1.
Greek language
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Greek is an independent branch of the Indo-European family of languages, native to Greece and other parts of the Eastern Mediterranean. It has the longest documented history of any living language, spanning 34 centuries of written records and its writing system has been the Greek alphabet for the major part of its history, other systems, such as Linear B and the Cypriot syllabary, were used previously. The alphabet arose from the Phoenician script and was in turn the basis of the Latin, Cyrillic, Armenian, Coptic, Gothic and many other writing systems. Together with the Latin texts and traditions of the Roman world, during antiquity, Greek was a widely spoken lingua franca in the Mediterranean world and many places beyond. It would eventually become the official parlance of the Byzantine Empire, the language is spoken by at least 13.2 million people today in Greece, Cyprus, Italy, Albania, Turkey, and the Greek diaspora. Greek roots are used to coin new words for other languages, Greek. Greek has been spoken in the Balkan peninsula since around the 3rd millennium BC, the earliest written evidence is a Linear B clay tablet found in Messenia that dates to between 1450 and 1350 BC, making Greek the worlds oldest recorded living language. Among the Indo-European languages, its date of earliest written attestation is matched only by the now extinct Anatolian languages, the Greek language is conventionally divided into the following periods, Proto-Greek, the unrecorded but assumed last ancestor of all known varieties of Greek. The unity of Proto-Greek would have ended as Hellenic migrants entered the Greek peninsula sometime in the Neolithic era or the Bronze Age, Mycenaean Greek, the language of the Mycenaean civilisation. It is recorded in the Linear B script on tablets dating from the 15th century BC onwards, Ancient Greek, in its various dialects, the language of the Archaic and Classical periods of the ancient Greek civilisation. It was widely known throughout the Roman Empire, after the Roman conquest of Greece, an unofficial bilingualism of Greek and Latin was established in the city of Rome and Koine Greek became a first or second language in the Roman Empire. The origin of Christianity can also be traced through Koine Greek, Medieval Greek, also known as Byzantine Greek, the continuation of Koine Greek in Byzantine Greece, up to the demise of the Byzantine Empire in the 15th century. Much of the written Greek that was used as the language of the Byzantine Empire was an eclectic middle-ground variety based on the tradition of written Koine. Modern Greek, Stemming from Medieval Greek, Modern Greek usages can be traced in the Byzantine period and it is the language used by the modern Greeks, and, apart from Standard Modern Greek, there are several dialects of it. In the modern era, the Greek language entered a state of diglossia, the historical unity and continuing identity between the various stages of the Greek language is often emphasised. Greek speakers today still tend to regard literary works of ancient Greek as part of their own rather than a foreign language and it is also often stated that the historical changes have been relatively slight compared with some other languages. According to one estimation, Homeric Greek is probably closer to demotic than 12-century Middle English is to modern spoken English, Greek is spoken by about 13 million people, mainly in Greece, Albania and Cyprus, but also worldwide by the large Greek diaspora. Greek is the language of Greece, where it is spoken by almost the entire population
2.
Logical consequence
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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
3.
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
4.
Socrates
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Socrates was a classical Greek philosopher credited as one of the founders of Western philosophy. He is a figure known chiefly through the accounts of classical writers, especially the writings of his students Plato and Xenophon. Platos dialogues are among the most comprehensive accounts of Socrates to survive from antiquity, though it is unclear the degree to which Socrates himself is hidden behind his best disciple, nothing written by Socrates remains extant. As a result, information about him and his philosophies depends upon secondary sources, furthermore, close comparison between the contents of these sources reveals contradictions, thus creating concerns about the possibility of knowing in-depth the real Socrates. This issue is known as the Socratic problem, or the Socratic question, to understand Socrates and his thought, one must turn primarily to the works of Plato, whose dialogues are thought the most informative source about Socrates life and philosophy, and also Xenophon. These writings are the Sokratikoi logoi, or Socratic dialogues, which consist of reports of conversations apparently involving Socrates, as for discovering the real-life Socrates, the difficulty is that ancient sources are mostly philosophical or dramatic texts, apart from Xenophon. There are no straightforward histories, contemporary with Socrates, that dealt with his own time, a corollary of this is that sources that do mention Socrates do not necessarily claim to be historically accurate, and are often partisan. For instance, those who prosecuted and convicted Socrates have left no testament, historians therefore face the challenge of reconciling the various evidence from the extant texts in order to attempt an accurate and consistent account of Socrates life and work. The result of such an effort is not necessarily realistic, even if consistent, amid all the disagreement resulting from differences within sources, two factors emerge from all sources pertaining to Socrates. It would seem, therefore, that he was ugly, also, Xenophon, being an historian, is a more reliable witness to the historical Socrates. It is a matter of debate over which Socrates it is whom Plato is describing at any given point—the historical figure. As British philosopher Martin Cohen has put it, Plato, the idealist, offers an idol, a Saint, a prophet of the Sun-God, a teacher condemned for his teachings as a heretic. It is also clear from other writings and historical artefacts, that Socrates was not simply a character, nor an invention, the testimony of Xenophon and Aristotle, alongside some of Aristophanes work, is useful in fleshing out a perception of Socrates beyond Platos work. The problem with discerning Socrates philosophical views stems from the perception of contradictions in statements made by the Socrates in the different dialogues of Plato and these contradictions produce doubt as to the actual philosophical doctrines of Socrates, within his milieu and as recorded by other individuals. Aristotle, in his Magna Moralia, refers to Socrates in words which make it patent that the virtue is knowledge was held by Socrates. Within the Metaphysics, he states Socrates was occupied with the search for moral virtues, however, in The Clouds, Aristophanes portrays Socrates as accepting payment for teaching and running a sophist school with Chaerephon. Also, in Platos Apology and Symposium, as well as in Xenophons accounts, more specifically, in the Apology, Socrates cites his poverty as proof that he is not a teacher. Two fragments are extant of the writings by Timon of Phlius pertaining to Socrates, although Timon is known to have written to ridicule, details about the life of Socrates can be derived from three contemporary sources, the dialogues of Plato and Xenophon, and the plays of Aristophanes
5.
History of logic
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The history of logic deals with the study of the development of the science of valid inference. Formal logics developed in ancient times in China, India, Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic, christian and Islamic philosophers such as Boethius and William of Ockham further developed Aristotles logic in the Middle Ages, reaching a high point in the mid-fourteenth century. The period between the fourteenth century and the beginning of the century saw largely decline and neglect. Empirical methods ruled the day, as evidenced by Sir Francis Bacons Novum Organon of 1620, valid reasoning has been employed in all periods of human history. However, logic studies the principles of reasoning, inference. It is probable that the idea of demonstrating a conclusion first arose in connection with geometry, the ancient Egyptians discovered geometry, including the formula for the volume of a truncated pyramid. Ancient Babylon was also skilled in mathematics, while the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative proof. Both Thales and Pythagoras of the Pre-Socratic philosophers seem aware of geometrys methods, fragments of early proofs are preserved in the works of Plato and Aristotle, and the idea of a deductive system was probably known in the Pythagorean school and the Platonic Academy. The proofs of Euclid of Alexandria are a paradigm of Greek geometry, the three basic principles of geometry are as follows, Certain propositions must be accepted as true without demonstration, such a proposition is known as an axiom of geometry. Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry, the proof must be formal, that is, the derivation of the proposition must be independent of the particular subject matter in question. Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called dissoi logoi and this is part of a protracted debate about truth and falsity. Thales was said to have had a sacrifice in celebration of discovering Thales Theorem just as Pythagoras had the Pythagorean Theorem, Indian and Babylonian mathematicians knew his theorem for special cases before he proved it. It is believed that Thales learned that an angle inscribed in a semicircle is a right angle during his travels to Babylon, before 520 BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c.54 years older Thales. The systematic study of proof seems to have begun with the school of Pythagoras in the sixth century BC. Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize rather than matter. He is known for his obscure sayings and this logos holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. But other people fail to notice what they do when awake, in contrast to Heraclitus, Parmenides held that all is one and nothing changes
6.
Prior Analytics
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The Prior Analytics is Aristotles work on deductive reasoning, which is known as his syllogistic. Being one of the six extant Aristotelian writings on logic and scientific method, modern work on Aristotles logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm. The term analytics comes from the Greek words ἀναλυτός and ἀναλύω, however, in Aristotles corpus, there are distinguishable differences in the meaning of ἀναλύω and its cognates. There is also the possibility that Aristotle may have borrowed his use of the analysis from his teacher Plato. Therefore, Analysis is the process of finding the reasoned facts, Aristotles Prior Analytics represents the first time in history when Logic is scientifically investigated. On those grounds alone, Aristotle could be considered the Father of Logic for as he says in Sophistical Refutations. When it comes to this subject, it is not the case that part had been worked out before in advance and part had not, instead, some scholars prefer to use the word deduction instead as the meaning given by Aristotle to the Greek word συλλογισμός syllogismos. In the Analytics then, Prior Analytics is the first theoretical part dealing with the science of deduction, Prior Analytics gives an account of deductions in general narrowed down to three basic syllogisms while Posterior Analytics deals with demonstration. In the Prior Analytics, Aristotle defines syllogism as, a deduction in a discourse in which, certain things being supposed, something different from the things supposed results of necessity because these things are so. In modern times, this definition has led to a debate as to how the word syllogism should be interpreted, scholars Jan Lukasiewicz, Józef Maria Bocheński and Günther Patzig have sided with the Protasis-Apodosis dichotomy while John Corcoran prefers to consider a syllogism as simply a deduction. In the third century AD, Alexander of Aphrodisiass commentary on the Prior Analytics is the oldest extant, in the sixth century, Boethius composed the first known Latin translation of the Prior Analytics. No Westerner between Boethius and Bernard of Utrecht is known to have read the Prior Analytics, the so-called Anonymus Aurelianensis III from the second half of the twelfth century is the first extant Latin commentary, or rather fragment of a commentary. The Prior Analytics represents the first formal study of logic, where logic is understood as the study of arguments, an argument is a series of true or false statements which lead to a true or false conclusion. In the Prior Analytics, Aristotle identifies valid and invalid forms of arguments called syllogisms, a syllogism is an argument that consists of at least three sentences, at least two premises and a conclusion. Although Aristotles does not call them categorical sentences, tradition does, he deals with them briefly in the Analytics, each proposition of a syllogism is a categorical sentence which has a subject and a predicate connected by a verb. In his formulation of syllogistic propositions, instead of the copula, belongs to/does not belong to all/some. Is said/is not said of all/some, there are four different types of categorical sentences, universal affirmative, particular affirmative, universal negative and particular negative. Depending on the position of the term, Aristotle divides the syllogism into three kinds, Syllogism in the first, second and third figure
7.
First-order logic
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First-order logic – also known as first-order predicate calculus and predicate logic – is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. This distinguishes it from propositional logic, which does not use quantifiers, Sometimes theory is understood in a more formal sense, which is just a set of sentences in first-order logic. In first-order theories, predicates are associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets, There are many deductive systems for first-order logic which are both sound and complete. Although the logical relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem, first-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, no first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures can be obtained in stronger logics such as second-order logic, for a history of first-order logic and how it came to dominate formal logic, see José Ferreirós. While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates, a predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences Socrates is a philosopher and Plato is a philosopher, in propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate is a philosopher occurs in both sentences, which have a structure of a is a philosopher. The variable a is instantiated as Socrates in the first sentence and is instantiated as Plato in the second sentence, while first-order logic allows for the use of predicates, such as is a philosopher in this example, propositional logic does not. Relationships between predicates can be stated using logical connectives, consider, for example, the first-order formula if a is a philosopher, then a is a scholar. This formula is a statement with a is a philosopher as its hypothesis. The truth of this depends on which object is denoted by a. Quantifiers can be applied to variables in a formula, the variable a in the previous formula can be universally quantified, for instance, with the first-order sentence For every a, if a is a philosopher, then a is a scholar. The universal quantifier for every in this sentence expresses the idea that the if a is a philosopher. The negation of the sentence For every a, if a is a philosopher, then a is a scholar is logically equivalent to the sentence There exists a such that a is a philosopher and a is not a scholar
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.
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
10.
Term logic
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This entry is an introduction to the term logic needed to understand philosophy texts written before predicate logic came to be seen as the only formal logic of interest. Readers lacking a grasp of the terminology and ideas of term logic can have difficulty understanding such texts. Aristotles logical work is collected in the six texts that are known as the Organon. Modern work on Aristotles logic builds on the tradition started in 1951 with the establishment by Jan Lukasiewicz of a revolutionary paradigm, the proposition consists of two terms, in which one term is affirmed or denied of the other, and which is capable of truth or falsity. The syllogism is an inference in which one proposition follows of necessity from two others, a proposition may be universal or particular, and it may be affirmative or negative. Aristotles original square of opposition, however, does not lack existential import and this is the theory of two-premised arguments in which the premises and conclusion share three terms among them, with each proposition containing two of them. It is distinctive of this enterprise that everybody agrees on which syllogisms are valid, the theory of the syllogism partly constrains the interpretation of the forms. For example, it determines that the A form has existential import, at least if the I form does. For one of the patterns is, Every C is B Every C is A So, some A is B This is invalid if the A form lacks existential import. It is held to be valid, and so we know how the A form is to be interpreted, one then naturally asks about the O form, what do the syllogisms tell us about it. The answer is that they tell us nothing and this is because Aristotle did not discuss weakened forms of syllogisms, in which one concludes a particular proposition when one could already conclude the corresponding universal. But the weakened forms were typically ignored, one other piece of subject-matter bears on the interpretation of the O form. People were interested in Aristotles discussion of infinite negation, which is the use of negation to form a term from a term instead of a proposition from a proposition. In modern English we use non for this, we make non-horse, in medieval Latin non and not are the same word, and so the distinction required special discussion. It became common to use infinite negation, and logicians pondered its logic, some writers in the twelfth century and thirteenth centuries adopted a principle called conversion by contraposition. For in the case it leads directly from the truth, Every man is a being to the falsehood. Unfortunately, by Buridans time the principle of contraposition had been advocated by a number of authors, a term is the basic component of the proposition. The original meaning of the horos is extreme or boundary, the two terms lie on the outside of the proposition, joined by the act of affirmation or denial
11.
John Buridan
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Jean Buridan was a French priest who sowed the seeds of the Copernican revolution in Europe. He developed the concept of impetus, the first step toward the concept of inertia. His name is most familiar through the thought experiment known as Buridans ass, born, most probably, in Béthune, France, Buridan studied and later taught at the University of Paris. Unusually, he spent his life in the faculty of arts. He further maintained his independence by remaining a secular cleric. By 1340, his confidence had grown sufficiently for him to launch an attack on his predecessor, Buridan also wrote on solutions to paradoxes such as the liar paradox. An ordinance of Louis XI of France in 1473, directed against the nominalists, the bishop Albert of Saxony, himself renowned as a logician, was among the most notable of his students. The concept of inertia was alien to the physics of Aristotle, Aristotle, and his peripatetic followers held that a body was only maintained in motion by the action of a continuous external force. Thus, in the Aristotelian view, a projectile moving through the air would owe its continuing motion to eddies or vibrations in the surrounding medium, in the absence of a proximate force, the body would come to rest almost immediately. Jean Buridan, following in the footsteps of John Philoponus and Avicenna, proposed that motion was maintained by some property of the body, Buridan named the motion-maintaining property impetus. Moreover, he rejected the view that the impetus dissipated spontaneously, asserting that a body would be arrested by the forces of air resistance and gravity which might be opposing its impetus. Buridan further held that the impetus of a body increased with the speed with which it was set in motion, clearly, Buridans impetus is closely related to the modern concept of momentum. Buridan saw impetus as causing the motion of the object, Buridan anticipated Isaac Newton when he wrote. The theory of impetus was also adapted to explain phenomena in terms of circular impetus. Apocryphal stories abound about his amorous affairs and adventures which are enough to show that he enjoyed a reputation as a glamorous and mysterious figure in Paris life. In particular, a rumour held that he was sentenced to be thrown in a sack into the River Seine, françois Villon alludes to this in his famous poem Ballade des Dames du Temps Jadis. Buridan also seems to have had a facility for attracting academic funding which suggests that he was indeed a charismatic figure. Hughes, G. E. John Buridan on Self-Reference, Chapter Eight of Buridans Sophismata, an edition and translation with an introduction, and philosophical commentary
12.
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
13.
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
14.
Immanuel Kant
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Immanuel Kant was a German philosopher who is considered a central figure in modern philosophy. Kant took himself to have effected a Copernican revolution in philosophy and his beliefs continue to have a major influence on contemporary philosophy, especially the fields of metaphysics, epistemology, ethics, political theory, and aesthetics. Politically, Kant was one of the earliest exponents of the idea that peace could be secured through universal democracy. He believed that this will be the outcome of universal history. Kant wanted to put an end to an era of futile and speculative theories of human experience, Kant argued that our experiences are structured by necessary features of our minds. In his view, the shapes and structures experience so that, on an abstract level. Among other things, Kant believed that the concepts of space and time are integral to all human experience, as are our concepts of cause, Kant published other important works on ethics, religion, law, aesthetics, astronomy, and history. These included the Critique of Practical Reason, the Metaphysics of Morals, which dealt with ethics, and the Critique of Judgment, Immanuel Kant was born in 1724 in Königsberg, Prussia. His mother, Anna Regina Reuter, was born in Königsberg to a father from Nuremberg. His father, Johann Georg Kant, was a German harness maker from Memel, Immanuel Kant believed that his paternal grandfather Hans Kant was of Scottish origin. Kant was the fourth of nine children, baptized Emanuel, he changed his name to Immanuel after learning Hebrew. Young Kant was a solid, albeit unspectacular, student and he was brought up in a Pietist household that stressed religious devotion, humility, and a literal interpretation of the Bible. His education was strict, punitive and disciplinary, and focused on Latin and religious instruction over mathematics, despite his religious upbringing and maintaining a belief in God, Kant was skeptical of religion in later life, various commentators have labelled him agnostic. Common myths about Kants personal mannerisms are listed, explained, and refuted in Goldthwaits introduction to his translation of Observations on the Feeling of the Beautiful and Sublime. It is often held that Kant lived a strict and disciplined life. He never married, but seemed to have a social life — he was a popular teacher. He had a circle of friends whom he met, among them Joseph Green. A common myth is that Kant never traveled more than 16 kilometres from Königsberg his whole life, in fact, between 1750 and 1754 he worked as a tutor in Judtschen and in Groß-Arnsdorf
15.
Logic in Islamic philosophy
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Logic plays an important role in Islamic philosophy. The works of Hellenistic-influenced Islamic philosophers were crucial in the reception of Aristotelian logic in medieval Europe, in the area of formal logical analysis, they elaborated upon the theory of terms, propositions and syllogisms as formulated in Aristotles Categories, De interpretatione and Prior Analytics. In the spirit of Aristotle, they considered the syllogism to be the form to all rational argumentation could be reduced. Even poetics was considered as an art in some fashion by most of the major Islamic Aristotelians. Important developments made by Muslim logicians included the development of Avicennian logic as a replacement of Aristotelian logic, avicennas system of logic was responsible for the introduction of hypothetical syllogism, temporal modal logic and inductive logic. On the other hand, al-Ghazali argued that Qiyas refers to analogical reasoning in a real sense, other Islamic scholars at the time, however, argued that the term Qiyas refers to both analogical reasoning and categorical syllogism in a real sense. The first original Arabic writings on logic were produced by al-Kindi and he is also credited for categorizing logic into two separate groups, the first being idea and the second being proof. Averroes was the last major logician from al-Andalus, who wrote the most elaborate commentaries on Aristotelian logic, Avicenna developed his own system of logic known as Avicennian logic as an alternative to Aristotelian logic. By the 12th century, Avicennian logic had replaced Aristotelian logic as the dominant system of logic in the Islamic world, the first criticisms of Aristotelian logic were written by Avicenna, who produced independent treatises on logic rather than commentaries. He criticized the school of Baghdad for their devotion to Aristotle at the time. He investigated the theory of definition and classification and the quantification of the predicates of categorical propositions and its premises included modifiers such as at all times, at most times, and at some time. While Avicenna often relied on deductive reasoning in philosophy, he used a different approach in medicine, ibn Sina contributed inventively to the development of inductive logic, which he used to pioneer the idea of a syndrome. In his medical writings, Avicenna was the first to describe the methods of agreement, difference and concomitant variation which are critical to inductive logic, ibn Hazm wrote the Scope of Logic, in which he stressed on the importance of sense perception as a source of knowledge. Al-Ghazali had an important influence on the use of logic in theology, fakhr al-Din al-Razi criticised Aristotles first figure and developed a form of inductive logic, foreshadowing the system of inductive logic developed by John Stuart Mill. Islamic philosophy History of logic Rescher, Nicholas 1964, studies in the History of Arabic Logic, Pittsburgh, University of Pittsburgh Press. In, Gabbay, Dov & Woods, John, Greek, Indian and Arabic Logic, Volume I of the Handbook of the History of Logic, Amsterdam, Elsevier, pp. 523–596
16.
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
17.
Bernard Bolzano
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Bernard Bolzano was a Bohemian mathematician, logician, philosopher, theologian and Catholic priest of Italian extraction, also known for his antimilitarist views. Bolzano wrote in German, his mother tongue, for the most part, his work came to prominence posthumously. Bolzano was the son of two pious Catholics and his father, Bernard Pompeius Bolzano, was an Italian who had moved to Prague, where he married Maria Cecilia Maurer who came from Pragues German-speaking family Maurer. Only two of their children lived to adulthood. Bolzano entered the University of Prague in 1796 and studied mathematics, philosophy, starting in 1800, he also began studying theology, becoming a Catholic priest in 1804. He was appointed to the newly created chair of philosophy of religion at Prague University in 1805. He proved to be a popular lecturer not just in religion but also in philosophy, Bolzano alienated many faculty and church leaders with his teachings of the social waste of militarism and the needlessness of war. He urged a total reform of the educational, social, upon his refusal to recant his beliefs, Bolzano was dismissed from the university in 1819. His political convictions eventually proved to be too liberal for the Austrian authorities and he was exiled to the countryside and at that point devoted his energies to his writings on social, religious, philosophical, and mathematical matters. Although forbidden to publish in journals as a condition of his exile, Bolzano continued to develop his ideas. In 1842 he moved back to Prague, where he died in 1848, Bolzano made several original contributions to mathematics. His overall philosophical stance was that, contrary to much of the mathematics of the era, it was better not to introduce intuitive ideas such as time. These works presented. a sample of a new way of developing analysis, to the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε–δ definition of a mathematical limit. Bolzano was the first to recognize the greatest lower bound property of the real numbers, like several others of his day, he was skeptical of the possibility of Gottfried Leibnizs infinitesimals, that had been the earliest putative foundation for differential calculus. Bolzano also gave the first purely analytic proof of the theorem of algebra. He also gave the first purely analytic proof of the intermediate value theorem, the logical theory that Bolzano developed in this work has come to be acknowledged as ground-breaking. Other works are a four-volume Lehrbuch der Religionswissenschaft and the metaphysical work Athanasia, Bolzano also did valuable work in mathematics, which remained virtually unknown until Otto Stolz rediscovered many of his lost journal articles and republished them in 1881. Bolzano begins his work by explaining what he means by theory of science, human knowledge, he states, is made of all truths that men know or have known
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Bohemia
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Bohemia is the westernmost and largest historical region of the Czech lands in the present-day Czech Republic. Bohemia was a duchy of Great Moravia, later an independent principality, a kingdom in the Holy Roman Empire, and subsequently a part of the Habsburg Monarchy, after World War I and the establishment of an independent Czechoslovak state, Bohemia became a part of Czechoslovakia. Between 1938 and 1945, border regions with sizeable German-speaking minorities of all three Czech lands were joined to Nazi Germany as the Sudetenland, in 1990, the name was changed to the Czech Republic, which become a separate state in 1993 with the dissolution of Czechoslovakia. Until 1948, Bohemia was a unit of Czechoslovakia as one of its lands. Bohemia was bordered in the south by Upper and Lower Austria, in the west by Bavaria and in the north by Saxony and Lusatia, in the northeast by Silesia, and in the east by Moravia. In the 2nd century BC, the Romans were competing for dominance in northern Italy, the Romans defeated the Boii at the Battle of Placentia and the Battle of Mutina. After this, many of the Boii retreated north across the Alps, much later Roman authors refer to the area they had once occupied as Boiohaemum. The earliest mention was by Tacitus Germania 28, and later mentions of the name are in Strabo. The name appears to include the tribal name Boi- plus the Germanic element *haimaz home and this Boiohaemum was apparently isolated to the area where King Marobods kingdom was centred, within the Hercynian forest. The Czech name Čechy is derived from the name of the Slavic ethnic group, the Czechs, Bohemia, like neighbouring Bavaria, is named after the Boii, who were a large Celtic nation known to the Romans for their migrations and settlement in northern Italy and other places. Another part of the nation moved west with the Helvetii into southern France, to the south, over the Danube, the Romans extended their empire, and to the southeast in Hungaria, were Sarmatian peoples. In the area of modern Bohemia the Marcomanni and other Suebic groups were led by their king Marobodus and he took advantage of the natural defenses provided by its mountains and forests. In late classical times and the early Middle Ages, two new Suebic groupings appeared to the west of Bohemia in southern Germany, the Alemanni, many Suebic tribes from the Bohemian region took part in such movements westwards, even settling as far away as Spain and Portugal. With them were also tribes who had pushed from the east, such as the Vandals, other groups pushed southwards towards Pannonia. These are precursors of todays Czechs, though the amount of Slavic immigration is a subject of debate. The Slavic influx was divided into two or three waves, the first wave came from the southeast and east, when the Germanic Lombards left Bohemia. Soon after, from the 630s to 660s, the territory was taken by Samos tribal confederation and his death marked the end of the old Slavonic confederation, the second attempt to establish such a Slavonic union after Carantania in Carinthia. Other sources divide the population of Bohemia at this time into the Merehani, Marharaii, Beheimare, Christianity first appeared in the early 9th century, but only became dominant much later, in the 10th or 11th century
19.
Sentential logic
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Logical connectives are found in natural languages. In English for example, some examples are and, or, not”, the following is an example of a very simple inference within the scope of propositional logic, Premise 1, If its raining then its cloudy. Both premises and the conclusion are propositions, the premises are taken for granted and then with the application of modus ponens the conclusion follows. Not only that, but they will also correspond with any other inference of this form, Propositional logic may be studied through a formal system in which formulas of a formal language may be interpreted to represent propositions. A system of rules and axioms allows certain formulas to be derived. These derived formulas are called theorems and may be interpreted to be true propositions, a constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the represented by the theorem. When a formal system is used to represent formal logic, only statement letters are represented directly, usually in truth-functional propositional logic, formulas are interpreted as having either a truth value of true or a truth value of false. Truth-functional propositional logic and systems isomorphic to it, are considered to be zeroth-order logic, although propositional logic had been hinted by earlier philosophers, it was developed into a formal logic by Chrysippus in the 3rd century BC and expanded by his successor Stoics. The logic was focused on propositions and this advancement was different from the traditional syllogistic logic which was focused on terms. However, later in antiquity, the propositional logic developed by the Stoics was no longer understood, consequently, the system was essentially reinvented by Peter Abelard in the 12th century. Propositional logic was eventually refined using symbolic logic, the 17th/18th-century mathematician Gottfried Leibniz has been credited with being the founder of symbolic logic for his work with the calculus ratiocinator. Although his work was the first of its kind, it was unknown to the larger logical community, consequently, many of the advances achieved by Leibniz were recreated by logicians like George Boole and Augustus De Morgan completely independent of Leibniz. Just as propositional logic can be considered an advancement from the earlier syllogistic logic, one author describes predicate logic as combining the distinctive features of syllogistic logic and propositional logic. Consequently, predicate logic ushered in a new era in history, however, advances in propositional logic were still made after Frege, including Natural Deduction. Natural deduction was invented by Gerhard Gentzen and Jan Łukasiewicz, Truth-Trees were invented by Evert Willem Beth. The invention of truth-tables, however, is of controversial attribution, within works by Frege and Bertrand Russell, are ideas influential to the invention of truth tables. The actual tabular structure, itself, is credited to either Ludwig Wittgenstein or Emil Post
20.
Predicate logic
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First-order logic – also known as first-order predicate calculus and predicate logic – is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. This distinguishes it from propositional logic, which does not use quantifiers, Sometimes theory is understood in a more formal sense, which is just a set of sentences in first-order logic. In first-order theories, predicates are associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets, There are many deductive systems for first-order logic which are both sound and complete. Although the logical relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem, first-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Peano arithmetic and Zermelo–Fraenkel set theory are axiomatizations of number theory and set theory, respectively, no first-order theory, however, has the strength to uniquely describe a structure with an infinite domain, such as the natural numbers or the real line. Axioms systems that do fully describe these two structures can be obtained in stronger logics such as second-order logic, for a history of first-order logic and how it came to dominate formal logic, see José Ferreirós. While propositional logic deals with simple declarative propositions, first-order logic additionally covers predicates, a predicate takes an entity or entities in the domain of discourse as input and outputs either True or False. Consider the two sentences Socrates is a philosopher and Plato is a philosopher, in propositional logic, these sentences are viewed as being unrelated and might be denoted, for example, by variables such as p and q. The predicate is a philosopher occurs in both sentences, which have a structure of a is a philosopher. The variable a is instantiated as Socrates in the first sentence and is instantiated as Plato in the second sentence, while first-order logic allows for the use of predicates, such as is a philosopher in this example, propositional logic does not. Relationships between predicates can be stated using logical connectives, consider, for example, the first-order formula if a is a philosopher, then a is a scholar. This formula is a statement with a is a philosopher as its hypothesis. The truth of this depends on which object is denoted by a. Quantifiers can be applied to variables in a formula, the variable a in the previous formula can be universally quantified, for instance, with the first-order sentence For every a, if a is a philosopher, then a is a scholar. The universal quantifier for every in this sentence expresses the idea that the if a is a philosopher. The negation of the sentence For every a, if a is a philosopher, then a is a scholar is logically equivalent to the sentence There exists a such that a is a philosopher and a is not a scholar
21.
Congregation for the Doctrine of the Faith
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The Congregation for the Doctrine of the Faith is the oldest among the nine congregations of the Roman Curia. It was founded to defend the church from heresy, today and its headquarters are at the Palace of the Holy Office, just outside Vatican City. The congregation employs a board including cardinals, bishops, priests, lay theologians. The current Prefect is Cardinal Gerhard Ludwig Müller and it served as the final court of appeal in trials of heresy and served as an important part of the Counter-Reformation. This body was renamed the Supreme Sacred Congregation of the Holy Office in 1908 by Pope Saint Pius X, in many Catholic countries, the body is often informally called the Holy Office. The Congregations name was changed to Sacred Congregation for the Doctrine of the Faith on 7 December 1965 and this includes investigations into grave delicts, i. e. These crimes, in a motu proprio of 2001, Sacramentorum sanctitatis tutela, in effect, it is the promoter of justice who deals with, among other things, the question of priests accused of paedophilia. Within the CDF are the International Theological Commission, the Pontifical Biblical Commission, the Prefect of the CDF is ex officio president of these commissions. Until 1968, the Pope himself held the title of prefect, instead, he appointed one of the cardinals to preside over the meetings, first as Secretary, then as Pro-Prefect. Since 1968, the Cardinal head of the dicastery has borne the title of Prefect, therefore, from 1968 onwards, the title of Secretary refers to the second highest-ranking officer of the Congregation. The Congregation has a membership of some 18 other cardinals and a number of non-cardinal bishops, a staff of some 38 priests, religious and lay men and women. The work of the CDF is divided into four sections, the doctrinal, disciplinary, matrimonial, staff, Prefect, Cardinal Gerhard Ludwig Müller Secretary, Archbishop Luis Ladaria Ferrer, S. J. Assistant Secretary, Archbishop Joseph Augustine Di Noia, O. P and they refused to recant the doctrines of the Community of the Lady of All Nations. The nuns are members of the Good Shepherd Monastery of Our Lady of Charity, sister Mary Theresa Dionne,82, one of 6, said they will still live at the convent property, which they own. The sect believes that its 86-year-old founder, Marie Paule Giguere, is the reincarnation of the Virgin Mary, notification on the works of the Reverend Father Jon Sobrino, S. J. Notification regarding the book Jesus Symbol of God of the Reverend Father Roger Haight, notification on the book Toward a Christian Theology of Religious Pluralism by the Reverend Father Jacques Dupuis, S. J. Notification concerning some writings of Professor Dr. J, notification concerning the Text Mary and Human Liberation by the Reverend Father Tissa Balasuriya, O. M. I. Notification on the writings and activities of Mrs. M. I. P, essay on militant Ecclesiology by Leonardo Boff, O. F. M
22.
Roman Rota
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An appeal may be had to the Pope himself, who is the supreme ecclesiastical judge. The Catholic Church has a legal system, which is the oldest in the West still in use today. The court is named Rota because the judges, called auditors, the Rota was established in the 13th century. The Pope appoints the auditors of the Rota and designates one of them the dean, on Saturday, September 22,2012, Pope Benedict XVI accepted the resignation as Dean, for reasons of age, of Bishop Antoni Stankiewicz, and appointed in his place Msgr. Pio Vito Pinto, until then serving as a prelate Auditor of the Court of first instance, the Rota issues its decrees and sentences in Latin. The auditors of the Rota are selected from among recognized ecclesiastical judges serving various Dioceses around the world, the Rotas official records begin in 1171. Until the Risorgimento and the loss of the Papal States in 1870, until the 14th century the court was formally known as the Apostolic Court of Audience. Its first usage in a bull is in 1418. It is also possible that the term Rota comes from the wheel that was centered in the marble floor of Avignon. The Rota serves as a tribunal of first instance in cases such as any contentious case in which a Bishop of the Latin Church is a defendant. If the case can still be appealed after a Rotal decision, the Rota is the highest appeals court, or Supreme court, for all judicial trials in the Catholic Church. The Roman Rota proceedings are governed by a set of rules. Only advocates who are registered in a specific list are allowed to represent the parties before the Tribunal, since Pope Benedict XVI issued the motu proprio Quaerit semper the Rota has had exclusive competence to dispense from marriages ratum sed non consummatum. The Dean of the Rota, even if not already consecrated a Bishop, is to be addressed as Your Excellency, all Prelate Auditor Judges of the Rota are styled, Most Reverend Monsignor. The active auditors of the Rota, with their dates of appointment by the Pope, are, domenico Teti, Dean of the Roman Rota Tribunal of the Roman Rota Pontifical Council for Legislative Texts GCatholic. org Herbermann, Charles, ed. Sacra Romana Rota
23.
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
24.
John Corcoran (logician)
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John Corcoran is a logician, philosopher, mathematician, and historian of logic. Nine of Corcoran’s papers have been translated into Spanish, Portuguese, Persian, fourteen of his papers have been reprinted, one was reprinted twice. His work on Aristotle’s logic of the Prior Analytics is regarded as being highly faithful both to the Greek text and to the historical context and it is the basis for many subsequent investigations. It was adopted for the 1989 translation of the Prior Analytics by Robin Smith, a bibliography of Corcoran’s publications on Aristotles logic is available at ResearchGate. A current list of all of Corcoran’s publications is available at ResearchGate, Corcoran studied engineering at the Baltimore Polytechnic Institute, Advanced Curriculum Engineering 1956, and the Johns Hopkins University, BES Mechanical Engineering 1959. After briefly working in engineering, he studied philosophy at the Johns Hopkins University, MA Philosophy 1962, post-doctoral study, Yeshiva University, Mathematics 1964 and University of California Berkeley, Mathematics 1965. Dissertation, Generative Structure of Two-valued Logics, Supervisor Robert McNaughton, Corcoran’s student years, the late 1950s and early 1960s, were wonderful times to be learning logic, its history, and its philosophy. Corcoran studied Plato and Aristotle with Ludwig Edelstein, the historian of Greek science and his next two logic teachers were both accomplished and knowledgeable symbolic logicians, Joseph Ullian, a Quine PhD, and Richard Wiebe, a Mates PhD who had studied with Carnap and Tarski. McNaughton encouraged Corcoran to do studies at Yeshiva University in New York City with Raymond Smullyan and Martin Davis. McNaughton later encouraged Corcoran to go to UC Berkeley, the center for logic and methodology. He was also instrumental in Corcoran’s move to his first tenure-track position, at the University of Pennsylvania, Corcoran often mentions his teachers with great respect and warmth. Corcoran’s work in history of logic involves most of the discipline’s productive periods and it was adopted for the 1989 translation of the Prior Analytics by Robin Smith and for the 2009 translation of the Prior Analytics Book A by Gisela Striker. A2003 article provides a comparison and critical evaluation of Aristotelian logic and Boolean logic. According to Corcoran, Boole fully accepted and endorsed Aristotle’s logic. g, from propositions having only two terms to those having arbitrarily many. More specifically, Boole agreed with what Aristotle said, Boole’s ‘disagreements’, if they might be called that, first, in the realm of foundations, Boole reduced Aristotle’s four propositional forms of to one form, that of equations—-by itself a revolutionary idea. Third, in the realm of applications, Boole’s system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions, although several of his philosophical papers presuppose little history or mathematics, his historical papers often involve either original philosophy or original mathematics. He has referred to the dimension of his approach to history as mathematical archaeology. His philosophical papers often involve original historical research, many of Corcoran’s articles and reviews are co-authored and many of his single-author publications acknowledge involvement of colleagues and students
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Square of opposition
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The square of opposition is a diagram representing the relations between four categorical propositions. The origin of the square can be traced back to Aristotle making the distinction between two oppositions, contradiction and contrariety, but Aristotle did not draw any diagram. This was done several centuries laters by Apuleius and Boethius, in traditional logic, a proposition is a spoken assertion, not the meaning of an assertion, as in modern philosophy of language and logic. A categorical proposition is a simple proposition containing two terms, subject and predicate, in which the predicate is either asserted or denied of the subject, every categorical proposition can be reduced to one of four logical forms. These are, The so-called A proposition, the universal affirmative, whose form in Latin is omne S est P, the E proposition, the universal negative, Latin form nullum S est P, usually translated as no S are P. The I proposition, the particular affirmative, Latin quoddam S est P, the O proposition, the particular negative, Latin quoddam S non est P, usually translated as some S are not P. Aristotle states, that there are certain logical relationships between four kinds of proposition. He says that to every affirmation there corresponds exactly one negation, and that every affirmation and its negation are opposed such that one of them must be true. A pair of affirmative and negative statements he calls a contradiction, examples of contradictories are every man is white and not every man is white, no man is white and some man is white. Contrary statements, are such that both cannot at the time be true. Examples of these are the universal affirmative every man is white, and these cannot be true at the same time. However, these are not contradictories because both of them may be false, for example, it is false that every man is white, since some men are not white. Yet it is false that no man is white, since there are some white men. Since subcontraries are negations of universal statements, they were called particular statements by the medieval logicians, another logical opposition implied by this, though not mentioned explicitly by Aristotle, is alternation, consisting of subalternation and superalternation. Alternation is a relation between a statement and a universal statement of the same quality such that the particular is implied by the other. The particular is the subaltern of the universal, which is the particulars superaltern, for example, if every man is white is true, its contrary no man is white is false. Therefore the contradictory some man is white is true, similarly the universal no man is white implies the particular not every man is white. In summary, Universal statements are contraries, every man is just and no man is just cannot be together, although one may be true and the other false
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Alpha
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Alpha is the first letter of the Greek alphabet. In the system of Greek numerals it has a value of 1 and it was derived from the Phoenician and Hebrew letter aleph - an ox or leader. Letters that arose from alpha include the Latin A and the Cyrillic letter А, in English, the noun alpha is used as a synonym for beginning, or first, reflecting its Greek roots. In Ancient Greek, alpha was pronounced and could be either long or short. Where there is ambiguity, long and short alpha are sometimes written with a macron and breve today, Ᾱᾱ, ὥρα = ὥρᾱ hōrā a time γλῶσσα = γλῶσσᾰ glôssa tongue In Modern Greek, vowel length has been lost, and all instances of alpha simply represent. It can also combine with the iota subscript, in the Attic-Ionic dialect of Ancient Greek, long alpha fronted to. In Ionic, the took place in all positions. In Attic, the shift did not take place after epsilon, iota, in Doric and Aeolic, long alpha is preserved in all positions. It originates from the Proto-Indo-European *n̥- and is cognate with English un-, copulative a is the Greek prefix ἁ- or ἀ- ha-, a-. The letter alpha represents various concepts in physics and chemistry, including radiation, angular acceleration, alpha particles, alpha carbon. Alpha also stands for thermal expansion coefficient of a compound in physical chemistry and it is also commonly used in mathematics in algebraic solutions representing quantities such as angles. Furthermore, in mathematics, the alpha is used to denote the area underneath a normal curve in statistics to denote significance level when proving null. In zoology, it is used to name the dominant individual in a wolf or dog pack, the proportionality operator ∝ is sometimes mistaken for alpha. The uppercase letter alpha is not generally used as a symbol because it tends to be rendered identically to the uppercase Latin A, in the International Phonetic Alphabet, a letter based on the lower case of alpha represents the open back unrounded vowel. Alpha was derived from aleph, which in Phoenician means ox, plutarch, in Moralia, presents a discussion on why the letter alpha stands first in the alphabet. He then added that he would rather be assisted by Lamprias, his own grandfather, than by Dionysus grandfather, according to Plutarchs natural order of attribution of the vowels to the planets, alpha was connected with the Moon. Alpha, both as a symbol and term, is used to refer to or describe a variety of things, the New Testament has God declaring himself to be the Alpha and Omega, the beginning and the end, the first and the last. The term alpha has been used to position in social hierarchy
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Gamma
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Gamma is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3, in Ancient Greek, the letter gamma represented a voiced velar stop /ɡ/. In Modern Greek, this represents either a voiced velar fricative or a voiced palatal fricative. In the International Phonetic Alphabet and other modern Latin-alphabet based phonetic notations, the Greek letter Gamma Γ was derived from the Phoenician letter for the /g/ phoneme, and as such is cognate with Hebrew gimel ג. In Archaic Greece, the shape of gamma was closer to a classical lambda, letters that arose from the Greek gamma include Etruscan
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Infix notation
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Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands – infixed operators – such as the sign in 2 +2. Infix notation is more difficult to parse by computers than prefix notation or postfix notation, however many programming languages use it due to its familiarity. It is more used in arithmetic, e. g. 2+2, in the absence of parentheses, certain precedence rules determine the order of operations. Infix notation may also be distinguished from function notation, where the name of a function suggests a particular operation, an example of such a function notation would be S in which the function S denotes addition, S = 1+3 =4. Tree traversal, Infix is also a tree traversal order and it is described in a more detailed manner on this page. A brief analysis of Reverse Polish Notation against Direct Algebraic Logic Infix to postfix convertor
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Scholasticism
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It originated as an outgrowth of and a departure from Christian monastic schools at the earliest European universities. The Scholastic thought is known for rigorous conceptual analysis and the careful drawing of distinctions. Because of its emphasis on rigorous dialectical method, scholasticism was eventually applied to other fields of study. Some of the figures of scholasticism include Anselm of Canterbury, Peter Abelard, Alexander of Hales, Albertus Magnus, Duns Scotus, William of Ockham, Bonaventure. Important work in the tradition has been carried on well past Aquinass time, for instance by Francisco Suárez and Luis de Molina. The terms scholastic and scholasticism derive from the Latin word scholasticus and the latter from the Greek σχολαστικός, forerunners of Christian scholasticism were Islamic Ilm al-Kalām, literally science of discourse, and Jewish philosophy, especially Jewish Kalam. The first significant renewal of learning in the West came with the Carolingian Renaissance of the Early Middle Ages, charlemagne, advised by Peter of Pisa and Alcuin of York, attracted the scholars of England and Ireland. By decree in AD787, he established schools in every abbey in his empire and these schools, from which the name scholasticism is derived, became centers of medieval learning. During this period, knowledge of Ancient Greek had vanished in the west except in Ireland, Irish scholars had a considerable presence in the Frankish court, where they were renowned for their learning. Among them was Johannes Scotus Eriugena, one of the founders of scholasticism, Eriugena was the most significant Irish intellectual of the early monastic period and an outstanding philosopher in terms of originality. He had considerable familiarity with the Greek language and translated works into Latin, affording access to the Cappadocian Fathers. The other three founders of scholasticism were the 11th-century scholars Peter Abelard, Archbishop Lanfranc of Canterbury and Archbishop Anselm of Canterbury and this period saw the beginning of the rediscovery of many Greek works which had been lost to the Latin West. As early as the 10th century, scholars in Spain had begun to gather translated texts and, in the half of that century. After a successful burst of Reconquista in the 12th century, Spain opened even further for Christian scholars, as these Europeans encountered Islamic philosophy, they opened a wealth of Arab knowledge of mathematics and astronomy. Scholars such as Adelard of Bath traveled to Spain and Sicily, translating works on astronomy and mathematics, at the same time, Anselm of Laon systematized the production of the gloss on Scripture, followed by the rise to prominence of dialectic in the work of Abelard. Peter Lombard produced a collection of Sentences, or opinions of the Church Fathers and other authorities The 13th, the early 13th century witnessed the culmination of the recovery of Greek philosophy. Schools of translation grew up in Italy and Sicily, and eventually in the rest of Europe, powerful Norman kings gathered men of knowledge from Italy and other areas into their courts as a sign of their prestige. His work formed the basis of the commentaries that followed
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Mnemonic
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A mnemonic device, or memory device is any learning technique that aids information retention in the human memory. Mnemonics make use of encoding, retrieval cues, and imagery as specific tools to encode any given information in a way that allows for efficient storage. Mnemonics aid original information in becoming associated with something more meaningful—which, in turn, the word mnemonic is derived from the Ancient Greek word μνημονικός, meaning of memory, or relating to memory and is related to Mnemosyne, the name of the goddess of memory in Greek mythology. Both of these words are derived from μνήμη, remembrance, memory, mnemonics in antiquity were most often considered in the context of what is today known as the art of memory. Ancient Greeks and Romans distinguished between two types of memory, the memory and the artificial memory. The former is inborn, and is the one that everyone uses instinctively, the latter in contrast has to be trained and developed through the learning and practice of a variety of mnemonic techniques. Mnemonic systems are techniques or strategies consciously used to improve memory and they help use information already stored in long-term memory to make memorisation an easier task. Mnemonic devices were much cultivated by Greek sophists and philosophers and are referred to by Plato. In later times the poet Simonides was credited for development of these techniques, the Romans valued such helps in order to support facility in public speaking. The Greek and the Roman system of mnemonics was founded on the use of mental places and signs or pictures, to recall these, an individual had only to search over the apartments of the house until discovering the places where images had been placed by the imagination. Except that the rules of mnemonics are referred to by Martianus Capella, among the voluminous writings of Roger Bacon is a tractate De arte memorativa. Ramon Llull devoted special attention to mnemonics in connection with his ars generalis, about the end of the 15th century, Petrus de Ravenna provoked such astonishment in Italy by his mnemonic feats that he was believed by many to be a necromancer. His Phoenix artis memoriae went through as many as nine editions, about the end of the 16th century, Lambert Schenkel, who taught mnemonics in France, Italy and Germany, similarly surprised people with his memory. He was denounced as a sorcerer by the University of Louvain, the most complete account of his system is given in two works by his pupil Martin Sommer, published in Venice in 1619. In 1618 John Willis published Mnemonica, sive ars reminiscendi, containing a statement of the principles of topical or local mnemonics. Giordano Bruno included a memoria technica in his treatise De umbris idearum, other writers of this period are the Florentine Publicius, Johannes Romberch, Hieronimo Morafiot, Ars memoriae, and B. The philosopher Gottfried Wilhelm Leibniz adopted a very similar to that of Wennsshein for his scheme of a form of writing common to all languages. Wennssheins method was adopted with slight changes afterward by the majority of subsequent original systems and it was modified and supplemented by Richard Grey, a priest who published a Memoria technica in 1730
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Syllogism
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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
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Therefore sign
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In mathematical proof, the therefore sign is generally used before a logical consequence, such as the conclusion of a syllogism. The symbol consists of three dots placed in a triangle and is read therefore. It is encoded at U+2234 ∴ therefore, for common use in Microsoft Office hold the ALT key and type 8756. While it is not generally used in writing, it is used in mathematics. It is complementary to U+2235 ∵ because, in this century, the three-dot notation for therefore became very rare in continental Europe, but it remains popular in the British Isles. Used in a syllogism, All gods are immortal, X +1 =6 ∴ x =5 The inverted form ∵, known as the because sign, is sometimes used as a shorthand form of because. The therefore sign is used as a substitute for an asterism ⁂. In meteorology, the sign is used to indicate moderate rain on a station model. To denote logical implication or entailment, various signs are used in mathematical logic and these symbols are then part of a mathematical formula, and are not considered to be punctuation. In contrast, the sign is traditionally used as a punctuation mark. The graphically identical sign ∴ serves as a Japanese map symbol on the maps of the Geographical Survey Institute of Japan, on other maps the sign, often with thicker dots, is sometimes used to signal the presence of a national monument or ruins. The character ஃ in the Tamil script represents the āytam, a sound of the Tamil language. In Masonic traditions the symbol is used for abbreviation, instead of the usual period, for example R∴W∴ John Smith is an abbreviation for Right Worshipful John Smith