Valuation (logic)

<h2 id="mathematical-logic">Mathematical logic</h2>
<p>In mathematical logic (especially model theory), a valuation is an assignment of truth values to formal sentences that follows a <a href="/facts/T-schema/zNkFFSHD">truth schema</a>. Valuations are also called truth assignments.
</p><p>In propositional logic, there are no quantifiers, and formulas are built from propositional variables using logical connectives. In this context, a valuation begins with an assignment of a truth value to each propositional variable. This assignment can be uniquely extended to an assignment of truth values to all propositional formulas.
</p><p>In first-order logic, a language consists of a collection of constant symbols, a collection of function symbols, and a collection of relation symbols. Formulas are built out of <a href="/facts/Atomic_formula/s9WMnbqT">atomic formulas</a> using logical connectives and quantifiers.  A <a href="/facts/Structure_(mathematical_logic)/GsMx1fEb">structure</a> consists of a set (<a href="/facts/Domain_of_discourse/T1WRQxx6">domain of discourse</a>) that determines the range of the quantifiers, along with interpretations of the constant, function, and relation symbols in the language. Corresponding to each structure is a unique truth assignment for all <a href="/facts/Sentence_(mathematical_logic)/lnsLzvTx">sentences</a> (formulas with no <a href="/facts/Free_variables/ndWnXXYs">free variables</a>) in the language.
</p>
<h2 id="notation">Notation</h2>
<p>If 
  
    
      
        v
      
    
    {\displaystyle v}
  
 is a valuation, that is, a mapping from the atoms to the set 
  
    
      
        {
        t
        ,
        f
        }
      
    
    {\displaystyle \{t,f\}}
  
, then the double-bracket notation is commonly used to denote a valuation; that is, 
  
    
      
        v
        (
        ϕ
        )
        =
        [
        
        [
        ϕ
        ]
        
        
          ]
          
            v
          
        
      
    
    {\displaystyle v(\phi )=[\![\phi ]\!]_{v}}
  
 for a proposition 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
  
.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Algebraic_semantics_(mathematical_logic)/6fLp6gBo">Algebraic semantics</a></li></ul>

<ul><li><a href="/facts/Helena_Rasiowa/XVNygStk">Rasiowa, Helena</a>; <a href="/facts/Roman_Sikorski/PC4UY9xH">Sikorski, Roman</a> (1970), <i>The Mathematics of Metamathematics</i> (3rd ed.), Warsaw: PWN, chapter 6 <i>Algebra of formalized languages</i>.</li>
<li>J. Michael Dunn; Gary M. Hardegree (2001). <a href="https://books.google.com/books?id=LTOfZn728-EC&pg=PA155"><i>Algebraic methods in philosophical logic</i></a>. Oxford University Press. p. 155. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-19-853192-0.</li></ul>

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

<ol>
<li id="fn:1"><p>Dirk van Dalen, (2004) Logic and Structure, Springer Universitext, (see section 1.2) ISBN 978-3-540-20879-2 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Valuation (logic) open-in-new

Valuation (logic)