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Quantifier Shift
A quantifier shift is a logical fallacy in which the quantifiers of a statement are erroneously transposed during the rewriting process. The change in the logical nature of the statement may not be obvious when it is stated in a natural language like English. Definition The fallacious deduction is that: ''For every A, there is a B, such that C. Therefore, there is a B, such that for every A, C.'' :\forall x \,\exists y \,Rxy \vdash \exists y \,\forall x \,Rxy However, an inverse switching: :\exist y \,\forall x \,Rxy \vdash \forall x \,\exist y\, Rxy is logically valid. Examples 1. Every person has a woman that is their mother. Therefore, there is a woman that is the mother of every person. :\forall x \,\exists y \,(Px \to (Wy \land M(yx))) \vdash \exists y \,\forall x \,(Px \to (Wy \land M(yx))) It is fallacious to conclude that there is ''one woman'' who is the mother of ''all people''. However, if the major premise ("every person has a woman that is their mother") ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Fallacy
In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept called mathematical fallacy. There is a distinction between a simple ''mistake'' and a ''mathematical fallacy'' in a proof, in that a mistake in a proof leads to an invalid proof while in the best-known examples of mathematical fallacies there is some element of concealment or deception in the presentation of the proof. For example, the reason why validity fails may be attributed to a division by zero that is hidden by algebraic notation. There is a certain quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way. Therefore, these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifiers (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The mostly commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic, they are interdefinable using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has the property P. Othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rewriting
In mathematics, computer science, and logic, rewriting covers a wide range of methods of replacing subterms of a well-formed formula, formula with other terms. Such methods may be achieved by rewriting systems (also known as rewrite systems, rewrite engines, or reduction systems). In their most basic form, they consist of a set of objects, plus relations on how to transform those objects. Rewriting can be non-deterministic algorithm, non-deterministic. One rule to rewrite a term could be applied in many different ways to that term, or more than one rule could be applicable. Rewriting systems then do not provide an algorithm for changing one term to another, but a set of possible rule applications. When combined with an appropriate algorithm, however, rewrite systems can be viewed as computer programs, and several automated theorem proving, theorem provers and declarative programming languages are based on term rewriting. Example cases Logic In logic, the procedure for obtaini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages can take different forms, such as speech or signing. They are distinguished from constructed and formal languages such as those used to program computers or to study logic. Defining natural language Natural language can be broadly defined as different from * artificial and constructed languages, e.g. computer programming languages * constructed international auxiliary languages * non-human communication systems in nature such as whale and other marine mammal vocalizations or honey bees' waggle dance. All varieties of world languages are natural languages, including those that are associated with linguistic prescriptivism or language regulation. ( Nonstandard dialects can be viewed as a wild type in comparison with standard l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
English Language
English is a West Germanic language of the Indo-European language family, with its earliest forms spoken by the inhabitants of early medieval England. It is named after the Angles, one of the ancient Germanic peoples that migrated to the island of Great Britain. Existing on a dialect continuum with Scots, and then closest related to the Low Saxon and Frisian languages, English is genealogically West Germanic. However, its vocabulary is also distinctively influenced by dialects of France (about 29% of Modern English words) and Latin (also about 29%), plus some grammar and a small amount of core vocabulary influenced by Old Norse (a North Germanic language). Speakers of English are called Anglophones. The earliest forms of English, collectively known as Old English, evolved from a group of West Germanic (Ingvaeonic) dialects brought to Great Britain by Anglo-Saxon settlers in the 5th century and further mutated by Norse-speaking Viking settlers starting in the 8th and 9th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fallacious Induction
A fallacy is reasoning that is logically invalid, or that undermines the logical validity of an argument. All forms of human communication can contain fallacies. Because of their variety, fallacies are challenging to classify. They can be classified by their structure (formal fallacies) or content (informal fallacies). Informal fallacies, the larger group, may then be subdivided into categories such as improper presumption, faulty generalization, and error in assigning causation and relevance, among others. The use of fallacies is common when the speaker's goal of achieving common agreement is more important to them than utilizing sound reasoning. When fallacies are used, the premise should be recognized as not well-grounded, the conclusion as unproven (but not necessarily false), and the argument as unsound. Formal fallacies A formal fallacy is an error in the argument's form. All formal fallacies are types of . * Appeal to probability – a statement that takes something f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Audi
Robert N. Audi (born November 1941) is an American philosopher whose major work has focused on epistemology, ethics (especially on ethical intuitionism), rationality and the theory of action. He is O'Brien Professor of Philosophy at the University of Notre Dame, and previously held a Chair in the Business School there. His 2005 book, ''The Good in the Right'', updates and strengthens Rossian intuitionism and develops the epistemology of ethics. He has also written important works of political philosophy, particularly on the relationship between church and state. He is a past president of the American Philosophical Association and the Society of Christian Philosophers. Audi's contributions to epistemology include his defense of '' fallibilistic foundationalism''. Audi has expanded his theory of justification to non-doxastic states, e.g. desires and intentions, by developing a ''comprehensive account of rationality''. A mental state is rational if it is "well-grounded" in a ''source ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Cambridge Dictionary Of Philosophy
''The Cambridge Dictionary of Philosophy'' (1995; second edition 1999; third edition 2015) is a dictionary of philosophy published by Cambridge University Press and edited by the philosopher Robert Audi Robert N. Audi (born November 1941) is an American philosopher whose major work has focused on epistemology, ethics (especially on ethical intuitionism), rationality and the theory of action. He is O'Brien Professor of Philosophy at the University .... There are 28 members on the Board of Editorial Advisors and 440 contributors. Publication history ''The Cambridge Dictionary of Philosophy'' was first published in 1995 by Cambridge University Press. A second edition followed in 1999, and a third edition in 2015. References Bibliography ;Books * External links More information at the Cambridge University Press website 1995 non-fiction books Books by Robert Audi Cambridge University Press books Dictionaries of philosophy Encyclopedias of philosophy English-language book ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barnes & Noble
Barnes & Noble Booksellers is an American bookseller. It is a Fortune 1000 company and the bookseller with the largest number of retail outlets in the United States. As of July 7, 2020, the company operates 614 retail stores across all 50 U.S. states. Barnes & Noble operates mainly through its Barnes & Noble Booksellers chain of bookstores. The company's headquarters are at 33 E. 17th Street on Union Square in New York City. After a series of mergers and bankruptcies in the American bookstore industry since the 1990s, Barnes & Noble stands alone as the United States' largest national bookstore chain. Previously, Barnes & Noble operated the chain of small B. Dalton Bookseller stores in malls until they announced the liquidation of the chain. The company was also one of the nation's largest manager of college textbook stores located on or near many college campuses when that division was spun off as a separate public company called Barnes & Noble Education in 2015. During the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antony Flew
Antony Garrard Newton Flew (; 11 February 1923 – 8 April 2010) was a British philosopher. Belonging to the analytic and evidentialist schools of thought, Flew worked on the philosophy of religion. During the course of his career he taught at the universities of Oxford, Aberdeen, Keele and Reading, and at York University in Toronto. For much of his career Flew was known as a strong advocate of atheism, arguing that one should presuppose atheism until evidence suggesting a God surfaces. He also criticised the idea of life after death, the free will defence to the problem of evil, and the meaningfulness of the concept of God. In 2003, he was one of the signatories of the Humanist Manifesto III. However, in 2004 he changed his position, and stated that he now believed in the existence of an Intelligent Creator of the universe, shocking colleagues and fellow atheists. In order to further clarify his personal concept of God, Flew openly made an allegiance to Deism, more specifica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
