Multiplicity of infection

2013-08-18T15:18:56Z

75.156.164.124: No MOI can guarantee an infection of ALL cells by at least one virion. Therefore, stating that an MOI of 8 would infect 100% of cells is false; I changed it to approximately 100%.

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.

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>

==Atomic formula in first-order logic==
The well-formed terms and propositions of ordinary [[first-order logic]] have the following [[syntax]]:

[[Term algebra|Terms]]:
* <math>t \equiv c \ | \ x \ | \ f (t_{1}, ..., t_{n})</math>,

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''''k''. Functions map [[tuple]]s of objects to objects.

Propositions:
* <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>,

that is, a proposition is recursively defined to be an ''n''-ary [[predicate (mathematics)|predicate]] ''P'' whose arguments are terms ''t''''k'', or an expression composed of [[logical connective]]s (and, or) and [[quantifier]]s (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''''k'' 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
* <math> P (x) </math>
* <math>Q (y, f (x))</math>
* <math>R (z)</math>

When all of the terms in an atom are [[ground term]]s, then the atom is called a [[ground atom]] or ''ground predicate''.

== See also ==

* In [[model theory]], [[Structure (mathematical logic)|structures]] assign an interpretation to the atomic formulas.
* In [[proof theory]], [[Polarity (proof theory)|polarity]] assignment for atomic formulas is an essential component of [[focusing (proof theory)|focusing]].
* [[Atomic sentence]]

== References ==
{{reflist}}
* {{cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = A K Peters | year = 2005 | isbn = 1-56881-262-0}}

[[Category:Predicate logic]]
[[Category:Logical expressions]]

[[de:Aussage (Logik)#einfache Aussagen - zusammengesetzte Aussagen]]

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Multiplicity of infection