Scope (logic)

In logic, the scope of a quantifier or connective is the range in the formula where the quantifier or connective "engages in".^[1]^[2]^[3] The notions of a free variable and bound variable are defined in terms of whether that formula is within the scope of a quantifier,^[1]^[4] and the notions of a dominant connective and subordinate connective are defined in terms of whether a connective includes another within its scope.^[5]^[6]

Connectives[edit]

The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question.^[1]^[5]^[7] The connective with the largest scope in a formula is called its dominant connective,^[8]^[9] main connective,^[5]^[7]^[6] main operator,^[1] major connective,^[3] or principal connective;^[3] a connective within the scope of another connective is said to be subordinate to it.^[5]

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 →.^[5] 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.^[5]

Quantifiers[edit]

The scope of a quantifier is the part of a logical expression over which the quantifier exerts control.^[2] It is the shortest full sentence^[4] written right after the quantifier,^[2]^[4] often in parentheses;^[2] some authors^[10] describe this as including the variable written right after the universal or existential quantifier. In the formula $\forall xP$ , for example, $P$ ^[4] (or $xP$ )^[10] is the scope of the quantifier $\forall x$ ^[4] (or $\forall$ ).^[10]

This gives rise to the following definitions:

A variable in the formula is free if, and only if, it does not occur in the scope of any quantifier for that variable; otherwise, it is said to be bound by that quantifier.^[4]
A term is free for a variable in the formula (i.e. free to substitute that variable that occurs free), if and only if that variable does not occur free in the scope of any quantifier for any variable in the term.^[11]
A formula is closed if, and only if, all of its variables are bound; otherwise, it is open.^[11]

References[edit]

^ ^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 "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), Zalta, Edward N.; Nodelman, Uri (eds.), "Quantifiers and Quantification", The Stanford Encyclopedia of Philosophy (Winter 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved June 10, 2024

[:0-1] Cook, Roy T. (March 20, 2009). Dictionary of Philosophical Logic. Edinburgh University Press. pp. 99, 180, 254. ISBN 978-0-7486-3197-1.

[:1-2] Rich, Elaine; Cline, Alan Kaylor. Quantifier Scope.

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

[:3-4] ^ ^a ^b ^c ^d ^e ^f "3.3.2: Quantifier Scope, Bound Variables, and Free Variables". Humanities LibreTexts. January 21, 2017. Retrieved June 10, 2024.

[LemmonLogic-5] ^ ^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.

[:4-6] Gillon, Brendan S. (March 12, 2019). Natural Language Semantics: Formation and Valuation. MIT Press. pp. 250–253. ISBN 978-0-262-03920-8.

[:5-7] "Examples | Logic Notes - ANU". users.cecs.anu.edu.au. Retrieved June 10, 2024.

[8] Suppes, Patrick; Hill, Shirley (April 30, 2012). First Course in Mathematical Logic. Courier Corporation. pp. 23–26. ISBN 978-0-486-15094-9.

[9] Kirk, Donna (March 22, 2023). "2.2. Compound Statements". Contemporary Mathematics. OpenStax.

[:6-10] 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.

[:7-11] Uzquiano, Gabriel (2022), Zalta, Edward N.; Nodelman, Uri (eds.), "Quantifiers and Quantification", The Stanford Encyclopedia of Philosophy (Winter 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved June 10, 2024

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Connectives[edit]

Quantifiers[edit]

See also[edit]

References[edit]