Proof by example - Revision history

en>NukeofEarl: It appears that no such proposal was ever made.

2014-03-07T15:12:51Z

It appears that no such proposal was ever made.

← Older revision		Revision as of 17:12, 7 March 2014
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	~~The notion of '''cylindric algebra''', invented~~ by [[Alfred Tarski]], arises naturally in the [[Algebraic logic\|algebraization]] of [[first-order logic]] with [[First-order_logic#Equality_and_its_axioms\|equality]]. This is comparable to the role [[Boolean algebra (structure)\|Boolean algebra]]s play for [[propositional logic]]. Indeed, cylindric algebras are Boolean algebras equipped with additional cylindrification operations that ~~model [[quantification]] and equality~~. ~~They differ from [[polyadic algebra]]~~s in ~~that the latter do not model equality~~.		Royal Votaw is my title but I by no means truly favored that name. For years she's been living in Kansas. The thing she adores most is flower arranging and she is attempting to make it a occupation. Bookkeeping is what she does.<br><br>Feel free to surf to my webpage ... [http://www.Carelion.com/UserProfile/tabid/61/userId/107768/Default.aspx www.Carelion.com]

	~~== Definition of a cylindric algebra ==~~

	~~A '''cylindric algebra of dimension''' <math>\alpha</math> (where <math>\alpha</math>~~ is any [[ordinal number]]) is an algebraic structure <math>(A,+,\cdot,-,0,1,c_\kappa,d_{\kappa\lambda})_{\kappa,\lambda<\alpha}</math> such that <math>(A,+,\cdot,-,0,1)</math> is a [[Boolean algebra (structure)\|Boolean algebra]], <math>c_\kappa</math> a unary operator on <math>A</math> for every <math>\kappa</math>, and <math>d_{\kappa\lambda}</math> a distinguished element of <math>A</math> for every <math>\kappa</math> and ~~<math>\lambda</math>, such that the following hold:~~

	~~(C1) <math>c_\kappa 0=0</math>~~

	~~(C2) <math>x\leq c_\kappa x</math>~~

	~~(C3) <math>c_\kappa(x\cdot c_\kappa y)=c_\kappa x\cdot c_\kappa y</math>~~

	~~(C4) <math>c_\kappa c_\lambda x=c_\lambda c_\kappa x</math>~~

	~~(C5) <math>d_{\kappa\kappa}=1</math>~~

	~~(C6) If <math>\kappa\neq\lambda\mu</math>,{{clarify\|reason=What~~ is ~~the juxtaposition 'λμ' of variables, or ordinals, supposed~~ to ~~mean? One possible reformulation in standard logical notation see below.\|date=August 2013}} then <math>d_{\lambda\mu}=c_\kappa(d_{\lambda\kappa}\cdot d_{\kappa\mu})</math>~~

	~~(C7) If <math>\kappa\neq\lambda</math>, then <math>c_\kappa(d_{\kappa\lambda}\cdot x)\cdot c_\kappa(d_{\kappa\lambda}\cdot -x)=0</math>~~

	~~Assuming~~ a ~~presentation of first-order logic [[Functional predicate#Doing without functional predicates\|without function symbol]]s,~~
	the operator <math>c_\kappa x</math> models [[existential quantification]] over variable <math>\kappa</math> in formula <math>x</math> while the operator <math>d_{\kappa\lambda}</math> models the equality of variables <math>\kappa</math> and <math>\lambda</math>. Henceforth, reformulated using standard logical notations, the axioms read as

	~~(C1) <math>\exists \kappa. \mathit{false} \Leftrightarrow \mathit{false}</math>~~

	~~(C2) <math>x \Rightarrow \exists \kappa~~. ~~x</math>~~

	~~(C3) <math>\exists \kappa. (x\wedge \exists \kappa. y) \Leftrightarrow (\exists\kappa. x) \wedge (\exists\kappa. y)</math>~~

	~~(C4) <math>\exists\kappa \exists\lambda. x \Leftrightarrow \exists \lambda \exists\kappa. x</math>~~

	~~(C5) <math>\kappa=\kappa \Leftrightarrow \mathit{true}</math>~~

	~~(C6) If <math>\kappa</math>~~ is ~~a variable different from both <math>\lambda</math> and <math>\mu</math>, {{clarify\|reason=See above~~.~~\|date=August 2013}} then~~ <~~math~~>~~\lambda=\mu \Leftrightarrow \exists\kappa. (\lambda=\kappa \wedge \kappa=\mu)~~<~~/math~~>

	(C7) If <math>\kappa</math> and <math>\lambda</math> are different variables, then <math>\exists\kappa. (\kappa=\lambda \wedge x) \wedge \exists\kappa. (\kappa=\lambda\wedge \neg x) \Leftrightarrow \mathit{false}</math>

	~~== Generalizations ==~~

	~~Recently, cylindric algebras have been generalized~~ to ~~the [[Many-sorted logic\|many-sorted]] case, which allows for a better modeling of the duality between first-order formulas and terms~~.

	~~==See also==~~
	*[[Abstract algebraic logic]]
	*[[Lambda calculus]] and [[Combinatory logic]], other approaches to modelling quantification and eliminating variables
	*[[Hyperdoctrine]]s are a [[Category theory\|categorical]] formulation of cylindric algebras
	*[[Relation algebra]]s (RA)
	*[[Polyadic algebra]]

	~~==References==~~
	* [[Leon Henkin]], Monk, J.D.~~, and~~ [~~[Alfred Tarski]] (1971) ''Cylindric Algebras, Part I''. North-Holland. ISBN 978-0-7204-2043-2.~~
	* -------- (1985) ''Cylindric Algebras, Part II''. North-Holland.
	* {{cite book\| author=Carlos Caleiro, Ricardo Gonçalves\| chapter=On the algebraization of many-sorted logics\| title=Proc. 18th int. conf. on Recent trends in algebraic development techniques (WADT)\|editor=J. Fiadeiro and P.-Y. Schobbens\| year=2006\| volume=4409\| pages=21-36\| publisher=Springer\| series=LNCS\| isbn=978-3-540-71997-7\| url=http://~~sqig~~.~~math~~.~~ist.utl.pt~~/~~pub~~/~~CaleiroC~~/~~06-CG-manysorted~~.~~pdf}}~~

	~~== Further reading ==~~
	* {{cite doi\|10.~~1016/0022-0000(84)90077-1}}~~

	~~[[Category:Algebraic logic]~~]

24.213.201.195 at 17:16, 11 December 2013

2013-12-11T17:16:22Z

← Older revision		Revision as of 19:16, 11 December 2013
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	~~Alyson~~ is the ~~name people use to call me~~ and ~~I believe it sounds quite great when you say it~~. ~~My day job~~ is a ~~travel agent. It's not~~ a ~~common factor but what I like doing~~ is to ~~climb but I don't have~~ the ~~time lately~~. ~~Alaska~~ is ~~exactly where I've usually been residing~~.<br><br>~~Feel free~~ to ~~visit my homepage~~ [http://~~cartoonkorea~~.~~com~~/~~ce002~~/~~1093612 psychic chat online~~]		The notion of '''cylindric algebra''', invented by [[Alfred Tarski]], arises naturally in the [[Algebraic logic\|algebraization]] of [[first-order logic]] with [[First-order_logic#Equality_and_its_axioms\|equality]]. This is comparable to the role [[Boolean algebra (structure)\|Boolean algebra]]s play for [[propositional logic]]. Indeed, cylindric algebras are Boolean algebras equipped with additional cylindrification operations that model [[quantification]] and equality. They differ from [[polyadic algebra]]s in that the latter do not model equality.

			== Definition of a cylindric algebra ==

			A '''cylindric algebra of dimension''' <math>\alpha</math> (where <math>\alpha</math> is any [[ordinal number]]) is an algebraic structure <math>(A,+,\cdot,-,0,1,c_\kappa,d_{\kappa\lambda})_{\kappa,\lambda<\alpha}</math> such that <math>(A,+,\cdot,-,0,1)</math> is a [[Boolean algebra (structure)\|Boolean algebra]], <math>c_\kappa</math> a unary operator on <math>A</math> for every <math>\kappa</math>, and <math>d_{\kappa\lambda}</math> a distinguished element of <math>A</math> for every <math>\kappa</math> and <math>\lambda</math>, such that the following hold:

			(C1) <math>c_\kappa 0=0</math>

			(C2) <math>x\leq c_\kappa x</math>

			(C3) <math>c_\kappa(x\cdot c_\kappa y)=c_\kappa x\cdot c_\kappa y</math>

			(C4) <math>c_\kappa c_\lambda x=c_\lambda c_\kappa x</math>

			(C5) <math>d_{\kappa\kappa}=1</math>

			(C6) If <math>\kappa\neq\lambda\mu</math>,{{clarify\|reason=What is the juxtaposition 'λμ' of variables, or ordinals, supposed to mean? One possible reformulation in standard logical notation see below.\|date=August 2013}} then <math>d_{\lambda\mu}=c_\kappa(d_{\lambda\kappa}\cdot d_{\kappa\mu})</math>

			(C7) If <math>\kappa\neq\lambda</math>, then <math>c_\kappa(d_{\kappa\lambda}\cdot x)\cdot c_\kappa(d_{\kappa\lambda}\cdot -x)=0</math>

			Assuming a presentation of first-order logic [[Functional predicate#Doing without functional predicates\|without function symbol]]s,
			the operator <math>c_\kappa x</math> models [[existential quantification]] over variable <math>\kappa</math> in formula <math>x</math> while the operator <math>d_{\kappa\lambda}</math> models the equality of variables <math>\kappa</math> and <math>\lambda</math>. Henceforth, reformulated using standard logical notations, the axioms read as

			(C1) <math>\exists \kappa. \mathit{false} \Leftrightarrow \mathit{false}</math>

			(C2) <math>x \Rightarrow \exists \kappa. x</math>

			(C3) <math>\exists \kappa. (x\wedge \exists \kappa. y) \Leftrightarrow (\exists\kappa. x) \wedge (\exists\kappa. y)</math>

			(C4) <math>\exists\kappa \exists\lambda. x \Leftrightarrow \exists \lambda \exists\kappa. x</math>

			(C5) <math>\kappa=\kappa \Leftrightarrow \mathit{true}</math>

			(C6) If <math>\kappa</math> is a variable different from both <math>\lambda</math> and <math>\mu</math>, {{clarify\|reason=See above.\|date=August 2013}} then <math>\lambda=\mu \Leftrightarrow \exists\kappa. (\lambda=\kappa \wedge \kappa=\mu)</math>

			(C7) If <math>\kappa</math> and <math>\lambda</math> are different variables, then <math>\exists\kappa. (\kappa=\lambda \wedge x) \wedge \exists\kappa. (\kappa=\lambda\wedge \neg x) \Leftrightarrow \mathit{false}</math>

			== Generalizations ==

			Recently, cylindric algebras have been generalized to the [[Many-sorted logic\|many-sorted]] case, which allows for a better modeling of the duality between first-order formulas and terms.

			==See also==
			*[[Abstract algebraic logic]]
			*[[Lambda calculus]] and [[Combinatory logic]], other approaches to modelling quantification and eliminating variables
			*[[Hyperdoctrine]]s are a [[Category theory\|categorical]] formulation of cylindric algebras
			*[[Relation algebra]]s (RA)
			*[[Polyadic algebra]]

			==References==
			* [[Leon Henkin]], Monk, J.D., and [[Alfred Tarski]] (1971) ''Cylindric Algebras, Part I''. North-Holland. ISBN 978-0-7204-2043-2.
			* -------- (1985) ''Cylindric Algebras, Part II''. North-Holland.
			* {{cite book\| author=Carlos Caleiro, Ricardo Gonçalves\| chapter=On the algebraization of many-sorted logics\| title=Proc. 18th int. conf. on Recent trends in algebraic development techniques (WADT)\|editor=J. Fiadeiro and P.-Y. Schobbens\| year=2006\| volume=4409\| pages=21-36\| publisher=Springer\| series=LNCS\| isbn=978-3-540-71997-7\| url=http://sqig.math.ist.utl.pt/pub/CaleiroC/06-CG-manysorted.pdf}}

			== Further reading ==
			* {{cite doi\|10.1016/0022-0000(84)90077-1}}

			[[Category:Algebraic logic]]

en>BattyBot: changed {{Unreferenced}} to {{Refimprove}} & general fixes using AWB (7940)

2012-02-02T03:28:11Z

changed {{Unreferenced}} to {{Refimprove}} & general fixes using AWB (7940)

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Alyson is the name people use to call me and I believe it sounds quite great when you say it. My day job is a travel agent. It's not a common factor but what I like doing is to climb but I don't have the time lately. Alaska is exactly where I've usually been residing.<br><br>Feel free to visit my homepage [http://cartoonkorea.com/ce002/1093612 psychic chat online]