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Intuitionistic Logic
Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems of intuitionistic logic do not assume the law of the excluded middle and double negation elimination, which are fundamental inference rules in classical logic. Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for L. E. J. Brouwer's programme of intuitionism. From a proof-theoretic perspective, Heyting’s calculus is a restriction of classical logic in which the law of excluded middle and double negation elimination have been removed. Excluded middle and double negation elimination can still be proved for some propositions on a case by case basis, however, but do not hold universally as they do with classical logic. The standard explanation of intuitionistic logic is the BHK inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Symbolic Logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Realizability
In mathematical logic, realizability is a collection of methods in proof theory used to study constructive proofs and extract additional information from them. Formulas from a formal theory are "realized" by objects, known as "realizers", in a way that knowledge of the realizer gives knowledge about the truth of the formula. There are many variations of realizability; exactly which class of formulas is studied and which objects are realizers differ from one variation to another. Realizability can be seen as a formalization of the BHK interpretation of intuitionistic logic; in realizability the notion of "proof" (which is left undefined in the BHK interpretation) is replaced with a formal notion of "realizer". Most variants of realizability begin with a theorem that any statement that is provable in the formal system being studied is realizable. The realizer, however, usually gives more information about the formula than a formal proof would directly provide. Beyond giving insight ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Mathematical Constructivism
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories such as CZF ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Existence Property
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005). Disjunction property The disjunction property is satisfied by a theory if, whenever a sentence ''A'' ∨ ''B'' is a theorem, then either ''A'' is a theorem, or ''B'' is a theorem. Existence property The existence property or witness property is satisfied by a theory if, whenever a sentence is a theorem, where ''A''(''x'') has no other free variables, then there is some term ''t'' such that the theory proves . Related properties Rathjen (2005) lists five properties that a theory may possess. These include the disjunction property (DP), the existence property (EP), and three additional properties: * The numerical existence property (NEP) states that if the theory proves (\exists x \in \mathbb)\varphi(x), where ''φ'' has no other free variables, then the theory proves \varphi(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory). Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century. Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic. Life Early li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Brouwer–Hilbert Controversy
In a controversy over the foundations of mathematics, in twentieth-century mathematics, L. E. J. Brouwer, a proponent of the constructivist school of intuitionism, opposed David Hilbert, a proponent of formalism. The debate concerned fundamental questions about the consistency of axioms and the role of semantics and syntax in mathematics. Much of the controversy took place while both were involved with the prestigious '' Mathematische Annalen'' journal, with Hilbert as editor-in-chief and Brouwer as a member of its editorial board. In 1920 Hilbert succeeded in having Brouwer, whom he considered a threat to mathematics, removed from the editorial board of Mathematische Annalen, the leading mathematical journal of the time. Background The background for the controversy was set with David Hilbert's axiomatization of geometry in the late 1890s. In his biography of Kurt Gödel, John W. Dawson, Jr summarizes the result as follows: "At issue in the sometimes bitter disputes was t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Constructivism (mathematics)
In the philosophy of mathematics, constructivism asserts that it is necessary to find (or "construct") a specific example of a mathematical object in order to prove that an example exists. Contrastingly, in classical mathematics, one can prove the existence of a mathematical object without "finding" that object explicitly, by assuming its non-existence and then deriving a contradiction from that assumption. Such a proof by contradiction might be called non-constructive, and a constructivist might reject it. The constructive viewpoint involves a verificational interpretation of the existential quantifier, which is at odds with its classical interpretation. There are many forms of constructivism. These include the program of intuitionism founded by Brouwer, the finitism of Hilbert and Bernays, the constructive recursive mathematics of Shanin and Markov, and Bishop's program of constructive analysis. Constructivism also includes the study of constructive set theories suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Theory Of Justification
Justification (also called epistemic justification) is the property of belief that qualifies it as knowledge rather than mere opinion. Epistemology is the study of reasons that someone holds a rationally admissible belief (although the term is also sometimes applied to other propositional attitudes such as doubt). Epistemologists are concerned with various epistemic features of belief, which include the ideas of warrant (a proper justification for holding a belief), knowledge, rationality, and probability, among others. Debates surrounding epistemic justification often involve the ''structure'' of justification, including whether there are foundational justified beliefs or whether mere coherence is sufficient for a system of beliefs to qualify as justified. Another major subject of debate is the sources of justification, which might include perceptual experience (the evidence of the senses), reason, and authoritative testimony, among others. Justification and knowledge "Justi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Curry–Howard Correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and the logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Inhabited Set
In constructive mathematics, a set A is inhabited if there exists an element a \in A. In classical mathematics, this is the same as the set being nonempty; however, this equivalence is not valid in intuitionistic logic (or constructive logic). Comparison with nonempty sets In classical mathematics, a set is inhabited if and only if it is not the empty set. These definitions diverge in constructive mathematics, however. A set A is if \forall z (z \not \in A) while A is if it is not empty, that is, if \lnot forall z (z \not \in A) It is if \exists z (z \in A). Every inhabited set is a nonempty set (because if a \in A is an inhabitant of A then a \not\in A is false and consequently so is \forall z (z \not \in A)). In intuitionistic logic, the negation of a universal quantifier is weaker than an existential quantifier, not equivalent to it as in classical logic so a nonempty set is not automatically guaranteed to be inhabited. Example Because inhabited sets are the sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Evidence
Evidence for a proposition is what supports this proposition. It is usually understood as an indication that the supported proposition is true. What role evidence plays and how it is conceived varies from field to field. In epistemology, evidence is what justifies beliefs or what makes it rational to hold a certain doxastic attitude. For example, a perceptual experience of a tree may act as evidence that justifies the belief that there is a tree. In this role, evidence is usually understood as a private mental state. Important topics in this field include the questions of what the nature of these mental states is, for example, whether they have to be propositional, and whether misleading mental states can still qualify as evidence. In phenomenology, evidence is understood in a similar sense. Here, however, it is limited to intuitive knowledge that provides immediate access to truth and is therefore indubitable. In this role, it is supposed to provide ultimate justifications fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Truth Value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or ''false''). Computing In some programming languages, any expression can be evaluated in a context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero, the empty string, empty lists, and null evaluate to false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called "truthy" and "falsy" / "false". Classical logic In classical logic, with its intended semantics, the truth values are '' true'' (denoted by ''1'' or the verum ⊤), and '' untrue'' or ''false'' (denoted by ''0'' or the falsum ⊥); that is, classical logic is a two-valued logic. This set of two values is also called the Boolean domain. Corresponding semantic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |