- Formula (mathematical logic)
In

mathematical logic , a**formula**is a type ofabstract object a token of which is asymbol or string of symbols which may be interpreted as any meaningful unit (i.e. aname , anadjective , aproposition , aphrase , a string of names, a string of phrases, etcetera) in aformal language . Two different strings of symbols may be tokens of the same formula. It is not necessary for the existence of a formula that there be any tokens of it. The exact definition of a formula depends on the particular formal language in question. [*Hunter, Geoffrey, Metalogic: An Introduction to the Metatheory of Standard First-Order Logic*]A fairly typical definition (specific to

first-order logic ) goes as follows: Formulas are defined relative to a particular formal language and "relation symbols", where each of the function and relation symbols comes supplied with anarity that indicates the number of arguments it takes.Then a

**term**is defined recursively as

#A variable,

#A constant, or

#"f"("t"_{1},...,"t"_{"n"}), where "f" is an "n"-ary function symbol, and "t"_{1},...,"t"_{"n"}are terms.An

atomic formula is one of the form:

#"t"_{1}="t"_{2}, where "t"_{1}and "t"_{2}are terms, or

#"R"("t"_{1},...,"t"_{"n"}), where "R" is an "n"-ary relation symbol, and "t"_{1},...,"t"_{"n"}are terms.Finally, the set of formulae is defined to be the smallest set containing the set of atomic formulae such that the following holds:

#$egphi$ is a formula when $phi$ is a formula;

#$(phi\; land\; psi)$ and $(phi\; lor\; psi)$ are formulae when $phi$ and $psi$ are formulae;

#$exists\; x,\; phi$ is a formula when "x" is a variable and $phi$ is a formula;

#$forall\; x,\; phi$ is a formula when $x$ is a variable and $phi$ is a formula (alternatively, $forall\; x,\; phi$ could be defined as an abbreviation for $egexists\; x,\; egphi$).If a formula has no occurrences of $exists\; x$ or $forall\; x$, for any variable $x$, then it is called "quantifier-free". An "existential formula" is a string of existential quantification followed by a quantifier-free formula.

**ee also***

Well-formed formula

*Theorem **References***cite book | author = Hinman, P. | title = Fundamentals of Mathematical Logic | publisher = A K Peters | year = 2005 | id = ISBN 1-568-81262-0

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