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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

Logical connectives
NOT ${\displaystyle \neg A,-A,{\overline {A}},\sim A}$ AND ${\displaystyle A\land B,A\cdot B,AB,A\ \&\ B,A\ \&\&\ B}$ NAND ${\displaystyle A{\overline {\land }}B,A\uparrow B,A\mid B,{\overline {A\cdot B}}}$ OR ${\displaystyle A\lor B,A+B,A\mid B,A\parallel B}$ NOR ${\displaystyle A{\overline {\lor }}B,A\downarrow B,{\overline {A+B}}}$ XNOR ${\displaystyle A\odot B,{\overline {A{\overline {\lor }}B}}}$ └ equivalent ${\displaystyle A\equiv B,A\Leftrightarrow B,A\leftrightharpoons B}$ XOR ${\displaystyle A{\underline {\lor }}B,A\oplus B}$ └nonequivalent ${\displaystyle A\not \equiv B,A\not \Leftrightarrow B,A\nleftrightarrow B}$ implies ${\displaystyle A\Rightarrow B,A\supset B,A\rightarrow B}$ nonimplication (NIMPLY) ${\displaystyle A\not \Rightarrow B,A\not \supset B,A\nrightarrow B}$ converse ${\displaystyle A\Leftarrow B,A\subset B,A\leftarrow B}$ converse nonimplication ${\displaystyle A\not \Leftarrow B,A\not \subset B,A\nleftarrow B}$
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Propositional calculus Predicate logic Boolean algebra Truth table Truth function Boolean function Functional completeness Scope (logic)
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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 ${\displaystyle (\left(\left(P\rightarrow Q\right)\lor \lnot Q\right)\leftrightarrow \left(\lnot \lnot P\land Q\right))}$, 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 ${\displaystyle \left(P\rightarrow Q\right)\lor \lnot Q\leftrightarrow \lnot \lnot P\land Q}$, 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 ${\displaystyle \xi }$, as in ${\displaystyle \forall \xi }$ or ${\displaystyle \exists \xi }$, is said to be ${\displaystyle \xi }$-binding.
• An occurrence of a variable ${\displaystyle \xi }$ in a formula ${\displaystyle \phi }$ is free in ${\displaystyle \phi }$ if, and only if, it is not in the scope of any ${\displaystyle \xi }$-binding quantifier in ${\displaystyle \phi }$; otherwise it is bound in ${\displaystyle \phi }$.
• A closed formula is one in which no variable occurs free; a formula which is not closed is open.
• An occurrence of a quantifier ${\displaystyle \forall \xi }$ or ${\displaystyle \exists \xi }$ is vacuous if, and only if, its scope is ${\displaystyle \forall \xi \psi }$ or ${\displaystyle \exists \xi \psi }$, and the variable ${\displaystyle \xi }$ does not occur free in ${\displaystyle \psi }$.
• A variable ${\displaystyle \zeta }$ is free for a variable ${\displaystyle \xi }$ if, and only if, no free occurrences of ${\displaystyle \xi }$ lie within the scope of a quantification on ${\displaystyle \zeta }$.
• 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

1. These definitions follow the common practice of using Greek letters as metalogical symbols which may stand for symbols in a formal language for propositional or predicate logic. In particular, ${\displaystyle \phi }$ and ${\displaystyle \psi }$ are used to stand for any formulae whatsoever, whereas ${\displaystyle \xi }$ and ${\displaystyle \zeta }$ are used to stand for propositional variables.

References

1. ^ Bostock, David (1997). Intermediate logic. Oxford : New York: Clarendon Press; Oxford University Press. pp. 8, 79. ISBN 978-0-19-875141-0.
2. ^ Cook, Roy T. (March 20, 2009). Dictionary of Philosophical Logic. Edinburgh University Press. pp. 99, 180, 254. ISBN 978-0-7486-3197-1.
3. ^ Rich, Elaine; Cline, Alan Kaylor. Quantifier Scope.
4. ^ Makridis, Odysseus (February 21, 2022). Symbolic Logic. Springer Nature. pp. 93–95. ISBN 978-3-030-67396-3.
5. ^ "3.3.2: Quantifier Scope, Bound Variables, and Free Variables". Humanities LibreTexts. January 21, 2017. Retrieved June 10, 2024.
6. ^ Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. 45–48. ISBN 978-0-412-38090-7.
7. ^ Gillon, Brendan S. (March 12, 2019). Natural Language Semantics: Formation and Valuation. MIT Press. pp. 250–253. ISBN 978-0-262-03920-8.
8. ^ "Examples | Logic Notes - ANU". users.cecs.anu.edu.au. Retrieved June 10, 2024.
9. Suppes, Patrick; Hill, Shirley (April 30, 2012). First Course in Mathematical Logic. Courier Corporation. pp. 23–26. ISBN 978-0-486-15094-9.
10. Kirk, Donna (March 22, 2023). "2.2. Compound Statements". Contemporary Mathematics. OpenStax.
11. ^ Bell, John L.; Machover, Moshé (April 15, 2007). "Chapter 1. Beginning mathematical logic". A Course in Mathematical Logic. Elsevier Science Ltd. p. 17. ISBN 978-0-7204-2844-5.
12. ^ Uzquiano, Gabriel (2022), "Quantifiers and Quantification", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Winter 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved June 10, 2024
13. Allen, Colin; Hand, Michael (2001). Logic primer (2nd ed.). Cambridge, Mass: MIT Press. p. 66. ISBN 978-0-262-51126-1.

• Quantifier (logic)
• Predicate logic
• Mathematical terminology