In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, the scope of a
quantifier or
connective is the shortest formula in which it occurs,
determining the range in the
formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
to which the quantifier or connective is applied.
The notions of a
free variable and bound variable are defined in terms of whether that formula is ''within the scope'' of a quantifier,
and the notions of a and are defined in terms of whether a connective includes another ''within its scope''.
Connectives
The scope of a logical connective occurring within a formula is the smallest well-formed formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language.
The abbreviation wf ...
that contains the connective in question. The connective with the largest scope in a formula is called its ''dominant connective,'' ''main connective'',[ ''main operator'',][ ''major connective'',][ or ''principal connective'';][ a connective within the scope of another connective is said to be ''subordinate'' to it.][
For instance, in the formula , the dominant connective is ↔, and all other connectives are subordinate to it; the → is subordinate to the ∨, but not to the ∧; the first ¬ is also subordinate to the ∨, but not to the →; the second ¬ is subordinate to the ∧, but not to the ∨ or the →; and the third ¬ is subordinate to the second ¬, as well as to the ∧, but not to the ∨ or the →.][ If an ]order of precedence
An order of precedence is a sequential hierarchy of importance applied to individuals, groups, or organizations. For individuals, it is most often used for diplomats in attendance at very formal occasions. It can also be used in the context of ...
is adopted for the connectives, viz., with ¬ applying first, then ∧ and ∨, then →, and finally ↔, this formula may be written in the less parenthesized form , which some may find easier to read.[
]
Quantifiers
The scope of a quantifier is the part of a logical expression over which the quantifier exerts control.[ It is the shortest full sentence][ written right after the quantifier,][ often in parentheses;][ some authors][ describe this as including the variable written right after the universal or existential quantifier. In the formula , for example, ][ (or )] is the scope of the quantifier [ (or ).][
This gives rise to the following definitions:
* An occurrence of a quantifier or , immediately followed by an occurrence of the variable , as in or , is said to be -binding.][
* An occurrence of a variable in a formula is ''free in'' if, and only if, it is not in the scope of any -binding quantifier in ; otherwise it is ''bound in'' .][
* A ''closed'' formula is one in which no variable occurs free; a formula which is not closed is '']open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gerd Dudek, Buschi Niebergall, and Edward Vesala album), 1979
* ''Open'' (Go ...
''.[
* An occurrence of a quantifier or is ''vacuous'' if, and only if, its scope is or , and the variable does not occur free in .][
* A variable is ''free for'' a variable if, and only if, no free occurrences of lie within the scope of a quantification on .][
* A quantifier whose scope contains another quantifier is said to have ''wider scope'' than the second, which, in turn, is said to have ''narrower scope'' than the first.]
See also
* Modal scope fallacy
* Prenex form
* Glossary of logic
This is a glossary of logic. Logic is the study of the principles of valid reasoning and argumentation.
A
...
* Free variables and bound variables
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a variable may be said to be either free or bound. Some older books use the terms real variable and apparent variable for f ...
* Open formula
Notes
References
{{reflist
Quantifier (logic)
Predicate logic
Mathematical terminology