1.
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

2.
Propositional calculus
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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

3.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker

4.
Terence Tao
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Terence Terry Chi-Shen Tao FAA FRS, is an Australian-American mathematician who has worked in various areas of mathematics. He currently focuses on analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, compressed sensing. As of 2015, he holds the James and Carol Collins chair in mathematics at the University of California, Tao was a co-recipient of the 2006 Fields Medal and the 2014 Breakthrough Prize in Mathematics. Taos father, Dr. Billy Tao, was a pediatrician who was born in Shanghai, Taos mother, Grace, is from Hong Kong, she received a first-class honours degree in physics and mathematics at the University of Hong Kong. She was a school teacher of mathematics and physics in Hong Kong. Billy and Grace met as students at the University of Hong Kong and they then emigrated from Hong Kong to Australia. Tao has two living in Australia, both of whom represented Australia at the International Mathematical Olympiad. Nigel Tao was part of the team at Google Australia that created Google Wave and he now works on the Go programming language. Trevor Tao is an International Master in Chess and he has a double degree in mathematics and music and is an autistic savant. Taos wife, Laura, is an engineer at NASAs Jet Propulsion Laboratory and they live with their son and daughter in Los Angeles, California. Tao exhibited extraordinary mathematical abilities from an age, attending university level mathematics courses at the age of 9. In 1986,1987, and 1988, Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten, winning a bronze, silver, and gold medal. He remains the youngest winner of each of the three medals in the Olympiads history, winning the gold medal shortly after his thirteenth birthday, at age 14, Tao attended the Research Science Institute. When he was 15 he published his first assistant paper and he received his bachelors and masters degrees at the age of 16 from Flinders University under Garth Gaudry. In 1992 he won a Fulbright Scholarship to undertake study in the United States. From 1992 to 1996, Tao was a student at Princeton University under the direction of Elias Stein. He joined the faculty of the University of California, Los Angeles in 1996, when he was 24, he was promoted to full professor at UCLA and remains the youngest person ever appointed to that rank by the institution. Within the field of mathematics, Tao is known for his collaboration with Ben J. Green of Oxford University, known for his collaborative mindset, by 2006 Tao had worked with over 30 others in his discoveries, reaching 68 co-authors by October 2015

5.
Logic
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Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times

6.
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

7.
Metamathematics
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Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories, emphasis on metamathematics owes itself to David Hilberts attempt to secure the foundations of mathematics in the early part of the 20th Century. Metamathematics provides a mathematical technique for investigating a great variety of foundation problems for mathematics. An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system, an informal illustration of this is categorizing the proposition 2+2=4 as belonging to mathematics while categorizing the proposition 2+2=4 is valid as belonging to metamathematics. Something similar can be said around the well-known Russells paradox, Metamathematics was intimately connected to mathematical logic, so that the early histories of the two fields, during the late 19th and early 20th centuries, largely overlap. More recently, mathematical logic has often included the study of new pure mathematics, such as set theory, recursion theory and pure model theory, serious metamathematical reflection began with the work of Gottlob Frege, especially his Begriffsschrift. David Hilbert was the first to invoke the term metamathematics with regularity, in his hands, it meant something akin to contemporary proof theory, in which finitary methods are used to study various axiomatized mathematical theorems. Today, metalogic and metamathematics are largely synonymous with each other, the discovery of hyperbolic geometry had important philosophical consequences for Metamathematics. Before its discovery there was just one geometry and mathematics, the idea that another geometry existed was considered improbable, the uproar of the Boeotians came and went, and gave an impetus to metamathematics and great improvements in mathematical rigour, analytical philosophy and logic. Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, Begriffsschrift is usually translated as concept writing or concept notation, the full title of the book identifies it as a formula language, modeled on that of arithmetic, of pure thought. 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. As such, this project is of great importance in the history of mathematics and philosophy. One of the inspirations and motivations for PM was the earlier work of Gottlob Frege on logic. PM sought to avoid this problem by ruling out the creation of arbitrary sets. This was achieved by replacing the notion of a set with notion of a hierarchy of sets of different types. Contemporary mathematics, however, avoids paradoxes such as Russells in less unwieldy ways, gödels completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. It makes a link between model theory that deals with what is true in different models, and proof theory that studies what can be formally proven in particular formal systems. More formally, the theorem says that if a formula is logically valid then there is a finite deduction of the formula

8.
Set theory
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Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics, the language of set theory can be used in the definitions of nearly all mathematical objects. The modern study of set theory was initiated by Georg Cantor, Set theory is commonly employed as a foundational system for mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals. Mathematical topics typically emerge and evolve through interactions among many researchers, Set theory, however, was founded by a single paper in 1874 by Georg Cantor, On a Property of the Collection of All Real Algebraic Numbers. Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, especially notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1867–71, with Cantors work on number theory, an 1872 meeting between Cantor and Richard Dedekind influenced Cantors thinking and culminated in Cantors 1874 paper. Cantors work initially polarized the mathematicians of his day, while Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. This utility of set theory led to the article Mengenlehre contributed in 1898 by Arthur Schoenflies to Kleins encyclopedia, in 1899 Cantor had himself posed the question What is the cardinal number of the set of all sets. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics, in 1906 English readers gained the book Theory of Sets of Points by William Henry Young and his wife Grace Chisholm Young, published by Cambridge University Press. The momentum of set theory was such that debate on the paradoxes did not lead to its abandonment, the work of Zermelo in 1908 and Abraham Fraenkel in 1922 resulted in the set of axioms ZFC, which became the most commonly used set of axioms for set theory. The work of such as Henri Lebesgue demonstrated the great mathematical utility of set theory. Set theory is used as a foundational system, although in some areas category theory is thought to be a preferred foundation. Set theory begins with a binary relation between an object o and a set A. If o is a member of A, the notation o ∈ A is used, since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the relation, also called set inclusion. If all the members of set A are also members of set B, then A is a subset of B, for example, is a subset of, and so is but is not. As insinuated from this definition, a set is a subset of itself, for cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined

9.
Definition
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A definition is a statement of the meaning of a term. Definitions can be classified into two categories, intensional definitions and extensional definitions. Another important category of definitions is the class of ostensive definitions, a term may have many different senses and multiple meanings, and thus require multiple definitions. In mathematics, a definition is used to give a meaning to a new term. Definitions and axioms are the basis on all of mathematics is constructed. In modern usage, a definition is something, typically expressed in words, the word or group of words that is to be defined is called the definiendum, and the word, group of words, or action that defines it is called the definiens. In the definition An elephant is a large gray animal native to Asia and Africa, the elephant is the definiendum. Note that the definiens is not the meaning of the word defined, there are many sub-types of definitions, often specific to a given field of knowledge or study. An intensional definition, also called a connotative definition, specifies the necessary, any definition that attempts to set out the essence of something, such as that by genus and differentia, is an intensional definition. An extensional definition, also called a denotative definition, of a concept or term specifies its extension and it is a list naming every object that is a member of a specific set. An extensional definition would be the list of wrath, greed, sloth, pride, lust, envy, a genus–differentia definition is a type of intensional definition that takes a large category and narrows it down to a smaller category by a distinguishing characteristic. The differentia, The portion of the new definition that is not provided by the genus, for example, consider the following genus-differentia definitions, a triangle, A plane figure that has three straight bounding sides. A quadrilateral, A plane figure that has four straight bounding sides and those definitions can be expressed as a genus and two differentiae. It is possible to have two different genus-differentia definitions that describe the same term, especially when the term describes the overlap of two large categories, for instance, both of these genus-differentia definitions of square are equally acceptable, a square, a rectangle that is a rhombus. A square, a rhombus that is a rectangle, thus, a square is a member of both the genus rectangle and the genus rhombus. One important form of the definition is ostensive definition. This gives the meaning of a term by pointing, in the case of an individual, to the thing itself, or in the case of a class, to examples of the right kind. So one can explain who Alice is by pointing her out to another, or what a rabbit is by pointing at several, the process of ostensive definition itself was critically appraised by Ludwig Wittgenstein

10.
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

11.
Logical truth
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Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants and it is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, Logical truths are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and it must be true in every sense of intuition, practices, and bodies of beliefs. However, it is not universally agreed that there are any statements which are necessarily true, a logical truth is considered by some philosophers to be a statement which is true in all possible worlds. This is contrasted with facts which are true in this world, as it has historically unfolded, later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations. Empiricists commonly respond to this objection by arguing that logical truths, are analytic, Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a class of analytic statements. The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate, can be turned into No unmarried man is married. By substituting unmarried man for its synonym bachelor, in his essay, Two Dogmas of Empiricism, the philosopher W. V. O. Quine called into question the distinction between analytic and synthetic statements, in his conclusion, Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, considering different interpretations of the same statement leads to the notion of truth value. The simplest approach to truth values means that the statement may be true in one case, in one sense of the term tautology, it is any type of formula or proposition which turns out to be true under any possible interpretation of its terms. This is synonymous to logical truth, however, the term tautology is also commonly used to refer to what could more specifically be called truth-functional tautologies. Not all logical truths are tautologies of such a kind, Logical constants, including logical connectives and quantifiers, can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible if, and only if their conjunction is logically false, one statement logically implies another when it is logically incompatible with the negation of the other. A statement is true if, and only if its opposite is logically false. The opposite statements must contradict one another, in this way all logical connectives can be expressed in terms of preserving logical truth

12.
Name
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A name is a term used for identification. Names can identify a class or category of things, or a thing, either uniquely. A personal name identifies, not necessarily uniquely, an individual human. The name of an entity is sometimes called a proper name and is, when consisting of only one word. Other nouns are sometimes called names or general names. A name can be given to a person, place, or thing, for example, caution must be exercised when translating, for there are ways that one language may prefer one type of name over another. Also, claims to preference or authority can be refuted, the British did not refer to Louis-Napoleon as Napoleon III during his rule. The word name comes from Old English nama, cognate with Old High German namo, Sanskrit नामन्, Latin nomen, Greek ὄνομα, perhaps connected to non-Indo-European terms such as Tamil namam and Proto-Uralic *nime. In the ancient world, particularly in the ancient near-east names were thought to be powerful and to act, in some ways. By invoking a god or spirit by name, one was thought to be able to summon that spirits power for some kind of miracle or magic, in the Old Testament, the names of individuals are meaningful, and a change of name indicates a change of status. For example, the patriarch Abram and his wife Sarai are renamed Abraham, simon was renamed Peter when he was given the Keys to Heaven. This is recounted in the Gospel of Matthew chapter 16, which according to Roman Catholic teaching was when Jesus promised to Saint Peter the power to take binding actions. Throughout the Bible, characters are given names at birth that reflect something of significance or describe the course of their lives, for example, Solomon meant peace, and the king with that name was the first whose reign was without war. Likewise, Joseph named his firstborn son Manasseh, when Joseph also said, “God has made me all my troubles. However, they were known as the child of their father. For example, דוד בן ישי meaning, David, son of Jesse, the Talmud also states that all those who descend to Gehenna will rise in the time of Messiah. However, there are three exceptions, one of which is he who calls another by a derisive nickname, Street names within a city may follow a naming convention, some examples include, In Manhattan, roads that cross the island from east to west are called Streets. Those that run the length of the island are called Avenues, in Ontario, numbered concession roads are east–west whereas lines are north–south routes

13.
Necessity and sufficiency
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In logic, necessity and sufficiency are implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the statement is true if. That is, the two statements must be simultaneously true or simultaneously false. In ordinary English, necessary and sufficient indicate relations between conditions or states of affairs, not statements, being a male sibling is a necessary and sufficient condition for being a brother. Freds being a male sibling is necessary and sufficient for the truth of the statement that Fred is a brother, in the conditional statement, if S then N, the expression represented by S is called the antecedent and the expression represented by N is called the consequent. This conditional statement may be written in many equivalent ways, for instance, in the above situation, we also say that N is a necessary condition for S. In common language this is saying that if the statement is a true statement. Phrased differently, the antecedent S can not be true without N being true, for example, in order for someone to be called Socrates, it is necessary for that someone to be Named. We also say that S is a sufficient condition for N, if the conditional statement is true, then if S is true, N must be true. In common terms, S guarantees N, continuing the example, knowing that someone is called Socrates is sufficient to know that that someone has a Name. A necessary and sufficient condition requires that both of the implications S ⇒ N and N ⇒ S hold, from the first of these we see that S is a sufficient condition for N, and from the second that S is a necessary condition for N. This is expressed as S is necessary and sufficient for N, S if and only if N, the assertion that Q is necessary for P is colloquially equivalent to P cannot be true unless Q is true or if Q is false, then P is false. By contraposition, this is the thing as whenever P is true. The logical relation between them is expressed as if P, then Q and denoted P ⇒ Q and it may also be expressed as any of P only if Q, Q, if P, Q whenever P, and Q when P. One often finds, in prose for instance, several necessary conditions that, taken together, constitute a sufficient condition. Example 2 For the whole numbers greater than two, being odd is necessary to being prime, since two is the whole number that is both even and prime. Example 3 Consider thunder, the caused by lightning. We say that thunder is necessary for lightning, since lightning never occurs without thunder, the thunder does not cause the lightning, but because lightning always comes with thunder, we say that thunder is necessary for lightning

14.
Paradox
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A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time, some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in definitions assumed to be rigorous, others, such as Currys paradox, are not yet resolved. Examples outside logic include the Ship of Theseus from philosophy, paradoxes can also take the form of images or other media. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly. In common usage, the word often refers to statements that may be both true and false i. e. ironic or unexpected, such as the paradox that standing is more tiring than walking. Common themes in paradoxes include self-reference, infinite regress, circular definitions, patrick Hughes outlines three laws of the paradox, Self-reference An example is This statement is false, a form of the liar paradox. The statement is referring to itself, another example of self-reference is the question of whether the barber shaves himself in the barber paradox. One more example would be Is the answer to this question No, contradiction This statement is false, the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldnt come true, vicious circularity, or infinite regress This statement is false, if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the group of statements. Other paradoxes involve false statements or half-truths and the biased assumptions. This form is common in howlers, for example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree and the father is killed, the boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the suite, the surgeon says, I cant operate on this boy. The apparent paradox is caused by a hasty generalization, for if the surgeon is the boys father, the paradox is resolved if it is revealed that the surgeon is a woman — the boys mother. Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. Russells paradox, which shows that the notion of the set of all sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic

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Probability
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Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1, the higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair coin, since the coin is unbiased, the two outcomes are both equally probable, the probability of head equals the probability of tail. Since no other outcomes are possible, the probability is 1/2 and this type of probability is also called a priori probability. Probability theory is used to describe the underlying mechanics and regularities of complex systems. For example, tossing a coin twice will yield head-head, head-tail, tail-head. The probability of getting an outcome of head-head is 1 out of 4 outcomes or 1/4 or 0.25 and this interpretation considers probability to be the relative frequency in the long run of outcomes. A modification of this is propensity probability, which interprets probability as the tendency of some experiment to yield a certain outcome, subjectivists assign numbers per subjective probability, i. e. as a degree of belief. The degree of belief has been interpreted as, the price at which you would buy or sell a bet that pays 1 unit of utility if E,0 if not E. The most popular version of subjective probability is Bayesian probability, which includes expert knowledge as well as data to produce probabilities. The expert knowledge is represented by some prior probability distribution and these data are incorporated in a likelihood function. The product of the prior and the likelihood, normalized, results in a probability distribution that incorporates all the information known to date. The scientific study of probability is a development of mathematics. Gambling shows that there has been an interest in quantifying the ideas of probability for millennia, there are reasons of course, for the slow development of the mathematics of probability. Whereas games of chance provided the impetus for the study of probability. According to Richard Jeffrey, Before the middle of the century, the term probable meant approvable. A probable action or opinion was one such as people would undertake or hold. However, in legal contexts especially, probable could also apply to propositions for which there was good evidence, the sixteenth century Italian polymath Gerolamo Cardano demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes

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Reason
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Reason, or an aspect of it, is sometimes referred to as rationality. Reasoning is associated with thinking, cognition, and intellect, along these lines, a distinction is often drawn between discursive reason, reason proper, and intuitive reason, in which the reasoning process—however valid—tends toward the personal and the opaque. Reason, like habit or intuition, is one of the ways by which thinking comes from one idea to a related idea. For example, it is the means by which rational beings understand themselves to think about cause and effect, truth and falsehood, and what is good or bad. It is also identified with the ability to self-consciously change beliefs, attitudes, traditions, and institutions. In contrast to reason as a noun, a reason is a consideration which explains or justifies some event, phenomenon. The field of logic studies ways in which human beings reason formally through argument, the field of automated reasoning studies how reasoning may or may not be modeled computationally. Animal psychology considers the question of whether animals other than humans can reason, the original Greek term was λόγος logos, the root of the modern English word logic but also a word which could mean for example speech or explanation or an account. As a philosophical term logos was translated in its non-linguistic senses in Latin as ratio and this was originally not just a translation used for philosophy, but was also commonly a translation for logos in the sense of an account of money. French raison is derived directly from Latin, and this is the source of the English word reason. Some philosophers, Thomas Hobbes for example, also used the word ratiocination as a synonym for reasoning, Philosophy can be described as a way of life based upon reason, and in the other direction reason has been one of the major subjects of philosophical discussion since ancient times. Reason is often said to be reflexive, or self-correcting, and it has been defined in different ways, at different times, by different thinkers about human nature. Perhaps starting with Pythagoras or Heraclitus, the cosmos is even said to have reason, Reason, by this account, is not just one characteristic that humans happen to have, and that influences happiness amongst other characteristics. Within the human mind or soul, reason was described by Plato as being the monarch which should rule over the other parts, such as spiritedness. Aristotle, Platos student, defined human beings as rational animals and he defined the highest human happiness or well being as a life which is lived consistently, excellently and completely in accordance with reason. The conclusions to be drawn from the discussions of Aristotle and Plato on this matter are amongst the most debated in the history of philosophy. For example, in the neo-platonist account of Plotinus, the cosmos has one soul, which is the seat of all reason, Reason is for Plotinus both the provider of form to material things, and the light which brings individuals souls back into line with their source. The early modern era was marked by a number of significant changes in the understanding of reason, one of the most important of these changes involved a change in the metaphysical understanding of human beings

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Reference
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Reference is a relation between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to refer to the second object, the second object, the one to which the first object refers, is called the referent of the first object. In some cases, methods are used that intentionally hide the reference from some observers, References feature in many spheres of human activity and knowledge, and the term adopts shades of meaning particular to the contexts in which it is used. Some of them are described in the sections below, the word reference is derived from Middle English referren, from Middle French référer, from Latin referre, to carry back, formed from the prefix re- and ferre, to bear. A number of words derive from the root, including refer, referee, referential, referent. The verb refer and its derivatives may carry the sense of link to or connect to, another sense is consult, this is reflected in such expressions as reference work, reference desk, job reference, etc. In semantics, reference is generally construed as the relationships between nouns or pronouns and objects that are named by them, hence, the word John refers to the person John. The word it refers to some previously specified object, the object referred to is called the referent of the word. Sometimes the word-object relation is called denotation, the word denotes the object, the converse relation, the relation from object to word, is called exemplification, the object exemplifies what the word denotes. In syntactic analysis, if a word refers to a previous word and this problem led Frege to distinguish between the sense and reference of a word. Some cases seem to be too complicated to be classified within this framework, words can often be meaningful without having a concrete here-and-now referent. Fictional and mythological names such as Bo-Peep and Hercules illustrate this possibility, sign links with absent referents also allow for discussing abstract ideas as well as people and events of the past and future. For those who argue that one cannot directly experience the divine, additionally, certain sects of Judaism and other religions consider it sinful to write, discard, or deface the name of the divine. To avoid this problem, the signifier G-d is sometimes used, the very concept of the linguistic sign is the combination of content and expression, the former of which may refer entities in the world or refer more abstract concepts, e. g. thought. Certain parts of speech exist only to reference, namely anaphora such as pronouns. The subset of reflexives expresses co-reference of two participants in a sentence and these could be the agent and patient, as in The man washed himself, the theme and recipient, as in I showed Mary to herself, or various other possible combinations. In computer science, references are data types that refer to an object elsewhere in memory and are used to construct a variety of data structures. Generally, a reference is a value that enables a program to access the particular data item

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Syntax (logic)
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In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the used for constructing, or transforming the symbols and words of a language. Syntax is usually associated with the governing the composition of texts in a formal language that constitute the well-formed formulas of a formal system. In computer science, the term refers to the rules governing the composition of well-formed expressions in a programming language. As in mathematical logic, it is independent of semantics and interpretation, a symbol is an idea, abstraction or concept, tokens of which may be marks or a configuration of marks which form a particular pattern. Symbols of a formal language need not be symbols of anything, for instance there are logical constants which do not refer to any idea, but rather serve as a form of punctuation in the language. A symbol or string of symbols may comprise a well-formed formula if the formulation is consistent with the rules of the language. Symbols of a formal language must be capable of being specified without any reference to any interpretation of them, a formal language is a syntactic entity which consists of a set of finite strings of symbols which are its words. Which strings of symbols are words is determined by fiat by the creator of the language, usually by specifying a set of formation rules. Such a language can be defined without reference to any meanings of any of its expressions, it can exist before any interpretation is assigned to it – that is, Formation rules are a precise description of which strings of symbols are the well-formed formulas of a formal language. It is synonymous with the set of strings over the alphabet of the language which constitute well formed formulas. However, it does not describe their semantics, a proposition is a sentence expressing something true or false. A proposition is identified ontologically as an idea, concept or abstraction whose token instances are patterns of symbols, marks, sounds, propositions are considered to be syntactic entities and also truthbearers. A formal theory is a set of sentences in a formal language, a formal system consists of a formal language together with a deductive apparatus. The deductive apparatus may consist of a set of rules or a set of axioms. A formal system is used to derive one expression from one or more other expressions, Formal systems, like other syntactic entities may be defined without any interpretation given to it. A formula A is a syntactic consequence within some formal system F S of a set Г of formulas if there is a derivation in formal system F S of A from the set Г. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, a formal system S is syntactically complete iff for each formula A of the language of the system either A or ¬A is a theorem of S