logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...

, the

formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...

s used to create expressions consist of

symbol A symbol is a mark, Sign (semiotics), sign, or word that indicates, signifies, or is understood as representing an idea, physical object, object, or wikt:relationship, relationship. Symbols allow people to go beyond what is known or seen by cr ...

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 called predicate logic, predicate calculus, or quantificational logic, is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over ...

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. Connectives can be used to connect logical formulas. For instance in the syntax of propositional logic, the ...

s, 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 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 ...

) 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.

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 In logic, mathematics, and computer science, arity () is the number of arguments or operands taken by a function, operation or relation. In mathematics, arity may also be called rank, but this word can have many other meanings. In logic and ...

''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 or universe of discourse (borrowing from the mathematical concept of ''universe'') is the set of entities over which certain variables of interest in some formal treatment may range. It is also ...

) 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;

that is, a function

D^n \to D

from the

n

-fold

cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of the domain to the domain itself. Every

n

-ary relation symbol is interpreted by an

n

-ary relation on the domain; that is, by a subset of

D^n.

An example of a structure over the signature mentioned above is the ordered group of

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

s. Its domain is the set

\Z = \

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 and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the foundation of knowledge in ma ...

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.

References

;Notes *

External links