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
[edit]Logical connectives | ||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
||||||||||||||||||||||
Related concepts | ||||||||||||||||||||||
Applications | ||||||||||||||||||||||
| ||||||||||||||||||||||
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 , 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 , which some may find easier to read.[6]
Quantifiers
[edit]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 or , immediately followed by an occurrence of the variable , as in or , is said to be -binding.[1][5]
- An occurrence of a variable in a formula is free in if, and only if, it is not in the scope of any -binding quantifier in ; otherwise it is bound in .[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 or is vacuous if, and only if, its scope is or , and the variable does not occur free in .[1]
- A variable is free for a variable if, and only if, no free occurrences of lie within the scope of a quantification on .[12]
See also
[edit]Notes
[edit]- ^ 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, and are used to stand for any formulae whatsoever, whereas and are used to stand for propositional variables.[1]
References
[edit]- ^ 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