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Predicate (mathematics)
In mathematical logic , a PREDICATE is commonly understood to be a Boolean-valued function P: X→ {true, false}, called the predicate on X. However, predicates have many different uses and interpretations in mathematics and logic, and their precise definition, meaning and use will vary from theory to theory. So, for example, when a theory defines the concept of a relation , then a predicate is simply the characteristic function or the indicator function of a relation. However, not all theories have relations, or are founded on set theory , and so one must be careful with the proper definition and semantic interpretation of a predicate. CONTENTS * 1 Simplified overview * 2 Formal definition * 3 See also * 4 References * 5 External links SIMPLIFIED OVERVIEWInformally, a PREDICATE is a statement that may be true or false depending on the values of its variables
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Formal System
A FORMAL SYSTEM or LOGICAL CALCULUS is any well-defined system of abstract thought based on the model of mathematics . A formal system need not be mathematical as such; for example, Spinoza's Ethics imitates the form of Euclid's Elements.. Spinoza employed Euclidiean elements such as "axioms" or "primitive truths", rules of inferences etc. so that a calculus can be built using these. For nature of such primitive truths, one can consult Tarski's "Concept of truth for a formalized language". Some theorists use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style of notation , for example, Paul Dirac 's bra–ket notation
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Formation Rule
In mathematical logic , FORMATION RULES are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. These rules only address the location and manipulation of the strings of the language. It does not describe anything else about a language, such as its semantics (i.e. what the strings mean). (See also formal grammar ). CONTENTS * 1 Formal language
Formal language
* 2 Formal systems * 3 Propositional and predicate logic * 4 See also FORMAL LANGUAGE Main article: Formal language
Formal language
A formal language is an organized set of symbols the essential feature being that it can be precisely defined in terms of just the shapes and locations of those symbols
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Deductive System
A FORMAL SYSTEM or LOGICAL CALCULUS is any well-defined system of abstract thought based on the model of mathematics . A formal system need not be mathematical as such; for example, Spinoza's Ethics imitates the form of Euclid's Elements.. Spinoza employed Euclidiean elements such as "axioms" or "primitive truths", rules of inferences etc. so that a calculus can be built using these. For nature of such primitive truths, one can consult Tarski's "Concept of truth for a formalized language". Some theorists use the term formalism as a rough synonym for formal system, but the term is also used to refer to a particular style of notation , for example, Paul Dirac 's bra–ket notation
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Formal Proof
A FORMAL PROOF or DERIVATION is a finite sequence of sentences (called well-formed formulas in the case of a formal language ), each of which is an axiom , an assumption, or follows from the preceding sentences in the sequence by a rule of inference . The last sentence in the sequence is a theorem of a formal system . The notion of theorem is not in general effective , therefore there may be no method by which we can always find a proof of a given sentence or determine that none exists. The concept of natural deduction is a generalization of the concept of proof. The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof. For a well-formed formula to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus of some formal system to the previous well-formed formulae in the proof sequence. Formal proofs often are constructed with the help of computers in interactive theorem proving
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Formal Semantics (logic)
In logic , the SEMANTICS OF LOGIC is the study of the semantics , or interpretations , of formal and (idealizations of) natural languages usually trying to capture the pre-theoretic notion of entailment . OVERVIEWThe truth conditions of various sentences we may encounter in arguments will depend upon their meaning, and so logicians cannot completely avoid the need to provide some treatment of the meaning of these sentences. The semantics of logic refers to the approaches that logicians have introduced to understand and determine that part of meaning in which they are interested; the logician traditionally is not interested in the sentence as uttered but in the proposition , an idealised sentence suitable for logical manipulation
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Formal Language
In mathematics , computer science , and linguistics , a FORMAL LANGUAGE is a set of strings of symbols together with a set of rules that are specific to it. The alphabet of a formal language is the set of symbols, letters, or tokens from which the strings of the language may be formed. The strings formed from this alphabet are called words, and the words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas . A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar , also called its formation rule . The field of FORMAL LANGUAGE THEORY studies primarily the purely syntactical aspects of such languages—that is, their internal structural patterns. Formal language
Formal language
theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages
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Special
SPECIAL or SPECIALS may refer to: CONTENTS * 1 Music * 2 Film and television * 3 Other uses * 4 See also MUSIC * Special (album) , a 1992 album by Vesta Williams * "Special" (Garbage song) , 1998 * "Special" (Mew song) , 2005 * "Special" (Stephen Lynch song) , 2000 * The Specials
The Specials
, a British band * "Special", a song by Violent Femmes on The Blind Leading the Naked * "Special", a song on
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Opaque Predicate
In computer programming , an OPAQUE PREDICATE is a predicate —an expression that evaluates to either "true" or "false"—for which the outcome is known by the programmer a priori, but which, for a variety of reasons, still needs to be evaluated at run time . Opaque predicates have been used as watermarks , as it will be identifiable in a program's executable
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Classifying Topos
In mathematics , a CLASSIFYING TOPOS for some sort of structure is a topos T such that there is a natural equivalence between geometric morphisms from a cocomplete topos E to T and the category of models for the structure in E
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Binary Relation
In mathematics , a BINARY RELATION on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product
Cartesian product
A2 = A × A. More generally, a binary relation between two sets A and B is a subset of A × B. The terms CORRESPONDENCE, DYADIC RELATION and 2-PLACE RELATION are synonyms for binary relation. An example is the "divides " relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p (but with no integer that is not a multiple of p). In this relation, for instance, the prime 2 is associated with numbers that include −4, 0, 6, 10, but not 1 or 9; and the prime 3 is associated with numbers that include 0, 6, and 9, but not 4 or 13
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International Standard Book Number
The INTERNATIONAL STANDARD BOOK NUMBER (ISBN) is a unique numeric commercial book identifier. 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. 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 (SBN) created in 1966. The 10-digit ISBN format was developed by the International Organization for Standardization (ISO) and was published in 1970 as international standard ISO 2108 (the SBN code can be converted to a ten digit ISBN by prefixing it with a zero)
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Well-formed Formula
In mathematical logic , propositional logic and predicate logic , a WELL-FORMED FORMULA, abbreviated WFF or WFF, often simply FORMULA, is a finite sequence of symbols from a given alphabet that is part of a formal language . A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic. CONTENTS * 1 Introduction * 2 Propositional calculus * 3 Predicate logic * 4 Atomic and open formulas * 5 Closed formulas * 6 Properties applicable to formulas * 7 Usage of the terminology * 8 See also * 9 Notes * 10 References * 11 External links INTRODUCTIONA key use of formulas is in propositional logic and predicate logics such as first-order logic
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Set (mathematics)
In mathematics , a SET is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education , elementary topics such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree. The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano
Bernard Bolzano
in his work The Paradoxes of the Infinite
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Logical Consequence
LOGICAL CONSEQUENCE (also ENTAILMENT) 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 , because the conclusions are consequences 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? 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 logical consequence of a set of sentences, for a given language , if and only if , using only logic (i.e
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Theorem
In mathematics , a THEOREM is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms . A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system . The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive , in contrast to the notion of a scientific law , which is experimental . Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called HYPOTHESES or premises
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