Rows represent valuations
In classical two-valued logic each letter is true or false. Repeated occurrences of one letter use the same value within a row, so p∧p does not require four rows. With n distinct letters, a complete table contains 2ⁿ valuations.
Evaluate smaller subformulas before applying their enclosing connectives. The column beneath the main connective gives the whole formula’s value; intermediate columns are bookkeeping. Parentheses identify that structure, and changing them can change the result.
References: [1]
Different questions use the same rows
A formula is satisfiable if some row is true, a tautology if every row is true, and unsatisfiable if no row is true. Thus p∨¬p is always true, p∧¬p always false, and p∧q true only when both letters are true. These are exhaustive semantic tests, not measurements of real-world frequency.
To test equivalence, compare the formulas’ final columns under the same valuations. To test an argument, look for a row with every premise true and the conclusion false. One countervaluation refutes validity; the absence of such a row establishes validity of the represented form.
References: [2]
Exhaustive guarantees and costs
A complete table exhausts the finite valuations and therefore supplies a decision method for propositional formulas. A few sample rows do not establish tautology or unsatisfiability, whereas one satisfying row is enough for satisfiability. The direction of the claim determines the evidence required.
Exponential row growth motivates propagation and structured search. Moreover, statement letters are treated as independently assignable units; objects, quantifiers and causal interpretation require further representation. Enlarging a table cannot recover information omitted during formalization.