Scope (logic)#Quantifiers

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The scope of a logical connective occurring within a formula is the smallest well-formed formula that contains the connective in question.{{Cite book |last=Lemmon |first=Edward John |title=Beginning logic |date=1998 |publisher=Chapman & Hall/CRC |isbn=978-0-412-38090-7 |location=Boca Raton, FL |pages=45–48}}{{Cite web |title=Examples {{!}} Logic Notes - ANU |url=https://users.cecs.anu.edu.au/~jks/LogicNotes/connectives.html |access-date=June 10, 2024 |website=users.cecs.anu.edu.au}} The connective with the largest scope in a formula is called its dominant connective,{{Cite book |last1=Suppes |first1=Patrick |url=https://books.google.com/books?id=38LCAgAAQBAJ |title=First Course in Mathematical Logic |last2=Hill |first2=Shirley |date=April 30, 2012 |publisher=Courier Corporation |isbn=978-0-486-15094-9 |pages=23–26 |language=en}}{{Cite book |last=Kirk |first=Donna |url=https://openstax.org/books/contemporary-mathematics/pages/2-2-compound-statements |title=Contemporary Mathematics |date=March 22, 2023 |publisher=OpenStax |chapter=2.2. Compound Statements}} 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 $(\backslash left(\; \backslash left(\; P\; \backslash rightarrow\; Q\; \backslash right)\; \backslash lor\; \backslash lnot\; Q\; \backslash right)\; \backslash leftrightarrow\; \backslash left(\; \backslash lnot\; \backslash lnot\; P\; \backslash land\; Q\; \backslash right))$, 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 $\backslash left\; (\; P\; \backslash rightarrow\; Q\; \backslash right)\; \backslash lor\; \backslash lnot\; Q\; \backslash leftrightarrow\; \backslash lnot\; \backslash lnot\; P\; \backslash land\; Q$

, which some may find easier to read.

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 {{math|∀xP}}, for example, {{math|P}} (or {{math|xP}}){{cite book |last1=Bell |first1=John L. |title=A Course in Mathematical Logic |last2=Machover |first2=Moshé |date=April 15, 2007 |publisher=Elsevier Science Ltd |isbn=978-0-7204-2844-5 |page=[https://archive.org/details/courseinmathemat0000bell/page/17 17] |chapter=Chapter 1. Beginning mathematical logic |chapter-url=https://archive.org/details/courseinmathemat0000bell/page/17 |authorlink1=John Lane Bell |authorlink2=Moshé Machover}} is the scope of the quantifier {{math|∀x}} (or {{math|∀}}).

This gives rise to the following definitions:{{refn|group=lower-alpha|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, $\backslash phi$ and $\backslash psi$ are used to stand for any formulae whatsoever, whereas $\backslash xi$ and $\backslash zeta$ are used to stand for propositional variables.}}

• An occurrence of a quantifier $\backslash forall$ or $\backslash exists$, immediately followed by an occurrence of the variable $\backslash xi$, as in $\backslash forall\; \backslash xi$ or $\backslash exists\; \backslash xi$, is said to be $\backslash xi$-binding.
• An occurrence of a variable $\backslash xi$ in a formula $\backslash phi$ is free in $\backslash phi$ if, and only if, it is not in the scope of any $\backslash xi$-binding quantifier in $\backslash phi$; otherwise it is bound in $\backslash phi$.
• A closed formula is one in which no variable occurs free; a formula which is not closed is open.{{Citation |last=Uzquiano |first=Gabriel |title=Quantifiers and Quantification |date=2022 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/archives/win2022/entries/quantification/ |access-date=June 10, 2024 |edition=Winter 2022 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}
• An occurrence of a quantifier $\backslash forall\; \backslash xi$ or $\backslash exists\; \backslash xi$ is vacuous if, and only if, its scope is $\backslash forall\; \backslash xi\; \backslash psi$ or $\backslash exists\; \backslash xi\; \backslash psi$, and the variable $\backslash xi$ does not occur free in $\backslash psi$.
• A variable $\backslash zeta$ is free for a variable $\backslash xi$ if, and only if, no free occurrences of $\backslash xi$ lie within the scope of a quantification on $\backslash zeta$.
• 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.{{Cite book |last=Allen |first=Colin |title=Logic primer |last2=Hand |first2=Michael |date=2001 |publisher=MIT Press |isbn=978-0-262-51126-1 |edition=2nd |location=Cambridge, Mass |pages=66}}

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

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Category:Quantifier (logic)

Category:Predicate logic

Category:Mathematical terminology