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Concept
Non-logical symbol
Summary
In logic, the formal languages used to create expressions consist of symbols, which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols (sometimes also called logical and non-logical constants).
The non-logical symbols of a language of first-order logic consist of predicates and individual constants. These include symbols that, in an interpretation, may stand for individual constants, variables, functions, or predicates. A language of first-order logic is a formal language over the alphabet consisting of its non-logical symbols and its logical symbols. The latter include logical connectives, quantifiers, and variables that stand for statements.
A non-logical symbol only has meaning or semantic content when one is assigned to it by means of an interpretation. Consequently, a sentence 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 article on first-order logic, and in particular the section on syntax.
The logical constants, by contrast, have the same meaning in all interpretations. They include the symbols for truth-functional connectives (such as "and", "or", "not", "implies", and logical equivalence) and the symbols for the quantifiers "for all" and "there exists".
The equality 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 equivalence relation.
signature (logic)
A signature is a set of non-logical constants together with additional information identifying each symbol as either a constant symbol, or a function symbol of a specific arity n (a natural number), or a relation symbol of a specific arity.
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Related concepts (10)
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier in the first order formula expresses that everything in the domain satisfies the property denoted by . On the other hand, the existential quantifier in the formula expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula.
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic 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 formal semantics. The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation.
Symbol (formal)
A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in the formal languages studied in mathematics and logic, the term "symbol" refers to the idea, and the marks are considered to be a token instance of the symbol.