THE LOGIC NOTES

Main connective Glossary

Definition

In a formula of propositional logic, the main connective is the one whose scope is the whole formula—that is, the one which is not inside the scope of any other.

By extension, in first order logic we may speak of the main operator of a formula, rather than the main connective, since the main operator may be a quantifier.

Comments

All rules of inference in the natural deduction calculus or in the tableau system or in the sequent calculus operate on main operators only (well, two of them at once in the case of NOTNOTE, but they must be in main position). You cannot apply a rule inside a subformula, even if it would be valid reasoning.

Examples

  1. In the formula
              NOT(p IMP (q IMP r))
    the main connective is the 'NOT'.
  2. In the formula
              ((p AND NOTq) IMP (q OR r)) OR (r IMP p)
    the main connective is the second occurrence of 'OR'.
  3. In the formula
              SOMEx(Fx IMP ALLy Fy)
    the main operator is the existential quantifier 'SOMEx'.

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