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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>.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
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==
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
{| class="wikitable"
* Only registered users will be able to execute this rendering mode.
|-
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.
! ''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 law
|-
| p∨q≡q∨p<br />p∧q≡q∧p || Commutative laws
|-
| (p∨q)∨r≡p∨(q∨r)<br />(p∧q)∧r≡p∧(q∧r)  || Associative 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 />
Registered users will be able to choose between the following three rendering modes:  
:#p→q≡¬p∨q
:#p→q≡¬q→¬p
:#p∨q≡¬p→q
:#p∧q≡¬(p→¬q)
:#¬(p→q)≡p∧¬q
:#(p→q)∧(p→r)≡p→(q∧r)
:#(p→q)∨(p→r)≡p→(q∨r)
:#(p→r)∧(q→r)≡(p∨q)→r
:#(p→r)∨(q→r)≡(p∧q)→r


Logical equivalences involving biconditionals：<br />
'''MathML'''
:#p↔q≡(p→q)∧(q→p)
:<math forcemathmode="mathml">E=mc^2</math>
:#p↔q≡¬p↔¬q
:#p↔q≡(p∧q)∨(¬p∧¬q)
:#¬(p↔q)≡p↔¬q


==Example==
<!--'''PNG'''  (currently default in production)
The following statements are logically equivalent:
:<math forcemathmode="png">E=mc^2</math>


#If Lisa is in [[France]], then she is in [[Europe]].  (In symbols, <math>f \rightarrow e</math>.)
'''source'''
#If Lisa is not in Europe, then she is not in France.  (In symbols, <math>\neg e \rightarrow \neg f</math>.)
:<math forcemathmode="source">E=mc^2</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.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


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


==Relation to material equivalence==
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


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''.
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


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]]).
==Test pages ==


==See also==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
{{Portal|Thinking}}
*[[Displaystyle]]
* [[Logical consequence|Entailment]]
*[[MathAxisAlignment]]
* [[Equisatisfiability]]
*[[Styling]]
* [[If and only if]]
*[[Linebreaking]]
* [[Logical biconditional]]
*[[Unique Ids]]
* [[Logical equality]]
*[[Help:Formula]]


== References ==
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
* Elliot Mendelson, ''Introduction to Mathematical Logic'', second edition, 1979.
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
{{DEFAULTSORT:Logical Equivalence}}
[[Category:Mathematical logic]]
[[Category:Metalogic]]
[[Category:Logical consequence]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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