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Universal Instantiation
In predicate logic universal instantiation[1][2][3] (UI, also called universal specification or universal elimination, and sometimes confused with Dictum de omni) is a valid rule of inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom. It is one of the basic principles used in quantification theory. Example: "All dogs are mammals. Fido is a dog
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Irving Copi
Irving Marmer Copi (born Copilovich, July 28, 1917, Duluth, Minnesota – August 19, 2002, Honolulu, Hawaii) was an American philosopher, logician, and university textbook author. Copi studied under nobel laureate Bertrand Russell
Bertrand Russell
while at the University of Chicago.[1] Copi taught at the University of Illinois, the United States
United States
Air Force Academy, Princeton University, and the Georgetown University
Georgetown University
Logic
Logic
Institute, before teaching logic at the University of Michigan, 1958–69, and at the University of Hawaii
Hawaii
at Manoa, 1969-90. Copi is probably best known as the author of Introduction to Logic
Logic
and Symbolic Logic, both widely used, with the former currently in its 14th edition. While a professor emeritus at the University of Hawaiʻi at Mānoa, Copi acknowledged David A
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Predicate Logic
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. First-order logic
First-order logic
uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists X such that X is Socrates and X is a man" and there exists is a quantifier while X is a variable.[1] This distinguishes it from propositional logic, which does not use quantifiers or relations.[2] A theory about a topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold for those things
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Willard Van Orman Quine
Willard Van Orman Quine
Willard Van Orman Quine
(/kwaɪn/; known to intimates as "Van";[2] June 25, 1908 – December 25, 2000) was an American philosopher
American philosopher
and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century."[3] From 1930 until his death 70 years later, Quine was continually affiliated with Harvard University
Harvard University
in one way or another, first as a student, then as a professor of philosophy and a teacher of logic and set theory, and finally as a professor emeritus who published or revised several books in retirement. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978
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Special
Special
Special
or specials may refer to:Contents1 Music 2 Film and television 3 Other uses 4 See alsoMusic[edit] Special
Special
(album), a 1992
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International Standard Book Number
"ISBN" redirects here. For other uses, see ISBN (other).International Standard Book
Book
NumberA 13-digit ISBN, 978-3-16-148410-0, as represented by an EAN-13 bar codeAcronym ISBNIntroduced 1970; 48 years ago (1970)Managing organisation International ISBN AgencyNo. of digits 13 (formerly 10)Check digit Weighted sumExample 978-3-16-148410-0Website www.isbn-international.orgThe International Standard Book
Book
Number (ISBN) is a unique[a][b] numeric commercial book identifier. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.[1] An ISBN is assigned to each edition and variation (except reprintings) 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, and 10 digits long if assigned before 2007
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Quantification (logic)
In logic, quantification specifies the quantity of specimens in the domain of discourse that satisfy an open formula. The two most common quantifiers mean "for all" and "there exists". For example, in arithmetic, quantifiers allow one to say that the natural numbers go on for ever, by writing that for all n (where n is a natural number), there is another number (say, the successor of n) which is one bigger than n. A language element which generates a quantification (such as "every") is called a quantifier. The resulting expression is a quantified expression, it is said to be quantified over the predicate (such as "the natural number x has a successor") whose free variable is bound by the quantifier. In formal languages, quantification is a formula constructor that produces new formulas from old ones. The semantics of the language specifies how the constructor is interpreted
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Stanisław Jaśkowski
Stanisław Jaśkowski
Stanisław Jaśkowski
(22 April 1906, Warsaw
Warsaw
– 16 November 1965, Warsaw) was a Polish logician who made important contributions to proof theory and formal semantics. He was a student of Jan Łukasiewicz and a member of the Lwów– Warsaw
Warsaw
School of Logic. Upon his death his name was added to the Genius Wall of Fame. He was the President (rector) of the Nicolaus Copernicus University
Nicolaus Copernicus University
in Toruń. Jaśkowski is considered to be one of the founders of natural deduction, which he discovered independently of Gerhard Gentzen
Gerhard Gentzen
in the 1930s. (Gentzen's approach initially became more popular with logicians because it could be used to prove the cut-elimination theorem
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Gerhard Gentzen
Gerhard Karl Erich Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician. He made major contributions to the foundations of mathematics, proof theory, especially on natural deduction and sequent calculus. He died in 1945 after the Second World War, because he was deprived of food after being arrested in Prague.Contents1 Life and career 2 Work 3 Publications3.1 Posthumous4 See also 5 Notes 6 References 7 External linksLife and career[edit] Gentzen was a student of Paul Bernays
Paul Bernays
at the University of Göttingen. Bernays was fired as "non-Aryan" in April 1933 and therefore Hermann Weyl formally acted as his supervisor. Gentzen joined the Sturmabteilung
Sturmabteilung
in November 1933 although he was by no means compelled to do so.[1] Nevertheless he kept in contact with Bernays until the beginning of the Second World War
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Natural Deduction
In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning
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Logical Consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of 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?[1] All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.[2] Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation.[1] A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e
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Axiom Schema
In mathematical logic, an axiom schema (plural: axiom schemata or axiom schemas) generalizes the notion of axiom.Contents1 Formal definition 2 Finite axiomatization 3 Examples 4 Finitely axiomatized theories 5 In higher-order logic 6 See also 7 Notes 8 ReferencesFormal definition[edit] An axiom schema is a formula in the metalanguage of an axiomatic system, in which one or more schematic variables appear. These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which may or may not be required to satisfy certain conditions. Often, such conditions require that certain variables be free, or that certain variables not appear in the subformula or term. Finite axiomatization[edit] Given that the number of possible subformulas or terms that can be inserted in place of a schematic variable is countably infinite, an axiom schema stands for a countably infinite set of axioms. This set can usually be defined recursively
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Quantification Theory
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. First-order logic
First-order logic
uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists X such that X is Socrates and X is a man" and there exists is a quantifier while X is a variable.[1] This distinguishes it from propositional logic, which does not use quantifiers or relations.[2] A theory about a topic is usually a first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions from that domain to itself, finitely many predicates defined on that domain, and a set of axioms believed to hold for those things
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Axiom
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'[1][2] The term has subtle differences in definition when used in the context of different fields of study
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Validity
In logic, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false.[1] It is not required that a valid argument have premises that are actually true,[2] but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. A formula is valid if and only if it is true under every interpretation, and an argument form (or schema) is valid if and only if every argument of that logical form is valid.Contents1 Arguments 2 Valid formula 3 Statements 4 Soundness 5 Satisfiability 6 Preservation 7 See also 8 References 9 Further readingArguments[edit] An argument is valid if and only if the truth of its premises entails the truth of its conclusion and each step, sub-argument, or logical operation in the argument is valid. Under such conditions it would be self-contradictory to affirm the premises and deny the conclusion
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Tautology (rule Of Inference)
In propositional logic, tautology is one of two commonly used rules of replacement.[1][2][3] The rules are used to eliminate redundancy in disjunctions and conjunctions when they occur in logical proofs
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