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Topic: Higher order logic

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	Leivant. Higher Order Logic. (Site not responding. Last check: 2007-08-19)
		Finite order logic (omega-order logic, or type theory) [Church 1940] is introduced in the form of a relational variant [Schutte 1960] which is essentially Andrews F-omega.
		In first order, we have Fraisse's Theorem which states that properties of a model are first order definable iff "it can be recognized by a computation with a finite number of alternations between existential (nondeterministic) and universal (co-nondeterministic) guesses" [compare this to "finitely isomorphic" iff elementary equivalence for finite symbol sets].
		It follows that full second order logic cannot be interpreted in weak second order logic, and that f-validity of a second order formula is reducible to truth in N of a second order formula.
	www.andrew.cmu.edu /~cebrown/notes/leivant.html (4896 words)

	Higher-order logic - Wikipedia, the free encyclopedia
		In mathematics, higher-order logic is distinguished from first-order logic in a number of ways.
		One of these is the scope of quantifiers; in first-order logic, roughly speaking, it is forbidden to quantify over predicates.
		Another way in which higher-order logic differs from first-order logic is in the constructions allowed in the underlying type theory.
	www.encyclopedia-online.info /Higher_order_logic (149 words)

	Logic, Higher-order (Site not responding. Last check: 2007-08-19)
		Of course, first-order logic is very strong and it is possible to encode such a statement into it.
		This interpretation of higher-order logic as denoting truth in a standard model is often used by those studying the mathematical properties of integers and structures that can be built from them (Shapiro, 1985).
		Since logical connectives within substitutions are possible in higher-order logic, as this example shows, atomic formula unification does not suggest enough substitution terms.
	www.lix.polytechnique.fr /Labo/Dale.Miller/papers/AIencyclopedia (1791 words)

	Higher-order logic -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-08-19)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, higher-order logic is distinguished from (Click link for more info and facts about first-order logic) first-order logic in a number of ways.
		A higher-order predicate is a ((logic) what is predicated of the subject of a proposition; the second term in a proposition is predicated of the first term by means of the copula) predicate that takes one or more other predicates as arguments.
		Higher-order logics are more expressive, but their properties, in particular with respect to (Click link for more info and facts about model theory) model theory, make them less (Click link for more info and facts about well-behaved) well-behaved for many applications.
	www.absoluteastronomy.com /encyclopedia/h/hi/higher-order_logic.htm (218 words)

	HOL 4 Kananaskis 3
		It is the latest version of the HOL automated proof system for higher order logic: a programming environment in which theorems can be proved and proof tools implemented.
		HOL 4 is particularly suitable as a platform for implementing combinations of deduction, execution and property checking.
		HOL 4 is an open source project with a BSD-style licence that allows its free use in commercial products.
	hol.sourceforge.net (214 words)

	Peter Suber, "Glossary of First-Order Logic"
		A wff A of propositional logic created from a wff B of predicate logic by (1) removing the quantifiers from B, and (2) replacing each predicate symbol (and its arguments) in B with a propositional symbol.
		A property possessed by all the wffs in a set is logically hereditary iff the accepted rules of inference pass it on (transmit it) to all the conclusions derivable from that set by those rules.
		In propositional logic, an interpretation is just such a function; in predicate logic, it is some set (the domain) together with such a function defined for members of that domain.
	www.earlham.edu /~peters/courses/logsys/glossary.htm (9715 words)

	Higher-order logic - Wikipedia, the free encyclopedia
		One of these is the type of variables appearing in quantifications; in first-order logic, roughly speaking, it is forbidden to quantify over predicates.
		By a result of Gödel, classical higher-order logic does not admit a (recursively axiomatized) sound and complete proof calculus; this defect can be repaired by using Henkin models.
		Lambek, P. Scott: Introduction to Higher Order Categorical Logic.
	en.wikipedia.org /wiki/Higher-order_logic (231 words)

	Higher-order logic (Site not responding. Last check: 2007-08-19)
		One of these is the scope of quantifiers ; in first-order logic roughly speaking it forbidden to quantify over predicates.
		Higher-order logics are more expressive but their in particular with respect to model theory make them less well-behaved for many applications.
		Higher Order Logic Theorem Proving and Its Applications: Proceedings of the Ifip Tc10/Wg10.2 International Workshiop on Higher Order Logic Theorem P (...
	www.freeglossary.com /Higher-order_logic (359 words)

	Design and Application of Strategies/Tactics in Higher Order Logics (Site not responding. Last check: 2007-08-19)
		Higher order logic theorem provers support this style of reasoning by providing the user with a set of basic proof commands.
		User-defined strategies permit special-purpose user interfaces to higher order logic theorem provers to be designed that help liberate developers from dependence on specialists in formal methods and theorem proving.
		Thus, the expected audience includes users of PVS and other higher order logic theorem provers at all levels, particularly those with an interest in making higher order logic theorem proving both easier for themselves and more accessible to others outside the narrow theorem proving community.
	research.nianet.org /fm-at-nia/STRATA2003 (550 words)

	Amazon.com: Books: Introduction to HOL : A Theorem-Proving Environment for Higher-Order Logic (Site not responding. Last check: 2007-08-19)
		HOL is a proof development system intended for applications to both hardware and software.
		HOL is currently being applied to a wide variety of problems, including the specification and verification of critical systems.
		Introduction to HOL provides a coherent and self-contained description of HOL containing both a tutorial introduction and most of the material that is needed for day-to-day work with the system.
	www.amazon.com /exec/obidos/tg/detail/-/0521441897?v=glance (591 words)

	Mechanized Reasoning Systems (Site not responding. Last check: 2007-08-19)
		E is a theorem prover for clausal logic with equality.
		MONA is an implementation of decision procedures for weak second-order logics of successors.
		Nqthm is a prover for quantifier free logic for recursive functions over the integers and other finitely generated structures, combining rewriting, heuristics for induction, and other techniques.
	www-formal.stanford.edu /clt/ARS/systems.html (1545 words)

	The World Wide Web Virtual Library: HOL (Site not responding. Last check: 2007-08-19)
		Introduction to HOL: A theorem proving environment for higher order logic, edited by M.J.C. Gordon and T.F. Melham, 1993.
		Information on HOL and other automated reasoning systems in a standard format is available from Stanford University.
		Higher Order Logic Theorem Proving and its Applications special issue of The Computer Journal by Tom Melham.
	vl.fmnet.info /hol (466 words)

	Read about Higher-order logic at WorldVillage Encyclopedia. Research Higher-order logic and learn about Higher-order ... (Site not responding. Last check: 2007-08-19)
		quantifiers; in first-order logic, roughly speaking, it is forbidden to quantify over predicates.
		second-order logic for systems in which this is permitted.
		Another way in which higher-order logic differs from first-order logic is in the constructions allowed in the underlying
	encyclopedia.worldvillage.com /s/b/Higher-order_logic (129 words)

	Higher Order Logic (Suprema Group 1) - Introduction (Site not responding. Last check: 2007-08-19)
		We are all familiar with propositional and predicate calculus (zeroth and first order logic respectively), however if we stick only to these logics we soon encounter limits in expressibility.
		The general principle of induction expresses that this is true for all properties, not just a particular one, and to state this, we need to be able to quantify over properties (by properties, we mean relations or predicates).
		So in some (naive) higher order logic where this is possible, we might express the principle of induction over the natural numbers as:
	www.doc.ic.ac.uk /~rah03/suprema (213 words)

	Mike Gordon's Curriculum Vitae
		I am interested in exploring modern realisations of the principle "Computation = Logic + Control" developed by Kowalski and Hayes in the 1970s using higher order logic as a unifying language, and the HOL system as an experimental integration platform.
		A next step is to extend the HOL prover with a compiler to automate the generation of specialised property checkers from definitions of properties and models in higher order logic.
		Linking higher order logic to binary decision diagrams, to appear in the Proceedings of the Symposium in Celebration of the work of C.A.R. Hoare (editors Jim Davies, Jim Woodcock and Bill Roscoe), to be published in the series "Cornerstones in Computing" (series editor Richard Bird), MacMillan.
	www.cl.cam.ac.uk /users/mjcg/NewCV.html (4807 words)

	Bachelor Thesis: Studies in Higher-Order Equational Logic (Site not responding. Last check: 2007-08-19)
		Higher-order logic, also known as type theory, has been introduced in 1908 by Bertrand Russell [33] as a formal basis for mathematical reasoning, based on a functional view of logic originally developed by Gottlob Frege [13].
		In Proceedings of the Logic Colloquium 72-73 (1975), R. Parikh, Ed., vol.
		In To H. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J. Hindley and J. Seldin, Eds.
	www.ps.uni-sb.de /~kaminski/bthesis/bthesis.html (962 words)

	Amazon.ca: Books: Higher Order Logic and Hardware Verification (Site not responding. Last check: 2007-08-19)
		Melham shows here how formal logic can be used to specify the behavior of hardware designs and reason about their correctness.
		The author describes how certain fundamental abstraction mechanisms for hardware verification can be formalized in logic and used to express assertions about design correctness and the relative accuracy of models of hardware behavior.
		He also includes an introduction to higher-order logic, which is a widely used formalism in this subject, and describes how that formalism is actually used for hardware verification.
	www.amazon.ca /exec/obidos/ASIN/052141718X (274 words)

	Bibliography on Logical Frameworks
		A modular presentation of modal logics in a logical framework.
		Rewriting logic as a logical and semantical framework.
		Higher order quotients and their implementation in Isabelle HOL.
	www.cs.cmu.edu /afs/cs/user/fp/www/lfs-bib.html (8721 words)

	Many Sorted and Higher Order Logic
		This logic of points and sets is sometimes called second order logic expressing the idea that sets have a higher order than points.
		Ordinary unsorted logic is often called first order predicate calculus because one can think of its universe as consisting only of points, which are said to be first order.
		For the point, set logic for instance, we have two types, point and set, and each quantifier must be bounded to one of these types.
	www.math.psu.edu /melvin/logic/node9.html (704 words)

	Practical Foundations of Mathematics
		In higher order logic, predicates (or, by comprehension, subsets) are first class citizens, allowing quantification over predicates on predicates.
		In particular, first order logic is unable to characterise N or R up to isomorphism.
		The type of propositions Even though set theory can be presented in a first order meta-language, the subject itself is plainly intended to handle higher order logic.
	www.cs.man.ac.uk /~pt/Practical_Foundations/html/s28.html (945 words)

	Re: Higher-order linear logic (Site not responding. Last check: 2007-08-19)
		The earliest uses were with Horn clauses [nadathur90jacm] and intuitionistic logic [miller91apal].
		Later versions mixed higher-order quantification (as in Church's Simple Theory of Types) with intuitionistic linear logic [hodas94ic] and all of linear logic [miller96tcs].
		Numerous examples mixing higher-order quantification with linear logic are illustrated in these last two papers.
	www.seas.upenn.edu /~sweirich/types/archive/1999-2003/msg00638.html (140 words)

	LC '98 abstract: Wolfgang Degen (joint work with Jan Johannsen) (Site not responding. Last check: 2007-08-19)
		Our weakest logical systems $K_{\Delta}$, which are generalizations of simple type theory to cumulative types up to $Delta$, are complete w.r.t.\ CHSs.
		In order to embed set theory we define for each type $\alpha$ a type-homogeneous membership relation by a^{\alpha} \in b^{\alpha} := \exists x^{\alpha+1} (x^{\alpha+1}(a^{\alpha}) \wedge \forall y^{\alpha+2} (y^{\alpha+2}(x^{\alpha+1}) \to y^{\alpha+2}(b^{\alpha})))$.
		In this way, cumulative higher-order logic is a foundation of the set theory $Z$ via the mapping $\varphi \mapsto \varphi^{(\lambda)}$.
	www.math.cas.cz /~lc98/abstracts/Degen.html (518 words)

	Automated Reasoning Group HOL page
		The HOL System is an environment for interactive theorem proving in a higher-order logic.
		The name `HOL' is pronounced either as a word to rhyme with `doll' or letter by letter.
		The old HOL FTP area has the old HOL88 and HOL90 files.
	www.cl.cam.ac.uk /Research/HVG/HOL (212 words)

	Cetus Links: 18,452 Links on Objects and Components / Prolog
		Linear logic systems such as Lolli and LLP are supersets (more or less) of Prolog that define additional operations for these purposes.
		Closely related to LLP is higher-order logic (HOL), as embodied in Lambda Prolog.
		HOL concerns topics such as provability and correctness and other functional programming concepts as opposed to OO, but is also used in solving problems concerning mobile agents (Pi Calculus) as defined by Robin Milner and others.
	www.cetus-links.org /oo_prolog.html (2177 words)

	FIRST ORDER LOGIC - Definition
		First-order logic can only quantify over sets of atomic propositions.
		Second-order logic can quantify over functions on propositions, and higher-order logic can quantify over any type of entity.
		In first-order logic quantifiers always range over ALL the elements of the domain of discourse.
	www.hyperdictionary.com /dictionary/first+order+logic (259 words)

	Mind: Why higher-order vagueness is a pseudo-problem - Symposium: Higher-Order Vagueness
		The apparent lack of a sharp boundary to the predicate's extension is typically explained by the presence of border (borderline, or penumbral) cases for the predicate in question: cases which jointly constitute the border region (borderline region or penumbra) for the vague predicate.
		Even given qualifications to the notion of a border case in order to exclude cases of inexactness, meaninglessness, ambiguity, context sensitivity, etc. the notion of a border case still appears too broad to properly characterise the phenomenon of vagueness.
		I want to argue that the iterative conception captures a feature of vagueness that is real enough--the phenomenon of higher orders of vagueness--but that this phenomenon is ultimately an echo of a more basic feature of border cases.
	www.dynomind.com /p/articles/mi_m2346/is_n409_v103/ai_14916928 (1414 words)

	Higher-order logic (Site not responding. Last check: 2007-08-19)
		Theorem Proving in Higher Order Logics: 10th International Conference, Tphols '97, Murray Hill, Nj, Usa, August 19-22, 1997 : Proceedings (Lecture Notes in Computer Science, 1275)
		Theorem Proving in Higher Order Logics: 15th International Conference, Tphols 2002, Hampton, Va, Usa, August, 2002 : Proceedings (Lecture Notes in Computer Science, 2410)
		Theorem Proving in Higher Order Logics: 14th International Conference, Tphols 2001, Edinburgh, Scotland Uk, September 3-6, 2001 Proceedings (Lecture Notes in Computer Science, 2152)
	www.freeglossary.com /Higher-order_predicate (359 words)

	Isabelle/HOL. A Proof Assistant for Higher-Order Logic (Site not responding. Last check: 2007-08-19)
		This book is a self-contained introduction to interactive proof in higher-order logic (HOL), using the proof assistant Isabelle2002.
		Logic and Sets presents a collection of lower-level tactics that you can use to apply rules selectively.
		A slightly updated version of the book is part of the documentation of the latest Isabelle release.
	www4.in.tum.de /~nipkow/LNCS2283 (237 words)

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