atomic formula

TheInfoList

OR:

In
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 ordinary
first-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: * $t \equiv c \mid x \mid f \left(t_,\dotsc, t_\right)$, 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''''k''. Functions map
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: * $A, B, ... \equiv P \left(t_,\dotsc, t_\right) \mid A \wedge B \mid \top \mid A \vee B \mid \bot \mid A \supset B \mid \forall x.\ A \mid \exists x.\ A$, that is, a proposition is recursively defined to be an ''n''-ary predicate ''P'' whose arguments are terms ''t''''k'', or an expression composed of
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''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 * $P \left(x\right)$ * $Q \left(y, f \left(x\right)\right)$ * $R \left(z\right)$