Methods of Proof and Truth Tables
What this domain tests
Two decision procedures: truth tables (full, abbreviated and indirect) for validity, and formal proofs of validity using rules of inference and rules of replacement.
Truth tables
- A full truth table has 2ⁿ rows for n variables. An argument is valid when no row makes all premises true and the conclusion false.
- An indirect (shorthand) truth table assumes premises true and conclusion false; if that assumption forces a contradiction, the argument is valid.
Formal proof
Nine rules of inference: Modus Ponens, Modus Tollens, Hypothetical Syllogism, Disjunctive Syllogism, Constructive Dilemma, Absorption, Simplification, Conjunction, Addition. Replacement rules, usable in either direction on one line: De Morgan, Commutation, Association, Distribution, Double Negation, Transposition, Material Implication, Material Equivalence, Exportation, Tautology.
Worked example
- p ⊃ q
- q ⊃ r
- ~r / ∴ ~p
- p ⊃ r (1, 2, H.S.)
- ~p (4, 3, M.T.) Each step cites the lines used and the rule applied, which is exactly what the marking scheme looks for.
Traps that cost marks
Treating a replacement rule as though it consumed two lines - replacement rewrites a single line; forgetting Addition lets you introduce any disjunct; assuming the antecedent in conditional proof without flagging the assumption; concluding "invalid" when a truth table merely shows a false conclusion with a false premise.