individual constant

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In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premi ...
, the
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symb ...
s used to create expressions consist of
symbol A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise very different c ...
s, 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 First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
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 connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
s, quantifiers, and variables that stand for
statements Statement or statements may refer to: Common uses * Statement (computer science), the smallest standalone element of an imperative programming language *Statement (logic), declarative sentence that is either true or false *Statement, a declarativ ...
. 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 In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending on ...
) and the symbols for the quantifiers "for all" and "there exists". The
equality Equality may refer to: Society * Political equality, in which all members of a society are of equal standing ** Consociationalism, in which an ethnically, religiously, or linguistically divided state functions by cooperation of each group's elit ...
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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each 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 Arity () is the number of arguments or operands taken by a function, operation or relation in logic, mathematics, and computer science. In mathematics, arity may also be named ''rank'', but this word can have many other meanings in mathematics ...
''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''(''x'', ''c'') is a term, but ''c''(''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 <.

# 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 In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The domain ...
, together with interpretations 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 In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\tim ...
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
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. Its domain is the set  =  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.

# Descriptive signs

Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. ...
introduced a terminology distinguishing between logical and non-logical symbols (which he called ''descriptive signs'') of a
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...
under a certain type of interpretation, 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 to ''logical 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.Carnap, Rudolf (1958). ''Introduction to symbolic logic and its applications''. New York: Dover.

* Logical constant

;Notes *