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Topic: Intuitionistic logic

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	Logic - Wikipedia, the free encyclopedia
		The ambiguity is that "formal logic" is very often used with the alternate meaning of symbolic logic as we have defined it, with informal logic meaning any logical investigation that does not involve symbolic abstraction; it is this sense of 'formal' that is parallel to the received usages coming from "formal languages" or "formal theory".
		The boldest attempt to apply logic to mathematics was undoubtedly the logicism pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programme was to show this by means to a reduction of mathematics to logic.
		Again, relevance logic and dialetheism are the most important approaches here, though the concerns are different: the key issue that classical logic and some of its rivals, such as intuitionistic logic have is that they respect the principle of explosion, which means that the logic collapses if it is capable of deriving a contradiction.
	en.wikipedia.org /wiki/Logic (3434 words)

	Encyclopedia: Intuitionistic logic (Site not responding. Last check: 2007-11-07)
		Logic programming is a declarative programming paradigm in which a set of attributes that a solution should have are specified rather than set of steps to obtain such a solution.
		A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from a Boolean algebra, one uses values from a Heyting algebra, of which Boolean algebras are a special case.
		Intermediate logics are intermediate between intuitionistic logic and classical logic in the sense that they contain theorems that are not provable in intuitionistic logic, without giving rise to the whole of classical logic.
	www.nationmaster.com /encyclopedia/Intuitionistic-logic (2306 words)

	Intuitionistic logic - Open Encyclopedia (Site not responding. Last check: 2007-11-07)
		In order to formalize intuitionistic logic in a mathematically precise way, both a model theory (by semantics) and an appropriate proof theory are needed.
		The syntax of formulæ of intuitionistic logic is similar to propositional logic or first-order logic.
		A more familiar example of a classical tautology which is invalid in intuitionistic logic concerns the so-called double negation elimination.
	www.open-encyclopedia.com /Intuitionistic_logic (1031 words)

	Intuitionistic logic - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-07)
		More generally, the formula P ∨ ¬P is not a theorem of intuitionistic logic as it is of classical logic.
		In intuitionistic logic, only the first is a theorem: Double negation can be introduced, but it cannot be eliminated.
		Intuitionistic logic, model theory and forcing (Studies in logic and the foundations of mathematics)
	encyclopedia.worldsearch.com /intuitionistic_logic.htm (1133 words)

	Intuitionistic logic -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-07)
		Intuitionistic logic, or constructivist logic, is the (The branch of philosophy that analyzes inference) logic used in (Click link for more info and facts about mathematical intuitionism) mathematical intuitionism and other forms of (Click link for more info and facts about mathematical constructivism) mathematical constructivism.
		In a stricter sense, intuitionistic logic can be investigated as a very concrete and formal kind of (Any logical system that abstracts the form of statements away from their content in order to establish abstract criteria of consistency and validity) mathematical logic.
		The (The grammatical arrangement of words in sentences) syntax of formulæ of intuitionistic logic is similar to (A branch of symbolic logic dealing with propositions as units and with their combinations and the connectives that relate them) propositional logic or (Click link for more info and facts about first-order logic) first-order logic.
	www.absoluteastronomy.com /encyclopedia/I/In/Intuitionistic_logic.htm (1424 words)

	Intuitionistic logic (Site not responding. Last check: 2007-11-07)
		Intuitionistic logic, or constructivist logic, is the logic used in mathematicalintuitionism and other forms of mathematicalconstructivism.
		In classical logic, P ∨ ¬P means that oneof P or ¬P is true; in intuitionistic logic, P ∨ ¬P means that one ofP or ¬P can be proved, which is a much stronger statement, and which might not always be the case.
		Intuitionistic logic gave philosophical support to several schools of philosophy, most notably the Anti-realism of MichaelDummett.
	www.therfcc.org /intuitionistic-logic-25481.html (986 words)

	Intuitionistic logic (Site not responding. Last check: 2007-11-07)
		In intuitionistic logic it is permitted to assert a disjunction such as P ∨ ¬ P without also being able to say which one is true.
		In logic P ∨ ¬ P means that one of P or ¬ P is true ; in intuitionistic logic P ∨ ¬ P means that one of P or ¬ P can be proved which is a much stronger statement which might not always be the case.
		In classical logic ¬ P asserts that P is false; in intuitionistic logic ¬ P asserts that a proof of P is impossible.
	www.freeglossary.com /Intuitionistic_logic (1083 words)

	Intuitionistic logic (Site not responding. Last check: 2007-11-07)
		Intuitionistic Logic is the logical branch of Mathematical intuitionism.
		Instead, logic and math are the application of internally consistent methods to realize more complex mental constructs (really, a kind of game).
		In intuitionistic logic, epistemologically unclear steps in proofs are forbidden; in particular, the law of the excluded middle isn't valid because, in a logical calculus that allows it, it's possible to argue P ∨ ¬P without knowing which one specifically is the case.
	www.termsdefined.net /in/intuitionistic-logic.html (675 words)

	Intuitionistic Logic
		Intuitionistic logic encompasses the principles of logical reasoning which were used by L. Brouwer in developing his intuitionistic mathematics, beginning in [1907].
		Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics.
		While identity can of course be added to intuitionistic logic, for applications (e.g., to arithmetic) the equality symbol is generally treated as a distinguished predicate constant satisfying nonlogical axioms (e.g., the primitive recursive definitions of addition and multiplication) in addition to reflexivity, symmetry and transitivity.
	plato.stanford.edu /entries/logic-intuitionistic (6042 words)

	Intuitionistic logic (Site not responding. Last check: 2007-11-07)
		As an example of this difference law of the excluded middle while classically valid is not intuitionistically because in a logical calculus that allows it's possible to argue P ∨ ¬ P without knowing which one specifically is case.
		Intuitionistic logic gave philosophical support to several of philosophy most notably the Anti-realism of Michael Dummett.
		From a practical point of view there also a strong motivation for using intuitionistic Indeed if one goes for automated reasoning in logical programming then one obviously is not interested mere statements of existence.
	www.freeglossary.com /Intuitionistic (1083 words)

	CS 611, v3.0: Intuitionistic logic and the Halting Problem
		[Intuitionistic logic, intuitionistic logic at FOLDOC] Our discussions of a "Turing machine either halts or it doesn't, but I don't know which one" has lead to some disagreements of the form "yeah, but if you don't know which one then you don't know that it either halts or it doesn't.
		In intuitionistic logic, you don't get A \/ ~A as an axiom (like you do in classical logic, which is the logic we all learned).
		In intuitionistic logic, you get A \/ ~A if you can justify, or prove, A or you can justify, or prove, ~A. Until you have a proof of either A or ~A you only know that A \/ ~A is "undetermined." The language in the above link parallels this discussion very closely.
	lal.cs.byu.edu /cs611/archives/2003/09/intuitionistic.html (230 words)

	Abstracts (Site not responding. Last check: 2007-11-07)
		We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a self-dual logic; constructible duality.
		Truth is judged forward in the Kripke model, as in intuitionistic logic, while falsity is judged backwards, that is forward in the dual model.
		In particular, we show that this algebra is an instantiation of the $\Chu$ construction applied to a pseudo-Boolean algebra, the second Dialectica construction applied to a pseudo-Boolean algebra, and as posetal case of a suggestion by Pitts for the study of recursion and corecursion.
	www-formal.stanford.edu /annap/www/abstracts.html (1829 words)

	COMPUTABILITY LOGIC: a theory of interactive computation HOMEPAGE
		Technically CL is a game logic: it understands interactive computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of "always computable" problems.
		Correspondingly, classical logic is nothing but a special fragment of CL. One of the main -- so far rather abstract -- intuitions associated with intuitionistic logic is that it must be a logic of problems (Kolmogorov 1932); this is exactly what CL is, only in a much more expressive language than intuitionistic logic.
		The same appears to be the case for intuitionistic and linear logics (understood in a broad sense and not necessarily identified with the particular known axiomatic systems).
	www.cis.upenn.edu /~giorgi/cl.html (3951 words)

	Ksenija Simic's Web Page (Site not responding. Last check: 2007-11-07)
		Intuitionistic (or constructive) logic rejects the claim that a statement that is not false is true (or equivalently, that every statement is either true or false), a claim that we take for granted in mathematics, though not necessarily in everyday reasoning.
		Intuitionistic logic is often used in theoretical computer science.
		After this, we would explore the differences between classical and intuitionistic logic, find examples of facts that hold in one, but not the other, using tools of mathematical logic such as formal systems and model theory.
	math.arizona.edu /~ksimic/uresearch.html (252 words)

	Intuitionistic Logic in Business Systems (Site not responding. Last check: 2007-11-07)
		The purpose of this paper is to discuss how intuitionistic logic can be applied to develop and extensible reasoning system which supports the ideas of argument and refutation as a more natural way to address situational differences.
		Brouwer, the originator of intuitionistic logic, held that mathematical objects such as numbers, sets, functions, and so on exist because we construct them or show them the means for their construction.
		Intuitionistic logic rejects the law of exclude middle which says that every proposition P can be either confirmed or refuted.
	sern.ucalgary.ca /~moussavm/693/intuition.htm (1528 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		We also show how intuitionistic logic may be embedded in intuitionistic (and classical) linear logic, proving that we gain structure without a loss of expressive power.
		Logic programming in intuitionistic logic - Goal-directed proof-search - uniform proofs (uniformity/focusing) - the TOP, AND, IMPLIES, FORALL fragment - soundness and non-deterministic completeness - resolution - hereditary Harrop formulas - examples (lambda-Prolog) - further issues 3.
		A linear decoration of a proof in IL or CL is a proof in the corresponding linear logic, with the same structure as the original proof.
	www.cs.cmu.edu /afs/cs/user/fp/courses/95-linear/lectures.txt (2435 words)

	Intuitionistic logic (Site not responding. Last check: 2007-11-07)
		A valid sentence is then one which has valuation 1 in any valuation on any Heyting algebra.
		It can be shown that we need in fact consider only the Heyting algebra given by the open sets of the real plane with its usual topology — intuitionistic validities correspond precisely to Heyting formulae which evaluate to the entire plane for any assignment of open subsets to the variables.
		For example, the law of the excluded middle can then easily be seen not to be valid — let A be the strict upper plane, {(x,y)
	www.sciencedaily.com /encyclopedia/intuitionistic_logic (1044 words)

	Intuitionistic Logic (Site not responding. Last check: 2007-11-07)
		Intuitionists conclude that the meaning of a statement resides not in its truth conditions but in the means of proof or verification.
		In classical logic, disjunctive formulas of the form P∨¬P are provable without providing a proof of either P or of ¬P and some formulas of the form ∃x.F(x) (such as ∃x∀y(p(x) -> p(y)) is provable without providing a proof of F(t) for some particular t.
		Hence intuitionistic logic can be used for program synthesis.
	cs.wwc.edu /~aabyan/Logic/Intuitionistic.html (502 words)

	Introduction to Linear Logic (Site not responding. Last check: 2007-11-07)
		Linear Logic was introduced by J.-Y. Girard in 1987 and it has attracted much attention from computer scientists, as it is a logical way of coping with resources and resource control.
		Cut-elimination for Classical Logic is highly non-deterministic; it is shown how this can be remedied either by moving to Intuitionistic Logic or to Linear Logic.
		A Digression - Russell's Paradox and Linear Logic
	www.brics.dk /LS/96/6/BRICS-LS-96-6/BRICS-LS-96-6.html (210 words)

	Intuitionistic logic - Information
		Looking For intuitionistic logic - Find intuitionistic logic and more at Lycos Search.
		Find intuitionistic logic - Your relevant result is a click away!
		The meet and join operations in the Boolean algebra are identified with the ∧ and ∨ logical connectives, so that the value of a formula of the form A ∧ B is the meet of the value of A and the value of B in the Boolean algebra.
	www.logicjungle.com /wiki/Intuitionistic_logic (1294 words)

	Open Directory - Science: Math: Logic and Foundations: Nonstandard Logics and Extensions: Intuitionistic Logic (Site not responding. Last check: 2007-11-07)
		Intuitionistic Logic - A very concise introduction to the subject.
		Intuitionistic Logic - A short entry in the Stanford Encyclopaedia of Philosophy by Joan R. Moschovakis.
		Intuitionistic Logic - A very brief overview of the subject by Alex Sakharov from MathWorld.
	dmoz.org /Science/Math/Logic_and_Foundations/Nonstandard_Logics_and_Extensions/Intuitionistic_Logic (189 words)

	Wadler: Linear Logic
		This paper introduces a new way of attaching proof terms to proof trees for classical linear logic, which bears a close resemblance to the way that pattern matching is used in programming languages.
		The presentation of linear logic is simplified by basing it on the Logic of Unity.
		Surprisingly, there is not a good fit between a syntax for linear logic in the style of Abramsky, and a semantics in the style of Seely.
	homepages.inf.ed.ac.uk /wadler/topics/linear-logic.html (1065 words)

	A Taste of Intuitionistic Type Theory (Site not responding. Last check: 2007-11-07)
		In intuitionistic logic existential witnesses can be effectively computed from proofs, but this is an external procedure - you cannot reason about proofs or witnesses.
		Thus Type Theory is at the same time a logic and a programming language with a quite sophisticated type system.
		The course includes an overview of intuitionistic logic and its constructive semantics (proposed by Brouwer, Heyting and Kolmogorov).
	www.cs.nott.ac.uk /~txa/itt (189 words)

	Amazon.co.uk: Books: A Short Introduction to Intuitionistic Logic (University Series in Mathematics) (Site not responding. Last check: 2007-11-07)
		Intuitionistic logic is presented here as part of familiar classical logic which allows mechanical extraction of programs from proofs to make the material more accessible.
		Basic tchniques are presented first for propositional logic, then part II inroduces extensions to predicate logic.
		The presentation is based on natural deduction and the topics include programming; interpretation of intuitionistic logic by simply typed lambda-calculus; (Curry-Howard isomorphism); negative translation of classical into intuitionistic logic; normalization of natural deductions; applications to category theory; Kripke models; lagebraic and topological semantics; proof-search methods; and interpolation theorem.
	www.amazon.co.uk /exec/obidos/ASIN/0306463946 (354 words)

	Information and Computation Bibliography (Site not responding. Last check: 2007-11-07)
		When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses.
		While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic.
		Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting.
	theory.lcs.mit.edu /~iandc/References/hodasm1994:327.html (601 words)

	Citations: Combining classical and intuitionistic logic - Cerro, Herzig (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		It should be noted that it is necessary to be able to impose requirements in inference rules in order to be able to constrain their use.
		The following example presents the propositional intuitionistic logic system presentation and shows what happens when we fibre it with the classical propositional logic system....
		Fibring Of Logics As A Universal Construction - Caleiro, Carnielli, Rasga..
	citeseer.ist.psu.edu /context/849761/414270 (598 words)

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