In today's article we are going to explore the fascinating world of *Ground expression*. From its origin to its evolution today, *Ground expression* has been a topic of interest to many people in different fields. Through this article, we will dive into the history and importance of *Ground expression*, as well as its implications in modern society. Over time, *Ground expression* has captured the attention of researchers, academics, professionals and enthusiasts alike, and its relevance continues to grow in the contemporary world. Additionally, we will examine how *Ground expression* has influenced various aspects of everyday life, and how its impact remains significant today. Get ready to embark on a fascinating journey about *Ground expression* and discover everything this theme has to offer.

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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.

- are ground terms;
- are ground terms;
- are ground terms;
- and are terms, but not ground terms;
- and are ground formulae.

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):

- Elements of are ground terms;
- If is an -ary function symbol and are ground terms, then 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.

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:

- A ground atom is a ground formula.
- If and are ground formulas, then , , and are ground formulas.

Ground formulas are a particular kind of closed formulas.

- Open formula – formula that contains at least one free variable
- Sentence (mathematical logic) – In mathematical logic, a well-formed formula with no free variables

**^**Alex Sakharov. "Ground Atom".*MathWorld*. Retrieved October 20, 2022.

- Dalal, M. (2000), "Logic-based computer programming paradigms", in Rosen, K.H.; Michaels, J.G. (eds.),
*Handbook of discrete and combinatorial mathematics*, p. 68 - Hodges, Wilfrid (1997),
*A shorter model theory*, Cambridge University Press, ISBN 978-0-521-58713-6 - First-Order Logic: Syntax and Semantics