Transformation matrix - Revision history

194.39.218.10: /* Rotation */ The CW/CCW 2D rotation matrices were swapped

2014-12-16T20:26:33Z

Rotation: The CW/CCW 2D rotation matrices were swapped

← Older revision		Revision as of 22:26, 16 December 2014
Line 1:		Line 1:
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76.14.59.147: /* Rotation */

2014-02-22T16:00:04Z

Rotation

← Older revision		Revision as of 18:00, 22 February 2014
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	~~{{Unreferenced\|date=May 2008}}~~
	'''[[Non-monotonic logic\|Monotonicity]] of [[entailment]]''' is a property of many [[logical system]]s that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In [[sequent calculus\|sequent calculi]] this property can be captured by an inference rule called '''weakening''', or sometimes '''thinning''', and in such systems one may say that [[logical consequence\|entailment]] is monotone if and only if the rule is admissible. Logical systems with this property are occasionally called ''[[monotonic logics]]'' in order to differentiate them from [[non-monotonic logic]]s.

	~~==Weakening rule==~~

	~~To illustrate~~, ~~starting from~~ the [~~[natural deduction]] [[sequent]~~]:		Hi, I am Nicholas Bogdan and I completely dig that name. Horse using is the factor I love most. In my professional lifestyle I am a manufacturing and distribution officer but quickly my spouse and I will begin our own business. He's always cherished residing in [http://Www.sharkbayte.com/keyword/Delaware Delaware]. See what's new on my web site right here: http://bostonjin.com/MVJEL/Do_Nya_Online_Casinon_P%C3%A5_N%C3%A4tet_Better_Than_Seth_Godin<br><br>Look into my web site Vi listar alla senaste och nya casino 2014!!! ([http://bostonjin.com/MVJEL/Do_Nya_Online_Casinon_P%C3%A5_N%C3%A4tet_Better_Than_Seth_Godin disse detaljene])

	~~<center>Γ <math>\vdash<~~/~~math> C<~~/~~center>~~

	~~weakening allows one to conclude:~~

	<~~center~~>~~Γ, A~~ <~~math~~>~~\vdash<~~/~~math> C<~~/~~center>~~

	~~==Non-monotonic logics==~~
	~~{{main\|Non-monotonic logic}}~~
	~~In most logics, weakening is either an inference rule or a [[metatheorem]] if the logic doesn't have an explicit rule~~. ~~Notable exceptions are:~~

	* [[Strict logic]] ~~or [[relevant logic]], where every hypothesis must be necessary for the conclusion.~~
	* [[Linear logic]] which disallows arbitrary [[Contraction (operator theory)~~\|contraction]] in addition to arbitrary weakening.~~
	* [[Bunched logic\|Bunched implications]] where weakening is restricted to additive composition.
	* Various types of default reasoning.
	* [[Abductive reasoning]], the process of deriving the most likely explanations of the known facts.
	* Reasoning about knowledge, where statements specifying that something is not known need to be retracted when that thing is learned.

	~~== See also ==~~

	* [[Idempotency of entailment\|Contraction]]
	* [[Exchange rule]]
	* [[Substructural logic]]
	~~{{logic-stub}}~~

	~~[[Category:Logical consequence]]~~
	~~[[Category:Theorems in propositional logic]]~~

131.227.184.76: N is a vector not a surface

2014-01-28T13:51:57Z

N is a vector not a surface

← Older revision		Revision as of 15:51, 28 January 2014
Line 1:		Line 1:
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			'''[[Non-monotonic logic\|Monotonicity]] of [[entailment]]''' is a property of many [[logical system]]s that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In [[sequent calculus\|sequent calculi]] this property can be captured by an inference rule called '''weakening''', or sometimes '''thinning''', and in such systems one may say that [[logical consequence\|entailment]] is monotone if and only if the rule is admissible. Logical systems with this property are occasionally called ''[[monotonic logics]]'' in order to differentiate them from [[non-monotonic logic]]s.

			==Weakening rule==

			To illustrate, starting from the [[natural deduction]] [[sequent]]:

			<center>Γ <math>\vdash</math> C</center>

			weakening allows one to conclude:

			<center>Γ, A <math>\vdash</math> C</center>

			==Non-monotonic logics==
			{{main\|Non-monotonic logic}}
			In most logics, weakening is either an inference rule or a [[metatheorem]] if the logic doesn't have an explicit rule. Notable exceptions are:

			* [[Strict logic]] or [[relevant logic]], where every hypothesis must be necessary for the conclusion.
			* [[Linear logic]] which disallows arbitrary [[Contraction (operator theory)\|contraction]] in addition to arbitrary weakening.
			* [[Bunched logic\|Bunched implications]] where weakening is restricted to additive composition.
			* Various types of default reasoning.
			* [[Abductive reasoning]], the process of deriving the most likely explanations of the known facts.
			* Reasoning about knowledge, where statements specifying that something is not known need to be retracted when that thing is learned.

			== See also ==

			* [[Idempotency of entailment\|Contraction]]
			* [[Exchange rule]]
			* [[Substructural logic]]
			{{logic-stub}}

			[[Category:Logical consequence]]
			[[Category:Theorems in propositional logic]]

en>LarsPensjo: /* Composing and inverting transformations */ Add reference to associative property.

2012-08-07T12:01:16Z

Composing and inverting transformations: Add reference to associative property.

New page

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