The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic (or term logic) as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.
Christian and Islamic philosophers such as Boethius (died 524), Avicenna (died 1037), Thomas Aquinas (died 1274) and William of Ockham (died 1347) further developed Aristotle's logic in the Middle Ages, 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.Empirical methods ruled the day, as evidenced by Sir Francis Bacon's Novum Organon 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 used in 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, Frege, Russell, and Peano is the most significant in the two-thousand-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history.
Progress in mathematical logic in the first few decades of the twentieth century, particularly arising from the work of Gödel and Tarski, had a significant impact on analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic.
Logic in India
Hindu logic
Origin
The Nasadiya Sukta of the Rigveda (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".
Who really knows? Who will here proclaim it? Whence was it produced? Whence is this creation? The gods came afterwards, with the creation of this universe. Who then knows whence it has arisen?
— Nasadiya Sukta, concerns the origin of the universe, Rig Veda, 10:129-6
Logic began independently in ancient India 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) of various sorts, comprising relevant experts, were regularly convened to deliberate on a variety of matters, including administrative, legal and religious matters.[citation needed]
Dattatreya
A philosopher named Dattatreya is stated in the Bhagavata Purana to have taught Anviksiki to Aiarka, Prahlada and others. It appears from the Markandeya purana that the Anviksiki-vidya expounded by him consisted of a mere disquisition on soul in accordance with the yoga philosophy. Dattatreya expounded the philosophical side of Anviksiki 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 school of logic. The Mahabharata (12.173.45), around the 5th century BC, refers to the anviksiki and tarka schools of logic.
Panini
Pāṇini (c. 5th century BC) developed a form of logic (to which Boolean logic has some similarities) for his formulation of Sanskrit grammar. Logic is described by Chanakya (c. 350–283 BC) in his Arthashastra as an independent field of inquiry.
Nyaya-Vaisheshika
Two of the six Indian schools of thought deal with logic: Nyaya and 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 Hindu philosophy. This realist school developed a rigid five-member schema of inference involving an initial premise, a reason, an example, an application, and a conclusion. The idealistBuddhist philosophy became the chief opponent to the Naiyayikas.
Jain logic
Umaswati (2nd century AD), author of first Jain work in Sanskrit, Tattvārthasūtra, expounding the Jain philosophy in a most systematized form acceptable to all sects of Jainism.
Jains 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 concepts in Jain philosophy made important contributions to the thought, especially in the areas of skepticism and relativity. [4]
Buddhist logic
Nagarjuna
Nagarjuna (c. 150–250 AD), the founder of the Madhyamaka ("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 four possibilities of a proposition, P:
P; that is, being.
not P; that is, not being.
Painting of Nāgārjuna from the Shingon Hassozō, a series of scrolls authored by the Shingon school of Buddhism. Japan, Kamakura period (13th–14th century)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, De Morgan's laws would imply that this case is equivalent to the third case (P and not P), and would be therefore superfluous, with only 3 actual 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", 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) 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.
Logic in China
In China, a contemporary of Confucius, Mozi, "Master Mo", is credited with founding the Mohist school, whose canons dealt with issues relating to valid inference and the conditions of correct conclusions. In particular, one of the schools that grew out of Mohism, the Logicians, are credited by some scholars for their early investigation of formal logic. Due to the harsh rule of Legalism in the subsequent Qin dynasty, this line of investigation disappeared in China until the introduction of Indian philosophy by Buddhists.
Logic in the ancient Mediterranean
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 with geometry, which originally meant the same as "land measurement". The ancient Egyptians discovered geometry, including the formula for the volume of a truncated pyramid.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 axioms and assumptions, while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems, an important contribution to the philosophy of science.
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 demonstrative proof. Both Thales and Pythagoras of the Pre-Socratic philosophers 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 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 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 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 Rhetoricians or Orators and the Sophists, who used arguments to defend or attack a thesis, both in legal and political contexts.
Thales Theorem
Thales
It is said Thales, most widely regarded as the first philosopher in the Greek tradition, measured the height of the pyramids 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.
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.Indian 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 is a right angle during his travels to Babylon.
Pythagoras
Proof of the Pythagorean Theorem in Euclid's Elements
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 of Heraclitus (c. 535 – c. 475 BC) was the first place where the word logos 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.
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
Parmenides has been called the discoverer of logic.
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. "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,
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
Zeno of Elea, 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 portrays Zeno as claiming to have written a book defending the monism of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. 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
Let no one ignorant of geometry enter here.
— Inscribed over the entrance to Plato's Academy.
Plato's Academy mosaic
None of the surviving works of the great fourth-century philosopher Plato (428–347 BC) include any formal logic, 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 Theaetetus, where Plato identifies thought or opinion with talk or discourse (logos). The second question is a result of Plato's theory of Forms. Forms are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called universals, namely an abstract entity common to each set of things that have the same name. In both the Republic and the Sophist, Plato suggests that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary connection between "forms". The third question is about definition. Many of Plato's dialogues concern the search for a definition of some important concept (justice, truth, the Good), and it is likely that Plato was impressed by the importance of definition in mathematics. What underlies every definition is a Platonic Form, the common nature present in different particular things. Thus, a definition reflects the ultimate object of understanding, and is the foundation of all valid inference. This had a great influence on Plato's student Aristotle, in particular Aristotle's notion of the essence of a thing.
Aristotle
Aristotle
The logic of Aristotle, and particularly his theory of the syllogism, 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.
Aristotle's logic was still influential in the Renaissance.
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.
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 fallacies), which is presented in Topics and Sophistical Refutations.
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.
Stoics
The other great school of Greek logic is that of the Stoics. 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.
Chrysippus of Soli
The Stoics adopted the Megarian logic and systemized it. The most important member of the school was Chrysippus (c. 278 – c. 206 BC), who was its third head, and who formalized much of Stoic doctrine. He is supposed to have written over 700 works, including at least 300 on logic, almost none of which survive. Unlike with Aristotle, we have no complete works by the Megarians or the early Stoics, and have to rely mostly on accounts (sometimes hostile) by later sources, including prominently Diogenes Laërtius, Sextus Empiricus, Galen, Aulus Gellius, Alexander of Aphrodisias, and Cicero.
Three significant contributions of the Stoic school were (i) their account of modality, (ii) their theory of the Material conditional, and (iii) their account of meaning and truth.
Modality. According to Aristotle, the Megarians of his day claimed there was no distinction between potentiality and actuality. Diodorus Cronus defined the possible as that which either is or will be, the impossible as what will not be true, and the contingent as that which either is already, or will be false. Diodorus is also famous for 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. 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." 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.
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. 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.
The works of Al-Kindi, Al-Farabi, Avicenna, Al-Ghazali, Averroes 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 (Alfarabi) (873–950) was an Aristotelian logician who discussed the topics of future contingents, the number and relation of the categories, the relation between logic and grammar, and non-Aristotelian forms of inference. Al-Farabi also considered the theories of conditional syllogisms and analogical inference, which were part of the Stoic tradition of logic rather than the Aristotelian.
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.
Ibn Sina (Avicenna) (980–1037) was the founder of Avicennian logic, 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. Avicenna wrote on the hypothetical syllogism and on the propositional calculus, which were both part of the Stoic logical tradition. He developed an original "temporally modalized" syllogistic theory, involving temporal logic and modal logic. He also made use of inductive logic, such as the methods of agreement, difference, and concomitant variation which are critical to the scientific method. One of Avicenna's ideas had a particularly important influence on Western logicians such as William of Ockham: Avicenna's word for a meaning or notion (ma'na), was translated by the scholastic logicians as the Latin intentio; in medieval logic and epistemology, this is a sign in the mind that naturally represents a thing. This was crucial to the development of Ockham's conceptualism: A universal term (e.g., "man") does not signify a thing existing in reality, but rather a sign in the mind (intentio in intellectu) which represents many things in reality; Ockham cites Avicenna's commentary on Metaphysics V in support of this view.
Fakhr al-Din al-Razi (b. 1149) criticised Aristotle's "first figure" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill (1806–1873). Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a Post-Avicennian logic. This was further elaborated by his student Afdaladdîn al-Khûnajî (d. 1249), who developed a form of logic revolving around the subject matter of 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.
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, 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).Ibn Taymiyyah (1263–1328), wrote the Ar-Radd 'ala al-Mantiqiyyin, where he argued against the usefulness, though not the validity, of the syllogism and in favour of inductive reasoning. Ibn Taymiyyah also argued against the certainty of syllogistic arguments and in favour of analogy; his argument is that concepts founded on induction are themselves not certain but only probable, and thus a syllogism based on such concepts is no more certain than an argument based on analogy. He further claimed that induction itself is founded on a process of analogy. His model of analogical reasoning was based on that of juridical arguments. This model of analogy has been used in the recent work of John F. Sowa.
The Sharh al-takmil fi'l-mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century is the last major Arabic work on logic that has been studied. However, "thousands upon thousands of pages" on logic were written between the 14th and 19th centuries, though only a fraction of the texts written during this period have been studied by historians, hence little is known about the original work on Islamic logic produced during this later period.
Logic in medieval Europe
Brito's questions on the Old Logic
"Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logic developed in medieval Europe 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, who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics. Until the twelfth century, the only works of Aristotle available in the West were the Categories, On Interpretation, and Boethius's translation of the Isagoge of Porphyry (a commentary on the Categories). These works were known as the "Old Logic" (Logica Vetus or Ars Vetus). An important work in this tradition was the Logica Ingredientibus of Peter Abelard (1079–1142). His direct influence was small, but his influence through pupils such as John of Salisbury was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed. The proof for the principle of explosion, also known as the principle of Pseudo-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, Posterior Analytics, and the Sophistical Refutations (collectively known as the Logica Nova 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 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. "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'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.
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
Dudley Fenner's Art of Logic (1584)
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. Published in 1662, it was the most influential work on logic after Aristotle until the nineteenth century. The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather than terms, for example) within a framework that is broadly derived from Aristotelian and medieval term logic. Between 1664 and 1700, there were eight editions, and the book had considerable influence after that. The 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."
Dudley Fenner helped popularize Ramist logic, a reaction against Aristotle. Another influential work was the Novum Organum by Francis Bacon, published in 1620. The title translates as "new instrument". This is a reference to Aristotle's work 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, 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, three 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'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 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
Georg Wilhelm Friedrich Hegel
G.W.F. Hegel indicated the importance of logic to his philosophical system when he condensed his extensive Science of Logic into a shorter work published in 1817 as the first volume of his Encyclopaedia of the Philosophical Sciences. The "Shorter" or "Encyclopaedia" Logic, as it is often known, lays out a series of transitions which leads from the most empty and abstract of categories—Hegel begins with "Pure Being" and "Pure Nothing"—to the "Absolute", the category which contains and resolves all the categories which preceded it. Despite the title, Hegel's Logic is not really a contribution to the science of valid inference. Rather than deriving conclusions about concepts through valid inference from premises, Hegel seeks to show that thinking about one concept compels thinking about another concept (one cannot, he argues, possess the concept of "Quality" without the concept of "Quantity"); 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.
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, and in the various schools of Marxism.
Logic and psychology
Between the work of Mill and Frege stretched half a century during which logic was widely treated as a descriptive science, an empirical study of the structure of reasoning, and thus essentially as a branch of psychology. The German psychologist Wilhelm Wundt, for example, discussed deriving "the logical from the psychological laws of thought", emphasizing that "psychological thinking is always the more comprehensive form of thinking." This view was widespread among German philosophers of the period:
Theodor Lipps described logic as "a specific discipline of psychology".
Christoph von Sigwart understood logical necessity as grounded in the individual's compulsion to think in a certain way.
Benno Erdmann argued that "logical laws only hold within the limits of our thinking".
Such was the dominant view of logic in the years following Mill's work. This psychological approach to logic was rejected by Gottlob Frege. It was also subjected to an extended and destructive critique by Edmund Husserl in the first volume of his Logical Investigations (1900), an assault which has been described as "overwhelming". Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven, and that skepticism and relativism were unavoidable consequences.
Such criticisms did not immediately extirpate 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.
Rise of modern logic
The period between the fourteenth century and the beginning of the nineteenth century had been largely one of decline and neglect, and is generally regarded as barren by historians of logic. The revival of logic occurred in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in 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 noted that even though a mistake in the evaluation of a definite integral by Laplace led to an error concerning the moon's orbit that persisted for nearly 50 years, the mistake, once spotted, was corrected without any serious dispute. Peirce contrasted this with the disputation and uncertainty surrounding traditional logic, and especially reasoning in metaphysics. He argued that a truly "exact" logic would depend upon mathematical, i.e., "diagrammatic" or "iconic" thought. "Those who follow such methods will ... escape all error except such as will be speedily corrected after it is once suspected". Modern logic is also "constructive" rather than "abstractive"; i.e., rather than abstracting and formalising theorems derived from ordinary language (or from psychological intuitions about validity), it constructs theorems by formal methods, then looks for an interpretation in ordinary language. It is entirely symbolic, meaning that even the logical constants (which the medieval logicians called "syncategoremata") and the categoric terms are expressed in symbols.
Modern logic
The development of modern logic falls into roughly five periods:
The embryonic period from Leibniz to 1847, when the notion of a logical calculus was discussed and developed, particularly by Leibniz, but no schools were formed, and isolated periodic attempts were abandoned or went unnoticed.
The algebraic period from Boole's Analysis to Schröder's Vorlesungen. In this period, there were more practitioners, and a greater continuity of development.
The logicist period from the Begriffsschrift of Frege to the Principia Mathematica 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, Russell, and the early Wittgenstein. 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 metalogic, 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 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 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 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. 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; 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 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
George Boole
Modern logic begins with what is known as the "algebraic school", originating with Boole and including Peirce, Jevons, Schröder, and Venn. Their objective was to develop a calculus to formalise reasoning in the area of classes, propositions, and probabilities. The school begins with Boole's seminal work Mathematical Analysis of Logic which appeared in 1847, although De Morgan (1847) is its immediate precursor. The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an usher in a private school in Lincoln, Lincolnshire. For example, let x and y stand for classes, let the symbol = signify that the classes have the same members, xy stand for the class containing all and only the members of x and y and so on. Boole calls these elective symbols, i.e. symbols which select certain objects for consideration. An expression in which elective symbols are used is called an elective function, and an equation of which the members are elective functions, is an elective equation. The theory of elective functions and their "development" is essentially the modern idea of truth-functions and their expression in disjunctive normal form.
Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between "primary propositions" which are the subject of syllogistic theory, and "secondary propositions", which are the subject of propositional logic, and showed how under different "interpretations" the same algebraic system could represent both. An example of a primary proposition is "All inhabitants are either Europeans or Asiatics." An example of a secondary proposition is "Either all inhabitants are Europeans or they are all Asiatics." These are easily distinguished in modern 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 following year. In 1885 Allan Marquand proposed an electrical version of the machine that is still extant (picture at the Firestone Library).
Charles Sanders Peirce
The defects in Boole's system (such as the use of the letter v for existential propositions) were all remedied by his followers. Jevons published Pure Logic, or the Logic of Quality apart from Quantity in 1864, where he suggested a symbol to signify exclusive or, which allowed Boole's system to be greatly simplified. This was usefully exploited by Schröder when he set out theorems in parallel columns in his Vorlesungen (1890–1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive binary operation, "neither ... nor ..." and equally well "not both ... and ...", however, like many of Peirce's innovations, this remained unknown or unnoticed until Sheffer rediscovered it in 1913. Boole's early work also lacks the idea of the logical sum which originates in Peirce (1867), Schröder (1877) and Jevons (1890), and the concept of inclusion, first suggested by Gergonne (1816) and clearly articulated by Peirce (1870).
Boolean multiples
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 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
Gottlob Frege.
After Boole, the next great advances were made by the German mathematician Gottlob Frege. Frege's objective was the program of Logicism, i.e. demonstrating that arithmetic is identical with logic. Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or Begriffsschrift is important. Frege also tried to show that the concept of number can be defined by purely logical means, so that (if he was right) logic includes arithmetic and all branches of mathematics that are reducible to arithmetic. He was not the first writer to suggest this. In his pioneering work Die Grundlagen der Arithmetik (The Foundations of Arithmetic), sections 15–17, he acknowledges the efforts of Leibniz, J. S. Mill as well as Jevons, citing the latter's claim that "algebra is a highly developed logic, and number but logical discrimination."
Frege's first work, the Begriffsschrift ("concept script") is a rigorously axiomatised system of propositional logic, relying on just two connectives (negational and conditional), two rules of inference (modus ponens and substitution), and six axioms. Frege referred to the "completeness" of this system, but was unable to prove this. The most significant innovation, however, was his explanation of the quantifier in terms of mathematical functions. Traditional logic regards the sentence "Caesar is a man" as of fundamentally the same form as "all men are mortal." Sentences with a proper name subject were regarded as universal in character, interpretable as "every Caesar is a man". 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.
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"
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 [the Boolean] division into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it."
As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient problem of multiple generality. 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
Peano
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.
Ernst Zermelo
This period overlaps with the work of what is known as the "mathematical school", which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. 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 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, a three-volume work on the foundations of mathematics, written by Russell and Alfred North Whitehead and published 1910–1913 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 axioms and inference rules in symbolic logic.
Metamathematical period
Kurt Gödel
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 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 sentence is deducible if and only if it is logically valid—i.e. it is true in every structure for its language. This is known as Gödel's completeness theorem. A year later, he proved two important theorems, which showed Hibert's program to be unattainable in its original form. The first is that no consistent system of axioms whose theorems can be listed by an effective procedure such as an algorithm or computer program is capable of proving all facts about the natural 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, 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
Alfred Tarski, a pupil of Łukasiewicz, is best known for his definition of truth and logical consequence, and the semantic concept of logical satisfaction. In 1933, he published (in Polish) The concept of truth in formalized languages, in which he proposed his semantic theory of truth: a sentence such as "snow is white" is true if and only if snow is white. Tarski's theory separated the metalanguage, which makes the statement about truth, from the object language, which contains the sentence whose truth is being asserted, and gave a correspondence (the T-schema) between phrases in the object language and elements of an interpretation. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of model theory. Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as completeness, decidability, consistency and definability. According to Anita Feferman, Tarski "changed the face of logic in the twentieth century".
Alonzo Church and Alan Turing proposed formal models of computability, giving independent negative solutions to Hilbert's Entscheidungsproblem in 1936 and 1937, respectively. The Entscheidungsproblem asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the halting problem as a key example of a mathematical problem without an algorithmic solution.
Church's system for computation developed into the modern λ-calculus, while the Turing machine became a standard model for a general-purpose computing device. It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the Church–Turing thesis that any deterministic algorithm that can be carried out by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both Peano arithmetic and first-order logic are undecidable. Later work by Emil Post and Stephen Cole Kleene in the 1940s extended the scope of computability theory and introduced the concept of degrees of unsolvability.
The results of the first few decades of the twentieth century also had an impact upon analytic philosophy and philosophical logic, particularly from the 1950s onwards, in subjects such as modal logic, temporal logic, deontic logic, and relevance logic.
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. His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic.
Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. It developed into a study of abstract computability, which became known as recursion theory. The priority method, discovered independently by Albert Muchnik and Richard Friedberg in the 1950s, led to major advances in the understanding of the degrees of unsolvability and related structures. Research into higher-order computability theory demonstrated its connections to set theory. The fields of constructive analysis and computable analysis were developed to study the effective content of classical mathematical theorems; these in turn inspired the program of reverse mathematics. A separate branch of computability theory, computational complexity theory, was also characterized in logical terms as a result of investigations into descriptive complexity.
Model theory applies the methods of mathematical logic to study models of particular mathematical theories. Alfred Tarski published much pioneering work in the field, which is named after a series of papers he published under the title Contributions to the theory of models. In the 1960s, Abraham Robinson used model-theoretic techniques to develop calculus and analysis based on infinitesimals, a problem that first had been proposed by Leibniz.
In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability method invented by Georg Kreisel and Gödel's Dialectica interpretation. This work inspired the contemporary area of proof mining. The Curry–Howard correspondence emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and typed lambda calculi used in computer science. As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. Advances were also made in ordinal analysis and the study of independence results in arithmetic such as the Paris–Harrington theorem.
This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, tense logic is a formalised system for representing, and reasoning about, propositions qualified in terms of time. The philosopher Arthur Prior played a significant role in its development in the 1960s. Modal logics extend the scope of formal logic to include the elements of modality (for example, possibility and necessity). The ideas of Saul Kripke, particularly about possible worlds, and the formal system now called Kripke semantics have had a profound impact on analytic philosophy. His best known and most influential work is Naming and Necessity (1980).Deontic 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 fuzzy logic by Azerbaijani mathematician Lotfi Asker Zadeh in 1965.
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Buroker xxiii
(Locke, An Essay Concerning Human Understanding, IV. 5. 6)
Farrington, 1964, 89
N. Abbagnano, "Psychologism" in P. Edwards (ed) The Encyclopaedia of Philosophy, MacMillan, 1967
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
Carl von Prantl (1855–1867), Geschichte von Logik in Abendland, Leipzig: S. Hirzl, anastatically reprinted in 1997, Hildesheim: Georg Olds.
See e.g. Psychologism, Stanford Encyclopedia of Philosophy
Wilhelm Wundt, Logik (1880–1883); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, pp. 115–116.
Theodor Lipps, Grundzüge der Logik (1893); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. 40
Christoph von Sigwart, Logik (1873–1878); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. 51
Benno Erdmann, Logik (1892); quoted in Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. 96
Dermot Moran, "Introduction"; Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. xxi
Michael Dummett, "Preface"; Edmund Husserl, Logical Investigations, translated J. N. Findlay, Routledge, 2008, Volume 1, p. xvii
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
Bochenski, p. 266
Peirce 1896
See Bochenski p. 269
Oxford Companion p. 499
Edith Sylla (1999), "Oxford Calculators", in The Cambridge Dictionary of Philosophy, Cambridge, Cambridgeshire: Cambridge.
El. philos. sect. I de corp 1.1.2.
Bochenski p. 274
Rutherford, Donald, 1995, "Philosophy and language" in Jolley, N., ed., The Cambridge Companion to Leibniz. Cambridge Univ. Press.
Essai de dialectique rationelle, 211n, quoted in Bochenski p. 277.
Bolzano, Bernard (1972). George, Rolf (ed.). The Theory of Science: Die Wissenschaftslehre oder Versuch einer Neuen Darstellung der Logik. Translated by Rolf, George. University of California Press. p. 209. ISBN978-0-52001787-0.
See e.g. Bochenski p. 296 and passim
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
Kneale p. 404
Kneale p. 407
Boole (1847) p. 16
Boole 1847 pp. 58–59
Beaney p. 11
Kneale p. 422
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.
Trans. Amer. Math. Soc., xiv (1913), pp. 481–488. This is now known as the Sheffer stroke
Bochenski 296
See CP III
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.
JOHN CORCORAN, Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol. 24 (2003), pp. 261–288.
Kneale p. 435
Jevons, The Principles of Science, London 1879, p. 156, quoted in Grundlagen 15
Beaney p. 10 – the completeness of Frege's system was eventually proved by Jan Łukasiewicz in 1934
See for example the argument by the medieval logician William of Ockham that singular propositions are universal, in Summa Logicae III. 8 (??)
See e.g. The Internet Encyclopedia of Philosophy, article "Frege"
Van Heijenoort 1967, p. 83
See e.g. Potter 2004
Zermelo 1908
Feferman 1999 p. 1
Girard, Jean-Yves; Taylor, Paul; Lafont, Yves (1990) [1989]. Proofs and Types. Cambridge University Press (Cambridge Tracts in Theoretical Computer Science, 7). ISBN0-521-37181-3.
Feferman and Feferman 2004, p. 122, discussing "The Impact of Tarski's Theory of Truth".
Feferman 1999, p. 1
See e.g. Barwise, Handbook of Mathematical Logic
Cohen, Paul J. (1964). "The Independence of the Continuum Hypothesis, II". Proceedings of the National Academy of Sciences of the United States of America. 51 (1): 105–110. Bibcode:1964PNAS...51..105C. doi:10.1073/pnas.51.1.105. JSTOR 72252. PMC300611. PMID 16591132.
Many of the foundational papers are collected in The Undecidable (1965) edited by Martin Davis
Jerry Fodor, "Water's water everywhere", London Review of Books, 21 October 2004
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
References
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Bolzano, BernardWissenschaftslehre, (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).
Bolzano, Bernard Theory of Science (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell – D. Reidel Publishing Company, Dordrecht and Boston 1973).
Boole, George (1847) The Mathematical Analysis of Logic (Cambridge and London); repr. in Studies in Logic and Probability, ed. R. Rhees (London 1952).
Boole, George (1854) The Laws of Thought (London and Cambridge); repr. as Collected Logical Works. Vol. 2, (Chicago and London: Open Court, 1940).
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Frege, G., Boole's Logical Calculus and the Concept Script, 1882, in Posthumous Writings transl. P. Long and R. White 1969, pp. 9–46.
Gergonne, Joseph Diaz, (1816) Essai de dialectique rationelle, in Annales de mathématiques pures et appliquées 7, 1816/1817, 189–228.
Jevons, W. S. The Principles of Science, London 1879.
Ockham's Theory of Terms: Part I of the Summa Logicae, translated and introduced by Michael J. Loux (Notre Dame, IN: University of Notre Dame Press 1974). Reprinted: South Bend, IN: St. Augustine's Press, 1998.
Ockham's Theory of Propositions: Part II of the Summa Logicae, translated by Alfred J. Freddoso and Henry Schuurman and introduced by Alfred J. Freddoso (Notre Dame, IN: University of Notre Dame Press, 1980). Reprinted: South Bend, IN: St. Augustine's Press, 1998.
Peirce, C. S., (1896), "The Regenerated Logic", The Monist, vol. VII, No. 1, p pp. 19–40, The Open Court Publishing Co., Chicago, IL, 1896, for the Hegeler Institute. Reprinted (CP 3.425–455). Internet ArchiveThe Monist 7.
Sextus Empiricus, Against the Logicians. (Adversus Mathematicos VII and VIII). Richard Bett (trans.) Cambridge: Cambridge University Press, 2005. ISBN0-521-53195-0.
Zermelo, Ernst (1908). "Untersuchungen über die Grundlagen der Mengenlehre I". Mathematische Annalen. 65 (2): 261–281. doi:10.1007/BF01449999. S2CID 120085563. Archived from the original on 2017-09-08. Retrieved 2013-09-30. English translation in van Heijenoort, Jean (1967). "Investigations in the foundations of set theory". From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Source Books in the History of the Sciences. Harvard Univ. Press. pp. 199–215. ISBN978-0-674-32449-7..
Frege, Gottlob (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. translated in van Heijenoort 1967.
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Barwise, Jon, (ed.), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam, North Holland, 1982 ISBN978-0-444-86388-1 .
Beaney, Michael, The Frege Reader, London: Blackwell 1997.
Bochenski, I. M., A History of Formal Logic, Indiana, Notre Dame University Press, 1961.
Boehner, Philotheus, Medieval Logic, Manchester 1950.
Boyer, C.B. (1991) [1989], A History of Mathematics (2nd ed.), New York: Wiley, ISBN978-0-471-54397-8
Buroker, Jill Vance (transl. and introduction), A. Arnauld, P. Nicole Logic or the Art of Thinking, Cambridge University Press, 1996, ISBN0-521-48249-6.
Church, Alonzo, 1936–1938. "A bibliography of symbolic logic". Journal of Symbolic Logic 1: 121–218; 3:178–212.
de Jong, Everard (1989), Galileo Galilei's "Logical Treatises" and Giacomo Zabarella's "Opera Logica": A Comparison, PhD dissertation, Washington, DC: Catholic University of America.
Farrington, B., The Philosophy of Francis Bacon, Liverpool 1964.
Feferman, Anita B. (1999). "Alfred Tarski". American National Biography. 21. Oxford University Press. pp. 330–332. ISBN978-0-19-512800-0.
Feferman, Anita B.; Feferman, Solomon (2004). Alfred Tarski: Life and Logic. Cambridge University Press. ISBN978-0-521-80240-6. OCLC 54691904.
Gabbay, Dov and John Woods, eds, Handbook of the History of Logic 2004. 1. Greek, Indian and Arabic logic; 2. Mediaeval and Renaissance logic; 3. The rise of modern logic: from Leibniz to Frege; 4. British logic in the Nineteenth century; 5. Logic from Russell to Church; 6. Sets and extensions in the Twentieth century; 7. Logic and the modalities in the Twentieth century; 8. The many-valued and nonmonotonic turn in logic; 9. Computational Logic; 10. Inductive logic; 11. Logic: A history of its central concepts; Elsevier, ISBN0-444-51611-5.
Geach, P. T. Logic Matters, Blackwell 1972.
Goodman, Lenn Evan (2003). Islamic Humanism. Oxford University Press, ISBN0-19-513580-6.
Paul Spade's "Thoughts Words and Things" – An Introduction to Late Mediaeval Logic and Semantic Theory (PDF)
Open Access pdf download; Insights, Images, Bios, and links for 178 logicians by David Marans
The history of logic deals with the study of the development of the science of valid inference logic Formal logics developed in ancient times in India China and Greece Greek methods particularly Aristotelian logic or term logic as found in the Organon found wide application and acceptance in Western science and mathematics for millennia The Stoics especially Chrysippus began the development of predicate logic Christian and Islamic philosophers such as Boethius died 524 Avicenna died 1037 Thomas Aquinas died 1274 and William of Ockham died 1347 further developed Aristotle s logic in the Middle Ages 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 Empirical methods ruled the day as evidenced by Sir Francis Bacon s Novum Organon 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 used in 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 Frege Russell and Peano is the most significant in the two thousand year history of logic and is arguably one of the most important and remarkable events in human intellectual history Progress in mathematical logic in the first few decades of the twentieth century particularly arising from the work of Godel and Tarski had a significant impact on analytic philosophy and philosophical logic particularly from the 1950s onwards in subjects such as modal logic temporal logic deontic logic and relevance logic Logic in IndiaHindu logic Origin The Nasadiya Sukta of the Rigveda 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 Who really knows Who will here proclaim it Whence was it produced Whence is this creation The gods came afterwards with the creation of this universe Who then knows whence it has arisen Nasadiya Sukta concerns the origin of the universe Rig Veda 10 129 6 Logic began independently in ancient India 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 sabha of various sorts comprising relevant experts were regularly convened to deliberate on a variety of matters including administrative legal and religious matters citation needed Dattatreya A philosopher named Dattatreya is stated in the Bhagavata Purana to have taught Anviksiki to Aiarka Prahlada and others It appears from the Markandeya purana that the Anviksiki vidya expounded by him consisted of a mere disquisition on soul in accordance with the yoga philosophy Dattatreya expounded the philosophical side of Anviksiki 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 school of logic The Mahabharata 12 173 45 around the 5th century BC refers to the anviksiki and tarka schools of logic Panini Paṇini c 5th century BC developed a form of logic to which Boolean logic has some similarities for his formulation of Sanskrit grammar Logic is described by Chanakya c 350 283 BC in his Arthashastra as an independent field of inquiry Nyaya Vaisheshika Two of the six Indian schools of thought deal with logic Nyaya and Vaisheshika The Nyaya Sutras of Aksapada Gautama c 2nd century AD constitute the core texts of the Nyaya school one of the six orthodox schools of Hindu philosophy This realist school developed a rigid five member schema of inference involving an initial premise a reason an example an application and a conclusion The idealist Buddhist philosophy became the chief opponent to the Naiyayikas Jain logic Umaswati 2nd century AD author of first Jain work in Sanskrit Tattvarthasutra expounding the Jain philosophy in a most systematized form acceptable to all sects of Jainism Jains 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 Anekantavada the theory of relative pluralism or manifoldness Syadvada the theory of conditioned predication and Nayavada The theory of partial standpoints These concepts in Jain philosophy made important contributions to the thought especially in the areas of skepticism and relativity 4 Buddhist logic Nagarjuna Nagarjuna c 150 250 AD the founder of the Madhyamaka 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 four possibilities of a proposition P P that is being not P that is not being Painting of Nagarjuna from the Shingon Hassozō a series of scrolls authored by the Shingon school of Buddhism Japan Kamakura period 13th 14th century 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 De Morgan s laws would imply that this case is equivalent to the third case P and not P and would be therefore superfluous with only 3 actual cases to consider Dignaga However Dignaga 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 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 Dignaga s famous wheel of reason Hetucakra 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 Dignaga s analysis is much like John Stuart Mill s Joint Method of Agreement and Difference which is inductive Logic in ChinaIn China a contemporary of Confucius Mozi Master Mo is credited with founding the Mohist school whose canons dealt with issues relating to valid inference and the conditions of correct conclusions In particular one of the schools that grew out of Mohism the Logicians are credited by some scholars for their early investigation of formal logic Due to the harsh rule of Legalism in the subsequent Qin dynasty this line of investigation disappeared in China until the introduction of Indian philosophy by Buddhists Logic in the ancient MediterraneanPrehistory 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 with geometry which originally meant the same as land measurement The ancient Egyptians discovered geometry including the formula for the volume of a truncated pyramid 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 axioms and assumptions while Babylonian astronomers in the 8th and 7th centuries BC employed an internal logic within their predictive planetary systems an important contribution to the philosophy of science 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 demonstrative proof Both Thales and Pythagoras of the Pre Socratic philosophers 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 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 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 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 Rhetoricians or Orators and the Sophists who used arguments to defend or attack a thesis both in legal and political contexts Thales TheoremThales It is said Thales most widely regarded as the first philosopher in the Greek tradition measured the height of the pyramids 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 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 Indian 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 is a right angle during his travels to Babylon Pythagoras Proof of the Pythagorean Theorem in Euclid s Elements 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 of Heraclitus c 535 c 475 BC was the first place where the word logos 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 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 Parmenides has been called the discoverer of logic 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 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 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 Zeno of Elea 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 portrays Zeno as claiming to have written a book defending the monism of Parmenides by demonstrating the absurd consequence of assuming that there is plurality 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 Let no one ignorant of geometry enter here Inscribed over the entrance to Plato s Academy Plato s Academy mosaic None of the surviving works of the great fourth century philosopher Plato 428 347 BC include any formal logic 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 Theaetetus where Plato identifies thought or opinion with talk or discourse logos The second question is a result of Plato s theory of Forms Forms are not things in the ordinary sense nor strictly ideas in the mind but they correspond to what philosophers later called universals namely an abstract entity common to each set of things that have the same name In both the Republic and the Sophist Plato suggests that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary connection between forms The third question is about definition Many of Plato s dialogues concern the search for a definition of some important concept justice truth the Good and it is likely that Plato was impressed by the importance of definition in mathematics What underlies every definition is a Platonic Form the common nature present in different particular things Thus a definition reflects the ultimate object of understanding and is the foundation of all valid inference This had a great influence on Plato s student Aristotle in particular Aristotle s notion of the essence of a thing Aristotle Aristotle The logic of Aristotle and particularly his theory of the syllogism 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 Aristotle s logic was still influential in the Renaissance 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 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 This diagram shows the contradictory relationships between categorical propositions in the square of opposition of Aristotelian 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 fallacies which is presented in Topics and Sophistical Refutations 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 Stoics The other great school of Greek logic is that of the Stoics 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 Chrysippus of Soli The Stoics adopted the Megarian logic and systemized it The most important member of the school was Chrysippus c 278 c 206 BC who was its third head and who formalized much of Stoic doctrine He is supposed to have written over 700 works including at least 300 on logic almost none of which survive Unlike with Aristotle we have no complete works by the Megarians or the early Stoics and have to rely mostly on accounts sometimes hostile by later sources including prominently Diogenes Laertius Sextus Empiricus Galen Aulus Gellius Alexander of Aphrodisias and Cicero Three significant contributions of the Stoic school were i their account of modality ii their theory of the Material conditional and iii their account of meaning and truth Modality According to Aristotle the Megarians of his day claimed there was no distinction between potentiality and actuality Diodorus Cronus defined the possible as that which either is or will be the impossible as what will not be true and the contingent as that which either is already or will be false Diodorus is also famous for 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 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 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 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 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 logicLogic in the Middle East A text by Avicenna founder of Avicennian logic The works of Al Kindi Al Farabi Avicenna Al Ghazali Averroes 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 Alfarabi 873 950 was an Aristotelian logician who discussed the topics of future contingents the number and relation of the categories the relation between logic and grammar and non Aristotelian forms of inference Al Farabi also considered the theories of conditional syllogisms and analogical inference which were part of the Stoic tradition of logic rather than the Aristotelian 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 Ibn Sina Avicenna 980 1037 was the founder of Avicennian logic 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 Avicenna wrote on the hypothetical syllogism and on the propositional calculus which were both part of the Stoic logical tradition He developed an original temporally modalized syllogistic theory involving temporal logic and modal logic He also made use of inductive logic such as the methods of agreement difference and concomitant variation which are critical to the scientific method One of Avicenna s ideas had a particularly important influence on Western logicians such as William of Ockham Avicenna s word for a meaning or notion ma na was translated by the scholastic logicians as the Latin intentio in medieval logic and epistemology this is a sign in the mind that naturally represents a thing This was crucial to the development of Ockham s conceptualism A universal term e g man does not signify a thing existing in reality but rather a sign in the mind intentio in intellectu which represents many things in reality Ockham cites Avicenna s commentary on Metaphysics V in support of this view Fakhr al Din al Razi b 1149 criticised Aristotle s first figure and formulated an early system of inductive logic foreshadowing the system of inductive logic developed by John Stuart Mill 1806 1873 Al Razi s work was seen by later Islamic scholars as marking a new direction for Islamic logic towards a Post Avicennian logic This was further elaborated by his student Afdaladdin al Khunaji 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 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 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 Ibn Taymiyyah 1263 1328 wrote the Ar Radd ala al Mantiqiyyin where he argued against the usefulness though not the validity of the syllogism and in favour of inductive reasoning Ibn Taymiyyah also argued against the certainty of syllogistic arguments and in favour of analogy his argument is that concepts founded on induction are themselves not certain but only probable and thus a syllogism based on such concepts is no more certain than an argument based on analogy He further claimed that induction itself is founded on a process of analogy His model of analogical reasoning was based on that of juridical arguments This model of analogy has been used in the recent work of John F Sowa The Sharh al takmil fi l mantiq written by Muhammad ibn Fayd Allah ibn Muhammad Amin al Sharwani in the 15th century is the last major Arabic work on logic that has been studied However thousands upon thousands of pages on logic were written between the 14th and 19th centuries though only a fraction of the texts written during this period have been studied by historians hence little is known about the original work on Islamic logic produced during this later period Logic in medieval Europe Brito s questions on the Old Logic Medieval logic also known as Scholastic logic generally means the form of Aristotelian logic developed in medieval Europe 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 who was familiar with some of Aristotle s logic but almost none of the work of the Stoics Until the twelfth century the only works of Aristotle available in the West were the Categories On Interpretation and Boethius s translation of the Isagoge of Porphyry a commentary on the Categories These works were known as the Old Logic Logica Vetus or Ars Vetus An important work in this tradition was the Logica Ingredientibus of Peter Abelard 1079 1142 His direct influence was small but his influence through pupils such as John of Salisbury was great and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed The proof for the principle of explosion also known as the principle of Pseudo 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 Posterior Analytics and the Sophistical Refutations collectively known as the Logica Nova 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 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 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 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 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 logicThe textbook tradition Dudley Fenner s Art of Logic 1584 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 Published in 1662 it was the most influential work on logic after Aristotle until the nineteenth century The book presents a loosely Cartesian doctrine that the proposition is a combining of ideas rather than terms for example within a framework that is broadly derived from Aristotelian and medieval term logic Between 1664 and 1700 there were eight editions and the book had considerable influence after that The 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 Dudley Fenner helped popularize Ramist logic a reaction against Aristotle Another influential work was the Novum Organum by Francis Bacon published in 1620 The title translates as new instrument This is a reference to Aristotle s work 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 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 three 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 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 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 Georg Wilhelm Friedrich Hegel G W F Hegel indicated the importance of logic to his philosophical system when he condensed his extensive Science of Logic into a shorter work published in 1817 as the first volume of his Encyclopaedia of the Philosophical Sciences The Shorter or Encyclopaedia Logic as it is often known lays out a series of transitions which leads from the most empty and abstract of categories Hegel begins with Pure Being and Pure Nothing to the Absolute the category which contains and resolves all the categories which preceded it Despite the title Hegel s Logic is not really a contribution to the science of valid inference Rather than deriving conclusions about concepts through valid inference from premises Hegel seeks to show that thinking about one concept compels thinking about another concept one cannot he argues possess the concept of Quality without the concept of Quantity 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 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 and in the various schools of Marxism Logic and psychology Between the work of Mill and Frege stretched half a century during which logic was widely treated as a descriptive science an empirical study of the structure of reasoning and thus essentially as a branch of psychology The German psychologist Wilhelm Wundt for example discussed deriving the logical from the psychological laws of thought emphasizing that psychological thinking is always the more comprehensive form of thinking This view was widespread among German philosophers of the period Theodor Lipps described logic as a specific discipline of psychology Christoph von Sigwart understood logical necessity as grounded in the individual s compulsion to think in a certain way Benno Erdmann argued that logical laws only hold within the limits of our thinking Such was the dominant view of logic in the years following Mill s work This psychological approach to logic was rejected by Gottlob Frege It was also subjected to an extended and destructive critique by Edmund Husserl in the first volume of his Logical Investigations 1900 an assault which has been described as overwhelming Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven and that skepticism and relativism were unavoidable consequences Such criticisms did not immediately extirpate 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 Rise of modern logicThe period between the fourteenth century and the beginning of the nineteenth century had been largely one of decline and neglect and is generally regarded as barren by historians of logic The revival of logic occurred in the mid nineteenth century at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in 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 noted that even though a mistake in the evaluation of a definite integral by Laplace led to an error concerning the moon s orbit that persisted for nearly 50 years the mistake once spotted was corrected without any serious dispute Peirce contrasted this with the disputation and uncertainty surrounding traditional logic and especially reasoning in metaphysics He argued that a truly exact logic would depend upon mathematical i e diagrammatic or iconic thought Those who follow such methods will escape all error except such as will be speedily corrected after it is once suspected Modern logic is also constructive rather than abstractive i e rather than abstracting and formalising theorems derived from ordinary language or from psychological intuitions about validity it constructs theorems by formal methods then looks for an interpretation in ordinary language It is entirely symbolic meaning that even the logical constants which the medieval logicians called syncategoremata and the categoric terms are expressed in symbols Modern logicThe development of modern logic falls into roughly five periods The embryonic period from Leibniz to 1847 when the notion of a logical calculus was discussed and developed particularly by Leibniz but no schools were formed and isolated periodic attempts were abandoned or went unnoticed The algebraic period from Boole s Analysis to Schroder s Vorlesungen In this period there were more practitioners and a greater continuity of development The logicist period from the Begriffsschrift of Frege to the Principia Mathematica 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 Russell and the early Wittgenstein 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 metalogic in the finitist system of Hilbert and the non finitist system of Lowenheim and Skolem the combination of logic and metalogic in the work of Godel and Tarski Godel s incompleteness theorem of 1931 was one of the greatest achievements in the history of logic Later in the 1930s Godel developed the notion 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 Leibniz 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 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 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 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 anticipated a fundamental idea of modern proof theory when he defined logical consequence or deducibility in terms of variables Hence I say that propositions M displaystyle M N displaystyle N O displaystyle O are deducible from propositions A displaystyle A B displaystyle B C displaystyle C D displaystyle D with respect to variable parts i displaystyle i j displaystyle j if every class of ideas whose substitution for i displaystyle i j displaystyle j makes all of A displaystyle A B displaystyle B C displaystyle C D displaystyle D true also makes all of M displaystyle M N displaystyle N O displaystyle O true Occasionally since it is customary I shall say that propositions M displaystyle M N displaystyle N O displaystyle O follow or can be inferred or derived from A displaystyle A B displaystyle B C displaystyle C D displaystyle D Propositions A displaystyle A B displaystyle B C displaystyle C D displaystyle D I shall call the premises M displaystyle M N displaystyle N O displaystyle O the conclusions This is now known as semantic validity Algebraic period George Boole Modern logic begins with what is known as the algebraic school originating with Boole and including Peirce Jevons Schroder and Venn Their objective was to develop a calculus to formalise reasoning in the area of classes propositions and probabilities The school begins with Boole s seminal work Mathematical Analysis of Logic which appeared in 1847 although De Morgan 1847 is its immediate precursor The fundamental idea of Boole s system is that algebraic formulae can be used to express logical relations This idea occurred to Boole in his teenage years working as an usher in a private school in Lincoln Lincolnshire For example let x and y stand for classes let the symbol signify that the classes have the same members xy stand for the class containing all and only the members of x and y and so on Boole calls these elective symbols i e symbols which select certain objects for consideration An expression in which elective symbols are used is called an elective function and an equation of which the members are elective functions is an elective equation The theory of elective functions and their development is essentially the modern idea of truth functions and their expression in disjunctive normal form Boole s system admits of two interpretations in class logic and propositional logic Boole distinguished between primary propositions which are the subject of syllogistic theory and secondary propositions which are the subject of propositional logic and showed how under different interpretations the same algebraic system could represent both An example of a primary proposition is All inhabitants are either Europeans or Asiatics An example of a secondary proposition is Either all inhabitants are Europeans or they are all Asiatics These are easily distinguished in modern 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 following year In 1885 Allan Marquand proposed an electrical version of the machine that is still extant picture at the Firestone Library Charles Sanders Peirce The defects in Boole s system such as the use of the letter v for existential propositions were all remedied by his followers Jevons published Pure Logic or the Logic of Quality apart from Quantity in 1864 where he suggested a symbol to signify exclusive or which allowed Boole s system to be greatly simplified This was usefully exploited by Schroder 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 Schroder 1877 and Jevons 1890 and the concept of inclusion first suggested by Gergonne 1816 and clearly articulated by Peirce 1870 Boolean multiples The success of Boole s algebraic system suggested that all logic must be capable of algebraic representation and there were attempts to express a logic of relations in such form of which the most ambitious was Schroder s monumental Vorlesungen uber 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 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 Gottlob Frege After Boole the next great advances were made by the German mathematician Gottlob Frege Frege s objective was the program of Logicism i e demonstrating that arithmetic is identical with logic Frege went much further than any of his predecessors in his rigorous and formal approach to logic and his calculus or Begriffsschrift is important Frege also tried to show that the concept of number can be defined by purely logical means so that if he was right logic includes arithmetic and all branches of mathematics that are reducible to arithmetic He was not the first writer to suggest this In his pioneering work Die Grundlagen der Arithmetik The Foundations of Arithmetic sections 15 17 he acknowledges the efforts of Leibniz J S Mill as well as Jevons citing the latter s claim that algebra is a highly developed logic and number but logical discrimination Frege s first work the Begriffsschrift concept script is a rigorously axiomatised system of propositional logic relying on just two connectives negational and conditional two rules of inference modus ponens and substitution and six axioms Frege referred to the completeness of this system but was unable to prove this The most significant innovation however was his explanation of the quantifier in terms of mathematical functions Traditional logic regards the sentence Caesar is a man as of fundamentally the same form as all men are mortal Sentences with a proper name subject were regarded as universal in character interpretable as every Caesar is a man 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 x A x B x 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 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 x I x M x W x 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 x I x M x x I x W x 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 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 x G x y B y K x y displaystyle forall x Big G x rightarrow exists y big B y land K x y big Big Peano means that to every girl there corresponds some boy any one will do who the girl kissed But x B x y G y K y x 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 Ernst Zermelo This period overlaps with the work of what is known as the mathematical school which included Dedekind Pasch Peano Hilbert Zermelo Huntington Veblen and Heyting Their objective was the axiomatisation of branches of mathematics like geometry arithmetic analysis and set theory 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 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 Schroder 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 Let R x x x then R R R R displaystyle text Let R x mid x not in x text then R in R iff R not in R The monumental Principia Mathematica a three volume work on the foundations of mathematics written by Russell and Alfred North Whitehead and published 1910 1913 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 axioms and inference rules in symbolic logic Metamathematical period Kurt Godel The names of Godel and Tarski dominate the 1930s a crucial period in the development of metamathematics the study of mathematics using mathematical methods to produce metatheories or mathematical theories about other mathematical theories Early investigations into metamathematics had been driven by Hilbert s program Work on metamathematics culminated in the work of Godel who in 1929 showed that a given first order sentence is deducible if and only if it is logically valid i e it is true in every structure for its language This is known as Godel s completeness theorem A year later he proved two important theorems which showed Hibert s program to be unattainable in its original form The first is that no consistent system of axioms whose theorems can be listed by an effective procedure such as an algorithm or computer program is capable of proving all facts about the natural 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 Godel s incompleteness theorems or simply Godel s Theorem Later in the decade Godel developed the concept of set theoretic constructibility as part of his proof that the axiom of choice and the continuum hypothesis are consistent with Zermelo Fraenkel set theory In proof theory Gerhard Gentzen developed natural deduction and the sequent calculus The former attempts to model logical reasoning as it naturally occurs in practice and is most easily applied to intuitionistic logic while the latter was devised to clarify the derivation of logical proofs in any formal system Since Gentzen s work natural deduction and sequent calculi have been widely applied in the fields of proof theory mathematical logic and computer science Gentzen also proved normalization and cut elimination theorems for intuitionistic and classical logic which could be used to reduce logical proofs to a normal form Alfred Tarski Alfred Tarski a pupil of Lukasiewicz is best known for his definition of truth and logical consequence and the semantic concept of logical satisfaction In 1933 he published in Polish The concept of truth in formalized languages in which he proposed his semantic theory of truth a sentence such as snow is white is true if and only if snow is white Tarski s theory separated the metalanguage which makes the statement about truth from the object language which contains the sentence whose truth is being asserted and gave a correspondence the T schema between phrases in the object language and elements of an interpretation Tarski s approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy especially in the development of model theory Tarski also produced important work on the methodology of deductive systems and on fundamental principles such as completeness decidability consistency and definability According to Anita Feferman Tarski changed the face of logic in the twentieth century Alonzo Church and Alan Turing proposed formal models of computability giving independent negative solutions to Hilbert s Entscheidungsproblem in 1936 and 1937 respectively The Entscheidungsproblem asked for a procedure that given any formal mathematical statement would algorithmically determine whether the statement is true Church and Turing proved there is no such procedure Turing s paper introduced the halting problem as a key example of a mathematical problem without an algorithmic solution Church s system for computation developed into the modern l calculus while the Turing machine became a standard model for a general purpose computing device It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing These results led to the Church Turing thesis that any deterministic algorithm that can be carried out by a human can be carried out by a Turing machine Church proved additional undecidability results showing that both Peano arithmetic and first order logic are undecidable Later work by Emil Post and Stephen Cole Kleene in the 1940s extended the scope of computability theory and introduced the concept of degrees of unsolvability The results of the first few decades of the twentieth century also had an impact upon analytic philosophy and philosophical logic particularly from the 1950s onwards in subjects such as modal logic temporal logic deontic logic and relevance logic Logic after WWII Saul Kripke After World War II mathematical logic branched into four inter related but separate areas of research model theory proof theory computability theory and set theory In set theory the method of forcing revolutionized the field by providing a robust method for constructing models and obtaining independence results Paul Cohen introduced this method in 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 priority method discovered independently by Albert Muchnik and Richard Friedberg in the 1950s led to major advances in the understanding of the degrees of unsolvability and related structures Research into higher order computability theory demonstrated its connections to set theory The fields of constructive analysis and computable analysis were developed to study the effective content of classical mathematical theorems these in turn inspired the program of reverse mathematics A separate branch of computability theory computational complexity theory was also characterized in logical terms as a result of investigations into descriptive complexity Model theory applies the methods of mathematical logic to study models of particular mathematical theories Alfred Tarski published much pioneering work in the field which is named after a series of papers he published under the title Contributions to the theory of models In the 1960s Abraham Robinson used model theoretic techniques to develop calculus and analysis based on infinitesimals a problem that first had been proposed by Leibniz In proof theory the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the realizability method invented by Georg Kreisel and Godel s Dialectica interpretation This work inspired the contemporary area of proof mining The Curry Howard correspondence emerged as a deep analogy between logic and computation including a correspondence between systems of natural deduction and typed lambda calculi used in computer science As a result research into this class of formal systems began to address both logical and computational aspects this area of research came to be known as modern type theory Advances were also made in ordinal analysis and the study of independence results in arithmetic such as the Paris Harrington theorem This was also a period particularly in the 1950s and afterwards when the ideas of mathematical logic begin to influence philosophical thinking For example tense logic is a formalised system for representing and reasoning about propositions qualified in terms of time The philosopher Arthur Prior played a significant role in its development in the 1960s Modal logics extend the scope of formal logic to include the elements of modality for example possibility and necessity The ideas of Saul Kripke particularly about possible worlds and the formal system now called Kripke semantics have had a profound impact on analytic philosophy His best known and most influential work is Naming and Necessity 1980 Deontic 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 fuzzy logic by Azerbaijani mathematician Lotfi Asker Zadeh in 1965 See alsoPhilosophy portalHistory 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 logicNotesBoehner p xiv Oxford Companion p 498 Bochenski Part I Introduction passim Frege Gottlob The Foundations of Arithmetic PDF p 1 Archived from the original PDF on 2018 09 20 Retrieved 2016 02 03 Oxford Companion p 500 Kramer Kenneth January 1986 World Scriptures An Introduction to Comparative Religions Paulist Press pp 34 ISBN 978 0 8091 2781 8 Christian David 2011 09 01 Maps of Time An Introduction to Big History University of California Press pp 18 ISBN 978 0 520 95067 2 Singh Upinder 2008 A History of Ancient and Early Medieval India From the Stone Age to the 12th Century Pearson Education India pp 206 ISBN 978 81 317 1120 0 Bochenski p 446 Vidyabhusana S C 1921 History Of Indian Logic p 11 Bhusana Satis Chandra Vidya 1921 A History Of Indian Logic S C Vidyabhusana 1971 A History of Indian Logic Ancient Mediaeval and Modern Schools pp 17 21 R P Kangle 1986 The Kautiliya Arthashastra 1 2 11 Motilal Banarsidass Bochenski p 417 and passim Ganeri Jonardon 2002 Jaina Logic and the Philosophical Basis of Pluralism History and Philosophy of Logic 23 4 267 281 doi 10 1080 0144534021000051505 ISSN 0144 5340 S2CID 170089234 Bochenski pp 431 437 Matilal Bimal Krishna 1998 The Character of Logic in India Albany New York USA State University of New York Press pp 12 18 ISBN 9780791437407 Bochenksi p 441 Matilal 17 Kneale p 2 Kneale p 3 H F J Horstmanshoff Marten Stol Cornelis Tilburg 2004 Magic and Rationality in Ancient Near Eastern and Graeco Roman Medicine p 99 Brill Publishers ISBN 90 04 13666 5 D Brown 2000 Mesopotamian Planetary Astronomy Astrology Styx Publications ISBN 90 5693 036 2 Heath Mathematics in Aristotle cited in Kneale p 5 Kneale p 16 History of logic britannica com Retrieved 2018 04 02 Aristotle Metaphysics Alpha 983b18 Smith William 1870 Dictionary of Greek and Roman biography and mythology Boston Little p 1016 T Patronis amp D Patsopoulos The Theorem of Thales A Study of the naming of theorems in school Geometry textbooks Patras University Archived from the original on 2016 03 03 Retrieved 2012 02 12 Boyer 1991 Ionia and the Pythagoreans p 43 de Laet Siegfried J 1996 History of Humanity Scientific and Cultural Development UNESCO Volume 3 p 14 ISBN 92 3 102812 X Boyer Carl B and Merzbach Uta C 2010 A History of Mathematics John Wiley and Sons Chapter IV ISBN 0 470 63056 6 C B Boyer 1968 Samuel Enoch Stumpf Socrates to Sartre p 11 F E Peters Greek Philosophical Terms New York University Press 1967 Cornford Francis MacDonald 1957 1939 Plato and Parmenides Parmenides Way of Truthand Plato sParmenidestranslated with an introduction and running commentary PDF Liberal Arts Press R J Hollingdale 1974 Western Philosophy an introduction p 73 Cornford Francis MacDonald 1912 From religion to philosophy A study in the origins of western speculation PDF Longmans Green and Co Kneale p 15 The Numismatic Circular 2018 04 02 Retrieved 2018 04 02 via Google Books Kneale p 17 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 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 Kneale p 21 Zalta Edward N Aristotle s Logic Stanford University 18 March 2000 Retrieved 13 March 2010 See e g Aristotle s logic Stanford Encyclopedia of Philosophy Sowa John F 2000 Knowledge representation logical philosophical and computational foundations Pacific Grove Brooks Cole p 2 ISBN 0 534 94965 7 OCLC 38239202 Bochenski p 63 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 Oxford Companion article Chrysippus p 134 1 Stanford Encyclopedia of Philosophy Susanne Bobzien Ancient Logic K Hulser Die Fragmente zur Dialektik der Stoiker 4 vols Stuttgart 1986 1987 Kneale 117 158 Metaphysics Eta 3 1046b 29 Boethius Commentary on the Perihermenias Meiser p 234 Epictetus Dissertationes ed Schenkel ii 19 I Alexander p 177 Sextus Empiricus Adv Math viii Section 113 Sextus Empiricus Hypotyp ii 110 comp Cicero Academica ii 47 de Fato 6 See e g Lukasiewicz p 21 Sextus Bk viii Sections 11 12 See e g Routledge Encyclopedia of Philosophy Online Version 2 0 Archived 2022 06 06 at the Wayback Machine article Islamic philosophy History of logic Arabic logic Encyclopaedia Britannica Feldman Seymour 1964 11 26 Rescher on Arabic Logic The Journal of Philosophy 61 22 Journal of Philosophy Inc 724 734 doi 10 2307 2023632 ISSN 0022 362X JSTOR 2023632 726 Long A A Sedley D N 1987 The Hellenistic Philosophers Vol 1 Translations of the principal sources with philosophical commentary Cambridge Cambridge University Press ISBN 0 521 27556 3 Hasse Dag Nikolaus 2008 09 19 Influence of Arabic and Islamic Philosophy on the Latin West Stanford Encyclopedia of Philosophy Retrieved 2009 10 13 Richard F Washell 1973 Logic Language and Albert the Great Journal of the History of Ideas 34 3 pp 445 450 445 Goodman Lenn Evan 2003 Islamic Humanism p 155 Oxford University Press ISBN 0 19 513580 6 Goodman Lenn Evan 1992 Avicenna p 188 Routledge ISBN 0 415 01929 X Kneale p 229 Kneale p 266 Ockham Summa Logicae i 14 Avicenna Avicennae Opera Venice 1508 f87rb Muhammad Iqbal The Reconstruction of Religious Thought in Islam The Spirit of Muslim Culture cf 2 and 3 Tony Street 2008 07 23 Arabic and Islamic Philosophy of Language and Logic Stanford Encyclopedia of Philosophy Retrieved 2008 12 05 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 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 See pp 253 254 of Street Tony 2005 Logic In Peter Adamson Richard C Taylor eds The Cambridge Companion to Arabic Philosophy Cambridge University Press pp 247 265 ISBN 978 0 521 52069 0 Ruth Mas 1998 Qiyas A Study in Islamic Logic PDF Folia Orientalia 34 113 128 ISSN 0015 5675 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 Nicholas Rescher and Arnold vander Nat The Arabic Theory of Temporal Modal Syllogistic in George Fadlo Hourani 1975 Essays on Islamic Philosophy and Science pp 189 221 State University of New York Press ISBN 0 87395 224 3 Kneale p 198 Stephen Dumont article Peter Abelard in Gracia and Noone p 492 Kneale pp 202 203 See e g Kneale p 225 Boehner p 1 Boehner pp 19 76 Boehner p 29 Boehner p 30 Ebbesen 1981 Boehner pp 54 55 Oxford Companion p 504 article Traditional logic Buroker xxiii Locke An Essay Concerning Human Understanding IV 5 6 Farrington 1964 89 N Abbagnano Psychologism in P Edwards ed The Encyclopaedia of Philosophy MacMillan 1967 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 amp Sons 1911 page 242 Carl von Prantl 1855 1867 Geschichte von Logik in Abendland Leipzig S Hirzl anastatically reprinted in 1997 Hildesheim Georg Olds See e g Psychologism Stanford Encyclopedia of Philosophy Wilhelm Wundt Logik 1880 1883 quoted in Edmund Husserl Logical Investigations translated J N Findlay Routledge 2008 Volume 1 pp 115 116 Theodor Lipps Grundzuge der Logik 1893 quoted in Edmund Husserl Logical Investigations translated J N Findlay Routledge 2008 Volume 1 p 40 Christoph von Sigwart Logik 1873 1878 quoted in Edmund Husserl Logical Investigations translated J N Findlay Routledge 2008 Volume 1 p 51 Benno Erdmann Logik 1892 quoted in Edmund Husserl Logical Investigations translated J N Findlay Routledge 2008 Volume 1 p 96 Dermot Moran Introduction Edmund Husserl Logical Investigations translated J N Findlay Routledge 2008 Volume 1 p xxi Michael Dummett Preface Edmund Husserl Logical Investigations translated J N Findlay Routledge 2008 Volume 1 p xvii 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 Bochenski p 266 Peirce 1896 See Bochenski p 269 Oxford Companion p 499 Edith Sylla 1999 Oxford Calculators in The Cambridge Dictionary of Philosophy Cambridge Cambridgeshire Cambridge El philos sect I de corp 1 1 2 Bochenski p 274 Rutherford Donald 1995 Philosophy and language in Jolley N ed The Cambridge Companion to Leibniz Cambridge Univ Press Wiener Philip 1951 Leibniz Selections Scribner Essai de dialectique rationelle 211n quoted in Bochenski p 277 Bolzano Bernard 1972 George Rolf ed The Theory of Science Die Wissenschaftslehre oder Versuch einer Neuen Darstellung der Logik Translated by Rolf George University of California Press p 209 ISBN 978 0 52001787 0 See e g Bochenski p 296 and passim 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 Kneale p 404 Kneale p 407 Boole 1847 p 16 Boole 1847 pp 58 59 Beaney p 11 Kneale p 422 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 Trans Amer Math Soc xiv 1913 pp 481 488 This is now known as the Sheffer stroke Bochenski 296 See CP III 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 JOHN CORCORAN Aristotle s Prior Analytics and Boole s Laws of Thought History and Philosophy of Logic vol 24 2003 pp 261 288 Kneale p 435 Jevons The Principles of Science London 1879 p 156 quoted in Grundlagen 15 Beaney p 10 the completeness of Frege s system was eventually proved by Jan Lukasiewicz in 1934 See for example the argument by the medieval logician William of Ockham that singular propositions are universal in Summa Logicae III 8 Frege 1879 in van Heijenoort 1967 p 7 On concept and object p 198 Geach p 48 BLC p 14 quoted in Beaney p 12 See e g The Internet Encyclopedia of Philosophy article Frege Van Heijenoort 1967 p 83 See e g Potter 2004 Zermelo 1908 Feferman 1999 p 1 Girard Jean Yves Taylor Paul Lafont Yves 1990 1989 Proofs and Types Cambridge University Press Cambridge Tracts in Theoretical Computer Science 7 ISBN 0 521 37181 3 Feferman and Feferman 2004 p 122 discussing The Impact of Tarski s Theory of Truth Feferman 1999 p 1 See e g Barwise Handbook of Mathematical Logic Cohen Paul J 1964 The Independence of the Continuum Hypothesis II Proceedings of the National Academy of Sciences of the United States of America 51 1 105 110 Bibcode 1964PNAS 51 105C doi 10 1073 pnas 51 1 105 JSTOR 72252 PMC 300611 PMID 16591132 Many of the foundational papers are collected in The Undecidable 1965 edited by Martin Davis Jerry Fodor Water s water everywhere London Review of Books 21 October 2004 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 2004ReferencesPrimary SourcesAlexander of Aphrodisias In 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the Concept Script 1882 in Posthumous Writings transl P Long and R White 1969 pp 9 46 Gergonne Joseph Diaz 1816 Essai de dialectique rationelle in Annales de mathematiques pures et appliquees 7 1816 1817 189 228 Jevons W S The Principles of Science London 1879 Ockham s Theory of Terms Part I of the Summa Logicae translated and introduced by Michael J Loux Notre Dame IN University of Notre Dame Press 1974 Reprinted South Bend IN St Augustine s Press 1998 Ockham s Theory of Propositions Part II of the Summa Logicae translated by Alfred J Freddoso and Henry Schuurman and introduced by Alfred J Freddoso Notre Dame IN University of Notre Dame Press 1980 Reprinted South Bend IN St Augustine s Press 1998 Peirce C S 1896 The Regenerated Logic The Monist vol VII No 1 p 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 Sextus Empiricus Against the Logicians Adversus Mathematicos VII and VIII Richard Bett trans Cambridge Cambridge University Press 2005 ISBN 0 521 53195 0 Zermelo Ernst 1908 Untersuchungen uber die Grundlagen der Mengenlehre I Mathematische Annalen 65 2 261 281 doi 10 1007 BF01449999 S2CID 120085563 Archived from the original on 2017 09 08 Retrieved 2013 09 30 English translation in van Heijenoort Jean 1967 Investigations in the foundations of set theory From Frege to Godel A Source Book in Mathematical Logic 1879 1931 Source Books in the History of the Sciences Harvard Univ Press pp 199 215 ISBN 978 0 674 32449 7 Frege Gottlob 1879 Begriffsschrift a formula language modeled upon that of arithmetic for pure thought translated in van Heijenoort 1967 Secondary SourcesBarwise Jon ed Handbook of Mathematical Logic Studies in Logic and the Foundations of Mathematics Amsterdam North Holland 1982 ISBN 978 0 444 86388 1 Beaney Michael The Frege Reader London Blackwell 1997 Bochenski I M A History of Formal Logic Indiana Notre Dame University Press 1961 Boehner Philotheus Medieval Logic Manchester 1950 Boyer C B 1991 1989 A History of Mathematics 2nd ed New York Wiley ISBN 978 0 471 54397 8 Buroker Jill Vance transl and introduction A Arnauld P Nicole Logic or the Art of Thinking Cambridge University Press 1996 ISBN 0 521 48249 6 Church Alonzo 1936 1938 A bibliography of symbolic logic Journal of Symbolic Logic 1 121 218 3 178 212 de Jong Everard 1989 Galileo Galilei s Logical Treatises and Giacomo Zabarella s Opera Logica A Comparison PhD dissertation Washington DC Catholic University of America Ebbesen Sten Early supposition theory 12th 13th Century Histoire Epistemologie Langage 3 1 35 48 1981 Farrington B The Philosophy of Francis Bacon Liverpool 1964 Feferman Anita B 1999 Alfred Tarski American National Biography 21 Oxford University Press pp 330 332 ISBN 978 0 19 512800 0 Feferman Anita B Feferman Solomon 2004 Alfred Tarski Life and Logic Cambridge University Press ISBN 978 0 521 80240 6 OCLC 54691904 Gabbay Dov and John Woods eds Handbook of the History of Logic 2004 1 Greek Indian and Arabic logic 2 Mediaeval and Renaissance logic 3 The rise of modern logic from Leibniz to Frege 4 British logic in the Nineteenth century 5 Logic from Russell to Church 6 Sets and extensions in the Twentieth century 7 Logic and the modalities in the Twentieth century 8 The many valued and nonmonotonic turn in logic 9 Computational Logic 10 Inductive logic 11 Logic A history of its central concepts Elsevier ISBN 0 444 51611 5 Geach P T Logic Matters Blackwell 1972 Goodman Lenn Evan 2003 Islamic Humanism Oxford University Press ISBN 0 19 513580 6 Goodman Lenn Evan 1992 Avicenna Routledge ISBN 0 415 01929 X Grattan Guinness Ivor 2000 The Search for Mathematical Roots 1870 1940 Princeton University Press Gracia J G and Noone T B A Companion to Philosophy in the Middle Ages London 2003 Haaparanta Leila ed 2009 The Development of Modern Logic Oxford University Press Heath T L 1949 Mathematics in Aristotle Oxford University Press Heath T L 1931 A Manual of Greek Mathematics Oxford Clarendon Press Honderich Ted ed The Oxford Companion to Philosophy New York Oxford University Press 1995 ISBN 0 19 866132 0 Kneale William and Martha 1962 The development of logic Oxford University Press ISBN 0 19 824773 7 Lukasiewicz Aristotle s Syllogistic Oxford University Press 1951 Potter Michael 2004 Set Theory and its Philosophy Oxford University Press External linksThe History of Logic from Aristotle to Godel with annotated bibliographies on the history of logic Bobzien Susanne Ancient Logic In Zalta Edward N ed Stanford Encyclopedia of Philosophy Chatti Saloua Avicenna Ibn Sina Logic Internet Encyclopedia of Philosophy Spruyt Joke Peter of Spain In Zalta Edward N ed Stanford Encyclopedia of Philosophy Paul Spade s Thoughts Words and Things An Introduction to Late Mediaeval Logic and Semantic Theory PDF Open Access pdf download Insights Images Bios and links for 178 logicians by David Marans