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Interpretation (logic)
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<p>An interpretation is an assignment of <a href="/facts/Meaning_(philosophy)/WvFm4kKM">meaning</a> to the <a href="/facts/Symbol_(formal)/NS2oUqEk">symbols</a> of a <a href="/facts/Formal_language/crDTyP8q">formal language</a>. Many formal languages used in <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, <a href="/facts/Logic/8JnpJLtt">logic</a>, and <a href="/facts/Theoretical_computer_science/jSkGGuvy">theoretical computer science</a> are defined in solely <a href="/facts/Syntax/6PxSV98p">syntactic</a> terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called <a href="/facts/Formal_semantics_(logic)/VdAKHszK">formal semantics</a>. </p><p>The most commonly studied formal logics are <a href="/facts/Propositional_logic/MNt5v31V">propositional logic</a>, <a href="/facts/Predicate_logic/lcISqWIg">predicate logic</a> and their <a href="/facts/Modal_logic/I7uXvYYT">modal</a> analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that provides the <a href="/facts/Extension_(predicate_logic)/ruBkmsJ6">extension</a> of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate <i>T</i> (for "tall") and assign it the extension {<i>a</i>} (for "Abraham Lincoln"). All our interpretation does is assign the extension {a} to the non-logical constant <i>T</i>, and does not make a claim about whether <i>T</i> is to stand for tall and 'a' for Abraham Lincoln. Nor does logical interpretation have anything to say about logical connectives like 'and', 'or' and 'not'. Though <i>we</i> may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function. </p><p>An interpretation often (but not always) provides a way to determine the <a href="/facts/Truth_value/SY4e2w8l">truth values</a> of <a href="/facts/Sentence_(mathematical_logic)/lnsLzvTx">sentences</a> in a language. If a given interpretation assigns the value True to a sentence or <a href="/facts/Theory_(mathematical_logic)/mYe5gQpI">theory</a>, the interpretation is called a <a href="/facts/Model_(model_theory)/GsMx1fEb">model</a> of that sentence or theory. </p>