Boolean domain

<h2 id="generalizations">Generalizations</h2>
<p>The Boolean domain {0, 1} can be replaced by the <a href="/facts/Unit_interval/W4dP8IJo">unit interval</a> [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 
  
    
      
        1
        −
        x
        ,
      
    
    {\displaystyle 1-x,}
  
 conjunction (AND) is replaced with multiplication (
  
    
      
        x
        y
      
    
    {\displaystyle xy}
  
), and disjunction (OR) is defined via <a href="/facts/De_Morgan%2527s_law/dj3F8xpt">De Morgan's law</a> to be 
  
    
      
        1
        −
        (
        1
        −
        x
        )
        (
        1
        −
        y
        )
        =
        x
        +
        y
        −
        x
        y
      
    
    {\displaystyle 1-(1-x)(1-y)=x+y-xy}
  
.
</p><p>Interpreting these values as logical <a href="/facts/Truth_value/SY4e2w8l">truth values</a> yields a <a href="/facts/Multi-valued_logic/qNI26RdF">multi-valued logic</a>, which forms the basis for <a href="/facts/Fuzzy_logic/1PvN4Att">fuzzy logic</a> and <a href="/facts/Probabilistic_logic/4MIPxPIJ">probabilistic logic</a>. In these interpretations, a value is interpreted as the "degree" of truth – to what extent a proposition is true, or the probability that the proposition is true.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Boolean-valued_function/vlJBGgUl">Boolean-valued function</a></li>
<li><a href="/facts/GF(2)/KIKKpYua">GF(2)</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li>Steinbach, Bernd [in German], ed. (2014-04-01) [2013-09-25].  Written at Freiberg, Germany. <a href="https://books.google.com/books?id=_pwxBwAAQBAJ"><i>Recent Progress in the Boolean Domain</i></a> (1 ed.). Newcastle upon Tyne, UK: <a href="/facts/Cambridge_Scholars_Publishing/tMl5b2ru">Cambridge Scholars Publishing</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4438-5638-6. Retrieved 2019-08-04. <a href="https://web.archive.org/web/20210228020207/https://www.tau.ac.il/~ilia1/publications/rpbd_book.pdf">[1]</a> (xxx+428 pages) <a href="https://web.archive.org/web/20200131123408/http://www.informatik.tu-freiberg.de/prof2/ws_bp10/index.html">[2]</a> (NB. Contains extended versions of the best manuscripts from the 10th International Workshop on Boolean Problems held at the <a href="/facts/Technische_Universit%25C3%25A4t_Bergakademie_Freiberg/Mjskj6vb">Technische Universität Bergakademie Freiberg</a>, Germany on 2012-09-19/21.)</li>
<li>Steinbach, Bernd [in German], ed. (2016-05-01).  Written at Freiberg, Germany. <a href="https://books.google.com/books?id=ODX5DAAAQBAJ&pg=PP1"><i>Problems and New Solutions in the Boolean Domain</i></a> (1 ed.). Newcastle upon Tyne, UK: <a href="/facts/Cambridge_Scholars_Publishing/tMl5b2ru">Cambridge Scholars Publishing</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4438-8947-6. Retrieved 2019-08-04. (xxxv+1+445+1 pages) <a href="https://web.archive.org/web/20200112133809/http://www.informatik.tu-freiberg.de/prof2/ws_bp11/index.html">[3]</a> (NB. Contains extended versions of the best manuscripts from the 11th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2014-09-17/19.)</li>
<li>Steinbach, Bernd [in German], ed. (2018-01-01).  Written at Freiberg, Germany. <a href="https://books.google.com/books?id=DnSFDwAAQBAJ"><i>Further Improvements in the Boolean Domain</i></a> (1 ed.). Newcastle upon Tyne, UK: <a href="/facts/Cambridge_Scholars_Publishing/tMl5b2ru">Cambridge Scholars Publishing</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-5275-0371-7. Retrieved 2019-08-04. <a href="https://www.cambridgescholars.com/download/sample/64246">[4]</a> <a href="https://web.archive.org/web/20190804164806/https://www.cambridgescholars.com/download/sample/64246">Archived</a> 2019-08-04 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a> (xli+1+494 pages) <a href="https://web.archive.org/web/20200112133313/http://www.informatik.tu-freiberg.de/prof2/ws_bp12/index.html">[5]</a> (NB. Contains extended versions of the best manuscripts from the 12th International Workshop on Boolean Problems held at the Technische Universität Bergakademie Freiberg, Germany on 2016-09-22/23.)</li>
<li><a href="/facts/Rolf_Drechsler/rH0tYlRc">Drechsler, Rolf</a>; Soeken, Mathias, eds. (2020) [March 2019].  Written at Bremen, Germany. <i>Advanced Boolean Techniques - Selected Papers from the 13th International Workshop on Boolean Problems</i> (1 ed.). Cham, Switzerland: <a href="/facts/Springer_Nature_Switzerland_AG/TpTgApAh">Springer Nature Switzerland AG</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-030-20323-8">10.1007/978-3-030-20323-8</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-030-20322-1. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:240782759">240782759</a>. (vii+265+7 pages) <a href="https://web.archive.org/web/20210515084442/http://www.informatik.uni-bremen.de/iwsbp/2018/">[6]</a> (NB. Contains extended versions of the best manuscripts from the 13th International Workshop on Boolean Problems (IWSBP 2018) held at the <a href="/facts/University_of_Bremen/jAAw3zTl">University of Bremen</a>, Bremen, Germany on 2018-09-19/21.)</li>
<li><a href="/facts/Rolf_Drechsler/rH0tYlRc">Drechsler, Rolf</a>; Große, Daniel, eds. (2021-04-30). <i>Recent Findings in Boolean Techniques - Selected Papers from the 14th International Workshop on Boolean Problems</i> (1 ed.). Cham, Switzerland: <a href="/facts/Springer_Nature_Switzerland_AG/TpTgApAh">Springer Nature Switzerland AG</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-030-68071-8">10.1007/978-3-030-68071-8</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-030-68070-1. (vii+1+197+5 pages) <a href="https://web.archive.org/web/20210515084207/http://www.informatik.uni-bremen.de/iwsbp/">[7]</a> (NB. Contains extended versions of the best manuscripts from the 14th International Workshop on Boolean Problems (IWSBP 2020) held <a href="/facts/COVID-19/YsUgvsaX">virtually</a> on 2020-09-24/25.)</li>
<li>Steinbach, Bernd [in German], ed. (2022-09-29).  Written at Freiberg, Germany. <a href="https://books.google.com/books?id=5K2REAAAQBAJ"><i>Advances in the Boolean Domain</i></a> (1 ed.). Newcastle upon Tyne, UK: <a href="/facts/Cambridge_Scholars_Publishing/tMl5b2ru">Cambridge Scholars Publishing</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1527-58872-1. Retrieved 2024-07-15. (xxii+231+1 pages)</li>
<li><a href="/facts/Rolf_Drechsler/rH0tYlRc">Drechsler, Rolf</a>; Huhn, Sebastian, eds. (2023-05-30).  Written at Bremen, Germany. <i>Advanced Boolean Techniques - Selected Papers from the 15th International Workshop on Boolean Problems</i> (1 ed.). Cham, Switzerland: <a href="/facts/Springer_Nature_Switzerland_AG/TpTgApAh">Springer Nature Switzerland AG</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-031-28916-3">10.1007/978-3-031-28916-3</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-031-28915-6. (viii+172+6 pages) <a href="https://web.archive.org/web/20240616190220/https://www.informatik.uni-bremen.de/iwsbp/2022/">[8]</a> (NB. Contains extended versions of the best manuscripts from the 15th International Workshop on Boolean Problems (IWSBP 2022) held at the University of Bremen, Bremen, Germany on 2022-09-22/23.)</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>van Dalen, Dirk (2004). Logic and Structure. Springer. p. 15. <a href="/wiki/Dirk_van_Dalen" target="_blank">/wiki/Dirk_van_Dalen</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Makinson, David (2008). Sets, Logic and Maths for Computing. Springer. p. 13. Bibcode:2008slmc.book.....M. <a href="/wiki/David_Makinson" target="_blank">/wiki/David_Makinson</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Boolos, George S.; Jeffrey, Richard C. (1980). Computability and Logic. Cambridge University Press. p. 99. <a href="/wiki/George_S._Boolos" target="_blank">/wiki/George_S._Boolos</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Mendelson, Elliott (1997). Introduction to Mathematical Logic (4 ed.). Chapman & Hall/CRC. p. 11. <a href="/wiki/Elliott_Mendelson" target="_blank">/wiki/Elliott_Mendelson</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Hehner, Eric C. R. (2010) [1993]. A Practical Theory of Programming. Springer. p. 3. <a href="/wiki/Eric_C._R._Hehner" target="_blank">/wiki/Eric_C._R._Hehner</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Parberry, Ian (1994). Circuit Complexity and Neural Networks. MIT Press. pp. 65. ISBN 978-0-262-16148-0. <a href="978-0-262-16148-0" target="_blank">978-0-262-16148-0</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Cortadella, Jordi; Kishinevsky, Michael; Kondratyev, Alex; Lavagno, Luciano; Yakovlev, Alex (2002). Logic Synthesis for Asynchronous Controllers and Interfaces. Springer Series in Advanced Microelectronics. Vol. 8. Springer-Verlag Berlin Heidelberg New York. p. 73. ISBN 3-540-43152-7. ISSN 1437-0387. <a href="3-540-43152-7" target="_blank">3-540-43152-7</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
</ol>

Boolean domain open-in-new

Boolean domain