In mathematical logic, an atomic formula or atom is a formula with no underlying propositional structure. The precise form of the atomic formula depends on the logic under consideration; for propositional logic, the atoms are the propositional variables, whereas for predicate logic, the atoms are of the form P (t₁, …, t_n), where the t_i are terms and P is a predicate symbol. Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... In mathematical logic, a formula is a formal syntactic object that expresses a proposition, except that the proposition may depend on the values of the free variables of the formula. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... Propositional logic or sentential logic is the logic of propositions, sentences, or clauses. ... In mathematical logic, a propositional variable (also called a sentential variable) is a variable which can either be true or false. ... ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...

The atomic formulas are the simplest propositions of the logic. More complex propositions are formed by combining simpler propositions using the connectives of the logic. One thus obtains a recursively defined language of well-formed formulas of the logic. To illustrate, the well-formed propositions (A, B, …) of ordinary first-order logic have the following syntax (written in BNF): In logic, a logical connective is a syntactic operation on sentences, or the symbol for such an operation, that corresponds to a logical operation on the logical values of those sentences. ... A visual form of recursion known as the Droste effect. ... In logic, WFF is an abbreviation for well-formed formula. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... For other uses, see Syntax (disambiguation). ... The Backus-Naur form (BNF) (also known as Backus normal form) is a metasyntax used to express context-free grammars: that is, a formal way to describe formal languages. ...

<wff>  ::= P(t₁, …, t_n) | <wff> | (<wff> <wff>) | (<wff> <wff>) | <variable> <wff>

where P(t₁, …, t_n) are the atomic formulas and <variable> is a variable (x, y, a, b, etc.).

Any proposition, such as ((x P(x) y Q(y, f(x))) z R(z)), is formed from the relevant atoms (P, Q and R in this case) and the syntax rules.

See also

• In model theory, structures assign an interpretation to the atomic formulas.
• In proof theory, polarity assignment for atomic formulas is an essential component of focusing.

In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the structures that underlie mathematical systems. ... In the mathematical discipline of model theory, a structure for a language (referred to as an -structure, and commonly written as a Gothic capital) is an ordered pair whose first member is the domain of discourse or universe set (taken to be a set with possibly relations and functions defined... Proof theory is a branch of mathematical logic that represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. ...

References

• Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-568-81262-0.