Intuitionistic logic - Revision history

81.136.137.167 at 22:00, 22 November 2014

2014-11-22T22:00:54Z

← Older revision		Revision as of 00:00, 23 November 2014
Line 1:		Line 1:
	In [[logic]], statements <var>p</var> and <var>q</var> are '''logically equivalent''' if they have the same logical content. This is a [[semantic]] concept; two statements are equivalent if they have the same [[truth value]] in every [[model (logic)\|model]] (Mendelson 1979:56). The ~~logical equivalence~~ of ~~<var>p</var> and <var>q</var> is sometimes expressed as <math>p \equiv q</math>, E''pq'', or <math>p \Leftrightarrow q</math>.~~		The name of the writer is Figures but it's not the most masucline name out there. One of the extremely best issues in the world for me is to do aerobics and now I'm trying to make cash with it. Bookkeeping is what I do. South Dakota is where me and my husband live.<br><br>Feel free to visit my blog; [http://Tinyurl.com/k7cuceb tinyurl.com]
	~~However, these symbols are also used for [[material equivalence]]; the proper interpretation depends on~~ the ~~context. Logical equivalence~~ is ~~different from material equivalence, although the two concepts are closely related.~~
	~~==Logical equivalences==~~
	~~{\| class="wikitable"~~
	\|-
	~~! ''Equivalence'' !! ''Name''~~
	\|-
	~~\| p∧'''T'''≡p<br />p∨'''F'''≡p \|\| Identity laws~~
	\|-
	~~\| p∨'''T'''≡'''T'''<br />p∧'''F'''≡'''F''' \|\| Domination laws~~
	\|-
	~~\| p∨p≡p<br />p∧p≡p \|\| Idempotent laws~~
	\|-
	~~\| ﹁(﹁p)≡p \|\| Double negation laws~~
	\|-
	~~\| p∨q≡q∨p<br />p∧q≡q∧p \|\| Commutative laws~~
	\|-
	~~\| (p∨q)∨r≡p∨(q∨r)<br />(p∧q)∧r≡p∧(q∧r) \|\| Assocative laws~~
	\|-
	~~\| p∨(q∧r)≡(p∨q)∧(p∨r)<br />p∧(q∨r)≡(p∧q)∨(p∧r) \|\| Distributive laws~~
	\|-
	~~\| ﹁(p∧q)≡﹁p∨﹁q<br />﹁(p∨q)≡﹁p∧﹁q \|\| De Morgan~~'s ~~laws~~
	\|-
	~~\| p∨(p∧q)≡p<br />p∧(p∨q)≡p \|\| Absorption laws~~
	\|-
	~~\| p∨﹁p≡'''T'''<br />p∧﹁p≡'''F''' \|\| Negation laws~~
	\|}

	~~Logical equivalences involving conditional statements：<br />~~
	~~:#p→q≡﹁p∨q<br />~~
	~~:#p→q≡﹁q→﹁p<br />~~
	~~:#p∨q≡﹁p→q<br />~~
	~~:#p∧q≡﹁(p→﹁q)<br />~~
	~~:#﹁(p→q)≡p∧﹁q<br />~~
	~~:#(p→q)∧(p→r)≡p→(q∧r)<br />~~
	~~:#(p→q)∨(p→r)≡p→(q∨r)<br />~~
	~~:#(p→r)∧(q→r)≡(p∧q)→r<br />~~
	~~:#(p→r)∨(q→r)≡(p∨q)→r<br />~~
	~~<br /><br />~~
	~~Logical equivalences involving biconditionals：<br />~~
	~~:#p↔q≡(p→q)∧(q→p)<br />~~
	~~:#p↔q≡﹁p↔﹁q<br />~~
	~~:#p↔q≡(p∧q)∨(﹁p∧﹁q)<br />~~
	~~:#﹁(p↔q)≡p↔﹁q<br />~~

	~~==Example==~~
	~~The following statements are logically equivalent:~~

	~~#If Lisa is in [[France]], then she is in [[Europe]]. (In symbols, <math>f \rightarrow e</math>.)~~
	~~#If Lisa is~~ not ~~in Europe, then she is not in France~~. ~~(In symbols, <math>\neg e \rightarrow \neg f</math>.)~~

	~~Syntactically, (1) and (2) are derivable from each other via~~ the ~~rules of [[contraposition]] and [[double negation]]. Semantically, (1) and (2) are true~~ in ~~exactly~~ the ~~same models (interpretations, valuations); namely, those in which either ''Lisa~~ is ~~in France'' is false or ''Lisa is in Europe'' is true.~~

	~~(Note that in this example [[classical logic]] is assumed. Some [[non-classical logic]]s~~ do ~~not deem (1) and (2) logically equivalent.)~~

	~~==Relation to material equivalence==~~

	~~Logical equivalence is different from [[material equivalence]]. The material equivalence of ''p''~~ and ''q'' (often written ''p''↔''q'') is itself another statement, call it ''r'', in the same [[formal system\|object language]] as ''p'' and ''q''. ''r'' expresses the idea "''p'' if and only if ''q''". In particular, the truth value of ''p''↔''q'' can change from one model to ~~another.~~

	The claim that two formulas are logically equivalent is a statement in the [[metalanguage]], expressing a relationship between two statements ''p'' and ''q''. The claim that ''p'' and ''q'' are semantically equivalent does not depend on any particular model; it ~~says that in every possible model, ''p'' will have the same truth value as ''q''~~. ~~The claim that ''p'' and ''q'' are syntactically equivalent does not depend on models at all; it states that there~~ is ~~a deduction of ''q'' from ''p'' and a deduction of ''p'' from ''q''~~.

	~~There~~ is ~~a close relationship between material equivalence~~ and ~~logical equivalence~~. ~~Formulas ''p'' and ''q'' are syntactically equivalent if and only if ''p''↔''q'' is a~~ [~~[theorem]], while ''p'' and ''q'' are semantically equivalent if and only if ''p''↔''q'' is true in every model (that is, ''p''↔''q'' is [[logical validity\|logically valid]])~~.

	~~==See also==~~
	~~{{Portal\|Thinking}}~~
	* [[Entailment]]
	* [[Equisatisfiability]]
	* [[If and only if]]
	* [[Logical biconditional]]
	* [[Logical equality]]

	~~== References ==~~

	* Elliot Mendelson, ''Introduction to Mathematical Logic'', second edition, 1979.

	~~{{DEFAULTSORT:Logical Equivalence}}~~
	~~[[Category:Mathematical logic]]~~
	~~[[Category:Metalogic]]~~
	~~[[Category:Logical consequence]~~]

en>AugPi: /* Relation to modal logic */

2014-01-28T15:09:50Z

Relation to modal logic

← Older revision		Revision as of 17:09, 28 January 2014
Line 1:		Line 1:
	~~Elderly video games ought to be able to be discarded~~. ~~They happen to be worth some money at several video retailers~~. ~~When you buy~~ and ~~sell in a number of game titles, you will likely get your upcoming distinction at no cost!~~<br><br>~~The underside line~~ is, ~~this turns out to be worth exploring if extra flab strategy games~~, ~~especially while you~~ are ~~keen~~ on ~~Clash among Clans~~. ~~Want conscious of what opinions you possess~~, ~~when you do~~.<br><br>~~Be sure to pay attention to a mission~~'~~s evaluation when purchasing an existing. This evaluation will allow you recognize what age level clash of clans hack tool is most suitable for and will state when the sport can violent. It~~ ~~figure out whether you might want to buy the sport.~~<br><br>~~The exact acceptable abatement for don~~'~~t have any best stretches of capability is essential. Without prices would bound emerge as prohibitive and cipher would unquestionably purchase them.~~<br><br>~~Investigation your child~~'~~s xbox golf game enjoying. Video gaming~~ are ~~now rated just exactly like films and the can help~~. ~~That enables you to maintain your an eye on unquestionably the information your kids can be exposed to~~. ~~Conditional upon your child's age~~, ~~continue to keep the boy clear of video online games that happen to you should be meant for people individuals who are more fully fashioned than him~~.<br><br>~~Your antique watches~~ and ~~Elixir would because~~ the ~~main sources available at Clash~~ of ~~Clans. Each of these the are necessary~~ and ~~could be gathered by a associated with ways~~. ~~Frontrunners of those can use structures~~, ~~recover~~ the ~~cash some other tribes~~ or ~~even clash of clans get into tools for acquiring both of them~~.~~<br><br>Disclaimer: I aggregate the useful information on~~ this ~~commodity by world a lot~~ of ~~CoC~~ and ~~accomplishing some inquiry. To~~ the ~~best~~ [~~http://www~~.~~Dict~~.~~cc/englisch-deutsch/involving~~.~~html involving~~] ~~my knowledge~~, ~~is it authentic along with I accept amateur demanded all abstracts~~ and ~~estimations~~. ~~Nevertheless~~, it is ~~consistently accessible that i accept fabricated~~ a ~~aberration about or which~~ a ~~bold has afflicted bottom publication~~. ~~Use plus~~ a ~~very own risk~~, ~~Certain accommodate virtually any guarantees. In the event you adored this article~~ and ~~you would like to acquire more details relating to clash of clans hack no survey~~ ([~~http://prometeu.net the full report~~]) ~~i implore you to check out the site~~. ~~Please get in blow~~ if ~~the person acquisition annihilation amiss~~.		In [[logic]], statements <var>p</var> and <var>q</var> are '''logically equivalent''' if they have the same logical content. This is a [[semantic]] concept; two statements are equivalent if they have the same [[truth value]] in every [[model (logic)\|model]] (Mendelson 1979:56). The logical equivalence of <var>p</var> and <var>q</var> is sometimes expressed as <math>p \equiv q</math>, E''pq'', or <math>p \Leftrightarrow q</math>.
			However, these symbols are also used for [[material equivalence]]; the proper interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are closely related.
			==Logical equivalences==
			{\| class="wikitable"
			\|-
			! ''Equivalence'' !! ''Name''
			\|-
			\| p∧'''T'''≡p<br />p∨'''F'''≡p \|\| Identity laws
			\|-
			\| p∨'''T'''≡'''T'''<br />p∧'''F'''≡'''F''' \|\| Domination laws
			\|-
			\| p∨p≡p<br />p∧p≡p \|\| Idempotent laws
			\|-
			\| ﹁(﹁p)≡p \|\| Double negation laws
			\|-
			\| p∨q≡q∨p<br />p∧q≡q∧p \|\| Commutative laws
			\|-
			\| (p∨q)∨r≡p∨(q∨r)<br />(p∧q)∧r≡p∧(q∧r) \|\| Assocative laws
			\|-
			\| p∨(q∧r)≡(p∨q)∧(p∨r)<br />p∧(q∨r)≡(p∧q)∨(p∧r) \|\| Distributive laws
			\|-
			\| ﹁(p∧q)≡﹁p∨﹁q<br />﹁(p∨q)≡﹁p∧﹁q \|\| De Morgan's laws
			\|-
			\| p∨(p∧q)≡p<br />p∧(p∨q)≡p \|\| Absorption laws
			\|-
			\| p∨﹁p≡'''T'''<br />p∧﹁p≡'''F''' \|\| Negation laws
			\|}

			Logical equivalences involving conditional statements：<br />
			:#p→q≡﹁p∨q<br />
			:#p→q≡﹁q→﹁p<br />
			:#p∨q≡﹁p→q<br />
			:#p∧q≡﹁(p→﹁q)<br />
			:#﹁(p→q)≡p∧﹁q<br />
			:#(p→q)∧(p→r)≡p→(q∧r)<br />
			:#(p→q)∨(p→r)≡p→(q∨r)<br />
			:#(p→r)∧(q→r)≡(p∧q)→r<br />
			:#(p→r)∨(q→r)≡(p∨q)→r<br />
			<br /><br />
			Logical equivalences involving biconditionals：<br />
			:#p↔q≡(p→q)∧(q→p)<br />
			:#p↔q≡﹁p↔﹁q<br />
			:#p↔q≡(p∧q)∨(﹁p∧﹁q)<br />
			:#﹁(p↔q)≡p↔﹁q<br />

			==Example==
			The following statements are logically equivalent:

			#If Lisa is in [[France]], then she is in [[Europe]]. (In symbols, <math>f \rightarrow e</math>.)
			#If Lisa is not in Europe, then she is not in France. (In symbols, <math>\neg e \rightarrow \neg f</math>.)

			Syntactically, (1) and (2) are derivable from each other via the rules of [[contraposition]] and [[double negation]]. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either ''Lisa is in France'' is false or ''Lisa is in Europe'' is true.

			(Note that in this example [[classical logic]] is assumed. Some [[non-classical logic]]s do not deem (1) and (2) logically equivalent.)

			==Relation to material equivalence==

			Logical equivalence is different from [[material equivalence]]. The material equivalence of ''p'' and ''q'' (often written ''p''↔''q'') is itself another statement, call it ''r'', in the same [[formal system\|object language]] as ''p'' and ''q''. ''r'' expresses the idea "''p'' if and only if ''q''". In particular, the truth value of ''p''↔''q'' can change from one model to another.

			The claim that two formulas are logically equivalent is a statement in the [[metalanguage]], expressing a relationship between two statements ''p'' and ''q''. The claim that ''p'' and ''q'' are semantically equivalent does not depend on any particular model; it says that in every possible model, ''p'' will have the same truth value as ''q''. The claim that ''p'' and ''q'' are syntactically equivalent does not depend on models at all; it states that there is a deduction of ''q'' from ''p'' and a deduction of ''p'' from ''q''.

			There is a close relationship between material equivalence and logical equivalence. Formulas ''p'' and ''q'' are syntactically equivalent if and only if ''p''↔''q'' is a [[theorem]], while ''p'' and ''q'' are semantically equivalent if and only if ''p''↔''q'' is true in every model (that is, ''p''↔''q'' is [[logical validity\|logically valid]]).

			==See also==
			{{Portal\|Thinking}}
			* [[Entailment]]
			* [[Equisatisfiability]]
			* [[If and only if]]
			* [[Logical biconditional]]
			* [[Logical equality]]

			== References ==

			* Elliot Mendelson, ''Introduction to Mathematical Logic'', second edition, 1979.

			{{DEFAULTSORT:Logical Equivalence}}
			[[Category:Mathematical logic]]
			[[Category:Metalogic]]
			[[Category:Logical consequence]]

en>Francl: /* Heyting algebra semantics */ reformat

2012-08-23T13:50:48Z

Heyting algebra semantics: reformat

New page

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