Description of non-logical symbols
In
logic
, especially
mathematical logic
, a
signature
lists and describes the
non-logical symbols
of a
formal language
. In
universal algebra
, a signature lists the operations that characterize an
algebraic structure
. In
model theory
, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic.
Definition
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Formally, a (single-sorted)
signature
can be defined as a 4-tuple
where
and
are disjoint
sets
not containing any other basic logical symbols, called respectively
- function symbols
(examples:
),
- relation symbol
s
or
predicates
(examples:
),
- constant symbols
(examples:
),
and a function
which assigns a natural number called
arity
to every function or relation symbol. A function or relation symbol is called
-ary if its arity is
Some authors define a nullary (
-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
.
[1]
A
finite signature
is a signature such that
and
are
finite
. More generally, the
cardinality
of a signature
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
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In universal algebra the word
type
or
similarity type
is often used as a synonym for "signature". In model theory, a signature
is often called a
vocabulary
, or identified with the
(first-order) language
to which it provides the
non-logical symbols
. However, the
cardinality
of the language
will always be infinite; if
is finite then
will be
.
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 groups
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
(which is binary),
(which is unary) and
(which is nullary), but in reality the usual names are used even in connection with this convention.
In
mathematical logic
, very often symbols are not allowed to be nullary,
[
citation needed
]
so that constant symbols must be treated separately rather than as nullary function symbols. They form a set
disjoint from
on which the arity function
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
.
An example for an infinite signature uses
and
to formalize expressions and equations about a
vector space
over an infinite scalar field
where each
denotes the unary operation of scalar multiplication by
This way, the signature and the logic can be kept single-sorted, with vectors being the only sort.
[2]
Use of signatures in logic and algebra
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In the context of
first-order logic
, 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 languages
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
-ary function symbol
in a structure
with
domain
is a function
and the interpretation of an
-ary relation symbol is a
relation
Here
denotes the
-fold
cartesian product
of the domain
with itself, and so
is in fact an
-ary function, and
an
-ary relation.
Many-sorted signatures
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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.
[3]
Symbol types
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Let
be a set (of sorts) not containing the symbols
or
The symbol types over
are certain words over the alphabet
: the relational symbol types
and the functional symbol types
for non-negative integers
and
(For
the expression
denotes the empty word.)
Signature
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A (many-sorted) signature is a triple
consisting of
- a set
of sorts,
- a set
of symbols, and
- a map
which associates to every symbol in
a symbol type over
See also
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- Term algebra
? Freely generated algebraic structure over a given signature
Notes
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References
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External links
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