas of research: model theory, proof theory, computability theory, and set theory.^[144]

In set theory, the method of forcing revolutionized the field by providing a robust method for constructing models and obtaining independence results. Paul Cohen introduced this method in 1963 to prove the independence of the continuum hypothesis and the axiom of choice from Zermelo–Fraenkel set theory.^[145] 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

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,

infinitesimals

, a problem that first had been proposed by Leibniz.

In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability method invented by Georg Kreisel and Gödel's Dialectica interpretation. This work inspired the contemporary area of proof mining. The Curry–Howard correspondence emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and typed lambda calculi used in computer science. As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. Advances were also made in ordinal analysis and the study of independence results in arithmetic such as the Paris–Harrington theorem.

This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example,

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.^[147] His best known and most influential work is Naming and Necessity (1980).^[148] Deontic logics are closely related to modal logics: they attempt to capture the logical features of obligation, permission and related concepts. Although some basic novelties syncretizing mathematical and philosophical logic were shown by 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

.

Another logical system founded after World War II was

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 logic

Notes

1. ^ ^{a b} Boehner p. xiv
2. ^ ^{a b} Oxford Companion p. 498; Bochenski, Part I Introduction, passim
3. ^ Frege, Gottlob. The Foundations of Arithmetic (PDF). p. 1. Archived from the original (PDF) on 2018-09-20. Retrieved 2016-02-03.
4. ^ ^{a b} Oxford Companion p. 500
5. ISBN 978-0-8091-2781-8
   .
6. ISBN 978-0-520-95067-2
   .
7. ISBN 978-81-317-1120-0
   .
8. ^ Bochenski p. 446
9. ^ Vidyabhusana, S. C. (1921). History Of Indian Logic. p. 11.
10. ^ Bhusana, Satis Chandra Vidya (1921). A History Of Indian Logic.
11. ^ S. C. Vidyabhusana (1971). A History of Indian Logic: Ancient, Mediaeval, and Modern Schools, pp. 17–21.
12. ^ R. P. Kangle (1986). The Kautiliya Arthashastra (1.2.11). Motilal Banarsidass.
13. ^ Bochenski p. 417 and passim
14. S2CID 170089234
    .
15. ^ Bochenski pp. 431–437
16. ISBN 9780791437407
    .
17. ^ Bochenksi p. 441
18. ^ Matilal, 17
19. ^ Kneale, p. 2
20. ^ ^{a b c d} Kneale p. 3
21. ISBN 90-04-13666-5
    .
22. ISBN 90-5693-036-2
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23. ^ Heath, Mathematics in Aristotle, cited in Kneale, p. 5
24. ^ Kneale, p. 16
25. ^ "History of logic". britannica.com. Retrieved 2018-04-02.
26. ^ Aristotle, Metaphysics Alpha, 983b18.
27. ^ Smith, William (1870). Dictionary of Greek and Roman biography and mythology. Boston, Little. p. 1016.
28. Patras University
    . Archived from the original on 2016-03-03. Retrieved 2012-02-12.
29. ^ (Boyer 1991, "Ionia and the Pythagoreans" p. 43)
30. ISBN 92-3-102812-X
31. ISBN 0-470-63056-6
32. ^ C. B. Boyer (1968)
33. ^ Samuel Enoch Stumpf. Socrates to Sartre. p. 11.
34. ^ F.E. Peters, Greek Philosophical Terms, New York University Press, 1967.
35. ^ Cornford, Francis MacDonald (1957) [1939]. Plato and Parmenides: Parmenides' Way of Truth and Plato's Parmenides translated with an introduction and running commentary (PDF). Liberal Arts Press.
36. ^ R. J. Hollingdale (1974). Western Philosophy: an introduction. p. 73.
37. ^ Cornford, Francis MacDonald (1912). From religion to philosophy: A study in the origins of western speculation (PDF). Longmans, Green and Co.
38. ^ Kneale p. 15
39. ^ "The Numismatic Circular". 2018-04-02. Retrieved 2018-04-02 – via Google Books.
40. ^ Kneale p. 17
41. ^ "forming an opinion is talking, and opinion is speech that is held not with someone else or aloud but in silence with oneself" Theaetetus 189E–190A
42. ^ Kneale p. 20. For example, the proof given in the Meno that the square on the diagonal is double the area of the original square presumably involves the forms of the square and the triangle, and the necessary relation between them
43. ^ Kneale p. 21
44. ^ Zalta, Edward N. "Aristotle's Logic". Stanford University, 18 March 2000. Retrieved 13 March 2010.
45. ^ See e.g. Aristotle's logic, Stanford Encyclopedia of Philosophy
46. OCLC 38239202
    .
47. ^ ^{a b} Bochenski p. 63
48. ^ "Throughout later antiquity two great schools of logic were distinguished, the Peripatetic which was derived from Aristotle, and the Stoic which was developed by Chrysippus from the teachings of the Megarians" – Kneale p. 113
49. ^ Oxford Companion, article "Chrysippus", p. 134
50. ^ [1] Stanford Encyclopedia of Philosophy: Susanne Bobzien, Ancient Logic
51. ^ K. Hülser, Die Fragmente zur Dialektik der Stoiker, 4 vols, Stuttgart 1986–1987
52. ^ Kneale 117–158
53. ^ Metaphysics Eta 3, 1046b 29
54. ^ Boethius, Commentary on the Perihermenias, Meiser p. 234
55. ^ Epictetus, Dissertationes ed. Schenkel ii. 19. I.
56. ^ Alexander p. 177
57. ^ ^{a b} Sextus Empiricus, Adv. Math. viii, Section 113
58. ^ Sextus Empiricus, Hypotyp. ii. 110, comp.
59. ^ Cicero, Academica, ii. 47, de Fato, 6.
60. ^ See e.g. Lukasiewicz p. 21
61. ^ Sextus Bk viii., Sections 11, 12
62. ^ See e.g. Routledge Encyclopedia of Philosophy Online Version 2.0 Archived 2022-06-06 at the Wayback Machine, article 'Islamic philosophy'
63. ^ ^{a b} History of logic: Arabic logic, Encyclopædia Britannica.
64. ISBN 0-521-27556-3
    .
65. ^ Hasse, Dag Nikolaus (2008-09-19). "Influence of Arabic and Islamic Philosophy on the Latin West". Stanford Encyclopedia of Philosophy. Retrieved 2009-10-13.
66. ^ Richard F. Washell (1973), "Logic, Language, and Albert the Great", Journal of the History of Ideas 34 (3), pp. 445–450 [445].
67. ^
    ISBN 0-19-513580-6
    .
68. ISBN 0-415-01929-X
    .
69. ^ Kneale p. 229
70. Summa Logicae
    i. 14; Avicenna: Avicennae Opera Venice 1508 f87rb
71. ^ ^{a b} Muhammad Iqbal, The Reconstruction of Religious Thought in Islam, "The Spirit of Muslim Culture" (cf. [2] and [3])
72. ^ ^{a b} Tony Street (2008-07-23). "Arabic and Islamic Philosophy of Language and Logic". Stanford Encyclopedia of Philosophy. Retrieved 2008-12-05.
73. ^ Lotfollah Nabavi, Sohrevardi's Theory of Decisive Necessity and kripke's QSS System Archived 2008-01-26 at the Wayback Machine, Journal of Faculty of Literature and Human Sciences.
74. ^ Abu Shadi Al-Roubi (1982), "Ibn Al-Nafis as a philosopher", Symposium on Ibn al-Nafis, Second International Conference on Islamic Medicine: Islamic Medical Organization, Kuwait (cf. Ibn al-Nafis As a Philosopher Archived 2008-02-06 at the Wayback Machine, Encyclopedia of Islamic World).
75. ISBN 978-0-521-52069-0
    .
76. ISSN 0015-5675
    .
77. ^ ^{a b} John F. Sowa; Arun K. Majumdar (2003). "Analogical reasoning". Conceptual Structures for Knowledge Creation and Communication, Proceedings of ICCS 2003. Berlin: Springer-Verlag., pp. 16–36
78. ISBN 0-87395-224-3
    .
79. ^ Kneale p. 198
80. ^ Stephen Dumont, article "Peter Abelard" in Gracia and Noone p. 492
81. ^ Kneale, pp. 202–203
82. ^ See e.g. Kneale p. 225
83. ^ Boehner p. 1
84. ^ Boehner pp. 19–76
85. ^ Boehner p. 29
86. ^ Boehner p. 30
87. ^ Ebbesen 1981
88. ^ Boehner pp. 54–55
89. ^ Oxford Companion p. 504, article "Traditional logic"
90. ^ ^{a b} Buroker xxiii
91. ^ (Locke, An Essay Concerning Human Understanding, IV. 5. 6)
92. ^ Farrington, 1964, 89
93. ^ N. Abbagnano, "Psychologism" in P. Edwards (ed) The Encyclopaedia of Philosophy, MacMillan, 1967
94. ^ Of the German literature in this period, Robert Adamson wrote "Logics swarm as bees in springtime..."; Robert Adamson, A Short History of Logic, Wm. Blackwood & Sons, 1911, page 242
95. ^ Carl von Prantl (1855–1867), Geschichte von Logik in Abendland, Leipzig: S. Hirzl, anastatically reprinted in 1997, Hildesheim: Georg Olds.
96. ^ See e.g. Psychologism, Stanford Encyclopedia of Philosophy
97. ^ Wilhelm Wundt, Logik (1880–1883); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, pp. 115–116.
98. ^ Theodor Lipps, Grundzüge der Logik (1893); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. 40
99. ^ Christoph von Sigwart, Logik (1873–1878); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. 51
100. ^ Benno Erdmann, Logik (1892); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. 96
101. ^ Dermot Moran, "Introduction"; Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. xxi
102. ^ Michael Dummett, "Preface"; Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. xvii
103. ^ Josiah Royce, "Recent Logical Enquiries and their Psychological Bearings" (1902) in John J. McDermott (ed) The Basic Writings of Josiah Royce Volume 2, Fordham University Press, 2005, p. 661
104. ^ Bochenski, p. 266
105. ^ Peirce 1896
106. ^ See Bochenski p. 269
107. ^ Oxford Companion p. 499
108. ^ Edith Sylla (1999), "Oxford Calculators", in The Cambridge Dictionary of Philosophy, Cambridge, Cambridgeshire: Cambridge.
109. ^ El. philos. sect. I de corp 1.1.2.
110. ^ Bochenski p. 274
111. ^ Rutherford, Donald, 1995, "Philosophy and language" in Jolley, N., ed., The Cambridge Companion to Leibniz. Cambridge Univ. Press.
112. ^ Wiener, Philip, 1951. Leibniz: Selections. Scribner.
113. ^ Essai de dialectique rationelle, 211n, quoted in Bochenski p. 277.
114. ISBN 978-0-52001787-0
     .
115. ^ See e.g. Bochenski p. 296 and passim
116. ^ Before publishing, he wrote to De Morgan, who was just finishing his work Formal Logic. De Morgan suggested they should publish first, and thus the two books appeared at the same time, possibly even reaching the bookshops on the same day. cf. Kneale p. 404
117. ^ Kneale p. 404
118. ^ ^{a b c} Kneale p. 407
119. ^ Boole (1847) p. 16
120. ^ Boole 1847 pp. 58–59
121. ^ Beaney p. 11
122. ^ Kneale p. 422
123. ^ Peirce, "A Boolian Algebra with One Constant", 1880 MS, Collected Papers v. 4, paragraphs 12–20, reprinted Writings v. 4, pp. 218–221. Google Preview.
124. ^ Trans. Amer. Math. Soc., xiv (1913), pp. 481–488. This is now known as the Sheffer stroke
125. ^ Bochenski 296
126. ^ See CP III
127. ^ George Boole. 1854/2003. The Laws of Thought, facsimile of 1854 edition, with an introduction by J. Corcoran. Buffalo: Prometheus Books (2003). Reviewed by James van Evra in Philosophy in Review. 24 (2004) 167–169.
128. ^ JOHN CORCORAN, Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol. 24 (2003), pp. 261–288.
129. ^ ^{a b} Kneale p. 435
130. ^ Jevons, The Principles of Science, London 1879, p. 156, quoted in Grundlagen 15
131. ^ Beaney p. 10 – the completeness of Frege's system was eventually proved by Jan Łukasiewicz in 1934
132. Summa Logicae
     III. 8 (??)
133. ^ Frege 1879 in van Heijenoort 1967, p. 7
134. ^ "On concept and object" p. 198; Geach p. 48
135. ^ BLC p. 14, quoted in Beaney p. 12
136. ^ See e.g. The Internet Encyclopedia of Philosophy, article "Frege"
137. ^ Van Heijenoort 1967, p. 83
138. ^ See e.g. Potter 2004
139. ^ Zermelo 1908
140. ^ Feferman 1999 p. 1
141. ISBN 0-521-37181-3
     .
142. ^ Feferman and Feferman 2004, p. 122, discussing "The Impact of Tarski's Theory of Truth".
143. ^ Feferman 1999, p. 1
144. ^ See e.g. Barwise, Handbook of Mathematical Logic
145. PMID 16591132
     .
146. ^ Many of the foundational papers are collected in The Undecidable (1965) edited by Martin Davis
147. ^ Jerry Fodor, "Water's water everywhere", London Review of Books, 21 October 2004
148. ^ 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

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