1.
Aristotle
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Aristotle was an ancient Greek philosopher and scientist born in the city of Stagira, Chalkidice, on the northern periphery of Classical Greece. His father, Nicomachus, died when Aristotle was a child, at seventeen or eighteen years of age, he joined Platos Academy in Athens and remained there until the age of thirty-seven. Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, teaching Alexander the Great gave Aristotle many opportunities and an abundance of supplies. He established a library in the Lyceum which aided in the production of many of his hundreds of books and he believed all peoples concepts and all of their knowledge was ultimately based on perception. Aristotles views on natural sciences represent the groundwork underlying many of his works, Aristotles views on physical science profoundly shaped medieval scholarship. Their influence extended from Late Antiquity and the Early Middle Ages into the Renaissance, some of Aristotles zoological observations, such as on the hectocotyl arm of the octopus, were not confirmed or refuted until the 19th century. His works contain the earliest known study of logic, which was incorporated in the late 19th century into modern formal logic. Aristotle was well known among medieval Muslim intellectuals and revered as The First Teacher and his ethics, though always influential, gained renewed interest with the modern advent of virtue ethics. All aspects of Aristotles philosophy continue to be the object of academic study today. Though Aristotle wrote many elegant treatises and dialogues – Cicero described his style as a river of gold – it is thought that only around a third of his original output has survived. Aristotle, whose means the best purpose, was born in 384 BC in Stagira, Chalcidice. His father Nicomachus was the physician to King Amyntas of Macedon. Aristotle was orphaned at a young age, although there is little information on Aristotles childhood, he probably spent some time within the Macedonian palace, making his first connections with the Macedonian monarchy. At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Platos Academy and he remained there for nearly twenty years before leaving Athens in 348/47 BC. Aristotle then accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor, there, he traveled with Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island. Aristotle married Pythias, either Hermiass adoptive daughter or niece and she bore him a daughter, whom they also named Pythias. Soon after Hermias death, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander in 343 BC, Aristotle was appointed as the head of the royal academy of Macedon. During that time he gave not only to Alexander
2.
Boethius
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Anicius Manlius Severinus Boëthius, commonly called Boethius, was a Roman senator, consul, magister officiorum, and philosopher of the early 6th century. Boethius was born in Rome to a family around 480 AD. His family, the Anicii, included emperors Petronius Maximus and Olybrius and his father, Manlius Boethius, who was appointed consul in 487, died while Boethius was young. Another patrician, Quintus Aurelius Memmius Symmachus, adopted and raised Boethius, instilling in him a love for literature and philosophy. Both Memmius Symmachus and Boethius were fluent in Greek, a rare skill at the time in the Western Empire, for this reason. The French scholar Pierre Courcelle has argued that Boethius studied at Alexandria with the Neo-Platonist philosopher Ammonius Hermiae, on account of his erudition, Boethius entered the service of Theodoric the Great at a young age and was already a senator by the age of 25. Boethius married his foster-fathers daughter, Rusticiana, their children included two boys, Symmachus and Boethius. In 522, the year his two sons were appointed joint consuls, Boethius accepted the appointment to the position of magister officiorum. This may have set in place a course of events that would lead to loss of royal favour, five hundred years later, this continuing disagreement led to the East-West Schism in 1054, in which communion between the Catholic Church and Eastern Orthodox Church was broken. In 523 Boethius fell from power, after a period of imprisonment in Pavia for what was deemed a treasonable offence, he was executed in 524. The primary sources are in agreement over the facts of what happened. At a meeting of the Royal Council in Verona, the referandarius Cyprianus accused the ex-consul Caecina Decius Faustus Albinus of treasonous correspondence with Justin I. Boethius leapt to his defense, crying, The charge of Cyprianus is false, but if Albinus did that, so also have I, Cyprianus then also accused Boethius of the same crime, and produced three men who claimed they had witnessed the crime. First the pair were detained in the baptistery of a church, then Boethius was exiled to the Ager Calventianus, a distant country estate, the basic facts in the case are not in dispute, writes Jeffrey Richards. What is disputed about this sequence of events is the interpretation that should be put on them, Boethius claims his crime was seeking the safety of the Senate. He describes the three witnesses against him as dishonorable, Basilius had been dismissed from Royal service for his debts, while Venantius Opilio, however, other sources depict these men in a far more positive light. For example, Cassiodorus describes Cyprianus and Opilio as utterly scrupulous, just and loyal and mentions they are brothers, Theodoric was feeling threatened by international events. The Acacian Schism had been resolved, and the Nicene Christian aristocrats of his kingdom were seeking to renew their ties with Constantinople, the Catholic Hilderic had become king of the Vandals and had put Theodorics sister Amalafrida to death, and Arians in the East were being persecuted
3.
Peter Abelard
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Peter Abelard was a medieval French scholastic philosopher, theologian and preeminent logician. His love for, and affair with, Héloïse dArgenteuil have become legendary, the Chambers Biographical Dictionary describes him as the keenest thinker and boldest theologian of the 12th Century. Abelard, originally called Pierre le Pallet, was born c. 1079 in Le Pallet, about 10 miles east of Nantes, in Brittany, as a boy, he learned quickly. Instead of entering a career, as his father had done. During his early academic pursuits, Abelard wandered throughout France, debating and learning and he first studied in the Loire area, where the nominalist Roscellinus of Compiègne, who had been accused of heresy by Anselm, was his teacher during this period. Around 1100, Abelards travels finally brought him to Paris, in the great cathedral school of Notre-Dame de Paris, he was taught for a while by William of Champeaux, the disciple of Anselm of Laon, a leading proponent of Realism. During this time he changed his surname to Abelard, sometimes written Abailard or Abaelardus, and William thought Abelard was too arrogant. It was during this time that Abelard would provoke quarrels with both William and Roscellinus and his teaching was notably successful, though for a time he had to give it up and spend time in Brittany, the strain proving too great for his constitution. Abelard was once more victorious, and Abelard was almost able to hold the position of master at Notre Dame, for a short time, however, William was able to prevent Abelard from lecturing in Paris. Abelard accordingly was forced to resume his school at Melun, which he was able to move, from c. 1110-12, to Paris itself, on the heights of Montagne Sainte-Geneviève. From his success in dialectic, he turned to theology and in 1113 moved to Laon to attend the lectures of Anselm on biblical exegesis. Unimpressed by Anselms teaching, Abelard began to offer his own lectures on the Book of Ezekiel, Anselm forbade him to continue this teaching, and Abelard returned to Paris where, in around 1115, he became master of Notre Dame and a canon of Sens. Distinguished in figure and manners, Abelard was seen surrounded by crowds – it is thousands of students – drawn from all countries by the fame of his teaching. Enriched by the offerings of his pupils, and entertained with universal admiration, he came, as he says, but a change in his fortunes was at hand. In his devotion to science, he had lived a very regular life, enlivened only by philosophical debate, now, at the height of his fame. Héloïse dArgenteuil lived within the precincts of Notre-Dame, under the care of her uncle and she was remarkable for her knowledge of classical letters, which extended beyond Latin to Greek and Hebrew. Abelard sought a place in Fulberts house and, in 1115 or 1116, the affair interfered with his career, and Abelard himself boasted of his conquest. Once Fulbert found out, he separated them, but they continued to meet in secret, Héloïse became pregnant and was sent by Abelard to be looked after by his family in Brittany, where she gave birth to a son whom she named Astrolabe after the scientific instrument
4.
George Boole
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George Boole was an English mathematician, educator, philosopher and logician. He worked in the fields of differential equations and algebraic logic, Boolean logic is credited with laying the foundations for the information age. Boole was born in Lincoln, Lincolnshire, England, the son of John Boole Sr and he had a primary school education, and received lessons from his father, but had little further formal and academic teaching. William Brooke, a bookseller in Lincoln, may have helped him with Latin and he was self-taught in modern languages. At age 16 Boole became the breadwinner for his parents and three siblings, taking up a junior teaching position in Doncaster at Heighams School. Boole participated in the Mechanics Institute, in the Greyfriars, Lincoln, without a teacher, it took him many years to master calculus. At age 19, Boole successfully established his own school in Lincoln, four years later he took over Halls Academy in Waddington, outside Lincoln, following the death of Robert Hall. In 1840 he moved back to Lincoln, where he ran a boarding school, Boole became a prominent local figure, an admirer of John Kaye, the bishop. He took part in the campaign for early closing. With E. R. Larken and others he set up a society in 1847. He associated also with the Chartist Thomas Cooper, whose wife was a relation, from 1838 onwards Boole was making contacts with sympathetic British academic mathematicians and reading more widely. He studied algebra in the form of symbolic methods, as far as these were understood at the time, Booles status as mathematician was recognised by his appointment in 1849 as the first professor of mathematics at Queens College, Cork in Ireland. He met his wife, Mary Everest, there in 1850 while she was visiting her uncle John Ryall who was Professor of Greek. They married some years later in 1855 and he maintained his ties with Lincoln, working there with E. R. Larken in a campaign to reduce prostitution. Boole was awarded the Keith Medal by the Royal Society of Edinburgh in 1855 and was elected a Fellow of the Royal Society in 1857 and he received honorary degrees of LL. D. from the University of Dublin and the University of Oxford. In late November 1864, Boole walked, in rain, from his home at Lichfield Cottage in Ballintemple to the university. He soon became ill, developing a cold and high fever. As his wife believed that remedies should resemble their cause, she put her husband to bed and poured buckets of water over him – the wet having brought on his illness, Booles condition worsened and on 8 December 1864, he died of fever-induced pleural effusion
5.
Algirdas Julien Greimas
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Algirdas Julien Greimas, was a French-Lithuanian literary scientist, known among other things for the Greimas Square. He is, along with Roland Barthes, considered the most prominent of the French semioticians, with his training in structural linguistics, he added to the theory of signification and laid the foundations for the Parisian school of semiotics. Among Greimass major contributions to semiotics are the concepts of isotopy, the model, the narrative program. He also researched Lithuanian mythology and Proto-Indo-European religion, and was influential in literary criticism. Greimass father, Julius Greimas, 1882-1942, a teacher and later school inspector, was from Liudvinavas in the Suvalkija region of present-day Lithuania and his mother Konstancija Greimienė, née Mickevičiūtė, 1886-1956, a secretary, was from Kalvarija. They lived in Tula, Russia, when he was born and they returned with him to Lithuania when he was two years old. His baptismal names are Algirdas Julius but he used the French version of his name, Julien. In 1944 he enrolled for study at the Sorbonne in Paris and specialized in lexicography, namely taxonomies of exact. He wrote a thesis on the vocabulary of fashion, for which he received a PhD in 1949, early on, he also met Roland Barthes, with whom he remained close for the next 15 years. In 1959 he moved on to universities in Ankara and Istanbul in Turkey, in 1965 he became professor at the École des Hautes Études en Sciences Sociales in Paris, where he taught for almost 25 years. He co-founded and became Secretary General of the International Association for Semiotic Studies, Greimas died in 1992 in Paris, and was buried at his mothers resting place, Petrašiūnai Cemetery in Kaunas, Lithuania. He was survived by his wife, Teresa Mary Keane, the first work of direct significance to his subsequent research was his doctoral thesis La Mode en 1830. Essai de description du vocabulaire vestimentaire d après les journaux de modes de lépoque and he published three dictionaries throughout his career. During his decade in Alexandria, the discussions in his circle of friends helped broaden his interests, Greimas proposed an original method for discourse semiotics that evolved over a thirty-year period. His starting point began with a profound dissatisfaction with the linguistics of the mid-century that studied only phonemes and morphemes. These grammatical units could generate a number of sentences, the sentence remaining the largest unit of analysis. Such a molecular model did not permit the analysis of units beyond the sentence, the descriptive procedures of narratology and the notion of narrativity are at the very base of Greimassian semiotics of discourse. His initial hypothesis is that meaning is only if it is articulated or narrativized
6.
Apuleius
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Apuleius was a Latin-language prose writer, platonist philosopher and rhetorian. He was a Numidian who lived under the Roman Empire and was from Madauros and he studied Platonism in Athens, travelled to Italy, Asia Minor, and Egypt and was an initiate in several cults or mysteries. The most famous incident in his life was when he was accused of using magic to gain the attentions of a wealthy widow. He declaimed and then distributed a witty tour de force in his own defense before the proconsul and this is known as the Apologia. His most famous work is his bawdy picaresque novel, the Metamorphoses and it is the only Latin novel that has survived in its entirety. It relates the adventures of one Lucius, who experiments with magic and is accidentally turned into a donkey. Apuleius was born in Madauros, a colonia in Numidia on the North African coast bordering Gaetulia, as to his first name, no praenomen is given in any ancient source, late-medieval manuscripts began the tradition of calling him Lucius from the name of the hero of his novel. Details regarding his life come mostly from his speech and his work Florida. His father was a magistrate who bequeathed at his death the sum of nearly two million sesterces to his two sons. Apuleius studied with a master at Carthage and later at Athens and he subsequently went to Rome to study Latin rhetoric and, most likely, to speak in the law courts for a time before returning to his native North Africa. He also travelled extensively in Asia Minor and Egypt, studying philosophy and religion, Apuleius was an initiate in several Greco-Roman mysteries, including the Dionysian Mysteries. He was a priest of Asclepius and, according to Augustine, not long after his return home he set out upon a new journey to Alexandria. On his way there he was ill at the town of Oea and was hospitably received into the house of Sicinius Pontianus. The mother of Pontianus, Pudentilla, was a rich widow. With her sons consent – indeed encouragement – Apuleius agreed to marry her, the case was heard at Sabratha, near Tripoli, c.158 AD, before Claudius Maximus, proconsul of Africa. The accusation itself seems to have been ridiculous, and the spirited and this is known as the Apologia. Apuleius accused an extravagant personal enemy of turning his house into a brothel, of his subsequent career we know little. Judging from the works of which he was author, he must have devoted himself diligently to literature
7.
Venn diagram
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A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves, a Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. In Venn diagrams the curves are overlapped in every possible way and they are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn and they are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram in which in addition the area of each shape is proportional to the number of elements it contains is called an area-proportional or scaled Venn diagram and this example involves two sets, A and B, represented here as coloured circles. The orange circle, set A, represents all living creatures that are two-legged, the blue circle, set B, represents the living creatures that can fly. Each separate type of creature can be imagined as a point somewhere in the diagram, living creatures that both can fly and have two legs—for example, parrots—are then in both sets, so they correspond to points in the region where the blue and orange circles overlap. That region contains all such and only living creatures. Humans and penguins are bipedal, and so are then in the circle, but since they cannot fly they appear in the left part of the orange circle. Mosquitoes have six legs, and fly, so the point for mosquitoes is in the part of the circle that does not overlap with the orange one. Creatures that are not two-legged and cannot fly would all be represented by points outside both circles, the combined region of sets A and B is called the union of A and B, denoted by A ∪ B. The union in this case contains all living creatures that are either two-legged or that can fly, the region in both A and B, where the two sets overlap, is called the intersection of A and B, denoted by A ∩ B. For example, the intersection of the two sets is not empty, because there are points that represent creatures that are in both the orange and blue circles. They are rightly associated with Venn, however, because he comprehensively surveyed and formalized their usage, Venn himself did not use the term Venn diagram and referred to his invention as Eulerian Circles. Of these schemes one only, viz. that commonly called Eulerian circles, has met with any general acceptance, the first to use the term Venn diagram was Clarence Irving Lewis in 1918, in his book A Survey of Symbolic Logic. Venn diagrams are similar to Euler diagrams, which were invented by Leonhard Euler in the 18th century. Baron has noted that Leibniz in the 17th century produced similar diagrams before Euler and she also observes even earlier Euler-like diagrams by Ramon Lull in the 13th Century. In the 20th century, Venn diagrams were further developed, D. W. Henderson showed in 1963 that the existence of an n-Venn diagram with n-fold rotational symmetry implied that n was a prime number
8.
Gottlob Frege
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Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. Considered a major figure in mathematics, he is responsible for the development of modern logic and he is also understood by many to be the father of analytic philosophy, where he concentrated on the philosophy of language and mathematics. Though largely ignored during his lifetime, Giuseppe Peano and Bertrand Russell introduced his work to generations of logicians. Frege was born in 1848 in Wismar, Mecklenburg-Schwerin and his father Carl Alexander Frege was the co-founder and headmaster of a girls high school until his death. In childhood, Frege encountered philosophies that would guide his future scientific career, Frege studied at a gymnasium in Wismar and graduated in 1869. His teacher Gustav Adolf Leo Sachse, who was a poet, played the most important role in determining Freges future scientific career, Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation. In the four semesters of his studies he attended approximately twenty courses of lectures and his most important teacher was Ernst Karl Abbe. Abbe was more than a teacher to Frege, he was a trusted friend, after Freges graduation, they came into closer correspondence. His other notable university teachers were Christian Philipp Karl Snell, Hermann Karl Julius Traugott Schaeffer, Frege married Margarete Katharina Sophia Anna Lieseberg on 14 March 1887. Though his education and early work focused primarily on geometry. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a/S, the Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. Previous logic had dealt with the constants and, or. Freges conceptual notation however can represent such inferences, one of Freges stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to intuition. If there was an element, it was to be isolated and represented separately as an axiom, from there on. Already in the 1879 Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic and this idea was formulated in non-symbolic terms in his The Foundations of Arithmetic. Later, in his Basic Laws of Arithmetic, Frege attempted to derive, by use of his symbolism, most of these axioms were carried over from his Begriffsschrift, though not without some significant changes. The one truly new principle was one he called the Basic Law V, the crucial case of the law may be formulated in modern notation as follows. Let denote the extension of the predicate Fx, i. e. the set of all Fs, then Basic Law V says that the predicates Fx and Gx have the same extension iff ∀x
9.
Edward N. Zalta
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Edward N. Zalta is a Senior research scholar at the Center for the Study of Language and Information. He received his Ph. D. in philosophy from the University of Massachusetts Amherst in 1980, Zalta has taught courses at Stanford University, Rice University, the University of Salzburg, and the University of Auckland. Zalta is also the Principal Editor of the Stanford Encyclopedia of Philosophy, Zaltas most notable philosophical position is descended from the position of Alexius Meinong and Ernst Mally, who suggested that there are many non-existent objects. On Zaltas account, some objects exemplify properties, while others merely encode them, while the objects that exemplify properties are discovered through traditional empirical means, a simple set of axioms allows us to know about objects that encode properties. For every set of properties, there is one object that encodes exactly that set of properties. This allows for a formalized ontology
10.
Theorem
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In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms. A theorem is a consequence of the axioms. The proof of a theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific law, many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises, however, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a symbolic form, for example, within the propositional calculus. In some cases, a picture alone may be sufficient to prove a theorem, because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being trivial, or difficult, or deep and these subjective judgments vary not only from person to person, but also with time, for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a theorem may be simply stated. Fermats Last Theorem is a particularly well-known example of such a theorem, logically, many theorems are of the form of an indicative conditional, if A, then B. Such a theorem does not assert B, only that B is a consequence of A. In this case A is called the hypothesis of the theorem and B the conclusion. The theorem If n is an natural number then n/2 is a natural number is a typical example in which the hypothesis is n is an even natural number. To be proved, a theorem must be expressible as a precise, nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader can produce a formal statement from the informal one. It is common in mathematics to choose a number of hypotheses within a given language and these hypotheses form the foundational basis of the theory and are called axioms or postulates. The field of known as proof theory studies formal languages, axioms. Some theorems are trivial, in the sense that they follow from definitions, axioms, a theorem might be simple to state and yet be deep
11.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
12.
Begriffsschrift
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Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. Begriffsschrift is usually translated as concept writing or concept notation, the title of the book identifies it as a formula language, modeled on that of arithmetic. Freges motivation for developing his formal approach to logic resembled Leibnizs motivation for his calculus ratiocinator, Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century. The calculus contains the first appearance of quantified variables, and is essentially classical bivalent second-order logic with identity and it is bivalent in that sentences or formulas denote either True or False, second order because it includes relation variables in addition to object variables and allows quantification over both. The modifier with identity specifies that the language includes the identity relation, Frege presents his calculus using idiosyncratic two-dimensional notation, connectives and quantifiers are written using lines connecting formulas, rather than the symbols ¬, ∧, and ∀ in use today. For example, that judgement B materially implies judgement A, i. e, let signify that the third of those possibilities does not obtain, but one of the three others does. So if we negate, that means the third possibility is valid, i. e. we negate A, Frege declared nine of his propositions to be axioms, and justified them by arguing informally that, given their intended meanings, they express self-evident truths. – govern material implication, – negation, and identity, expresses Leibnizs indiscernibility of identicals, and asserts that identity is a reflexive relation. This rule is much harder to articulate precisely than the two preceding rules, and Frege invokes it in ways that are not obviously legitimate. The main results of the chapter, titled Parts from a general series theory. Frege applied the results from the Begriffsschrifft, including those on the ancestral of a relation, thus, if we take xRy to be the relation y = x +1, then 0R*y is the predicate y is a natural number. Says that if x, y, and z are natural numbers, then one of the following must hold, x < y, x = y and this is the so-called law of trichotomy. For a careful recent study of how the Begriffsschrift was reviewed in the German mathematical literature, some reviewers, especially Ernst Schröder, were on the whole favorable. Some vestige of Freges notation survives in the turnstile symbol ⊢ derived from his Urteilsstrich │, Frege used these symbols in the Begriffsschrift in the unified form ├─ for declaring that a proposition is true. In his later Grundgesetze he revises slightly his interpretation of the ├─ symbol, in Begriffsschrift the Definitionsdoppelstrich │├─ indicates that a proposition is a definition. Furthermore, the negation sign ¬ can be read as a combination of the horizontal Inhaltsstrich with a vertical negation stroke and this negation symbol was reintroduced by Arend Heyting in 1930 to distinguish intuitionistic from classical negation. It also appears in Gerhard Gentzens doctoral dissertation, in the Tractatus Logico Philosophicus, Ludwig Wittgenstein pays homage to Frege by employing the term Begriffsschrift as a synonym for logical formalism. Freges 1892 essay, Sense and Reference, recants some of the conclusions of the Begriffsschrifft about identity, ancestral relation Freges propositional calculus Gottlob Frege
