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Meta-theorem
In logic, a metatheorem is a statement about a formal system proven in a metalanguage. Unlike theorems proved within a given formal system, a metatheorem is proved within a metatheory, and may reference concepts that are present in the metatheory but not the object theory. A formal system is determined by a formal language and a deductive system (axioms and rules of inference). The formal system can be used to prove particular sentences of the formal language with that system. Metatheorems, however, are proved externally to the system in question, in its metatheory. Common metatheories used in logic are set theory (especially in model theory) and primitive recursive arithmetic (especially in proof theory). Rather than demonstrating particular sentences to be provable, metatheorems may show that each of a broad class of sentences can be proved, or show that certain sentences cannot be proved. Examples Examples of metatheorems include: * The deduction theorem for first-order logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Deduction Theorem
In mathematical logic, a deduction theorem is a metatheorem that justifies doing conditional proofs from a hypothesis in systems that do not explicitly axiomatize that hypothesis, i.e. to prove an implication A \to B, it is sufficient to assume A as a hypothesis and then proceed to derive B. Deduction theorems exist for both propositional logic and first-order logic. The deduction theorem is an important tool in Hilbert-style deduction systems because it permits one to write more comprehensible and usually much shorter proofs than would be possible without it. In certain other formal proof systems the same conveniency is provided by an explicit inference rule; for example natural deduction calls it implication introduction. In more detail, the propositional logic deduction theorem states that if a formula B is deducible from a set of assumptions \Delta \cup \ then the implication A \to B is deducible from \Delta ; in symbols, \Delta \cup \ \vdash B implies \Delta \vdash A \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. Informal logic examines arguments expressed in natural language whereas formal logic uses formal language. When used as a countable noun, the term "a logic" refers to a specific logical formal system that articulates a proof system. Logic plays a central role in many fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Quantifier (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 most commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic: each can be defined in terms of the other 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Metalogic
Metalogic is the metatheory of logic. Whereas ''logic'' studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived ''about'' the languages and systems that are used to express truths. The basic objects of metalogical study are formal languages, formal systems, and their interpretations. The study of interpretation of formal systems is the branch of mathematical logic that is known as model theory, and the study of deductive systems is the branch that is known as proof theory. Overview Formal language A ''formal language'' is an organized set of symbols, the symbols of which precisely define it by shape and place. Such a language therefore can be defined without reference to the meanings of its expressions; it can exist before any interpretation is assigned to it—that is, befo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Geoffrey Hunter (logician)
Geoffrey Basil Bailey Hunter (14 December 1925 – 8 June 2000) was a British professor, philosopher, and logician. Hunter was Professor Emeritus of the University College of Wales, Bangor, where he was professor from 1978 until he retired in 1992. He also taught at Queen's University Kingston, Ontario (1950–1952) and was a lecturer in Philosophy at the University of Leeds (1952–1965), and reader in Logic at the University of St Andrews (1965–1978). Geoffrey was probably most known for his work titled ''Metalogic: An Introduction to the Metatheory of Standard First-Order Logic'', published in 1971. Published work *Hunter, Geoffrey (1971). "Metalogic: An Introduction to the Metatheory of Standard First Order Logic". Macmillan *Hunter, Geoffrey (1973). "Not Both P and not Q, therefore if P then Q" is not a valid form of argument Mind vol LXXXII:280-280 *Hunter, Geoffrey (1974). "Concepts and Meaning." in Hume and the Enlightenment: essays presented to Ernest Campbell M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Use–mention Distinction
In analytic philosophy, a fundamental distinction is made between the use of a term and the mere mention of it.Devitt and Sterelny (1999) pp. 40–1. W. V. O. Quine (1940) p. 24. Many philosophical works have been "vitiated by a failure to distinguish use and mention." The distinction can sometimes be pedantic, especially in simple cases where it is obvious. The distinction between use and mention can be illustrated with the word "cheese": # Use: Cheese is derived from milk. # Mention: "Cheese" is derived from the Old English word . The first sentence is a statement about the substance called "cheese": it the word "cheese" to refer to that substance. The second is a statement about the word "cheese" as a signifier: it the word without using it to refer to anything other than itself. Overview In written language, words or phrases often appear between single or double quotation marks or in italics. In philosophy, single quotation marks are typically used, while in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheory, metatheories, which are Mathematical theory, mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to David Hilbert's Hilbert program, attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic" (Kleene 1952, p. 59). 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. History Metamathematical metatheorems about mathematics itself were or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Completeness (logic)
In mathematical logic and metalogic, a formal system is called complete with respect to a particular property if every formula having the property can be derived using that system, i.e. is one of its theorems; otherwise the system is said to be incomplete. The term "complete" is also used without qualification, with differing meanings depending on the context, mostly referring to the property of semantical validity. Intuitively, a system is called complete in this particular sense, if it can derive every formula that is true. Other properties related to completeness The property converse to completeness is called soundness: a system is sound with respect to a property (mostly semantical validity) if each of its theorems has that property. Forms of completeness Expressive completeness A formal language is ''expressively complete'' if it can express the subject matter for which it is intended. Functional completeness A set of logical connectives associated with a formal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Gödel's Completeness Theorem
Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantics, semantic truth and syntactic Provability logic, provability in first-order logic. The completeness theorem applies to any first-order Theory (mathematical logic), theory: If ''T'' is such a theory, and φ is a sentence (in the same language) and every model of ''T'' is a model of φ, then there is a (first-order) proof of φ using the statements of ''T'' as axioms. One sometimes says this as "anything true in all models is provable". (This does not contradict Gödel's incompleteness theorem, which is about a formula φu that is unprovable in a certain theory ''T'' but true in the "standard" model of the natural numbers: φu is false in some other, "non-standard" models of ''T''.) The completeness theorem makes a close link between model theory, which deals with what is true in different models, and proof theory, which studies what can be formally prov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Peano Arithmetic
In mathematical logic, the Peano axioms (, ), also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete. The axiomatization of arithmetic provided by Peano axioms is commonly called Peano arithmetic. The importance of formalizing arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Consistency Proof
In deductive logic, a consistent theory (mathematical logic), theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no Formula (mathematical logic), formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences of T. Let A be a set of Closed-form expression, closed sentences (informally "axioms") and \langle A\rangle the set of closed sentences provable from A under some (specified, possibly implicitly) formal deductive system. The set of axioms A is consistent when there is no formula \varphi such that \varphi \in \langle A \rangle and \lnot \varphi \in \langle A \rangle. A ''trivial'' theory (i.e., one which proves every sentence in the language of the theory) is clearly inconsistent. Conversely, in an principle of explosion, explosive formal system (e.g., classical or intuitionistic propositional or first-order logics) every inconsistent theory is trivial. Consistency of a theory is a sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |