- Source: Scope (logic)
In logic, the scope of a quantifier or connective is the shortest formula in which it occurs, determining the range in the formula 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 dominant connective and subordinate connective 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 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
(
(
(
P
→
Q
)
∨
¬
Q
)
↔
(
¬
¬
P
∧
Q
)
)
{\displaystyle (\left(\left(P\rightarrow Q\right)\lor \lnot Q\right)\leftrightarrow \left(\lnot \lnot P\land Q\right))}
, 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 is adopted for the connectives, viz., with ¬ applying first, then ∧ and ∨, then →, and finally ↔, this formula may be written in the less parenthesized form
(
P
→
Q
)
∨
¬
Q
↔
¬
¬
P
∧
Q
{\displaystyle \left(P\rightarrow Q\right)\lor \lnot Q\leftrightarrow \lnot \lnot P\land Q}
, 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 ∀xP, for example, P (or xP) is the scope of the quantifier ∀x (or ∀).
This gives rise to the following definitions:
An occurrence of a quantifier
∀
{\displaystyle \forall }
or
∃
{\displaystyle \exists }
, immediately followed by an occurrence of the variable
ξ
{\displaystyle \xi }
, as in
∀
ξ
{\displaystyle \forall \xi }
or
∃
ξ
{\displaystyle \exists \xi }
, is said to be
ξ
{\displaystyle \xi }
-binding.
An occurrence of a variable
ξ
{\displaystyle \xi }
in a formula
ϕ
{\displaystyle \phi }
is free in
ϕ
{\displaystyle \phi }
if, and only if, it is not in the scope of any
ξ
{\displaystyle \xi }
-binding quantifier in
ϕ
{\displaystyle \phi }
; otherwise it is bound in
ϕ
{\displaystyle \phi }
.
A closed formula is one in which no variable occurs free; a formula which is not closed is open.
An occurrence of a quantifier
∀
ξ
{\displaystyle \forall \xi }
or
∃
ξ
{\displaystyle \exists \xi }
is vacuous if, and only if, its scope is
∀
ξ
ψ
{\displaystyle \forall \xi \psi }
or
∃
ξ
ψ
{\displaystyle \exists \xi \psi }
, and the variable
ξ
{\displaystyle \xi }
does not occur free in
ψ
{\displaystyle \psi }
.
A variable
ζ
{\displaystyle \zeta }
is free for a variable
ξ
{\displaystyle \xi }
if, and only if, no free occurrences of
ξ
{\displaystyle \xi }
lie within the scope of a quantification on
ζ
{\displaystyle \zeta }
.
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
Free variables and bound variables
Open formula
Notes
References
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