Building a Full Truth Table
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Building a full truth table means counting the distinct letters to fix the number of rows, filling the reference columns in standard order, then working out one column per subformula from the inside out until the main connective is reached.
Table this: the wedge of p and q, horseshoe, tilde p. Two letters means four rows. Fill p in standard order — true, true, false, false — and q alternating: true, false, true, false. Now the subformulas. The disjunction of p and q is false only when both are false, so its column reads true, true, true, false. The negation of p is the reverse of the p column: false, false, true, true. The main connective is the horseshoe, and it takes those two columns as antecedent and consequent. A conditional is false only when its antecedent is true and its consequent false. Row one: true antecedent, false consequent — false. Row two: the same — false. Row three: true antecedent, true consequent — true. Row four: false antecedent — true. The main column reads false, false, true, true, and that column is the statement's whole profile.
The main connective's column is the answer; the subformula columns are scaffolding. Reading a verdict off an inner column is the common slip, and it is easy to make because the inner columns are finished first and sit right there looking complete.