An open formula is a
formula that contains at least one
free variable
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
.
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 progr ...
assigned to it, in contrast with a
closed formula
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th ro ...
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 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 n ...
s, the formula "''x''+2 > ''y''" is open, since it contains the free variables ''x'' and ''y''. In contrast, the formula "
∃''y''
∀''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 number
In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form
:F_ = 2^ + 1,
where ''n'' is a non-negative integer. The first few Fermat numbers are:
: 3, 5, 17, 257, 65537, 42949672 ...
s
:
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
*
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 quantifie ...
*
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 express ...
*
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)
In logic, a predicate is a symbol which represents a property or a relation. For instance, in the first order formula P(a), the symbol P is a predicate which applies to the individual constant a. Similarly, in the formula R(a,b), R is a predicat ...
References
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Logical expressions
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