Once formulas have more than one connective, we need to know how they group. In ordinary arithmetic, $2 + 3 \times 4$ means $2 + (3 \times 4)$; in logic we avoid surprises by always writing parentheses around every binary part of a larger formula. This lesson builds on The Five Connectives.
Key idea
Why it works
Worked examples
Worked example: Find the main connective
- Cover what is inside parentheses: $p \lor q$ is one block.
- What is left is block $\to$ $\neg r$.
- The $\neg$ covers only $r$.
Answer: The conditional $\to$: the formula is a conditional.
Worked example: A negation of a whole formula
- The $\neg$ stands in front of a pair of parentheses that holds everything else.
- So it is applied last, to the whole conjunction.
Answer: Negation: the formula is a negation.
Worked example: Write the negation
- Put $\neg$ in front.
- Keep parentheses so the $\neg$ covers both disjuncts.
Answer: $\neg (p \lor q)$
Worked example: A longer formula
- The parentheses hold $q \leftrightarrow \neg r$ together as one block.
- Outside them is only $p$ $\to$ block.
- So the antecedent is $p$ and the consequent is $q \leftrightarrow \neg r$.
Answer: The conditional $\to$.
Why insist on parentheses around every binary part? Without them, $(p \land q) \lor r$ could mean $(p \land q) \lor r$ or $p \land (q \lor r)$, and those two disagree when $p$ is false and $r$ is true. Writing the parentheses every time removes the doubt, so two readers always see the same formula.
Common mistake
- Count the parentheses from the outside in.
- The connective outside every pair is the main one.
- If a $\neg$ stands in front of everything (with parentheses), the formula is a negation.
Knowing the main connective is the first step of almost every later skill: computing truth values in Computing a Truth Value, reading formulas aloud, and choosing which proof rule applies.