FORMULA (MATHEMATICAL LOGIC)

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In mathematical logic, a 'formula' is a formal syntactic object that expresses a proposition.
The exact definition of a formula depends on the particular development of formal logic in question, but a fairly typical one (specific to first-order logic) goes as follows: Formulas are defined relative to a particular 'language', which consists of a collection of 'variables', 'constants', 'logic symbols', 'function symbols', and 'relation symbols', where each of the function and relation symbols comes supplied with an arity 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.
If a formula has no occurrences of $exists$''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.

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References

See also

★ WFF

References

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