Definitions

# Non-logical symbol

Non-logical symbol is a technical term used in Logic

In logic, the non-logical symbols (sometimes also called non-logical constants) of a language of first-order logic are all of the symbols which are not part of symbolic logic, including symbols which, in an interpretation, may stand for 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. Main article: first order logic especially Syntax of first-order logic

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 nonlogical symbol, it may be interpreted by an arbitrary equivalence relation.

## Signatures

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. The additional information controls how the non-logical symbols can be used to form terms and formulas. For instance if f is a binary function symbol and c is a constant symbol, then f(xc) is a term, but c(xf) 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 <.

## Models

Structures over a signature, also known as models, provide formal semantics to a signature and the first-order language over it.

A structure over a signature consists of a set D, known as the domain of discourse, together with interpreations of the non-logical symbols: Every constant symbol is interpreted by an element of D, and the interpretation of an n-ary function symbol is an n-ary function on D, i.e. a function Dn → D from the n-fold cartesian product of the domain to the domain itself. Every n-ary relation symbol is interpreted by an n-ary relation on the domain, i.e. by a subset of Dn.

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

## Informal semantics

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