Signature (logic)

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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 ...
, especially
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, a signature lists and describes the
non-logical symbol 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 a ...
s of a
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 ...
. In
universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study ...
, a signature lists the operations that characterize an
algebraic structure In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.

# Definition

Formally, a (single-sorted) signature can be defined as a 4-tuple , where ''S''func and ''S''rel are disjoint sets not containing any other basic logical symbols, called respectively * ''function symbols'' (examples: +, ×, 0, 1), * ''relation symbols'' or ''predicates'' (examples: ≤, ∈), * ''constant symbols'' (examples: 0, 1), and a function ar: ''S''func $\cup$ ''S''rel$\mathbb N$ which assigns a natural number called ''
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 ...
'' to every function or relation symbol. A function or relation symbol is called ''n''-ary if its arity is ''n''. Some authors define a nullary (0-ary) function symbol as ''constant symbol'', otherwise constant symbols are defined separately. A signature with no function symbols is called a relational signature, and a signature with no relation symbols is called an algebraic signature. A finite signature is a signature such that ''S''func and ''S''rel are
finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb Traditionally, a finite verb (from la, fīnītus, past partici ...
. More generally, the cardinality of a signature ''σ'' = (''S''func, ''S''rel, ''S''const, ar) is defined as = + + . The language of a signature is the set of all well formed sentences built from the symbols in that signature together with the symbols in the logical system.

# Other conventions

In universal algebra the word type or similarity type is often used as a synonym for "signature". In model theory, a signature ''σ'' is often called a vocabulary, or identified with the (first-order) language ''L'' to which it provides the non-logical symbols. However, the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
of the language ''L'' will always be infinite; if ''σ'' is finite then will be 0. As the formal definition is inconvenient for everyday use, the definition of a specific signature is often abbreviated in an informal way, as in: :"The standard signature for
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s is , where − is a unary operator." Sometimes an algebraic signature is regarded as just a list of arities, as in: :"The similarity type for abelian groups is ." Formally this would define the function symbols of the signature as something like ''f''0 (which is binary), ''f''1 (which is unary) and ''f''2 (which is nullary), but in reality the usual names are used even in connection with this convention. In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, very often symbols are not allowed to be nullary, so that constant symbols must be treated separately rather than as nullary function symbols. They form a set ''S''const disjoint from ''S''func, on which the arity function ''ar'' is not defined. However, this only serves to complicate matters, especially in proofs by induction over the structure of a formula, where an additional case must be considered. Any nullary relation symbol, which is also not allowed under such a definition, can be emulated by a unary relation symbol together with a sentence expressing that its value is the same for all elements. This translation fails only for empty structures (which are often excluded by convention). If nullary symbols are allowed, then every formula of propositional logic is also a formula 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 ...
. An example for an infinite signature uses ''S''func = ∪ and ''S''rel = to formalize expressions and equations about a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ca ...
over an infinite scalar field ''F'', where each f''a'' denotes the unary operation of scalar multiplication by ''a''. This way, the signature and the logic can be kept single-sorted, with vectors being the only sort.

# Use of signatures in logic and algebra

In the context 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 ...
, the symbols in a signature are also known as the non-logical symbols, because together with the logical symbols they form the underlying alphabet over which two
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 are inductively defined: The set of ''terms'' over the signature and the set of (well-formed) ''formulas'' over the signature. In a structure, an ''interpretation'' ties the function and relation symbols to mathematical objects that justify their names: The interpretation of an ''n''-ary function symbol ''f'' in a structure ''A'' with ''domain'' ''A'' is a function ''fA'': ''An'' → ''A'', and the interpretation of an ''n''-ary relation symbol is a relation ''RA'' ⊆ ''An''. Here ''A''''n'' = ''A'' × ''A'' × ... × ''A'' denotes 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 ''A'' with itself, and so ''f'' is in fact an ''n''-ary function, and ''R'' an ''n''-ary relation.

# Many-sorted signatures

For many-sorted logic and for many-sorted structures signatures must encode information about the sorts. The most straightforward way of doing this is via symbol types that play the role of generalized arities.Many-Sorted Logic
the first chapter i

written b
Calogero G. Zarba

## Symbol types

Let ''S'' be a set (of sorts) not containing the symbols × or →. The symbol types over ''S'' are certain words over the alphabet $S \cup \$: the relational symbol types , and the functional symbol types , for non-negative integers ''n'' and . (For , the expression denotes the empty word.)

## Signature

A (many-sorted) signature is a triple (''S'', ''P'', type) consisting of * a set ''S'' of sorts, * a set ''P'' of symbols, and * a map type which associates to every symbol in ''P'' a symbol type over ''S''.

* Term algebra

# References

*
Free online edition
*

# External links

Stanford Encyclopedia of Philosophy

Model theory
—by
Wilfred Hodges Wilfrid Augustine Hodges, FBA (born 27 May 1941) is a British mathematician and logician known for his work in model theory. Life Hodges attended New College, Oxford (1959–65), where he received degrees in both '' Literae Humaniores'' and (C ...
.
PlanetMath:
Entry
Signature
describes the concept for the case when no sorts are introduced.

{{Mathematical logic Model theory Universal algebra