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Recipe

Find the main connective

Use when: you have a well-formed formula and need to know what kind of sentence it is, or which connective holds the whole thing together

Goal: the one connective whose scope is the whole formula, and the formula's name (negation, conjunction, disjunction, conditional, or biconditional)

Taught in Finding the main connective

Quick recipe

  1. Check for a single letter
  2. Strip brackets that wrap the whole formula
  3. Count bracket depth, left to right
  4. Pick the two-place connective at depth 0
  5. Name the sentence
  1. 1

    If the formula is one sentence letter, stop.

    • a single letter such as P — it has no main connective.
    • anything longer — go on to the next step.

    Why: only a compound formula was built with a connective.

  2. 2

    If the very first bracket closes at the very end, remove that pair. Repeat if another pair still wraps everything.

    Example

    (\neg(P \vee Q) \to R) becomes \neg(P \vee Q) \to R. The first bracket closed at the very end. It wrapped everything.

    Why: dropping the outermost pair never changes the formula, and it keeps the last-built connective from looking enclosed.

  3. 3

    Start a counter at 0. Add 1 at each "(" and subtract 1 at each ")". Write the counter's value under every connective.

    Example

    (P \wedge Q) \to \neg(R \vee S): the \wedge is at depth 1, the \to at 0, the \neg at 0, and the \vee at 1.

    Why: depth 0 means the connective is not inside any smaller box.

  4. 4

    Look only at \wedge, \vee, \to and \leftrightarrow sitting at depth 0.

    Example

    In (P \wedge Q) \to \neg(R \vee S) the only two-place connective at depth 0 is \to. The \neg at depth 0 doesn't count, since a two-place connective beats it.

    • just one — that is the main connective.
    • none — the formula begins with \neg, and that first \neg is the main connective.
    • several copies of one \wedge or one \vee — it's a chain; the last copy is the main connective, and the formula is named by that connective.
    • two different ones — the formula isn't well formed in this course; add brackets first.

    Why: rule 3 brackets every two-place compound except the one built last.

  5. 5

    Name the formula after its main connective: negation, conjunction, disjunction, conditional, or biconditional.

    Why: the name tells you which truth rule and which proof rules apply to the whole formula.

Common mistakes

  • Picking the first symbol because it's a \neg. In \neg P \vee Q the \neg covers only P.
  • Forgetting to strip brackets that wrap the whole formula. Then nothing seems to sit at depth 0.
  • Counting a \neg at depth 0 as the winner when a two-place connective also sits at depth 0.