Information about Formula (mathematical Logic)
- For help on Wikipedia math, see Wikipedia:Formula
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(t1,...,tn), where f is an n-ary function symbol, and t1,...,tn are terms.
An atomic formula is one of the form:
- t1=t2, where t1 and t2 are terms, or
- R(t1,...,tn), where R is an n-ary relation symbol, and t1,...,tn 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:
is a formula when 
is a formula;
and 
are formulae when and 
are formulae;
- is a formula when x is a variable and is a formula;
- is a formula when
is a variable and is a formula (alternatively, could be defined as an abbreviation for ).
If a formula has no occurrences of
or 
, for any variable , then it is called quantifier-free. An existential formula is a string of existential quantification followed by a quantifier-free formula.
See also
References
Hinman, P. (2005). Fundamentals of Mathematical Logic. A K Peters. ISBN 1-568-81262-0. Mathematical logic is a branch of mathematics, which grew out of symbolic logic. Subfields include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic has contributed to, and been motivated by, the study of foundations of mathematics, but
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proposition is the content of an assertion, that is, it is true-or-false and defined by the meaning of a particular piece of language. The proposition is independent of the of communication.
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First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. It goes by many names, including: first-order predicate calculus (FOPC), the lower predicate calculus,
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In logic, mathematics, and computer science, the arity (synonyms include type, adicity, and rank) of a function or operation is the number of arguments or operands that the function takes.
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In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula with no logical connectives or strict sub-formulas. Atoms are thus the simplest well-formed formulas of the logic.
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WFF (pronounced "wiff") is an abbreviation for well-formed formula. Given a formal grammar, a WFF is any string that is generated by that grammar. To say that a string is a WFF with respect to a given formal grammar is equivalent to saying that belongs to the language
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