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]
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]
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]