In mathematicallogic , independence refers to the unprovability of a sentence mathematicallogic sentence from other sentences. A sentence is independent of a given theory mathematicallogic first order theory T if T neither proves nor refutes &sigma that is, it is impossible to prove &sigma from T , and it is also impossible to prove from T that &sigma is false. Sometimes, &sigma is said synonymously to be undecidable from T this is not the same meaning of decidability as in a decision problem . A theory T is independent if each axiom in T is not provable from the remaining axioms in T . A theory for which there is an independent set of axioms is independently axiomatizable . Usage note Some authors say that is independent of T if T simply cannot prove &sigma , and do not necessarily assert by this that T cannot refute &sigma . These authors will sometimes say &sigma is independent of and consistent with T to indicate that T can neither prove nor refute . Independence results in set theory Many interesting statements in set theory are independent of Zermelo Fraenkel set theory ZF . The following statements in set theory are known to be independent of ZF, granting that ZF is consistent The axiom of choice The continuum hypothesis and the Continuum hypothesis The generalized continuum hypothesis generalised continuum hypothesis The Suslin s problem Suslin conjecture The existence of a Kurepa tree The following statements none of which have been proved false cannot be proved in ZFC to be independent of ZFC, even if the added hypothesis is granted that ZFC is consistent. However, they cannot be proved in ZFC granting that ZFC is consistent , and few working set theorists ... Elliott title An Introduction to MathematicalLogic publisher Chapman & Hall location London edition 4th isbn 978 0 412 80830 2 year 1997 Citation last1 Monk first1 J. Donald title MathematicalLogic ... 0 387 90170 1 year 1976 logic Category Proof theory cs Nez visl tvrzen it Decidibilit nl Onafhankelijkheid ... more details
Mathematicallogic also known as symbolic logic is a subfield of mathematics with close connections to foundations ... themes in mathematicallogic include the study of the expressive power of formal system s and the deductive power of formal mathematical proof proof systems. Mathematicallogic is often divided ... for those. Since its inception, mathematicallogic has contributed to, and has been motivated by, the study ... Mathematicallogic emerged in the mid 19th century as a subfield of mathematics independent of the traditional ... and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building ... is now an important tool for establishing independence result s in set theory. Symbolic logic Leopold ... theory, and that it implies any such formalization has a countable structure mathematicallogic model ... this history around 1950 Subfields and scope The Handbook of MathematicalLogic makes a rough division of contemporary mathematicallogic into four areas set theory model theory recursion theory , and proof ... between these fields, and the lines between mathematicallogic and other fields of mathematics ... a subfield of mathematicallogic. Because of its applicability in diverse fields of mathematics, mathematicians ... Formal logic At its core, mathematicallogic deals with mathematical concepts expressed using formal ... the models of various formal theories. Here a theory mathematicallogic theory is a set of formulas in a particular formal logic and signature logic signature , while a structure mathematicallogic ... of mathematicallogic, includes the study of systems in non classical logic such as intuitionistic ... in computer science is closely related to the study of computability in mathematicallogic. There is a difference ... and feasible computability , while researchers in mathematicallogic often focus on computability ... with classical mathematics. See also Portal Logic List of mathematicallogic topics List of computability ... title A mathematical introduction to logic publisher Academic Press location Boston, MA edition 2nd ... more details
Independence friendly logic IF logic , proposed by Jaakko Hintikka and Gabriel Sandu philosopher Gabriel Sandu , aims at being a more natural and intuitive alternative to classical first order logic FOL . IF logic is characterized by branching quantifier s. It is more expressive than FOL because it allows one to express independence relations between quantified variables. For example, the formula a b c b d a a,b,c,d x y should be read as x independent of y cannot be expressed in FOL. This is because c depends only on a and d depends only on b. First order logic cannot express these independences ... for the standard semantics of IF logic, except that the games are of imperfect information . Independence ... us squarely in full second order logic emphasis Feferman s . See also Game semantics Game Semantics Branching quantifier Branching Quantifiers Dependence logic Dependence Logic References reflist Solomon Feferman , What kind of logic is Independence Friendly logic? , in The Philosophy of Jaakko Hintikka ... Logic A New Approach to Independence Friendly Logic , ISBN 9780521876599 Wilfrid Hodges , 1997, Compositional ... and the interpretation of IF logic. Tero Tulenheimo, 2009. http plato.stanford.edu entries logic if Independence friendly logic . Stanford Encyclopedia of Philosophy . Dag Westerst hl, 2005. http plato.stanford.edu ... truth value s, it cannot be used for IF logic. Hintikka further argues that the standard Tarskian semantics semantics of FOL cannot accommodate IF logic because the principle of compositionality ... order logic . He argues contra Hintikka that while satisfiability might be a first order matter, the question ... 1 4020 3210 2 Matti Eklund and Daniel Kolak , Is Hintikka s Logic First Order? Synthese An International ... Hodges , 2004. http plato.stanford.edu entries logic games Logic and Games . Stanford Encyclopedia ... . External links http planetmath.org encyclopedia IFLogic.html Planet Math article on IF logic . Category Systems of formal logic Category Philosophical logic Category Non classical logic ... more details
The relative strength of two systems of formal logic can be defined via model theory . Specifically, a logic math Alpha math is said to be as strong as a logic math Beta math if every elementary class in math Alpha math is an elementary class in math Beta math . ref Heinz Dieter Ebbinghaus Extended logics the general framework in K. J. Barwise and S. Feferman, editors, Model theoretic logics , 1985 ISBN 0387909362 page 43 ref See also Abstract logic Lindstr m s theorem References Reflist Category Model theory Category Mathematicallogic Category Concepts in logic mathlogic stub nl Sterkte wiskundige logica ... more details
Infobox journal title Archive for MathematicalLogic cover abbreviation Arch. Math. Logic discipline Mathematicallogic editor nowrap 1 Ralf Schindler publisher Springer Science Business Media Springer frequency 8 year history 1950 present impact 0.349 impact year 2009 url http www.springer.com mathematics journal 153 ISSN 0933 5846 eISSN 1432 0665 CODEN AMLOEH LCCN 88645365 OCLC 18237511 formernames Archiv f r mathematische Logik und Grundlagenforschung link1 http www.springerlink.com content 0933 5846 link1 name Online access Archive for MathematicalLogic is a peer review peer reviewed mathematics journal published by Springer Science Business Media Springer . Founded in 1950, the journal publishes articles on mathematicallogic . The journal is indexed by Mathematical Reviews and Zentralblatt MATH . Its 2009 Mathematical Citation Quotient MCQ was 0.24, and its 2009 impact factor was 0.349. External links Official http www.springer.com mathematics journal 153 Category Mathematics journals Category Publications established in 1950 Category English language journals Category Springer academic journals Category Logic journals math journal stub ... more details
italictitle Infobox Journal title Journal of MathematicalLogic cover image JMLcover.jpg 180px discipline Mathematics abbreviation editor Chitat Chong, Qi Feng, Theodore A. Slaman, W. Hugh Woodin publisher World Scientific country Singapore impact 0.684 impact year 2008 history 2001 present website http www.worldscinet.com jml jml.shtml ISSN 0219 0613 eISSN 1793 6691 The Journal of MathematicalLogic was established in 2001 and is published by World Scientific . It covers the field of mathematicallogic and its applications. Abstracting and indexing The journal is abstracted and indexed in Current Mathematical Publications Mathematical Reviews Mathematical Reviews MathSciNet Zentralblatt MATH Science Citation Index Science Citation Index Expanded Current Contents Physical, Chemical and Earth Sciences Journal Citation Reports Science Edition External links Official http www.worldscinet.com jml jml.shtml Category English language journals Category Publications established in 2001 Category Mathematics journals Category World Scientific academic journals Category Logic journals ... more details
Principles of MathematicalLogic is the 1950 American translation of the 1938 second edition of David Hilbert s and Wilhelm Ackermann s classic text Grundz ge der theoretischen Logik , on elementary mathematicallogic. The 1928 first edition thereof is considered the first elementary text clearly grounded in the formalism now known as first order logic FOL . Hilbert and Ackermann also formalized FOL in a way that subsequently achieved canonical status. FOL is now a core formalism of mathematicallogic, and is presupposed by contemporary treatments of Peano arithmetic and nearly all treatments of axiomatic set theory . The 1928 edition included a clear statement of the Entscheidungsproblem decision problem for FOL, and also asked whether that logic was G del s completeness theorem complete i.e., whether all semantic truths of FOL were theorems derivable from the FOL axioms and rules . The first problem was answered in the negative by Alonzo Church in 1936. The second was answered affirmatively by Kurt G del in 1929. The text also touched on set theory and relational algebra as ways of going beyond FOL. Contemporary notation for logic owes more to this text than it does to the notation of Principia Mathematica , long popular in the English speaking world. References David Hilbert and Wilhelm Ackermann 1928 . Grundz ge der theoretischen Logik Principles of MathematicalLogic . Springer Verlag, ISBN 0 8218 2024 9. This text went into four subsequent German editions, the last in 1972. Hendricks, Neuhaus, Petersen, Scheffler and Wansing eds. 2004 . First order logic revisited . Logos Verlag, ISBN 3 8325 0475 3. Proceedings of a workshop, FOL 75, commemorating the 75th anniversary of the publication of Hilbert and Ackermann 1928 . logic stub Category 1928 books Category 1938 books Category Logic books Category Mathematics books Category History of logic fr Principes de logique th orique ... more details
This article is a technical mathematical article in the area of predicate logic. For the ordinary English language meaning see Sentence , for a less technical introductory article see Statement logic . In mathematicallogic , a sentence of a predicate logic is a well formed formula with no free variable s. A sentence can be viewed as expressing a Proposition mathematics proposition . It makes an assertion, potentially concerning any structure mathematicallogic structure of L . This assertion has a fixed truth value with respect to the structure. In contrast, the truth value of a formula with free variables may be indeterminate with respect to any structure. As the free variables of a formula can range over several values which could be members of a universe, relations or functions , its truth value may vary. Example The following example is in first order logic . math forall y exists x x 2 y math is a sentence. This sentence is true in the positive real numbers , false in the real numbers, and true in the complex numbers. In plain English, this sentence is interpreted to mean that every member of the structure concerned is the Square algebra square of a member of that particular structure. On the other hand, the formula math exists x x 2 y math is not a sentence, because of the presence of the free variable y . In the structure of the real numbers, this formula is true if we substitute arbitrarily y 2, but is false if y 2. See also Ground expression Atomic sentence Open sentence Statement logic Proposition References cite book author Hinman, P. title Fundamentals of MathematicalLogic publisher A K Peters year 2005 isbn 1 568 81262 0 Category Predicate logic Category Statements logic stub eo Vikipedio Projekto matematiko Kondamno matematika logiko fr Proposition logique math matique he pl Zdanie logiczne ru zh ... more details
Use dmy dates date October 2010 A timeline of mathematicallogic . See also History of logic . 19th century 1847 George Boole formalizes symbolic logic in The Mathematical Analysis of Logic , defining what is now called Boolean algebra logic Boolean algebra . 1874 Georg Cantor proves that the set of all real number s is uncountable uncountably infinite but the set of all real algebraic number s is countable countably infinite . Cantor s first uncountability proof His proof does not use his famous Cantor s diagonal argument diagonal argument , which he published in 1891. 1895 Georg Cantor publishes a book about set theory containing the arithmetic of infinite cardinal number s and the continuum hypothesis . 1899 Georg Cantor discovers a contradiction in his set theory. 20th century 1908 Ernst Zermelo axiomizes set theory , thus avoiding Cantor s contradictions. 1931 Kurt G del proves G del s incompleteness theorem his incompleteness theorem which shows that every axiomatic system for mathematics is either incomplete or inconsistent. 1940 Kurt G del shows that neither the continuum hypothesis nor the axiom of choice can be disproven from the standard axioms of set theory. 1961 Abraham Robinson creates non standard analysis . 1963 Paul Cohen mathematician Paul Cohen uses his technique of forcing mathematics forcing to show that neither the continuum hypothesis nor the axiom of choice can be proven from the standard axioms of set theory. Category Mathematics timelines Logic Category Mathematicallogic ... more details
changes . This is a list of mathematicallogic topics , by Wikipedia page. For traditional syllogistic logic, see the list of topics in logic . See also the list of computability and complexity topics for more theory of algorithm s. Working foundations Peano axioms Giuseppe Peano Mathematical ... s paradox G del s incompleteness theorems Structure mathematicallogic Interpretation logic Substructure Elementary substructure Skolem hull Non standard model Atomic model mathematicallogic Prime ... Hrushovski construction Potential isomorphism Theory mathematicallogic Complete theory Vaught ... extension Elementary class Pseudoelementary class Strength mathematicallogic Differentially ... lists Logic Category Mathematicallogic List ru ... method Formal system Mathematical proof Direct proof Reductio ad absurdum Proof by exhaustion Constructive proof Nonconstructive proof Tautology logic Tautology Consistency proof Arithmetization of analysis ... Definable real number Algebraic logic Boolean algebra logic Dialectica space categorical logic ... Kripke semantics General frame Predicate logic First order logic Infinitary logic Many sorted logic Higher order logic Lindstr m quantifier Second order logic Soundness theorem G del s completeness theorem ... elimination Reduct Signature logic Skolem normal form Type model theory Zariski geometry Set theory ... Recursion theory Entscheidungsproblem Decision problem Decidability logic Church Turing thesis Recursive ... calculus Church Rosser theorem Calculus of constructions Combinatory logic Post correspondence problem ... Tarski s indefinability theorem Diagonal lemma Provability logic Interpretability logic Sequent Sequent calculus Analytic proof Structural proof theory Self verifying theories Substructural logic s Structural rule Weakening Contraction Linear logic Intuitionistic linear logic Proof net Affine logic Strict logic Relevant logic Proof theoretic semantics Ludics System F Gerhard Gentzen Gentzen ... more details
about theories in a formal language, as studied in mathematicallogic Theory disambiguation In mathematicallogic , a theory also called a formal theory is a set of sentence mathematicallogic sentence s in a formal language . The individual sentences of a theory are called its theorem s. A first order theory is a set of first order logic first order sentences. Many authors require that the theory ... name curry Curry, Haskell, Foundations of MathematicalLogic ref Subtheories and extensions A theory ..., Foundations of MathematicalLogic p.48 ref Theories associated with a structure Each Structure mathematicallogic structure has several associated theories. The complete theory of a structure A is the set of all first order logic first order sentence mathematicallogic sentence s over the Signature logic signature of A which are satisfied by A . It is denoted by Th A . More generally, the theory ... of math mathcal QS math Interpretation of a first order theory Main Structure mathematicallogic ... mathematicallogic structure and then let the theory be the set of formulas that are satisfied ... order logic that satisfies the principle of explosion , this is equivalent to requiring ... theory. For first order logic , the most important case, it follows from the G del ... order logic , there are syntactically consistent theories that are not satisfiable, such as .... Interpretation of a theory Main Interpretation logic An interpretation of a theory is the relationship ... in a first order theory Main First order logic Deductive systems There are many formal derivation proof systems for first order logic. Syntactic consequence in a first order theory Main First order logic ... mathcal QS math is satisfied. First order theories with identity Main First order logic Equality ... enumerable set of axioms. The theory of R , , , 0, 1, was shown by Tarski to be Decidability logic ... University Press title A shorter model theory year 1997 isbn 0 521 58713 1 logic Category Model ... more details
otheruses Judgement disambiguation In mathematicallogic , a judgment can be for example an assertion about occurrence of a free variable in an expression of the object language, or about provability of a proposition either as a tautology logic tautology or from a given context but judgments can be also other inductively definable assertions in the metatheory . Judgments are used for example in formalizing deduction systems a logical axiom expresses a judgment, premises of a rule of inference are formed as a sequence of judgments, and their conclusion is a judgment as well. Also the result of a proof expresses a judgment, and the used hypotheses are formed as a sequence of judgments. A characteristic feature of the various variants of Hilbert style deduction system s is that the context is not changed in any of their rules of inference, while both natural deduction and sequent calculus contain some context changing rules. Thus, if we are interested only in the derivability of tautologies, no hypothetical judgments, then we can formalize the Hilbert style deduction system in such a way that its rules of inference contain only judgments of a rather simple form. The same cannot be done with the other two deductions systems as context is changed in some of their rules of inferences, they cannot be formalized so that hypothetical judgments could be avoided not even if we want to use them just for proving derivability of tautologies. This basic diversity among the various calculi allows such difference, that the same basic thought e.g. deduction theorem must be proven as a metatheorem in Hilbert style deduction system , while it can be declared explicitly as a rule of inference in natural deduction . In type theory , some analogous notions are used as in mathematicallogic giving rise to connections between the two fields, e.g. Curry Howard correspondence . The abstraction in the notion of judgment in mathematicallogic can exploited also in foundation of type theory as well ... more details
toida nerzic content logic pred logic predicate pred intro.html Introduction to predicates Logic Category Predicate logic cs Predik t logika de Pr dikat Logik et Predikaat es Predicado l gica matem tica ... more details
In mathematicallogic , a literal is an atomic formula atom or its negation. They mostly appear in the context of conjunctive normal form and the method of resolution logic resolution . Literals can be divided into two types A positive literal is just an atom. A negative literal is the negation of an atom. For a literal math l math , the complementary literal is a literal corresponding to the negation of math l math , we can write math bar l math to denote the complementary literal of math l math . More precisely, if math l equiv x math then math bar l math is math lnot x math and if math l equiv lnot x math then math bar l math is math x math . In the context of a formula in the conjunctive normal form , a literal is pure if the literal s complement does not appear in the formula. References cite conference first Samuel last Buss title An introduction to proof theory booktitle Handbook of proof theory pages 1 78 url http math.ucsd.edu sbuss ResearchWeb handbookI publisher Elsevier date 1998 id ISBN 0 444 89840 9 Category Propositional calculus logic stub de Literal es Literal l gica matem tica fr Litt ral logique nl Literal ja pl Litera pt Literal l gica ru sr zh ... more details
. A structure math mathcal M math is said to be a model of a Theory mathematicallogic theory ... last1 Hinman first1 P. title Fundamentals of MathematicalLogic publisher A K Peters isbn 978 1 ... An Introduction to Contemporary MathematicalLogic publisher Springer Verlag location Berlin, New York ... theoretic point of view, structures are the objects used to define the semantics of first order logic ... consisting of a domain A , a signature logic signature , and an interpretation function I that indicates ... important in logic, because several common inference rules are not sound when empty structures ... to confusion. ref Signature main Signature logic The signature logic signature of a structure consists ... order logic see also Model theory First order logic Model theory Axiomatizability, elimination ... . This is misleading, as nothing in their definition ties them to any specific logic, and in fact they are suitable as semantic objects both for very restricted fragments of first order logic such as that used in universal algebra, and for second order logic . In connection with first order logic ... the role of names for the different domains. Signature logic Many sorted signatures Many ... and tedious hence unrewarding to carry out the generalization explicitly. In most mathematical endeavours, not much attention is paid to the sorts. A many sorted logic however naturally leads to a type theory . As Bart Jacobs puts it A logic is always a logic over a type theory. This emphasis in turn leads to categorical logic because a logic over a type theory categorically corresponds to one total category, capturing the logic, being fibred category fibred over another base category, capturing the type theory. ref Citation first Bart last Jacobs title Categorical Logic and Type ... represented by each object of that type. Higher order languages main Second order logic There is more than one possible semantics for higher order logic , as discussed in the article on second ... more details
In model theory , an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula. Such types are called principal types , and the formulas that axiomatize them are called complete formulas . Definitions A complete type p x sub 1 sub , ..., x sub n sub is called principal or atomic if it is axiomatized by a single formula &phi x sub 1 sub , ..., x sub n sub &isin p x sub 1 sub , ..., x sub n sub . A formula in a complete theory T is called complete if for every other formula &psi x sub 1 sub , ..., x sub n sub , the formula &phi implies exactly one of &psi and ¬ &psi in T . ref Some authors refer to complete formulas as atomic formulas , but this is inconsistent with the purely syntactical notion of an atom or atomic formula as a formula that does not contain a proper subformula. ref It follows that a complete type is principal if and only if it contains a complete formula. A model M of the theory is called atomic if every n tuple of elements of M satisfies a complete formula. Examples The ordered field of real algebraic numbers is the unique atomic model of the theory of real closed field s. Any finite model is atomic A dense linear ordering without endpoints is atomic. Any prime model of a countable theory is atomic. Any countable atomic model is prime, but there are plenty of atomic models that are not prime, such as an uncountable dense linear order without endpoints. The theory of a countable number of independent unary relations is complete but has no completable formulas and no atomic models. Properties The back and forth method can be used to show that any two countable atomic models of a theory that are elementarily equivalent are isomorphic. Notes references References Citation last1 Chang first1 Chen Chung last2 Keisler first2 H. Jerome author2 link Howard Jerome Keisler title Model Theory publisher Elsevier edition 3rd series Studies in Logic and the Foundatio ... more details
Other uses Image Spirit of Independence.JPG right thumb 250px A personification of independence as represented ... s Epcot . Independence is a condition of a nation , country , or Sovereign state state in which its ... , over its territory. Attainment of independence should not be confused with revolution , which typically ... and achieve national independence, others aim only to redistribute power &mdash with or without an element .... Furthermore, some countries were granted independence without any revolutionary acts. The October Revolution Russian October Revolution , for example, was not intended to seek national independence ... refers to a kind of independence which has been granted by an overseeing authority that itself ... of established independence or, less commonly, the commencement of revolution , are typically celebrated as a national Public holiday holiday known as an independence day . Sometimes, a state wishing to achieve independence from a dominating power will issue a declaration of independence the earliest ... being Abkhazia s s Act of State Independence of the Republic of Abkhazia Act of State Independence in 1999. Declaring independence and attaining it however, are quite different. A well known successful example is the Declaration of Independence United States U.S. Declaration of Independence issued in 1776. Historically, there have been three major periods of declaring independence the years from ... content 18 3 61.full The Declaration of Independence in World Context , Organization of American ... wishing to seek independence are many. Disillusionment rising from the establishment is a cause ... independence movement , to a violent civil war . See also Independence constitution Independence referendum List of countries by Independence Day List of countries by statehood List of sovereign ... Committee on Decolonization War of Independence disambiguation Wars of independence References Reflist Article note The dates of independence, as they have been ported into many articles from the CIA ... more details
The Department of MathematicalLogic at the Bulgarian Academy of Sciences was created by the Institute of Mathematics and Informatics Bulgarian Academy of Sciences Institute of Mathematics and Informatics in implementation of Government Decree N0. 236 of November 3, 1959. Its first chairman was Boyan Petkanchin 1907 87 who worked to promote and disseminate the knowledge of mathematicallogic both in the professional mathematics mathematical community in Bulgaria and as popular science . Vladimir Sotirov and Radoslav Pavlov joined the department in 1970, followed by George Gargov, Anatoly Buda, Lyubomir Ivanov , Slavyan Radev and Solomon Passy in 1976 89. In 1996 2000 the department was joined by Dimiter Dobrev, Jordan Zashev and Dimitar Guelev. From 1971 to 1989 the department was merged with the corresponding division of the Faculty of Mathematics and Informatics at Sofia University , with Dimiter Skordev heading the integrated structure since 1971. In 1989 the institutional relationship with Sofia University was severed, and the department resumed as a division of the Institute of Mathematics and Informatics, headed since then by Lyubomir Ivanov . The logicians Bogdan Dyankov, Hristo Smolenov, Veselin Petrov and Marion Mircheva stayed with the department for various periods of time, all of them coming from the Institute of Philosophy at the Bulgarian Academy of Sciences once the latter was dissolved on account of the political dissidents dissident activities of its members in 1989. The research of the department is mostly in the area of algebra ic recursion theory , modal logic modal , temporal logic temporal and other classical logic non classical logics , as well as logic ... of Bulgaria Bulgarian Constitution . References http www.math.bas.bg logic Department of MathematicalLogic http www.fmi.uni sofia.bg fmi logic skordev history.htm Historical notes on the development of mathematicallogic in Sofia Andreev A., I. Derzhanski eds. http www.math.bas.bg report IMI 20Jubilee ... more details
Wiktionarypar independenceIndependence is the self government of a nation, country, or state by its residents and population. Independence may also mean Mathematics and technology Algebraic independence Bit sequence independence in telecommunications Independencemathematicallogic , logical independenceIndependence probability theory , statistical independence Conditional independence Independent variable Use in statistics Independence of variables in regression and controlled experiments Independent variable Use in mathematics Independence of variables in mathematics Linear independence Resolution independence in computing Place names United States Independence, Alabama Independence, California Independence, Inyo County, California Independence, Calaveras County, California Independence, Teller County, Colorado Independence, Pitkin County, Colorado Independence, Indiana Independence, Iowa Independence, Kansas Independence, Kentucky Independence, Louisiana Independence, Minnesota Independence, Hennepin County, Minnesota Independence, St. Louis County, Minnesota Independence, Mississippi Independence, Missouri Independence, New York Independence, Ohio , in Cuyahoga County Independence, Defiance County, Ohio Independence, Oregon Independence, Snyder County, Pennsylvania Independence, Washington County, Pennsylvania Independence, Texas Independence, Utah Independence, Virginia Independence, Barbour County, West Virginia Independence, Clay County, West Virginia Independence, Jackson County, West Virginia Independence, Preston County, West Virginia Independence, Wisconsin The Independence River , a tributary of the Black River in New York Elsewhere Independence and Mango Creek , adjacent villages considered as one community in Belize Independence, a former name of Niulakita , Tuvalu Other uses Independence Israeli political party , a political party in Israel. Independent living , a concept in disability rights Independence Air , a low cost airline Independence D , an independent ... more details
In proof theory and constructive mathematics , the principle of independence of premise states that if &phi and &exist x &theta are sentences in a formal theory and nowrap 1 &phi &rarr &exist x &theta is provable, then nowrap 1 &exist x &phi &rarr &theta is provable. Here x cannot be a free variable of &phi . The principle is valid in classical logic. Its main application is in the study of intuitionistic logic, where the principle is not always valid. In classical logic The principle of independence of premise is valid in classical logic because of the law of the excluded middle . Assume that nowrap 1 &phi &rarr &exist x &theta is provable. Then, if &phi holds, there is an x satisfying &phi but if &phi does not hold then any x satisfies span class nowrap &phi &rarr &theta span . In either case, there is some x such that &phi &rarr &theta . Thus nowrap 1 &exist x &phi &rarr &theta is provable. In intuitionistic logic The principle of independence of premise is not generally valid in intuitionistic logic Avigad and Feferman 1999 . This can illustrated the BHK interpretation , which says that in order to prove nowrap 1 &phi &rarr &exist x &theta intuitionistically, one must create a function that takes a proof of &phi and returns a proof of nowrap &exist x &theta . Here the proof itself is an input to the function and may be used to construct x . On the other hand, a proof of nowrap 1 &exist x &phi &rarr &theta must first demonstrate a particular x , and then provide a function that converts a proof of &phi into a proof of &theta in which x has that particular value. As a weak counterexample , suppose &theta x is some decidable predicate of a natural number such that it is not known whether any x satisfies &theta . For example, &theta may say that x is a formal proof of some mathematical conjecture whose provability is not known. Let &phi the formula nowrap 1 &exist z &theta z . Then nowrap 1 &phi &rarr &exist x &theta is trivially provable. However, to prove nowrap 1 ... more details
Basic Definition and Requirements An axiom P is independent if there are no other axioms Q such that Q implies P. In many cases independence is desired, either to reach the logical consequence conclusion of a reduced set of axioms, or to be able to replace an independent axiom to create a more concise system for example, the parallel postulate is independent of Euclid s Axioms, and can provide interesting results when a negated or manipulated form of the postulate is put into its place . Proving Independence If the original axioms Q are not consistent , then no new axiom is independent. If they are consistent, then P can be shown independent of them if adding P to them, or adding the negation of P, both yield consistent sets of axioms. ref Kenneth Kunen, Set Theory An Introduction to Independence Proofs , page xi. ref For example, Euclid s Axioms, with the parallel postulate included, yields Euclidean geometry, and with the parallel postulate negated, yields non Euclidean spherical or hyperbolic geometry. Both of these are consistent systems, showing that the parallel postulate is independent of the other axioms of geometry. ref Harold Scott Macdonald Coxeter, Non Euclidean Geometry , pages 1 15. ref Proving independence is often very difficult. Forcing mathematics Forcing is one commonly used technique. ref Kenneth Kunen, Set Theory An Introduction to Independence Proofs , pages 184 237. ref Reflist DEFAULTSORT Axiom Independence Category Logic ... more details
Independence School may refer to one or more of the following Baltimore Independence School Baltimore, Maryland Independence High School Independence, Ohio Independence High School Independence, Ohio disambig ... more details
Independence Stadium may also refer to Independence Stadium United States in Louisiana Independence Stadium Bakau in Gambia Independence Stadium Namibia in Windhoek Independence Stadium Zambia in Lusaka Stadium Merdeka Independence Stadium in Malaysia disambig de Independence Stadium fi Independence Stadium ... more details
Independence Rock may refer to Independence Rock Wyoming , a well known landmark on the Oregon Trail Independence Rock Oregon disambig ... more details
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