Non-logical symbol

In <a href="/facts/Logic/8JnpJLtt">logic</a>, the <a href="/facts/Formal_language/crDTyP8q">formal languages</a> used to create expressions consist of <a href="/facts/Symbol_(formal)/NS2oUqEk">symbols</a>, which can be broadly divided into <a href="/facts/Logical_constants/bvTHIVCM">constants</a> and <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a>. The constants of a language can further be divided into <a href="/facts/Logical_constant/bvTHIVCM">logical symbols</a> and non-logical symbols (sometimes also called logical and non-logical constants).
The non-logical symbols of a language of <a href="/facts/First-order_logic/lcISqWIg">first-order logic</a> consist of <a href="/facts/Predicate_(mathematical_logic)/5WazAvKF">predicates</a> and individual constants. These include symbols that, in an interpretation, may stand for individual constants, <a href="/facts/Variable_(mathematics)/VbQlrvKz">variables</a>, <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a>, or <a href="/facts/Predicate_(logic)/5WazAvKF">predicates</a>. A language of first-order logic is a formal language over the alphabet consisting of its non-logical symbols and its <a href="/facts/Logical_constants/bvTHIVCM">logical symbols</a>. The latter include <a href="/facts/Logical_connective/VkPhUs6M">logical connectives</a>, <a href="/facts/Quantifier_(logic)/bgmvwxoo">quantifiers</a>, and variables that stand for <a href="/facts/Statement_(logic)/CjpWAyxk">statements</a>.
A non-logical symbol only has meaning or semantic content when one is assigned to it by means of an <a href="/facts/Interpretation_(logic)/goPmjjQG">interpretation</a>. Consequently, a <a href="/facts/Sentence_(mathematical_logic)/lnsLzvTx">sentence</a> containing a non-logical symbol lacks meaning except under an interpretation, so a sentence is said to be true or false under an interpretation. These concepts are defined and discussed in the <a href="/facts/First_order_logic/lcISqWIg">article on first-order logic</a>, and in particular the <a href="/facts/First-order_logic/lcISqWIg">section on syntax</a>.
The <a href="/facts/Logical_constants/bvTHIVCM">logical constants</a>, by contrast, have the same meaning in all interpretations. They include the symbols for truth-functional connectives (such as "and", "or", "not", "implies", and <a href="/facts/Logical_equivalence/ysCWmsxA">logical equivalence</a>) and the symbols for the quantifiers "for all" and "there exists".
The <a href="/facts/Equality_(mathematics)/AQCuumLg">equality</a> symbol is sometimes treated as a non-logical symbol and sometimes treated as a symbol of logic. If it is treated as a logical symbol, then any interpretation will be required to interpret the equality sign using true equality; if interpreted as a non-logical symbol, it may be interpreted by an arbitrary <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a>.

Non-logical symbol open-in-new

Non-logical symbol