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Topic: Impredicative

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	[Haskell] Impredicative Types? (Site not responding. Last check: 2007-11-02)
		Simon PJ may have > explained it to me ("impredicative", he called it), but I don't remember > offhand.
		Whenever you instantiate type variables with type schemes, you are impredicative.
		Impredicative instantiation makes type checking (much) harder -- without extra constraints, you don't even have principal types.
	www.haskell.org /pipermail/haskell/2004-February/013629.html (348 words)

	[Phil-logic] Kaplan's 2d-order reading of G-K
		In particular, PA is seen to make the notion of number itself impredicative in that numbers are somehow taken as all existing already before we even begin to represent much about them.
		1.1 This impredicative quality is apparently not bothersome so long as we consider that the domain of AC (viewed, in extension, as a relation) is finite.
		But it seems to me that even SMACr(x, y) is impredicative as I formulated it and that is what was nagging me as a "technical matter" that might remain to be resolved.
	philo.at /pipermail/phil-logic/2004-February/003277.html (997 words)

	Polymorphism (computer science) - Wikipedia, the free encyclopedia
		Instead, the types of arguments and return value are observed in run-time to match the operations performed on them.
		In an impredicative system, the type being substituted may be any type whatsoever, including a type that is itself polymorphic; thus
		In languages where explicit type annotations are necessary when applying a polymorphic function, the predicativity restriction is less important; thus these languages are generally impredicative.
	en.wikipedia.org /wiki/Type_polymorphism (2592 words)

	Impredicative constructions (from foundations of mathematics) -- Encyclopædia Britannica
		Impredicative constructions (from foundations of mathematics) -- Encyclopædia Britannica
		In particular, the French mathematician Henri Poincaré (1854–1912) objected to impredicative constructions, which construct an entity of a certain type in terms of entities of the same or higher type—i.e., self-referencing…
		More results on "Impredicative constructions (from foundations of mathematics)" when you join.
	www.britannica.com /eb/article-35455 (745 words)

	Jules Henri Poincaré [Internet Encyclopedia of Philosophy]
		For Poincaré, impredicative definitions are the source of antinomies in set theory, and the prohibition of impredicative definitions will remove such antinomies.
		Poincaré's prohibition of impredicative definitions is also connected with his point of view on infinity.
		This dispute is not about the role of impredicative definitions in producing antinomies, but about the independence of mathematical entities from human thinking.
	www.utm.edu /research/iep/p/poincare.htm (3565 words)

	Utrecht-Muenster
		From the proof-theoretic point of view it is interesting, because it is one of the simplest examples of the so-called impredicative theories.
		After that, we give some equivalent definitions of the proof-theoretic ordinal of an impredicative theory as an absolute measure of its strength.
		Finally, we treat the reductive proof theory of Kripke-Platek set theory (with infinity), $KP\omega$, and Kreisels theory of inductive definitions, $ID_1$, since both are paradigms of impredicative theories, thereby giving an overview of history and state-of-the-art of ordinal analysis.
	www.phil.uu.nl /~lev/Muenster/mu-prog.html (600 words)

	query: models of CIC
		Moreover we have (dependent) inductive types both on the > predicative and the impredicative level and we have dependent > elimination on Set (which is stronger than the one for Prop).
		I am not sure what you mean by a "set theoretic model": there is an old result by Reynolds which shows that there are no non-trivial set-theoritic models of impredicativity (in classical set theory).
		To construct an impredicative universe one observes that the subcategory of modest omega-sets (where the - relation is injective) provide a good interpretation fro Prop.
	www.seas.upenn.edu /~sweirich/types/archive/1999-2003/msg01486.html (385 words)

	Parametricity and variants of Girard's J operator
		This is available by anon ftp from theory.stanford.edu, file pub/jcm/girard-j.dvi, or through the web page http://theory.stanford.edu/people/jcm/home.html ------------------------------------------------------------------- Parametricity and variants of Girard's $J$ operator Robert Harper and John Mitchell Dec 29, 1994 SUMMARY The impredicative polymorphic lambda calculus, or System F, is generally recognized as a calculus of parametric polymorphism.
		Intuitively, this means that the polymorphic functions definable in the language must use the same algorithm at all types.
		Our goal is to dispel the possible misconception that in impredicative systems, Girard's example is simply a ``typing trick'' that could not be carried out with other non-parametric operations.
	www.cis.upenn.edu /~bcpierce/types/archives/1994/msg00165.html (589 words)

	Re: query: models of CIC
		By set theoretical model I mean any extension of constructive set theories of Aczel, based on subset collection or different forms of power-set axiom [1].
		Proceedings of Types 98, LNCS 1257 > To construct an impredicative > universe one observes that the subcategory of modest omega-sets > (where the - relation is injective) provide a good interpretation > fro Prop.
		The elimination rules for types Prop and Set are not symmetric so I guess we need some modification of subcategory of modes-sets to interpret both.
	www.seas.upenn.edu /~sweirich/types/archive/1999-2003/msg01487.html (182 words)

	Amazon.com: Essays on Life Itself: Books: Robert Rosen (Site not responding. Last check: 2007-11-02)
		impredicative loops, enough entailment, nonadmissible environment, morphogenetic networks, inferential entailments, excitable networks, local state transition, relational biology, causal entailments, entailment structures, gravitational aspects, rote operations, purely predicative, structuring mind, given material system, generic perturbation, constructible universe, closed causal loops, chimera formation, adopted shell, order from order, forced behavior, pure syntax, biggest model, optimality problem
		Rosen builds his case against Church Thesis, arguing that contemporary mathematical and, more generally, scientific rigor, which bans impredicative loops from scientific discource, would not allow us to build what he calls "new science", which is needed to account for life and consciousness.
		More than once he mentiones Goedel Theorem, as well as various paradoxes, encountered by science over the centuries, emphasizing the fact, that they all are directly related to the impossibility to draw definite border between an observer and her subject (not just in quantum physics).
	www.amazon.com /exec/obidos/tg/detail/-/0231105118?v=glance (3783 words)

	University of New England - New England Institute - Past Conference 2002
		Impredicative abstract structures provide a rational way to understand endogenous natural processes.
		Endogenous and impredicative effects are, from the reductionist perspective, bizarre.
		Nevertheless, endogeny and impredicativity might potentially answer many perplexing questions in cognition, linguistics, protein folding, quantum entanglement, and "the measurement problem.
	www.une.edu /nei/conference/past02.asp (3552 words)

	University of New England - The New England Instute - Endogenous
		An impredicative mathematical structure is one that is defined by its relationship with itself.
		Recent discoveries in mathematics, such as hyperset formalism, show that such models are both theoretically legitimate and practically useful.
		The hypothesized correspondence between endogenous natural systems and impredicative formal systems is one of the objects of investigation by the NEI Endogenous Systems Research Group.
	www.une.edu /nei/endogenous.asp (475 words)

	Inductive Definition in Type Theory, by Nax P. Mendler (Site not responding. Last check: 2007-11-02)
		We also consider typing terms in the presence of type constraints,and present a condition on the constraints (of polynomial complexity in the size of the constraints) for determining if the terms will be strongly normalizable or there will be a diverging typed term.
		Second, we develop a semantic account of the basic type theory, then relativize it to account for the impredicativity inherent in the definition of the new type constructors.
		We also show how this model can justify other impredicative type constructors, such as an impredicative type abstraction operation.
	www.nuprl.org /documents/Mendler/InductiveDefinition.html (340 words)

	On Gödel's Philosophy of Mathematics, Chapter I
		While the observation that the paradoxes were created by impredicative definitions is correct, it may be argued that there are innocuous, perfectly clear, impredicative definitions deeply rooted in classical mathematics, and hence the question of paradox is not reducible to the question of predicativity.
		Unless one assumes the Continuity Axiom, there will be "gaps in the line," and there will not be any way of "filling them in."[23] Gödel justifies the inherent impredicativity in classical mathematics by arguing that it causes no inconsistency or imprecision when restricted to the domain of classical mathematics.
		The assumption of the existence of the entities found in the paradoxes leads to contradictions, owing to the instability of these entities in the presence of impredicativity.
	www.friesian.com /goedel/chap-1.htm (4807 words)

	Abstracts
		As a working definition I call a theory fully impredicative if it is at least as strong, in the proof theoretic sense, as full higher order arithmetic (or possibly just full second order arithmetic).
		One such principle is expressed by the rules for the impredicative type Prop in the calculus of constructions.
		This principle justifies full impredicativity and gives rise to theories stronger than various extensions of Zermelo set theory.
	www.mathematik.uni-muenchen.de /~gamma0/abstract.html (1249 words)

	An Indexed Model of Impredicative Polymorphism and Mutable References (ResearchIndex)
		Abstract: We present a semantic model of the polymorphic lambda calculus augmented with a higher-order store, allowing the storage of values of any type, including impredicative quantified types, mutable references, recursive types, and functions.
		Our model provides the first denotational semantics for a type system with updatable references to values of impredicative quantified types.
		The central idea behind our semantics is that instead of tracking the exact type of a mutable reference in a possible...
	citeseer.ist.psu.edu /661035.html (423 words)

	[No title] (Site not responding. Last check: 2007-11-02)
		(* To build an impredicative definition that simulates an inductive type following this technique: * construct a function that takes the same type of arguments as the inductive type and returns a type, obtained in the following manner.
		Theorem or_to_impredicative : forall A B, A \/ B -> impredicative_or A B. Proof.
		Theorem impredicative_to_or : forall A B, impredicative_or A B -> A \/ B. Proof.
	www.labri.u-bordeaux.fr /Perso/~casteran/CoqArt/inductive-prop-chap/SRC/impredicative.v (547 words)

	[No title]
		More on Impredicativity: Last time, I talked about the difference between predicate and impredicative polymorphism: At issue is that when we introduce type variables, the question is what do they range over?
		If they range over all types, then we inevitably get a circular definition of some sort and we must somehow break the circularity to get a well-formed interpretation of the types.
		For impredicative polymorphism, I started to give a model that was aimed at proving strong normalization of reduction.
	www.eecs.harvard.edu /~greg/cs256sp2005/lec14.txt (1515 words)

	A Model for Impredicative Type Systems, Universes, Intersection Types and Subtyping
		We introduce a new model based on coherence spaces for interpreting large impredicative type systems such as the Extended Calculus of Constructions (ECC).
		Moreover, we show that this model is well suited for interpreting intersection types and subtyping too, and we illustrate this by interpreting a variant of ECC with an additional intersection type binder.
		Furthermore, we propose a general method for interpreting the impredicative level in a non-syntactical way, by allowing the model to be parametrized by an arbitrarily large coherence space in order to interpret inhabitants of impredicative types.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/lics/2000/0725/00/0725toc.xml&DOI=10.1109/LICS.2000.855752 (234 words)

	[No title]
		That last word I do not understand at all, could not find impredicative in the little Oxford dictionary.
		> On further thought the 'impredicative' includes an unrestricted view, in > which what we call 'inside' is undefined, rather it is part of the total.
		I need to > better understand the 'impredicative', as something lacking the restrictive > predicat(iv)es which characterise OUR cut-off entity.
	necsi.org:8100 /Lists/complex-science/Message/4248-P.txt (1695 words)

	Re: The difference between "Causal" and "Impredicative" loops, etc.
		Sorry about the ghost message with only my (decidedly political) signature on it-- I clicked reply and must have triggered it to send as well, before I had a chance to either actually say something or delete the signature for the list.
		So, to answer John M.'s question: Impredicativities refer to loops of entailment in models of (the causal loops in) complex systems.
		The word can, and has been, applied to the quality of those causal loops in the natural world, but this is why it gets confusing.
	www.panmere.com /rosen/mhout/msg01979.html (733 words)

	Re: The difference between "Causal" and "Impredicative" loops, etc.
		One of the areas of confusion I'm seeing in many different settings, including the ISSS conference I attended in July, is caused by a difference my father specified in his use of the phrases "Causal loops" and "Impredicative loops".
		The ideas are connected because they both describe chains of entailment, which is why it gets confusing, but "Causal loops" exist in natural complex systems whereas "Impredicative loops" exist in formalisms (i.e., models).
		In this sense the real sense of 'impredicativity' SHOULD apply to formalism.
	www.panmere.com /rosen/mhout/msg01965.html (344 words)

	Stc / Impredicative Higher-RankTypes
		In the talk, I give examples of first-class modules and generic iterators.
		ML-F is the first (and only) type system that supports impredicative type inference.
		In practice this means that the system is compositional where data structures can hold polymorphic values and generic functions can be applied to polymorphic arguments.
	www.cs.uu.nl /wiki/bin/view/Stc/ImpredicativeHigher-RankTypes?rev=1.1 (131 words)

	A selection of papers on Ontology and the Theory of Objects
		Remarkable in its analytic power for both ontology and logic is the here developed Particularized Predicate Logic (PPL), the logic inherent in the realist version of the doctrine of unit or individuated predicates.
		The power of PPL is illustrated by its clarification of the self-referential nature of impredicative definitions and its distinguishing between legitimate and illegitimate forms.
		With a well-motivated refinement on the axiom of comprehension, PPL is, in effect, a higher-order logic without a forced stratification of predicates into types or the use of other ad hoc restrictions.
	www.formalontology.it /onto_papers.htm (5293 words)

	[Coq-Club] corecursion on impredicative coinductive definitions (Site not responding. Last check: 2007-11-02)
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		Milad Niqui wrote: > Dear All, > > I am trying to define a cofixpoint on an impredicative type.
	pauillac.inria.fr /pipermail/coq-club/2003/000956.html (275 words)

	Predicative Higher-Order PROPOSITIONAL Logic vs. the Impredicative Propositional Calculus (Site not responding. Last check: 2007-11-02)
		I will contrast the predicative (or ramified, Nuprl-like) higher-order PROPOSITIONAL logic with the impredicative (Coq-like) propositional calculus.
		I will relate this to the well-known polymorphic lambda calculus of Girard/Reynolds, and then discuss its extension to predicate logic and type theory.
		I hope to also discuss the relationship of foundational concerns to the on-going projects using Nuprl.
	www.cs.cornell.edu /Nuprl/PRLSeminar/PRLSeminar92_93/Constable/Apr19.html (100 words)

	A program from an A-translated impredicative proof of Higman's Lemma (Site not responding. Last check: 2007-11-02)
		The file Higman.v formalizes an A-translated version of Nash-Williams impredicative and classical proof of Higman's lemma for a two-letter alphabet.
		A constructive and impredicative program can be extracted from the proof.
		This page was automatically generated from this description file.
	www.iist.unu.edu /~alumni/software/other/inria/www/coq/contribs/higman.html (68 words)

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