Ground expression

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 a and b , the sentence Q(a) \lor P(b) is a ground formula. A ground expression is a ground term or ground formula.

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 a and b , the sentence Q(a) \lor P(b) is a ground formula. A ground expression is a ground term or ground formula. ==Examples== Consider the following expressions in first order logic over a signature containing the constant symbols 0 and 1 for the numbers 0 and 1, respectively, a unary function symbol s for the successor function and a binary function symbol + for addition. s(0), s(s(0)), s(s(s(0))), \ldots are ground terms; 0 + 1, \; 0 + 1 + 1, \ldots are ground terms; 0+s(0), \; s(0)+ s(0), \; s(0)+s(s(0))+0 are ground terms; x + s(1) and s(x) are terms, but not ground terms; s(0) = 1 and 0 + 0 = 0 are ground formulae. ==Formal definitions== What follows is a formal definition for first-order languages. Let a first-order language be given, with C the set of constant symbols, F the set of functional operators, and P the set of predicate symbols. ===Ground term=== A is a term that contains no variables. Ground terms may be defined by logical recursion (formula-recursion): Elements of C are ground terms; If f \in F is an n -ary function symbol and \alpha_1, \alpha_2, \ldots, \alpha_n are ground terms, then f\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right) is a ground term. Every ground term can be given by a finite application of the above two rules (there are no other ground terms; in particular, predicates cannot be ground terms). Roughly speaking, the Herbrand universe is the set of all ground terms. ===Ground atom=== A , or is an atomic formula all of whose argument terms are ground terms. If p \in P is an n -ary predicate symbol and \alpha_1, \alpha_2, \ldots, \alpha_n are ground terms, then p\left(\alpha_1, \alpha_2, \ldots, \alpha_n\right) is a ground predicate or ground atom. Roughly speaking, the Herbrand base is the set of all ground atoms, while a Herbrand interpretation assigns a truth value to each ground atom in the base. ===Ground formula=== A or is a formula without variables. Ground formulas may be defined by syntactic recursion as follows: A ground atom is a ground formula. If \varphi and \psi are ground formulas, then \lnot \varphi , \varphi \lor \psi , and \varphi \land \psi are ground formulas. Ground formulas are a particular kind of closed formulas. ==See also== == Notes== ==References==

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