Scope (logic)

In logic, the scope of a quantifier or connective is the shortest formula in which it occurs,^[1] determining the range in the formula to which the quantifier or connective is applied.^[2][3][4] The notions of a free variable and bound variable are defined in terms of whether that formula is within the scope of a quantifier,^[2][5] and the notions of a dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope.^[6][7]

Connectives

The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question.^[2][6][8] The connective with the largest scope in a formula is called its dominant connective,^[9][10] main connective,^[6][8][7] main operator,^[2] major connective,^[4] or principal connective;^[4] a connective within the scope of another connective is said to be subordinate to it.^[6]

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 →.^[6] 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.^[6]

Quantifiers

The scope of a quantifier is the part of a logical expression over which the quantifier exerts control.^[3] It is the shortest full sentence^[5] written right after the quantifier,^[3][5] often in parentheses;^[3] some authors^[11] describe this as including the variable written right after the universal or existential quantifier. In the formula ∀xP, for example, P^[5] (or xP)^[11] is the scope of the quantifier ∀x^[5] (or ∀).^[11]

This gives rise to the following definitions:^[a]

• 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.^[1][5]
• 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 ${\displaystyle \phi }$.^[1][5]
• A closed formula is one in which no variable occurs free; a formula which is not closed is open.^[12][1]
• 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 ${\displaystyle \psi }$.^[1]
• 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 ${\displaystyle \zeta }$.^[12]
• 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.^[13]

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.^[1]

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.