Term (logic)
Encyclopedia
In mathematical logic
, universal algebra
, and rewriting systems, terms are expressions which can be obtained
from constant symbols, variables and function symbols. Constant symbols are the 0-ary functions, so no special syntactic class is needed for them.
Terms that do not contain variables are known as ground terms; they are used to form ground expression
s.
Terms in first-order logic are essentially defined this way.
Given a signature
for the function symbols, the set of all possible terms that can be freely generated from the constants, variables and functions form a term algebra
.
An expression formed by applying a predicate to a sequence of terms, whose length matches the arity
of the predicate (or one of the allowed arities, in the case of a multigrade predicate
), is known as an atomic formula
. In bivalent logics
, given an interpretation
, this atomic formula will then be true or false.
,
That is, a term is recursively defined
to be a constant c (a named object from the domain of discourse
), or a variable x (ranging over the objects in the domain of discourse), or an n-ary function f whose arguments are terms tk. Functions map tuple
s of objects to objects.
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
, and rewriting systems, terms are expressions which can be obtained
Recursive definition
In mathematical logic and computer science, a recursive definition is used to define an object in terms of itself ....
from constant symbols, variables and function symbols. Constant symbols are the 0-ary functions, so no special syntactic class is needed for them.
Terms that do not contain variables are known as ground terms; they are used to form ground expression
Ground expression
In mathematical logic, a ground term of a formal system is a term that does not contain any variables at all, and a closed term is a term that has no free variables...
s.
Terms in first-order logic are essentially defined this way.
Given a signature
Signature (logic)
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.Signatures play the same...
for the function symbols, the set of all possible terms that can be freely generated from the constants, variables and functions form a term algebra
Term algebra
In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. For example, in a signature consisting of a single binary operation, the term algebra over a set X of variables is exactly the free magma generated by X...
.
An expression formed by applying a predicate to a sequence of terms, whose length matches the arity
Arity
In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...
of the predicate (or one of the allowed arities, in the case of a multigrade predicate
Multigrade operator
In logic and mathematics, a multigrade operator \Omega is a parametric operator with parameter k in the set N of non-negative integers....
), is known as an atomic formula
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...
. In bivalent logics
Principle of bivalence
In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...
, given an interpretation
Interpretation (logic)
An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation...
, this atomic formula will then be true or false.
Formal definition
A term may be defined as:,That is, a term is recursively defined
Recursive definition
In mathematical logic and computer science, a recursive definition is used to define an object in terms of itself ....
to be a constant c (a named object from the domain of discourse
Domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse , is the set of entities over which certain variables of interest in some formal treatment may range...
), or a variable x (ranging over the objects in the domain of discourse), or an n-ary function f whose arguments are terms tk. Functions map tuple
Tuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
s of objects to objects.