An open formula is a
formula that contains at least one
free variable.
An open formula does not have a
truth value 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 roo ...
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 numbers, 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''
∀''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 quantifi ...
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Higher-order logic
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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 everything i ...
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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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