en>BG19bot: WP:CHECKWIKI error fix. Category problem. Syntax fixes. Do general fixes if a problem exists. - using AWB (9421)

2013-08-17T08:52:17Z

WP:CHECKWIKI error fix. Category problem. Syntax fixes. Do general fixes if a problem exists. - using AWB (9421)

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	~~Hello~~ and ~~welcome~~. ~~My name~~ is ~~Figures Wunder~~. ~~One~~ of the ~~extremely best issues~~ in the ~~world~~ for me is to ~~do aerobics~~ and ~~now I'm attempting~~ to ~~make cash with it~~. ~~Her family lives~~ in ~~Minnesota~~. ~~Managing people~~ has ~~been his day job~~ for a ~~while~~.~~<br><br>Also visit my blog [~~http://~~chorokdeul~~.co.kr/~~index~~.~~php?document_srl~~=~~454418&mid~~=~~customer21 home std test kit~~]		'''Condensed detachment''' (Rule D) is a method of finding the most general possible conclusion given two formal logical statements.
			It was developed by the Irish [[logician]] [[Carew Meredith]] in the 1950s and inspired by the work of [[Jan Łukasiewicz\|Łukasiewicz]].

			==Informal description==

			A rule of detachment says:<br>
			"Given that <math>p</math> implies <math>q</math>, and given <math>p</math>, infer <math>q</math>."

			The condensed detachment goes a step further and says:<br>
			"Given that <math>p</math> implies <math>q</math>, and given an <math>r</math>, use a unifier of <math>p</math> and <math>r</math> to make <math>p</math> and <math>r</math> the same, then use a standard rule of detachment."

			A substitution ''A'' that when applied to <math>p</math> produces <math>t</math>, and substitution ''B'' that when applied to <math>r</math> produces <math>t</math>, are called the Unifiers of <math>p</math> and <math>r</math>.

			Various unifiers may produce expressions with varying numbers of [[free variable]]s. Some possible unifying expressions are [[substitution instance]]s of others. If one expression is a substitution instance of another (and not just a variable renaming), then that other is called "more general".

			If the most general unifier is used in condensed detachment, then the logical result is the most general conclusion that can be made in the given inference with the given second expression. (And since any weaker inference you can get is a substitution instance of the most general one, nothing less than the most general unifier is ever used in practice.)

			In some logics (such as standard PC) have a set of defining axioms with the "D-completeness" property. If a set of axioms is D-Complete, then any vaild theorems of the system can be generated by condensed detachment alone. Note that "D-completeness" is a property of an axiomatic basis for a system, not an intrinsic property of a logic system itself.

			J.A.Kalman proved that any conclusion that can be generated by a sequence of uniform substitution and ''[[modus ponens]]'' steps can either be generated by condensed detachment alone, or is a substitution instance of something that can be generated by condensed detachment alone.
			This makes condensed detachment useful for any logic system that has ''[[modus ponens]]'' and substitution, regardless of whether or not it is D-complete.

			==D-notation==
			Since a given major premise and a given minor premise uniquely determine the conclusion (up to variable renaming), [[Carew Meredith\|Meredith]] observed that it was only necessary to note which two statements were involved and that the condensed detachment can be used without any other notation required. This led to the "D-notation" for [[Mathematical proof\|proofs]]. This notation uses the "D" operator to mean condensed detachment, and takes 2 arguments, in a standard [[prefix notation]] string. For example, if you have four axioms a typical proof in D-notation might look like: DD12D34 which shows a condensed detachment step using the result of two prior condensed detachment steps, the first of which used axioms 1 and 2, and the second of which used axioms 3 and 4.

			This notation, besides being used in some automated theorem provers, sometimes appears in catalogs of proofs

			Condensed detachment's use of [[unification]] predates the [[Resolution (logic)\|resolution]] techniques of [[automated theorem proving]].

			==Advantages==

			For automated theorem proving condensed detachment has a number of advantages over raw ''[[modus ponens]]'' and uniform substitution.

			At a Modus Ponens and substitution proof you have an infinite number of choices for what you can substitute for variables. This means that you have an infinite number of possible next steps. With condensed detachment there are only a finite number of possible next steps in a proof.

			The D-notation for complete condensed detachment proofs allows easy description of proofs for cataloging and search. A typical complete 30 step proof is less than 60 characters long in D-notation (excluding the statement of the axioms.)

			==References==
			{{reflist}}
			* Condensed Detachment as a Rule of inference, J.A.Kalman, ''Studia Logica'', Volume 42, Number 4, December, 1983
			*{{citation
			\| last = Hindley \| first = J. Roger
			\| author-link = J. Roger Hindley
			\| title = BCK and BCI logics, condensed detachment and the $2$-property
			\| url = http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.ndjfl/1093634655
			\| mr = 1231287
			\| doi = 10.1305/ndjfl/1093634655
			\| journal = [[Notre Dame Journal of Formal Logic]]
			\| volume = 34
			\| issue = 2
			\| pages = 231–250
			\| year = 1993
			}}
			* Experiments in Automated Deduction with Condensed Detachment, William McCune and Larry Wos, Proceedings of the 11th International Conference on Automated Deduction (1992)

			{{logic}}

			[[Category:Logic]]

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2012-04-30T18:05:14Z

Alternative definition

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Unconditional convergence - Revision history

en>BG19bot: WP:CHECKWIKI error fix. Category problem. Syntax fixes. Do general fixes if a problem exists. - using AWB (9421)

46.126.107.181: /* Alternative definition */