In

_{''k''}. Functions map _{''k''}, or an expression composed of _{1} ,…, ''t''_{''n''}) for ''P'' a predicate, and the ''t''_{''n''} terms.
All other well-formed formulae are obtained by composing atoms with logical connectives and quantifiers.
For example, the formula ∀''x. P'' (''x'') ∧ ∃''y. Q'' (''y'', ''f'' (''x'')) ∨ ∃''z. R'' (''z'') contains the atoms
* $P\; (x)$
* $Q\; (y,\; f\; (x))$
* $R\; (z)$

mathematical logic
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, an atomic formula (also known as an atom or a prime formula) is a formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

with no deeper proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

al structure, that is, a formula that contains no logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

s or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...

s of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives.
The precise form of atomic formulas depends on the logic under consideration; for propositional logic
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...

, for example, a propositional variable
In mathematical logic, a propositional variable (also called a sentential variable or sentential letter) is an input variable (that can either be true or false) of a truth function. Propositional variables are the basic building-blocks of propos ...

is often more briefly referred to as an "atomic formula", but, more precisely, a propositional variable is not an atomic formula but a formal expression that denotes an atomic formula. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...

, atomic formulas are merely strings of symbols with a given signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...

, which may or may not be satisfiable
In mathematical logic, a formula is ''satisfiable'' if it is true under some assignment of values to its variables. For example, the formula x+3=y is satisfiable because it is true when x=3 and y=6, while the formula x+1=x is not satisfiable over ...

with respect to a given model.
Atomic formula in first-order logic

The well-formed terms and propositions of ordinaryfirst-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...

have the following syntax
In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), ...

:
Terms:
* $t\; \backslash equiv\; c\; \backslash mid\; x\; \backslash mid\; f\; (t\_,\backslash dotsc,\; t\_)$,
that is, a term is recursively defined to be a constant ''c'' (a named object from the domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
Overview
The domain ...

), or a variable ''x'' (ranging over the objects in the domain of discourse), or an ''n''-ary function ''f'' whose arguments are terms ''t''tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...

s of objects to objects.
Propositions:
* $A,\; B,\; ...\; \backslash equiv\; P\; (t\_,\backslash dotsc,\; t\_)\; \backslash mid\; A\; \backslash wedge\; B\; \backslash mid\; \backslash top\; \backslash mid\; A\; \backslash vee\; B\; \backslash mid\; \backslash bot\; \backslash mid\; A\; \backslash supset\; B\; \backslash mid\; \backslash forall\; x.\backslash \; A\; \backslash mid\; \backslash exists\; x.\backslash \; A$,
that is, a proposition is recursively defined to be an ''n''-ary predicate ''P'' whose arguments are terms ''t''logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...

s (and, or) and quantifiers (for-all, there-exists) used with other propositions.
An atomic formula or atom is simply a predicate applied to a tuple of terms; that is, an atomic formula is a formula of the form ''P'' (''t''See also

* Inmodel theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...

, structures assign an interpretation to the atomic formulas.
* In proof theory, polarity assignment for atomic formulas is an essential component of focusing.
* Atomic sentence
References

Further reading

* {{Mathematical logic Predicate logic Logical expressions de:Aussage (Logik)#einfache Aussagen - zusammengesetzte Aussagen