1.
Venn diagram
–
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
2.
Empty set
–
In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
3.
Aristotle
–
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
4.
Contradiction
–
In classical logic, a contradiction consists of a logical incompatibility between two or more propositions. It occurs when the propositions, taken together, yield two conclusions which form the logical, usually opposite inversions of each other. Illustrating a general tendency in applied logic, Aristotles law of noncontradiction states that One cannot say of something that it is, by extension, outside of classical logic, one can speak of contradictions between actions when one presumes that their motives contradict each other. By creation of a paradox, Platos Euthydemus dialogue demonstrates the need for the notion of contradiction, in the ensuing dialogue Dionysodorus denies the existence of contradiction, all the while that Socrates is contradicting him. I in my astonishment said, What do you mean Dionysodorus, the dictum is that there is no such thing as a falsehood, a man must either say what is true or say nothing. Indeed, Dionysodorus agrees that there is no such thing as false opinion, there is no such thing as ignorance and demands of Socrates to Refute me. Socrates responds But how can I refute you, if, as you say, note, The symbol ⊥ represents an arbitrary contradiction, with the dual tee symbol ⊤ used to denote an arbitrary tautology. Contradiction is sometimes symbolized by Opq, and tautology by Vpq, the turnstile symbol, ⊢ is often read as yields or proves. In classical logic, particularly in propositional and first-order logic, a proposition φ is a contradiction if, since for contradictory φ it is true that ⊢ φ → ψ for all ψ, one may prove any proposition from a set of axioms which contains contradictions. This is called the principle of explosion or ex falso quodlibet, in a complete logic, a formula is contradictory if and only if it is unsatisfiable. Therefore, a proof that ¬ φ ⊢ ⊥ also proves that φ is true, the use of this fact constitutes the technique of the proof by contradiction, which mathematicians use extensively. This applies only in a logic using the excluded middle A ∨ ¬ A as an axiom, in mathematics, the symbol used to represent a contradiction within a proof varies. A consistency proof requires an axiomatic system a demonstration that it is not the case both the formula p and its negation ~p can be derived in the system. Posts solution to the problem is described in the demonstration An Example of a Successful Absolute Proof of Consistency offered by Ernest Nagel and they too observe a problem with respect to the notion of contradiction with its usual truth values of truth and falsity. They observe that, The property of being a tautology has been defined in notions of truth, yet these notions obviously involve a reference to something outside the formula calculus. Therefore, the mentioned in the text in effect offers an interpretation of the calculus. This being so, the authors have not done what they promised, namely, proofs of consistency which are based on models, and which argue from the truth of axioms to their consistency, merely shift the problem. Given some primitive formulas such as PMs primitives S1 V S2, so what will be the definition of tautologous
5.
Apuleius
–
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
6.
Boethius
–
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
7.
Syllogism
–
A syllogism is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true. In its earliest form, defined by Aristotle, from the combination of a statement and a specific statement. For example, knowing that all men are mortal and that Socrates is a man, Syllogistic arguments are usually represented in a three-line form, All men are mortal. In antiquity, two theories of the syllogism existed, Aristotelian syllogistic and Stoic syllogistic. Aristotle defines the syllogism as. a discourse in which certain things having been supposed, despite this very general definition, in Aristotles work Prior Analytics, he limits himself to categorical syllogisms that consist of three categorical propositions. From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably and this article is concerned only with this traditional use. The use of syllogisms as a tool for understanding can be dated back to the logical reasoning discussions of Aristotle, the onset of a New Logic, or logica nova, arose alongside the reappearance of Prior Analytics, the work in which Aristotle develops his theory of the syllogism. Prior Analytics, upon re-discovery, was regarded by logicians as a closed and complete body of doctrine, leaving very little for thinkers of the day to debate. Aristotles theories on the syllogism for assertoric sentences was considered especially remarkable, Aristotles Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise. Aristotles terminology in this aspect of his theory was deemed vague and in many cases unclear and his original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of the day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use, boethius contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the twelfth century and his perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus. With the help of Abelards distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a coherent concept of Aristotles modal syllogism model. For two hundred years after Buridans discussions, little was said about syllogistic logic, the Aristotelian syllogism dominated Western philosophical thought for many centuries. In the 17th century, Sir Francis Bacon rejected the idea of syllogism as being the best way to draw conclusions in nature. Instead, Bacon proposed a more inductive approach to the observation of nature, in the 19th century, modifications to syllogism were incorporated to deal with disjunctive and conditional statements. Kant famously claimed, in Logic, that logic was the one completed science, though there were alternative systems of logic such as Avicennian logic or Indian logic elsewhere, Kants opinion stood unchallenged in the West until 1879 when Frege published his Begriffsschrift. This introduced a calculus, a method of representing categorical statements by the use of quantifiers, in the last 20 years, Bolzanos work has resurfaced and become subject of both translation and contemporary study
8.
Peter Abelard
–
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
9.
George Boole
–
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
10.
Gottlob Frege
–
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
11.
Begriffsschrift
–
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
12.
Algirdas Julien Greimas
–
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
13.
Semiotic square
–
Greimas first presented the square in Semantique Structurale, a book which was later published as Structural Semantics, An Attempt at a Method. He further developed the semiotic square with Francois Rastier in The Interaction of Semiotic Constraints, S1 is considered to be the assertion/positive element and S2 is the negation/negative element in the binary pair, The second binary relationship is now created on the ~S axis. ~S1 is considered to be the term, and ~S2 is the neutral term. This is where the principle of difference is brought into play, in most modes of interpretation, the S-axis is a hyponym of the ~S-axis. The ~S1 element combines aspects of S1 and S2 and is contradictory to S1. The ~S2 element contains aspects of neither S1 nor S2, finally, the ~S2 element can be identified. e. Hermaphrodite, bi-sexual neither S1 nor S2, neither masculine nor feminine, as such, one form of interpretation is to look at each of the elements, S1, S2, ~S1, and ~S2 as either developed by Ferdinand de Saussure or Charles Sanders Peirce sign. In the Peircean system of semiotics, the interpretant becomes the representamen for another, in this same way, each of the elements of the Semiotic Square can become an element in a new, interrelated square. Finally, Greimas suggests placing semiotic squares of associated meaning on top of one another to create an effect and another form of analysis. The square is a map of logical possibilities, as such, it can be used as a heuristic device, and in fact, attempting to fill it in stimulates the imagination. Aristotles Non-Logical Works and the Square of Oppositions in Semiotics, Logica Universalis, “The Interaction of Semiotic Constraints, ” Yale French Studies. Hébert, Louis, The Semiotic Square, in Louis Hébert, Signo, Rimouski Katilius-Boydstun, Greimas, An Introduction, ” Litanus, Lithuanian Quarterly Journal of Arts and Sciences. Was That Last Turn a Right Turn, “The Structural Study of Myth, ” The Journal of American Folklore. Greimas and the nature of meaning, linguistics, semiotics and discourse theory, “Disciplinarity and Collaboration in the Sciences and Humanities, ” College English. Cultural semiotics, Spenser, and the captive woman, sebeok, Thomas A. and Jean Umiker-Sebeok. Berlin, Walter de Gruyter & Co, modules on Greimas, On the semiotic square Timothy Lenoir, Was That Last Turn A Right Turn
14.
Logical hexagon
–
The logical hexagon is a conceptual model of the relationships between the truth values of six statements. It is an extension of Aristotles square of opposition and it was discovered independently by both Augustin Sesmat and Robert Blanché. This extension consists in introducing two statements U and Y, whereas U is the disjunction of A and E, Y is the conjunction of the two traditional particulars I and O. The traditional square of opposition demonstrates two sets of contradictories A and O, and E and I, two contraries A and E, and two subcontraries I and O according to Aristotle’s definitions, however, the logical hexagon provides that U and Y are also contradictory. The logical hexagon may be interpreted in ways, including as a model of traditional logic, quantifications, modal logic, order theory. For instance, the statement A may be interpreted as Whatever x may be, if x is a man, the statement E may be interpreted as Whatever x may be, if x is a man, then x is non-white. The statement I may be interpreted as There exists at least one x that is both a man and white. W The logical hexagon may be interpreted in ways, including as a model of traditional logic, quantifications, modal logic, order theory. Alessio Moretti Jean-Yves Béziau, New Light on the Square of Oppositions and its Nameless Corner Jean-Yves Beziau, The power of the hexagon, Logica Universalis 6,2012, 1-43. Doi,10. 1007/s11787-012-0046-9 Blanché Blanché Blanché Structures intellectuelles Gallais, P. Gottschalk Kalinowski Monteil, the logical square of Aristotle or square of Apuleius. The logical hexagon of Robert Blanché in Structures intellectuelles. The triangle of Indian logic mentioned by J. M Bochenski
15.
International Standard Book Number
–
The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
16.
Edward N. Zalta
–
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
17.
Formal language
–
In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols together with a set of rules that are specific to it. The alphabet of a language is the set of symbols, letters. The strings formed from this alphabet are called words, and the words belong to a particular formal language are sometimes called well-formed words or well-formed formulas. A formal language is defined by means of a formal grammar such as a regular grammar or context-free grammar. The field of language theory studies primarily the purely syntactical aspects of such languages—that is. Formal language theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages. The first formal language is thought to be the one used by Gottlob Frege in his Begriffsschrift, literally meaning concept writing, axel Thues early semi-Thue system, which can be used for rewriting strings, was influential on formal grammars. The elements of an alphabet are called its letters, alphabets may be infinite, however, most definitions in formal language theory specify finite alphabets, and most results only apply to them. A word over an alphabet can be any sequence of letters. The set of all words over an alphabet Σ is usually denoted by Σ*, the length of a word is the number of letters it is composed of. For any alphabet there is one word of length 0, the empty word. By concatenation one can combine two words to form a new word, whose length is the sum of the lengths of the original words, the result of concatenating a word with the empty word is the original word. A formal language L over an alphabet Σ is a subset of Σ*, that is, sometimes the sets of words are grouped into expressions, whereas rules and constraints may be formulated for the creation of well-formed expressions. In computer science and mathematics, which do not usually deal with natural languages, in practice, there are many languages that can be described by rules, such as regular languages or context-free languages. The notion of a formal grammar may be closer to the concept of a language. By an abuse of the definition, a formal language is often thought of as being equipped with a formal grammar that describes it. The following rules describe a formal language L over the alphabet Σ =, Every nonempty string that does not contain + or =, a string containing = is in L if and only if there is exactly one =, and it separates two valid strings of L. A string containing + but not = is in L if, no string is in L other than those implied by the previous rules
18.
Formal proof
–
A formal proof or derivation is a finite sequence of sentences, each of which is an axiom, an assumption, or follows from the preceding sentences in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system, the notion of theorem is not in general effective, therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of natural deduction is a generalization of the concept of proof, the theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. Formal proofs often are constructed with the help of computers in interactive theorem proving, significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually simple, while the problem of finding proofs is usually computationally intractable and/or only semi-decidable, a formal language is a set of finite sequences of symbols. Such a language can be defined without reference to any meanings of any of its expressions, it can exist before any interpretation is assigned to it – that is, Formal proofs are expressed in some formal language. A formal grammar is a description of the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the language which constitute well formed formulas. However, it does not describe their semantics, a formal system consists of a formal language together with a deductive apparatus. The deductive apparatus may consist of a set of rules or a set of axioms. A formal system is used to derive one expression from one or more other expressions, an interpretation of a formal system is the assignment of meanings to the symbols, and values to the sentences of a formal system. The study of interpretations is called formal semantics, giving an interpretation is synonymous with constructing a model. Proof Mathematical proof Proof theory Axiomatic system A Special Issue on Formal Proof, notices of the American Mathematical Society. 2πix. com, Logic Part of a series of articles covering mathematics and logic
19.
Well-formed formula
–
In mathematical logic, a well-formed formula, abbreviated wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language, a formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic, a key use of formulas is in propositional logic and predicate logics such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask is φ true, once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, although the term formula may be used for written marks, it is more precisely understood as the sequence of symbols being expressed, with the marks being a token instance of formula. Thus the same formula may be more than once. They are given meanings by interpretations, for example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not be interpreted, however, to be considered solely as a formula, the formulas of propositional calculus, also called propositional formulas, are expressions such as. Their definition begins with the choice of a set V of propositional variables. The alphabet consists of the letters in V along with the symbols for the propositional connectives and parentheses, the formulas will be certain expressions over this alphabet. The formulas are inductively defined as follows, Each propositional variable is, on its own, If φ is a formula, then ¬φ is a formula. If φ and ψ are formulas, and • is any binary connective, here • could be the usual operators ∨, ∧, →, or ↔. The sequence of symbols p)) is not a formula, because it does not conform to the grammar, a complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules are assumed among the operators, for example, assuming the precedence 1. Then the formula may be abbreviated as p → q ∧ r → s ∨ ¬q ∧ ¬s This is, however, If the precedence was assumed, for example, to be left-right associative, in following order,1. ∨4. →, then the formula above would be rewritten as → The definition of a formula in first-order logic Q S is relative to the signature of the theory at hand. This signature specifies the constant symbols, relation symbols, and function symbols of the theory at hand, the definition of a formula comes in several parts. First, the set of terms is defined recursively, terms, informally, are expressions that represent objects from the domain of discourse
20.
Set (mathematics)
–
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
21.
Element (mathematics)
–
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1,2,3 and 4, sets of elements of A, for example, are subsets of A. For example, consider the set B =, the elements of B are not 1,2,3, and 4. Rather, there are three elements of B, namely the numbers 1 and 2, and the set. The elements of a set can be anything, for example, C =, is the set whose elements are the colors red, green and blue. The relation is an element of, also called set membership, is denoted by the symbol ∈, writing x ∈ A means that x is an element of A. Equivalent expressions are x is a member of A, x belongs to A, x is in A and x lies in A, another possible notation for the same relation is A ∋ x, meaning A contains x, though it is used less often. The negation of set membership is denoted by the symbol ∉, writing x ∉ A means that x is not an element of A. The symbol ϵ was first used by Giuseppe Peano 1889 in his work Arithmetices principia nova methodo exposita, here he wrote on page X, Signum ϵ significat est. Ita a ϵ b legitur a est quoddam b. which means The symbol ϵ means is, so a ϵ b is read as a is a b. The symbol itself is a stylized lowercase Greek letter epsilon, the first letter of the word ἐστί, the Unicode characters for these symbols are U+2208, U+220B and U+2209. The equivalent LaTeX commands are \in, \ni and \notin, mathematica has commands \ and \. The number of elements in a set is a property known as cardinality, informally. In the above examples the cardinality of the set A is 4, an infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets, an example of an infinite set is the set of positive integers =. Using the sets defined above, namely A =, B = and C =,2 ∈ A ∈ B3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite, the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY, Springer-Verlag, ISBN 0-387-90092-6 - Naive means that it is not fully axiomatized, not that it is silly or easy. Jech, Thomas, Set Theory, Stanford Encyclopedia of Philosophy Suppes, Patrick, Axiomatic Set Theory, NY, Dover Publications, Inc
22.
Classical logic
–
Classical logic is an intensively studied and widely used class of formal logics. Classical logic was devised as a two-level logical system, with simple semantics for the levels representing true. These judgments find themselves if two pairs of two operators, and each operator is the negation of another, relationships that Aristotle summarised with his square of oppositions. George Booles algebraic reformulation of logic, his system of Boolean logic, with the advent of algebraic logic it became apparent that classical propositional calculus admits other semantics. In Boolean-valued semantics, the values are the elements of an arbitrary Boolean algebra, true corresponds to the maximal element of the algebra. Intermediate elements of the algebra correspond to truth values other than true, the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements. Many-valued logic, including logic, which rejects the law of the excluded middle. Graham Priest, An Introduction to Non-Classical Logic, From If to Is, 2nd Edition, CUP,2008, ISBN 978-0-521-67026-5 Warren Goldfard, Deductive Logic, 1st edition,2003, ISBN 0-87220-660-2
23.
Theorem
–
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
24.
Logical consequence
–
Logical consequence is a fundamental concept in logic, which describes the relationship between statements that holds true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusions are entailed by the premises, the philosophical analysis of logical consequence involves the questions, In what sense does a conclusion follow from its premises. And What does it mean for a conclusion to be a consequence of premises, All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth. Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a consequence of a set of sentences, for a given language, if and only if. The most widely prevailing view on how to best account for logical consequence is to appeal to formality and this is to say that whether statements follow from one another logically depends on the structure or logical form of the statements without regard to the contents of that form. Syntactic accounts of logical consequence rely on schemes using inference rules, for instance, we can express the logical form of a valid argument as, All A are B. All C are A. Therefore, all C are B and this argument is formally valid, because every instance of arguments constructed using this scheme are valid. This is in contrast to an argument like Fred is Mikes brothers son, if you know that Q follows logically from P no information about the possible interpretations of P or Q will affect that knowledge. Our knowledge that Q is a consequence of P cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, however, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a property of logical consequence is considered to be independent of formality. The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs, the study of the syntactic consequence is called proof theory whereas the study of semantic consequence is called model theory. A formula A is a syntactic consequence within some formal system F S of a set Γ of formulas if there is a proof in F S of A from the set Γ. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, or, in other words, the set of the interpretations that make all members of Γ true is a subset of the set of the interpretations that make A true. Modal accounts of logical consequence are variations on the basic idea, Γ ⊢ A is true if and only if it is necessary that if all of the elements of Γ are true. Alternatively, Γ ⊢ A is true if and only if it is impossible for all of the elements of Γ to be true, such accounts are called modal because they appeal to the modal notions of logical necessity and logical possibility. Consider the modal account in terms of the argument given as an example above, the conclusion is a logical consequence of the premises because we cant imagine a possible world where all frogs are green, Kermit is a frog, and Kermit is not green
25.
Symbol (formal)
–
A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern. In logic, symbols build literal utility to illustrate ideas, symbols of a formal language need not be symbols of anything. For instance there are constants which do not refer to any idea. Symbols of a formal language must be capable of being specified without any reference to any interpretation of them, a symbol or string of symbols may comprise a well-formed formula if it is consistent with the formation rules of the language. In a formal system a symbol may be used as a token in formal operations. The set of symbols in a formal language is referred to as an alphabet A formal symbol as used in first-order logic may be a variable, a constant. Formal symbols are thought of as purely syntactic structures, composed into larger structures using a formal grammar. The move to view units in natural language as formal symbols was initiated by Noam Chomsky, the generative grammar model looked upon syntax as autonomous from semantics. On this point I differ from a number of philosophers, but agree, I believe, with Chomsky and this is the philosophical premise underlying Montague grammar. List of mathematical symbols List of logic symbols
26.
Syntax (logic)
–
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the used for constructing, or transforming the symbols and words of a language. Syntax is usually associated with the governing the composition of texts in a formal language that constitute the well-formed formulas of a formal system. In computer science, the term refers to the rules governing the composition of well-formed expressions in a programming language. As in mathematical logic, it is independent of semantics and interpretation, a symbol is an idea, abstraction or concept, tokens of which may be marks or a configuration of marks which form a particular pattern. Symbols of a formal language need not be symbols of anything, for instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language. A symbol or string of symbols may comprise a well-formed formula if the formulation is consistent with the rules of the language. Symbols of a formal language must be capable of being specified without any reference to any interpretation of them, a formal language is a syntactic entity which consists of a set of finite strings of symbols which are its words. Which strings of symbols are words is determined by fiat by the creator of the language, usually by specifying a set of formation rules. Such a language can be defined without reference to any meanings of any of its expressions, it can exist before any interpretation is assigned to it – that is, Formation rules are a precise description of which strings of symbols are the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the language which constitute well formed formulas. However, it does not describe their semantics, a proposition is a sentence expressing something true or false. A proposition is identified ontologically as an idea, concept or abstraction whose token instances are patterns of symbols, marks, sounds, propositions are considered to be syntactic entities and also truthbearers. A formal theory is a set of sentences in a formal language, a formal system consists of a formal language together with a deductive apparatus. The deductive apparatus may consist of a set of rules or a set of axioms. A formal system is used to derive one expression from one or more other expressions, Formal systems, like other syntactic entities may be defined without any interpretation given to it. A formula A is a syntactic consequence within some formal system F S of a set Г of formulas if there is a derivation in formal system F S of A from the set Г. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, a formal system S is syntactically complete iff for each formula A of the language of the system either A or ¬A is a theorem of S
27.
Theory (mathematical logic)
–
In mathematical logic, a theory is a set of sentences in a formal language. Usually a deductive system is understood from context, an element ϕ ∈ T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms is called a theorem of the theory. Every axiom is also a theorem, a first-order theory is a set of first-order sentences. When defining theories for foundational purposes, additional care must be taken, the construction of a theory begins by specifying a definite non-empty conceptual class E, the elements of which are called statements. These initial statements are often called the elements or elementary statements of the theory. A theory T is a class consisting of certain of these elementary statements. The elementary statements which belong to T are called the elementary theorems of T, in this way, a theory is a way of designating a subset of E which consists entirely of true statements. This general way of designating a theory stipulates that the truth of any of its statements is not known without reference to T. Thus the same statement may be true with respect to one theory. This is as in language, where statements such as He is a terrible person. Cannot be judged to be true or false without reference to some interpretation of who He is, a theory S is a subtheory of a theory T if S is a subset of T. If T is a subset of S then S is an extension or supertheory of T A theory is said to be a theory if T is an inductive class. That is, that its content is based on some formal deductive system, in a deductive theory, any sentence which is a logical consequence of one or more of the axioms is also a sentence of that theory. A syntactically consistent theory is a theory from which not every sentence in the language can be proven. In a deductive system that satisfies the principle of explosion, this is equivalent to requiring that there is no sentence φ such that both φ and its negation can be proven from the theory, a satisfiable theory is a theory that has a model. This means there is a structure M that satisfies every sentence in the theory, any satisfiable theory is syntactically consistent, because the structure satisfying the theory will satisfy exactly one of φ and the negation of φ, for each sentence φ. A consistent theory is defined to be a syntactically consistent theory. For first-order logic, the most important case, it follows from the theorem that the two meanings coincide
