Inmathematical logic, especiallymodel theory,non-logical symbols are elements of aformal language whoseinterpretation may change depending on the model. Infirst-order logic, these usually consist of constant symbols,function symbols, andpredicates. This is in contrast tological constants which are required to have the same interpretation under every model, such aslogical connectives andquantifiers.

A non-logical symbol only has meaning or semantic content when one is assigned to it by means of aninterpretation. Consequently, asentence containing a non-logical symbol lacks meaning except under an interpretation, so a sentence is said to betrue or false under an interpretation. These concepts are defined and discussed in thearticle on first-order logic, and in particular thesection on syntax.

Theequality 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 arbitraryequivalence relation.

Asignature 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 specificarityn (a natural number), or a relation symbol of a specific arity. The additional information controls how the non-logical symbols can be used to form terms and formulas. For instance iff is a binary function symbol andc is a constant symbol, thenf(x, c) is a term, butc(x, f) is not a term. Relation symbols cannot be used in terms, but they can be used to combine one or more (depending on the arity) terms into an atomic formula.

For example a signature could consist of a binary function symbol +, a constant symbol 0, and a binary relation symbol <.

Structures over a signature, also known asmodels, provideformal semantics to a signature and thefirst-order language over it.

A structure over a signature consists of a set${\displaystyle D}$ (known as thedomain of discourse) together withinterpretations of the non-logical symbols: Every constant symbol is interpreted by an element of${\displaystyle D}$ and the interpretation of an${\displaystyle n}$-ary function symbol is an${\displaystyle n}$-ary function on${\displaystyle D;}$ that is, a function${\displaystyle D^n\to D}$ from the${\displaystyle n}$-foldcartesian product of the domain to the domain itself. Every${\displaystyle n}$-ary relation symbol is interpreted by an${\displaystyle n}$-ary relation on the domain; that is, by a subset of${\displaystyle D^n.}$

An example of a structure over the signature mentioned above is the ordered group ofintegers. Its domain is the set${\displaystyle \mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}}$ of integers. The binary function symbol${\displaystyle +}$ is interpreted by addition, the constant symbol 0 by the additive identity, and the binary relation symbol < by the relation less than.

Outside a mathematical context, it is often more appropriate to work with more informal interpretations.

Rudolf Carnap introduced a terminology distinguishing between logical and non-logical symbols (which he calleddescriptive signs) of aformal system under a certain type ofinterpretation, defined by what they describe in the world.

A descriptive sign is defined as any symbol of a formal language which designates things or processes in the world, or properties or relations of things. This is in contrast tological signs which do not designate any thing in the world of objects. The use of logical signs is determined by the logical rules of the language, whereas meaning is arbitrarily attached to descriptive signs when they are applied to a given domain of individuals.^[1]

• Logical constant

Notes

• Hinman, P. (2005),Fundamentals of Mathematical Logic,A K Peters,ISBN 978-1-56881-262-5

• Semantics section inClassical Logic (an entry ofStanford Encyclopedia of Philosophy)

• Logic symbols
• Formal languages

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