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Topic: Counterexamples

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	Counterexample - Wikipedia, the free encyclopedia
		In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule, i.e., a specific instance of the falsity of a universal quantification (a "for all" statement).
		A mathematical counterexample would be something like this: If you had a theorem that said "all numbers that are not negative are positive," and someone pointed out that zero is not negative, but is also not positive, then zero would be a counterexample.
		In philosophy, counterexamples are usually used to argue that a certain philosophical position is wrong by showing that it doesn't apply in certain cases.
	en.wikipedia.org /wiki/Counterexample (1108 words)

	Gettier problem - Wikipedia, the free encyclopedia
		The Gettier problem is a fundamental problem in contemporary epistemology (the philosophy of knowledge), issuing from counterexamples to the definition of knowledge as justified true belief.
		Gettier's paper used counterexamples to argue that there are cases of beliefs which are both true and justified - therefore satisfying all three conditions for knowledge on the JTB account - but which do not appear to be genuine cases of knowledge.
		He concludes that there will always be a counterexample to any definition of knowledge that does not require that the believer's evidence logically necessitates the truth of the belief.
	en.wikipedia.org /wiki/Gettier_problem (1845 words)

	EthicsNotesChapter8
		The importance of this distinction in connection with the technique of counterexamples lies in the fact that it is often more difficult to find a plausible counterexample to a moral statement when it is interpreted as expressing a prima facie moral rule than when it is interpreted as expressing as absolute moral rule.
		Nevertheless, this is still a very promising counterexample, and one that many people would view as sound, and the key to arriving at it was to construct a case where various factors that bear upon the morality of adultery are not present.
		In short, one good type of counterexample to claim 2 involves the case where you kill a guilty aggressor in self-defense, and the crucial thing that you needed to notice about claim 2 was that it did not specify that the person in question was innocent.
	www.austin.cc.tx.us /rloverin/Tooley'sEthicsNotes8.html (3211 words)

	Counterexamples in Clifford algebras
		Counterexamples are also given in the cases where the author failed to notice a general pattern after some dimension (typically at and above 4,5,6).
		After verification of the validity of a counterexample, further study in the same line, to rescue the "theorem", at whatever generality, is useless and erroneous activity.
		Lounesto: Counterexamples in Clifford algebras with CLICAL, pp.
	users.tkk.fi /~ppuska/mirror/Lounesto/counterexamples.htm (3687 words)

	TheMethodofCounterexamples
		Non-doomsday-style counterexamples have the advantage, then, of greater applicability, because they apply to prima facie moral rules, as well as absolute ones, and, as a consequence, someone who initially advances an absolute moral claim cannot escape your counterexample by shifting to the corresponding prima facie claim.
		The upshot is that in applying the technique of counterexamples to moral claims, it will often be unnecessary to point to a case of some actual action that falls under the moral generalization, but that fails to have the moral property which, according to the generalization, it ought to have.
		So that while the case that Mary proposes as a counterexample to the generalization may seem to Mary to be a plausible counterexample, it will not seem so to John, and so it cannot help their discussion to move ahead.
	www.austin.cc.tx.us /rloverin/Tooley'sCriticalThinkingNotes5.html (3203 words)

	FLoC '02 - LICS Monday July 22nd
		Although counterexamples are considered one of the main advantages of model checking, state-of the art model checkers are restricted to relatively simple counterexamples, and surprisingly little research effort has been put into counterexamples.
		To this end, we investigate the notion of counterexample and show that a large class of temporal logics beyond ACTL admits counterexamples with a simple tree-like transition relation.
		We show that the existence of tree-like counterexamples is related to a universal fragment of extended branching time logic based on omega-regular temporal operators.
	floc02.diku.dk /LICS/Monday.html (1356 words)

	Books : Counterexamples in Analysis (Dover Books on Mathematics) (Site not responding. Last check: 2007-11-07)
		These counterexamples, arranged according to difficulty or sophistication, deal mostly with the part of analysis known as 'real variables,' starting at the level of calculus.
		The counterexamples here are a wonderful aid to educating intuition about definitions in Real Variables.
		Some of the (counter)examples are well-known (ordered space which is cauchy complete but not complete) and some are highly contrived (two non homeomorphic topological spaces which is countinuous one-to-one image of each other).
	arabiadirectory.com /ItemId/0486428753 (492 words)

	[No title]
		One may argue that the first one of these found as not reasonable, but this led to the quite natural ones which are known.
		The obtaining of a rigorous treatment led to the type of previously known mathematical paradox which were the natural counterexamples.
		B.R. Gelbaum, J.H.M. Olmsted: Theorems and Counterexamples in Mathematics.
	www.math.niu.edu /~rusin/known-math/00_incoming/errors (681 words)

	The Geomblog: Counter-examples
		On the web, there are pages on counterexamples in functional analysis, Clifford algebras, and mathematical programming.
		In general, broken cryptosystems serve as counterexamples that push the development of theory in cryptography.
		These counterexamples might be closer to the spirit of the mathematical ones you mention.
	geomblog.blogspot.com /2004/10/counter-examples.html (920 words)

	DI & CoS - Counterexamples
		Finding counterexamples is one of the funniest sides of our activity.
		The same is not possible in the sequent calculus, as shown by the following counterexample due to Kai Brünnler.
		The following counterexample by Alwen Tiu shows that actually the fix does not work in general.
	alessio.guglielmi.name /res/cos/cex.html (964 words)

	Counterexamples in Analysis (Dover Books on Mathematics) (Site not responding. Last check: 2007-11-07)
		These counterexamples displayed features that were nowhere to be found in the physical universe.
		When Richard Feynman was a physics graduate student at Princeton, he would tease his mathematician friends that mathematics was so easy that he could instantly decide the truth or falsehood of any mathematical statement they could give him.
		When I was a graduate student in mathematics, "Counterexamples in Analysis" was my favorite book, and I had a lot of fun amazing my fellow graduate students by quoting from it.
	494062.onlinesportdiscount.com /3439343036322d312d30343836343238373533.html (1110 words)

	CoLab Document Server - Projection and proximal point methods: convergence results and counterexamples (Site not responding. Last check: 2007-11-07)
		We show how this construction results in a counterexample to norm convergence for iterates of averaged projections; hence, we give an affirmative answer to a question raised by Reich two decades ago.
		Furthermore, new counterexamples to norm convergence for iterates of firmly nonexpansive maps (\`a la Genel and Lindenstrauss) and for the proximal point algorithm (\`a la G\"uler) are provided.
		We also present a counterexample, along with some weak and norm convergence results, for the new framework of string-averaging projection methods introduced by Censor, Elfving, and Herman.
	eprints.cecm.sfu.ca /archive/00000030 (241 words)

	Philosophers' Imprint (Site not responding. Last check: 2007-11-07)
		I survey the range of counterexamples, or at least putative counterexamples, to the principles.
		I argue that the distinctive attitudes we have towards moral propositions cannot explain the counterexamples, since some of the examples don’t involve moral concepts.
		And I put forward a proposed explanation that turns on the role of ‘higher-level concepts’, concepts that if they are satisfied are satisfied in virtue of more fundamental facts about the world, in fiction, and in imagination.
	www.philosophersimprint.org /004003 (220 words)

	Counterexamples in Topology (Site not responding. Last check: 2007-11-07)
		A distinct characteristic of point set topology is that it builds on counterexamples.
		While textbooks usually present a counterexample to show why Theorem Three Point Five Oh will not work on a weaker assumption -- most students (and teachers) tend to skip these parts.
		A collection of counterexamples presented in this book (excellent organisation, by the way) is an essential supplement of a topology course; it enables one to 'see' between the points, so to speak.
	494062.onlinesportdiscount.com /3439343036322d312d30343836363837333558.html (735 words)

	Exploration Through Example
		(Eugene Wallingford confirms that the counterexamples were necessary.) In that world, an arch was two pillars of blocks with a crosspiece.
		The counterexamples included, if I remember correctly, arches without a top (just two pillars) and maybe a crosspiece balanced on a single pillar.
		In many ways it is opposite to a politician's approach, where they avoid counterexamples because they want you to read into their words what you want to.
	www.testing.com /cgi-bin/blog/ideas (2245 words)

	Mercer: What went wrong: explaining counterexamples (Site not responding. Last check: 2007-11-07)
		What went wrong: explaining counterexamples by Alex Groce and Willem Visser and published in the SPIN 2003 Workshop.
		Expand the single trace to the counterexample to generate sets of negative and positive traces.
		Generate the neg and pos sets by exploring backward from the original counterexample.
	lal.cs.byu.edu /~egm/blog/archives/000255.html (268 words)

	Counterexamples in Topology
		In the second part, the book provides "Counterexamples": quite a few topologies, both the predictable and the quirky kinds.
		Counterexamples in Topology is both useful and enjoyable, particularly for people who benefit from charts and outlines.
		Counterexamples in Analysis (Dover Books on Mathematics) von Bernard R. Gelbaum, John M. Olmsted
	www.bildungimkindergarten.de /Counterexamples/Topology/2review/048668735X.html (263 words)

	Examples And Counterexamples For Perles' Conjecture - Haase, Ziegler (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		Examples And Counterexamples For Perles' Conjecture - Haase, Ziegler (ResearchIndex)
		The combinatorial structure of a d{dimensional simple convex polytope { as given, for example, by the set of the (d 1){regular subgraphs of facets { can be reconstructed from its abstract graph [3] [10].
		Haase and G. Ziegler, Examples and counterexamples for Perles ' conjecture, Tech.
	citeseer.ist.psu.edu /384012.html (474 words)

	Gettier Counterexamples (Site not responding. Last check: 2007-11-07)
		In his 1963 paper, Gettier presented two counterexamples to the JTB account of knowledge.
		S knows that p if and only if: (1) S is justified in believing p, (2) p is true, and (3) S believes p (on the basis of her justification).
		Gettier gave two counterexamples which showed that this account is "too weak".
	www.unc.edu /~theis/Exp&R/gettier.html (433 words)

	EETimes.com - Counterexamples and metrics drive assertion-based verification (Site not responding. Last check: 2007-11-07)
		A counterexample is simply an input sequence that causes one of the design assertions to be violated.
		Many times, counterexamples point to assertions that are wrong, either because of a simple coding error in the assertion or because of a misunderstanding about what the assertion should check for.
		The effect is that the reported counterexample uses input sequences that are illegal, or that the RTL was not designed to handle.
	www.eetimes.com /news/design/showArticle.jhtml?articleID=16500114 (2185 words)

	validity and counterexamples
		So a possible situation where the argument has true premises but a false conclusion is all it takes to show that the argument is invalid.
		Validity counterexample: a possible situation where the premises of an argument are true, but the conclusion is false.
		So a validity counterexample is a possible situation which reveals the invalidity of the argument.
	www.ux1.eiu.edu /~cfbxb/class/1900/inf/validity.htm (402 words)

	Assumptions, Implications, Counterexamples
		To find possible sources of critique - one way of interrogating an argument is to identify counterexamples that do not fit with a set of assumptions.
		If you can find an important counterexample to a given position, this will often help one understand what is assumed in order for it to ignore such a counterexample.
		The degree to which an argument can withstand all the counterexamples you can think of throwing at it is a good indicator of its strength.
	www-rohan.sdsu.edu /~digger/305/assumptions2.html (1121 words)

	Learning Deterministic Finite Automata from Smallest Counterexamples
		An application to DFAs with at most $n$ states shows that $(1+o(1))\Sigman\log n$ smallest counterexamples are sufficient (if efficiency is not an issue).
		Furthermore, we present a general lower bound on the number of smallest counterexamples (required by any learning algorithm).
		Finally, we present a general conversion of algorithms learning from smallest counterexamples into algorithms performing self-directed learning.
	epubs.siam.org /sam-bin/dbq/article/34094 (337 words)

	Myths and Counterexamples (Site not responding. Last check: 2007-11-07)
		This has counterexamples to some mathematical statements that seem plausible.
		It serves as useful teaching material, and is referenced by my Mathematical Programming Glossary.
		Newton's method has converged when the change in the iterate value is less than some specified, small tolerance.
	carbon.cudenver.edu /~hgreenbe/myths/myths.html (1072 words)

	Beal's Conjecture: A Search for Counterexamples
		A wide array of sophisticated mathematical techniques could be used in the attempt to prove the conjecture true (and the majority of mathematicians competent to judge seem to believe that it likely is true).
		SETI@home where interested people could download the code, be assigned min and max values to search on, and a large network of people could together search for solutions.
		Ultimately, I think that if there are any counterexamples at all, they will probably be found through generating functions based on elliptical curves, or some other such approach.
	www.norvig.com /beal.html (1365 words)

	Amazon.ca: Books: Counterexamples in Topology (Site not responding. Last check: 2007-11-07)
		A distinct characteristic of point set topology is that it thrives on counterexamples.
		Students who really wants to master point set topology should know the various counterexamples (no matter how contrived or unnatural they seem), and while textbooks usually present a counterexample to show why Theorem Three Point Five Oh will not work on weaker assumption -- most students (and teachers, too) tend to skip these parts.
		A collection of counterexamples presented in this book (excellent organisationm, by the way) is an essential supplement of a topology course, and it might make people 'see' through the whole thing.
	www.amazon.ca /exec/obidos/ASIN/048668735X (813 words)

	Counterexamples in Chemical Ring Perception
		We then outline the basic properties of different cycle sets and provide numerous counterexamples to incorrect claims in the published literature.
		These counterexamples may have a serious practical impact because at least some of them are molecular graphs of well-known molecules.
		As a consequence, we propose a catalog of desirable properties for chemically useful sets of rings.
	pubs.acs.org /cgi-bin/abstract.cgi/jcisd8/2004/44/i02/abs/ci030405d.html (231 words)

	Abstract counterexamples
		), it represents a class of real counterexamples (where, for example, the value 1 is obtained instead of 0).
		In fact, the counterexample is abstract in another way.
		Notice that at step 5, a result returns on the the result bus
	www.cis.ksu.edu /santos/smv-doc/tutorial/node40.html (506 words)

	Amazon.com: Books: Counterexamples in Topology (Site not responding. Last check: 2007-11-07)
		Buy this book with Counterexamples in Analysis (Dover Books on Mathem...
		Counterexamples in Analysis (Dover Books on Mathematics) by Bernard R. Gelbaum
		Customers interested in Counterexamples in Topology may also be interested in
	www.amazon.com /exec/obidos/tg/detail/-/048668735X?v=glance (1341 words)

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