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Arend Heyting | A Wisdom Archive on Arend Heyting | | Arend Heyting A selection of articles related to Arend Heyting | |
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Arend Heyting
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ARTICLES RELATED TO Arend Heyting | |
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| | |  Arend Heyting: Encyclopedia II - Logic - Nature of logicBecause of its fundamental role in philosophy, the nature of logic has been the object of intense disputation; and it is not possible to give a clear delineation of the bounds of logic in terms acceptable to all rival viewpoints. Nonetheless, the study of logic has, despite this controversy, been very coherent and technically grounded. Here we characterise logic, first by introducing the fundamental ideas about form and then by outlining some of the different schools of thought as well as giving a brief overview of its history, an account of its relationship to other sciences, and--finally--an expositi ...
See also:Logic, Logic - Nature of logic, Logic - Informal formal and symbolic logic, Logic - Rival conceptions of logic, Logic - History of logic, Logic - Relation to other sciences, Logic - Deductive and inductive reasoning, Logic - Topics in logic, Logic - Syllogistic logic, Logic - Predicate logic, Logic - Modal logic, Logic - Deduction and reasoning, Logic - Mathematical logic, Logic - Philosophical logic, Logic - Logic and computation, Logic - Controversies in logic, Logic - Bivalence and the law of the excluded middle, Logic - Implication: strict or material?, Logic - Tolerating the impossible, Logic - Is logic empirical? Read more here: » Logic: Encyclopedia II - Logic - Nature of logic |
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| | | | Arend Heyting: Encyclopedia II - Constructivism mathematics - Constructivist mathematics Constructivist mathematics use constructivist logic, which is essentially a removal of the law of the excluded middle from classical logic. This is not to say that the law of the excluded middle is denied entirely; special cases of the law will be provable as theorems. It is just that the law is not assumed as an axiom. (The law of non-contradiction, on the other hand, is still valid.)
For instance, in Heyting arithmetic, one can prove that for any proposition p which does not contain quantifiers, is a theorem (where See also: Constructivism mathematics, Constructivism mathematics - Constructivist mathematics, Constructivism mathematics - Example from real analysis, Constructivism mathematics - Cardinality, Constructivism mathematics - Attitude of mathematicians, Constructivism mathematics - Mathematicians who have contributed to constructivism, Constructivism mathematics - Branches Read more here: » Constructivism mathematics: Encyclopedia II - Constructivism mathematics - Constructivist mathematics |
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| | | | Arend Heyting: Encyclopedia II - Logic - Nature of logic Because of its fundamental role in philosophy, the nature of logic has been the object of intense dispute: it is not possible clearly to delineate the bounds of logic in terms acceptable to all rival viewpoints. Despite that controversy, the study of logic has been very coherent and technically grounded. In this article, we first characterise logic by introducing fundamental ideas about form, then by outlining some schools of thought, as well as by giving a brief overview of logic's history, an account of its relationship to other sciences, and finally, an exposition of some of logic's essential concepts.
Logic - I ...
See also:Logic, Logic - Nature of logic, Logic - Informal formal and symbolic logic, Logic - Rival conceptions of logic, Logic - History of logic, Logic - Relation to other sciences, Logic - Deductive and inductive reasoning, Logic - Topics in logic, Logic - Syllogistic logic, Logic - Predicate logic, Logic - Modal logic, Logic - Deduction and reasoning, Logic - Mathematical logic, Logic - Philosophical logic, Logic - Logic and computation, Logic - Controversies in logic, Logic - Bivalence and the law of the excluded middle, Logic - Implication: strict or material?, Logic - Tolerating the impossible, Logic - Is logic empirical? Read more here: » Logic: Encyclopedia II - Logic - Nature of logic |
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| | | | Arend Heyting: Encyclopedia II - Heyting algebra - Heyting algebras as applied to intuitionistic logic Arend Heyting (1898-1980) was himself interested in clarifying the foundational status of intuitionistic logic, in introducing this type of structure. The case of Peirce's law illustrates the semantic role of Heyting algebras. No simple proof is known that Peirce's law cannot be deduced from the basic laws of intuitionistic logic.
A Heyting algebra, from the logical standpoint, is essentially a generalization of the usual system of truth values. Amongst other properties, the largest element, called in logic , is analogous to 'true'. T ...
See also:Heyting algebra, Heyting algebra - Formal definitions, Heyting algebra - Properties, Heyting algebra - Examples, Heyting algebra - Heyting algebras as applied to intuitionistic logic Read more here: » Heyting algebra: Encyclopedia II - Heyting algebra - Heyting algebras as applied to intuitionistic logic |
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| | | | Arend Heyting: Encyclopedia II - Logic - Topics in logic Throughout history, there has been interest in distinguishing good from bad arguments, and so logic has been studied in some more or less familiar form. Aristotelian logic has principally been concerned with teaching good argument, and is still taught with that end today, while in mathematical logic and analytical philosophy much greater emphasis is placed on logic as an object of study in its own right, and so l ...
See also:Logic, Logic - Nature of logic, Logic - Informal formal and symbolic logic, Logic - Rival conceptions of logic, Logic - History of logic, Logic - Relation to other sciences, Logic - Deductive and inductive reasoning, Logic - Topics in logic, Logic - Syllogistic logic, Logic - Predicate logic, Logic - Modal logic, Logic - Deduction and reasoning, Logic - Mathematical logic, Logic - Philosophical logic, Logic - Logic and computation, Logic - Controversies in logic, Logic - Bivalence and the law of the excluded middle, Logic - Implication: strict or material?, Logic - Tolerating the impossible, Logic - Is logic empirical? Read more here: » Logic: Encyclopedia II - Logic - Topics in logic |
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| | | | Arend Heyting: Encyclopedia II - Constructivism mathematics - Attitude of mathematicians Traditionally, mathematicians have been suspicious, if not downright antagonistic, towards mathematical constructivism, largely because of the limitations that it poses for constructive analysis. These views were forcefully expressed by David Hilbert in 1928, when he wrote in Die Grundlagen der Mathematik, "Taking the principle of excluded middle from the mathematician would be the same, say, as proscribing the telescope to the astronomer or to the boxer the use of his fists" [1]. (The law of excluded middle is not valid in cons ...
See also:Constructivism mathematics, Constructivism mathematics - Constructivist mathematics, Constructivism mathematics - Example from real analysis, Constructivism mathematics - Cardinality, Constructivism mathematics - Attitude of mathematicians, Constructivism mathematics - Mathematicians who have contributed to constructivism, Constructivism mathematics - Branches Read more here: » Constructivism mathematics: Encyclopedia II - Constructivism mathematics - Attitude of mathematicians |
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| | | | Arend Heyting: Encyclopedia II - Logic - Controversies in logic Just as we have seen there is disagreement over what logic is about, so there is disagreement about what logical truths there are.
Logic - Bivalence and the law of the excluded middle.
Main article: classical logic
The logics discussed above are all "bivalent" or "two-valued"; that is, they are most naturally understood as dividing propositions into the true and the false propositions. Systems which rej ...
See also:Logic, Logic - Nature of logic, Logic - Informal formal and symbolic logic, Logic - Rival conceptions of logic, Logic - History of logic, Logic - Relation to other sciences, Logic - Deductive and inductive reasoning, Logic - Topics in logic, Logic - Syllogistic logic, Logic - Predicate logic, Logic - Modal logic, Logic - Deduction and reasoning, Logic - Mathematical logic, Logic - Philosophical logic, Logic - Logic and computation, Logic - Controversies in logic, Logic - Bivalence and the law of the excluded middle, Logic - Implication: strict or material?, Logic - Tolerating the impossible, Logic - Is logic empirical? Read more here: » Logic: Encyclopedia II - Logic - Controversies in logic |
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