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Heyting
__NOTOC__ Arend Heyting (; 9 May 1898 – 9 July 1980) was a Dutch mathematician and logician. Biography Heyting was a student of Luitzen Egbertus Jan Brouwer at the University of Amsterdam, and did much to put intuitionistic logic on a footing where it could become part of mathematical logic. Heyting gave the first formal development of intuitionistic logic in order to codify Brouwer's way of doing mathematics. The inclusion of Brouwer's name in the Brouwer–Heyting–Kolmogorov interpretation is largely honorific, as Brouwer was opposed in principle to the formalisation of certain intuitionistic principles (and went as far as calling Heyting's work a "sterile exercise"). In 1942 he became a member of the Royal Netherlands Academy of Arts and Sciences. Heyting was born in Amsterdam, Netherlands, and died in Lugano, Switzerland ). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are installed in Bern, while other federal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Algebra
In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of ''implication'' such that (''c'' ∧ ''a'') ≤ ''b'' is equivalent to ''c'' ≤ (''a'' → ''b''). From a logical standpoint, ''A'' → ''B'' is by this definition the weakest proposition for which modus ponens, the inference rule ''A'' → ''B'', ''A'' ⊢ ''B'', is sound. Like Boolean algebras, Heyting algebras form a variety axiomatizable with finitely many equations. Heyting algebras were introduced by to formalize intuitionistic logic. As lattices, Heyting algebras are distributive. Every Boolean algebra is a Heyting algebra when ''a'' → ''b'' is defined as ¬''a'' ∨ ''b'', as is every complete distributive lattice satisfying a one-sided infinite distributive law when ''a'' → ''b'' is taken to be the supremum of the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 interpretati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 interpretati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Arithmetic
In mathematical logic, Heyting arithmetic is an axiomatization of arithmetic in accordance with the philosophy of intuitionism.Troelstra 1973:18 It is named after Arend Heyting, who first proposed it. Axiomatization As with first-order Peano arithmetic , the intended model of this theory are the natural numbers and the theories characterize addition and multiplication. Heyting arithmetic adopts the axioms of Peano arithmetic, including the signature with zero "0" and the successor "S", but uses intuitionistic logic for inference. In particular, the principle of the excluded middle does not hold in general. Metalogic and theorems As with other theories over intuitionistic logic, various instances of can be proven. For instance, proves equality "=" is decidable for all numbers, :\vdash \forall n. \forall m. \big((n = m)\lor\neg(n = m)\big) In fact, since equality is the only predicate symbol in Heyting arithmetic, it then follows that, for any quantifier-free formula \phi, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brouwer–Heyting–Kolmogorov Interpretation
In mathematical logic, the Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, of intuitionistic logic was proposed by L. E. J. Brouwer and Arend Heyting, and independently by Andrey Kolmogorov. It is also sometimes called the realizability interpretation, because of the connection with the realizability theory of Stephen Kleene. It is the standard explanation of intuitionistic logic. The interpretation The interpretation states what is intended to be a proof of a given formula. This is specified by induction on the structure of that formula: *A proof of P \wedge Q is a pair \langle a, b \rangle where a is a proof of P and b is a proof of Q. *A proof of P \vee Q is either \langle 0, a \rangle where a is a proof of P or \langle 1, b\rangle where b is a proof of Q. *A proof of P \to Q is a function f that converts a proof of P into a proof of Q. *A proof of (\exists x S) P is a pair \langle x, a \rangle where x is an element of S and a is a proof of P. *A proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intuitionism
In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality. Truth and proof The fundamental distinguishing characteristic of intuitionism is its interpretation of what it means for a mathematical statement to be true. In Brouwer's original intuitionism, the truth of a mathematical statement is a subjective claim: a mathematical statement corresponds to a mental construction, and a mathematician can ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Luitzen Egbertus Jan Brouwer
Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and complex analysis. Regarded as one of the greatest mathematicians of the 20th century, he is known as the founder of modern topology, particularly for establishing his fixed-point theorem and the topological invariance of dimension. Brouwer also became a major figure in the philosophy of intuitionism, a constructivist school of mathematics which argues that math is a cognitive construct rather than a type of objective truth. This position led to the Brouwer–Hilbert controversy, in which Brouwer sparred with his formalist colleague David Hilbert. Brouwer's ideas were subsequently taken up by his student Arend Heyting and Hilbert's former student Hermann Weyl. Biography Brouwer was born to Dutch Protestant parents. Early in his career, Brou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heyting Field
A Heyting field is one of the inequivalent ways in constructive mathematics to capture the classical notion of a field. It is essentially a field with an apartness relation. A commutative ring is a Heyting field if ¬(0=1), either a or 1-a is invertible for every a, and each noninvertible element is zero. The first two conditions say that the ring is local; the first and third conditions say that it is a field in the classical sense. The apartness relation is defined by writing a # b if a-b is invertible. This relation is often now written as with the warning that it is not equivalent to ¬(a=b). For example, the assumption ¬(a=0) is not generally sufficient to construct the inverse of a, but is. The prototypical Heyting field is the real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirk Van Dalen
Dirk van Dalen (born 20 December 1932, Amsterdam) is a Dutch mathematician and historian of science. Van Dalen studied mathematics and physics and astronomy at the University of Amsterdam. Inspired by the work of Brouwer and Heyting, he received his Ph.D. in 1963 from the University of Amsterdam for the thesis ''Extension problems in intuitionistic plane Projective geometry.'' From 1964 to 1966 Van Dalen taught logic and mathematics at MIT, and later Oxford. From 1967 he was professor at the University of Utrecht. In 2003 Dirk van Dalen was awarded the Academy Medal 2003 of the Royal Dutch Academy of Sciences for bringing the works of Brouwer to international attention. Works * 1958: (with Yehoshua Bar-Hillel and Azriel Levy) ''Foundations of Set Theory'', North Holland Publishing * 1963: Extension problems in intuitionistic plane projective geometry * 1978: (with H.C. Doets and H. De Swart) ''Sets: Naive, Axiomatic and Applied'', Pergamon Press * 1980: ''Logic and Structure'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Logic
Mathematical logic is the study of logic, 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 Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, 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 pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anne Sjerp Troelstra
Anne Sjerp Troelstra (10 August 1939 – 7 March 2019) was a professor of pure mathematics and foundations of mathematics at the Institute for Logic, Language and Computation (ILLC) of the University of Amsterdam. He was a Constructivism (mathematics), constructivist logician, who was influential in the development of intuitionistic logic With Georg Kreisel, he was a developer of the theory of choice sequences. He wrote one of the first texts on linear logic, and, with Helmut Schwichtenberg, he co-wrote an important book on proof theory. He became a member of the Royal Netherlands Academy of Arts and Sciences in 1976. Troelstra died on 7 March 2019. Notes External linksHomepage of A. S. Troelstra: Dead Link - Archived : Retrieved on 27 June 2018 * 1939 births 2019 deaths Dutch mathematicians Members of the Royal Netherlands Academy of Arts and Sciences People from De Bilt University of Amsterdam alumni University of Amsterdam faculty 20th-century Dutch people {{europ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logician
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises in a topic-neutral way. When used as a countable noun, the term "a logic" refers to a logical formal system that articulates a proof system. Formal logic contrasts with informal logic, which is associated with informal fallacies, critical thinking, and argumentation theory. While there is no general agreement on how formal and informal logic are to be distinguished, one prominent approach associates their difference with whether the studied arguments are expressed in formal or informal languages. Logic plays a central role in multiple fields, such as philosophy, mathematics, computer science, and linguistics. Logic studies arguments, which consist of a set of premises together with a conclusion. Premises and conclusions are usually und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
