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| In [[mathematical logic]], an '''atomic formula''' (also known simply as an '''atom''') is a [[formula (mathematical logic)|formula]] with no deeper [[proposition]]al structure, that is, a formula that contains no [[logical connective]]s or equivalently a formula that has no strict [[subformula]]s. Atoms are thus the simplest [[well-formed formula]]s of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives.
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| The precise form of atomic formulas depends on the logic under consideration; for [[propositional logic]], for example, the atomic formulas are the [[propositional variable]]s. For [[predicate logic]], the atoms are predicate symbols together with their arguments, each argument being a [[first-order logic#Formation rules|term]]. In [[model theory]], atomic formula are merely [[string (computer science)|strings]] of symbols with a given [[signature (logic)|signature]], which may or may not be [[satisfiable]] with respect to a given model.<ref>{{cite book|author1=Wilfrid Hodges|title=A Shorter Model Theory|year=1997|publisher=Cambridge University Press|isbn=0-521-58713-1|pages=11–14}}</ref>
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| ==Atomic formula in first-order logic==
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| The well-formed terms and propositions of ordinary [[first-order logic]] have the following [[syntax]]:
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| [[Term algebra|Terms]]:
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| * <math>t \equiv c \ | \ x \ | \ f (t_{1}, ..., t_{n})</math>,
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| that is, a term is [[recursive definition|recursively defined]] to be a constant ''c'' (a named object from the [[domain of discourse]]), or a variable ''x'' (ranging over the objects in the domain of discourse), or an ''n''-ary function ''f'' whose arguments are terms ''t''<sub>''k''</sub>. Functions map [[tuple]]s of objects to objects.
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| Propositions:
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| * <math>A, B, ... \equiv P (t_{1}, ..., t_{n}) \ | \ A \wedge B \ | \top | \ A \vee B \ | \perp | \ A \supset B \ | \ \forall x. A \ | \ \exists x. \ A </math>,
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| that is, a proposition is recursively defined to be an ''n''-ary [[predicate (mathematics)|predicate]] ''P'' whose arguments are terms ''t''<sub>''k''</sub>, or an expression composed of [[logical connective]]s (and, or) and [[quantifier]]s (for-all, there-exists) used with other propositions.
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| 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''<sub>1</sub>, …, ''t''<sub>''n''</sub>) for ''P'' a predicate, and the ''t''<sub>''k''</sub> terms.
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| All other well-formed formulae are obtained by composing atoms with logical connectives and quantifiers.
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| For example, the formula ∀''x. P'' (''x'') ∧ ∃''y. Q'' (''y'', ''f'' (''x'')) ∨ ∃''z. R'' (''z'') contains the atoms
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| * <math> P (x) </math>
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| * <math>Q (y, f (x))</math>
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| * <math>R (z)</math>
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| When all of the terms in an atom are [[ground term]]s, then the atom is called a [[ground atom]] or ''ground predicate''.
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| == See also ==
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| * In [[model theory]], [[Structure (mathematical logic)|structures]] assign an interpretation to the atomic formulas.
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| * In [[proof theory]], [[Polarity (proof theory)|polarity]] assignment for atomic formulas is an essential component of [[focusing (proof theory)|focusing]].
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| * [[Atomic sentence]]
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| == References == | |
| {{reflist}}
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| * {{cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = A K Peters | year = 2005 | isbn = 1-56881-262-0}}
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| [[Category:Predicate logic]]
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| [[Category:Logical expressions]]
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| [[de:Aussage (Logik)#einfache Aussagen - zusammengesetzte Aussagen]]
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Nice to meet you, I am Marvella Shryock. I used to be unemployed but now I am a librarian and the salary has been really satisfying. Puerto Rico is exactly where he and his wife reside. One of the issues she enjoys most is to do aerobics and now she is trying to earn money with it.
Take a look at my homepage ... std testing at home