View source for Truth value

One of the two values,  "true"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t0943901.png" /> or  "false"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t0943902.png" />, that can be taken by a given logical formula in an interpretation (model) considered. Sometimes the truth value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t0943903.png" /> is denoted in the literature by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t0943904.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t0943905.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t0943906.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t0943907.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t0943908.png" />. If the truth values of elementary formulas are defined in a model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t0943909.png" />, then the truth value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439010.png" /> of any formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439011.png" /> can be inductively determined in the following way (for classical logic):

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439012.png" /></td> </tr></table>

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One sometimes considers interpretations in which logical formulas may take, besides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439019.png" />, other  "intermediate"  truth values. In such interpretations, the truth values of formulas may be, e.g., elements of Boolean algebras (so-called Boolean-valued models for classical logic, cf. [[Boolean-valued model|Boolean-valued model]]), elements of pseudo-Boolean algebras (also known as Heyting algebras, cf. [[Pseudo-Boolean algebra|Pseudo-Boolean algebra]]) or open sets in topological spaces (for intuitionistic logic), or elements of topological Boolean algebras (for [[Modal logic|modal logic]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439020.png" />) (cf. [[#References|[2]]]). In a Boolean-valued model, if the truth values of elementary formulas are defined, then the truth values of compound formulas can be determined as follows

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439025.png" /> is the complement to the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439026.png" />. For example, in topological models for intuitionistic logic, the truth values of compound formulas can be determined as follows:

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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439031.png" /> denotes the interior of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094390/t09439032.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Novikov,   "Elements of mathematical logic" , Oliver &amp; Boyd and Acad. Press  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Rasiowa,   R. Sikorski,   "The mathematics of metamathematics" , Polska Akad. Nauk  (1963)</TD></TR></table>