Scope (logic) explained

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,^[5] and the notions of a and are defined in terms of whether a connective includes another within its scope.^[6]

Connectives

The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question.^{[7] [8]} The connective with the largest scope in a formula is called its dominant connective,^{[9] [10]} 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

(\left( \left( P → Q \right) \lor ln ot Q \right) \leftrightarrow \left( ln ot ln ot 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 \left ( P → Q \right) \lor ln ot Q \leftrightarrow ln ot ln ot 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, for example, (or)^[11] is the scope of the quantifier (or).

This gives rise to the following definitions:

• An occurrence of a quantifier

\forall or \exists , immediately followed by an occurrence of the variable \xi , as in \forall \xi or \exists \xi , is said to be \xi -binding.

• An occurrence of a variable

\xi in a formula \phi is free in \phi if, and only if, it is not in the scope of any \xi -binding quantifier in \phi ; otherwise it is bound in \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

\forall \xi or \exists \xi is vacuous if, and only if, its scope is \forall \xi \psi or \exists \xi \psi , and the variable \xi does not occur free in \psi .

• A variable

\zeta is free for a variable \xi if, and only if, no free occurrences of \xi lie within the scope of a quantification on \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.^[12]

See also

• Modal scope fallacy
• Prenex form
• Glossary of logic
• Free variables and bound variables
• Open formula

Notes and References

1. Book: Bostock, David . Intermediate logic . 1997 . Clarendon Press ; Oxford University Press . 978-0-19-875141-0 . Oxford : New York . 8,79.
2. Book: Cook, Roy T. . Dictionary of Philosophical Logic . March 20, 2009 . Edinburgh University Press . 978-0-7486-3197-1 . 99,180,254 . en.
3. Book: Rich . Elaine . Quantifier Scope . Cline . Alan Kaylor . en-US.
4. Book: Makridis, Odysseus . Symbolic Logic . February 21, 2022 . Springer Nature . 978-3-030-67396-3 . 93–95 . en.
5. Web site: January 21, 2017 . 3.3.2: Quantifier Scope, Bound Variables, and Free Variables . June 10, 2024 . Humanities LibreTexts . en.
6. Book: Gillon, Brendan S. . Natural Language Semantics: Formation and Valuation . March 12, 2019 . MIT Press . 978-0-262-03920-8 . 250–253 . en.
7. Book: Lemmon, Edward John . Beginning logic . 1998 . Chapman & Hall/CRC . 978-0-412-38090-7 . Boca Raton, FL . 45–48.
8. Web site: Examples Logic Notes - ANU . June 10, 2024 . users.cecs.anu.edu.au.
9. Book: Suppes . Patrick . First Course in Mathematical Logic . Hill . Shirley . April 30, 2012 . Courier Corporation . 978-0-486-15094-9 . 23–26 . en.
10. Book: Kirk, Donna . Contemporary Mathematics . March 22, 2023 . OpenStax . 2.2. Compound Statements.
11. Book: Bell . John L. . A Course in Mathematical Logic . Machover . Moshé . April 15, 2007 . Elsevier Science Ltd . 978-0-7204-2844-5 . 17 . Chapter 1. Beginning mathematical logic . https://archive.org/details/courseinmathemat0000bell/page/17 . John Lane Bell . Moshé Machover.
12. Book: Allen, Colin . Logic primer . Hand . Michael . 2001 . MIT Press . 978-0-262-51126-1 . 2nd . Cambridge, Mass . 66.