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In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables.
In first-order logic with identity with constant symbols and , the sentence is a ground formula. A ground expression is a ground term or ground formula.
Consider the following expressions in first order logic over a signature containing the constant symbols and for the numbers 0 and 1, respectively, a unary function symbol for the successor function and a binary function symbol for addition.
What follows is a formal definition for first-order languages. Let a first-order language be given, with the set of constant symbols, the set of functional operators, and the set of predicate symbols.
A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):
Roughly speaking, the Herbrand universe is the set of all ground terms.
A ground predicate, ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.
If is an -ary predicate symbol and are ground terms, then is a ground predicate or ground atom.
Roughly speaking, the Herbrand base is the set of all ground atoms, [1] while a Herbrand interpretation assigns a truth value to each ground atom in the base.
A ground formula or ground clause is a formula without variables.
Ground formulas may be defined by syntactic recursion as follows:
Ground formulas are a particular kind of closed formulas.
Part of a series on |
Formal languages |
---|
In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables.
In first-order logic with identity with constant symbols and , the sentence is a ground formula. A ground expression is a ground term or ground formula.
Consider the following expressions in first order logic over a signature containing the constant symbols and for the numbers 0 and 1, respectively, a unary function symbol for the successor function and a binary function symbol for addition.
What follows is a formal definition for first-order languages. Let a first-order language be given, with the set of constant symbols, the set of functional operators, and the set of predicate symbols.
A ground term is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion):
Roughly speaking, the Herbrand universe is the set of all ground terms.
A ground predicate, ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.
If is an -ary predicate symbol and are ground terms, then is a ground predicate or ground atom.
Roughly speaking, the Herbrand base is the set of all ground atoms, [1] while a Herbrand interpretation assigns a truth value to each ground atom in the base.
A ground formula or ground clause is a formula without variables.
Ground formulas may be defined by syntactic recursion as follows:
Ground formulas are a particular kind of closed formulas.