An open formula is a
formula that contains at least one
free variable.
An open formula does not have a
truth value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or ''false'').
Computing
In some prog ...
assigned to it, in contrast with a
closed formula which constitutes a proposition and thus can have a truth value like ''true'' or ''false''. An open formula can be transformed into a closed formula by applying quantifiers or specifying of the
domain of discourse
In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.
Overview
The dom ...
of individuals for each free variable denoted x, y, z....or x
1, x
2, x
3.... This transformation is called capture of the free variables to make them bound variables, bound to a domain of individual constants.
For example, when reasoning about
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
s, the formula "''x''+2 > ''y''" is open, since it contains the free variables ''x'' and ''y''. In contrast, the formula "
∃
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, whe ...
''y''
∀
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. ...
''x'': ''x''+2 > ''y''" is closed, and has truth value ''true''.
An example of closed formula with truth value ''false'' involves the sequence of
Fermat numbers
:
studied by Fermat in connection to the
primality
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. The attachment of the predicate letter P (''is prime'') to each number from the Fermat sequence gives a set of false closed formulae when the rank ''n'' of the Fermat number is greater than 4. Thus the closed formula ∀''n'' ''P''(''F''
''n'') is false.
See also
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First-order logic
First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quanti ...
*
Higher-order logic
mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics. Higher-order logics with their standard semantics are more expres ...
*
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first order formula \forall x P(x) expresses that everythin ...
*
Predicate (mathematical logic)
References
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Logical expressions
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