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 $(\left(\left(P\rightarrow Q\right)\lor \lnot Q\right)\leftrightarrow \left(\lnot \lnot P\land Q\right))$ , 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 $\left(P\rightarrow Q\right)\lor \lnot Q\leftrightarrow \lnot \lnot P\land Q$ , 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 $\forall xP$ , for example, $P$ ^[5] (or $xP$ )^[11] is the scope of the quantifier $\forall x$ ^[5] (or $\forall$ ).^[11]

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

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.^[1]^[5]
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 $\phi$ .^[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 $\forall \xi$ 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 $\psi$ .^[1]
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 $\zeta$ .^[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]

Notes

^ 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, $\phi$ and $\psi$ are used to stand for any formulae whatsoever, whereas $\xi$ and $\zeta$ are used to stand for propositional variables.^[1]

References

^ ^a ^b ^c ^d ^e ^f Bostock, David (1997). Intermediate logic. Oxford : New York: Clarendon Press; Oxford University Press. pp. 8, 79. ISBN 978-0-19-875141-0.
^ ^a ^b ^c ^d Cook, Roy T. (March 20, 2009). Dictionary of Philosophical Logic. Edinburgh University Press. pp. 99, 180, 254. ISBN 978-0-7486-3197-1.
^ ^a ^b ^c ^d Rich, Elaine; Cline, Alan Kaylor. Quantifier Scope.
^ ^a ^b ^c Makridis, Odysseus (February 21, 2022). Symbolic Logic. Springer Nature. pp. 93–95. ISBN 978-3-030-67396-3.
^ ^a ^b ^c ^d ^e ^f ^g "3.3.2: Quantifier Scope, Bound Variables, and Free Variables". Humanities LibreTexts. January 21, 2017. Retrieved June 10, 2024.
^ ^a ^b ^c ^d ^e ^f Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. 45–48. ISBN 978-0-412-38090-7.
^ ^a ^b Gillon, Brendan S. (March 12, 2019). Natural Language Semantics: Formation and Valuation. MIT Press. pp. 250–253. ISBN 978-0-262-03920-8.
^ ^a ^b "Examples | Logic Notes - ANU". users.cecs.anu.edu.au. Retrieved June 10, 2024.
^ Suppes, Patrick; Hill, Shirley (April 30, 2012). First Course in Mathematical Logic. Courier Corporation. pp. 23–26. ISBN 978-0-486-15094-9.
^ Kirk, Donna (March 22, 2023). "2.2. Compound Statements". Contemporary Mathematics. OpenStax.
^ ^a ^b ^c 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.
^ ^a ^b 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
^ Allen, Colin; Hand, Michael (2001). Logic primer (2nd ed.). Cambridge, Mass: MIT Press. p. 66. ISBN 978-0-262-51126-1.

[12] 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, $\phi$ and $\psi$ are used to stand for any formulae whatsoever, whereas $\xi$ and $\zeta$ are used to stand for propositional variables.^[1]

[BostockIntermediate-1] ^ ^a ^b ^c ^d ^e ^f Bostock, David (1997). Intermediate logic. Oxford : New York: Clarendon Press; Oxford University Press. pp. 8, 79. ISBN 978-0-19-875141-0.

[: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.

[:1-3] Rich, Elaine; Cline, Alan Kaylor. Quantifier Scope.

[:2-4] Makridis, Odysseus (February 21, 2022). Symbolic Logic. Springer Nature. pp. 93–95. ISBN 978-3-030-67396-3.

[:3-5] ^ ^a ^b ^c ^d ^e ^f ^g "3.3.2: Quantifier Scope, Bound Variables, and Free Variables". Humanities LibreTexts. January 21, 2017. Retrieved June 10, 2024.

[LemmonLogic-6] ^ ^a ^b ^c ^d ^e ^f Lemmon, Edward John (1998). Beginning logic. Boca Raton, FL: Chapman & Hall/CRC. pp. 45–48. ISBN 978-0-412-38090-7.

[:4-7] Gillon, Brendan S. (March 12, 2019). Natural Language Semantics: Formation and Valuation. MIT Press. pp. 250–253. ISBN 978-0-262-03920-8.

[:5-8] "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.

[:6-11] 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.

[:7-13] 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

[14] Allen, Colin; Hand, Michael (2001). Logic primer (2nd ed.). Cambridge, Mass: MIT Press. p. 66. ISBN 978-0-262-51126-1.

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Connectives

Quantifiers

See also

Notes

References