Determining the Main Operator (Connective)


Every symbolized statement is overall either a negation, a conjunction, a disjunction, a conditional, or a biconditional.  The operator which controls the entire statement is the main operator (or connective).  To determine which operator has control over the entire statement, follow these steps.

1.  Does the statement begin with a tilde?  The tilde only has scope over the letter following it, unless it is followed by a grouping symbol (parenthesis, bracket or brace).  Otherwise is has scope over the statement within that particular grouping symbol.  If the entire rest of the statement is bounded by that grouping symbol, then that first tilde is the main operator, and the entire statement is at root a negation.

Examples
1
2
3
4
~A ∨ B
~(A ∨ B)
~[(A ∨ B) ∨ (C ∨ D)]
~[F ∨ (G ∨ H)] ∨ I
not a negation
negation
negation
not a negation

In Example 1, the tilde is at the beginning of the statement, but it is followed immediately by a capital A, so the scope of the tilde is only the A, and the statement is not fundamentally a negation.

In Example 2, the tilde is followed by a parenthesis, and the entire rest of the statement is contained by the parentheses, so the statement is a negation, and the tilde is the main operator.

In Example 3, the opening tilde is followed by a bracket, and the corresponding end bracket comes at the end of the entire statement, so it is also a negation, and the tilde is the main operator.

Example 4 is similar to Example 3, but the bracket ends after the H and before the wedge, so the scope of the tilde is only the statement contained by the brackets, and it is not the main operator.

2. If the statement is not a negation, it is either a conjunction, disjunction, conditional, or biconditional.  In either case, the connective must have one complete statement to its left (perhaps bounded by grouping symbols) and one complete statement to its right, with no other symbols remaining .  You can simply go through each connective from left to right and see if it meets this condition.  If a connective does not split the entire statement into two separate statements, then move to the next one.  If you get to the last connective, and it does not meet this condition, then the statement cannot be a well-formed formula (of course, you should consider that you have made a mistake somewhere before abondonning your search).

Consider the statement to the right.  The statement does not begin with a tilde, but with a bracket.  The bracket does not encompass the entire statement, so we could skip any connective within the brackets, but we will consider them, just for the sake of example.  The wedge is bounded by brackets, and it has scope over the statement to its left and the statement in parentheses to its right within the brackets, and that is all (See the example next to the 2 ).  The dot has scope only over the statements within the parentheses (See 3).  Examine the horseshoe.  To its left it has a complete statement, and to its right there is a complete statement, and nothing else remains (See 4).  So the conditional is the main connective.   
1
  [A ∨(B • D)] ⊃ F
2

 [A ∨(B • D)] ⊃ F

3

 [A ∨(B • D)] ⊃ F
4
  [A ∨(B • D)] ⊃ F
Each red connective has a statement to its left (in green) and a statement to its right (in blue).