Maharashtra HSC Logic: Methods of Proof and Formal Deduction

Two kinds of rule

A formal proof derives the conclusion line by line from the premises using accepted rules. Rules of inference apply to whole lines: Modus Ponens (P ⊃ Q, P ∴ Q), Modus Tollens (P ⊃ Q, ~Q ∴ ~P), Hypothetical Syllogism, Disjunctive Syllogism, Constructive Dilemma, Simplification, Conjunction, Addition, Absorption. Rules of replacement (De Morgan, Double Negation, Commutation, Association, Distribution, Contraposition, Implication, Equivalence, Exportation, Tautology, Material Implication) may be applied to parts of a line.

Proof strategies

• Work backwards from the conclusion.
• Use Conditional Proof: assume the antecedent, derive the consequent, then discharge.
• Use Indirect Proof: assume the negation of what you want and derive a contradiction.

Worked example

1. A ⊃ B
2. ~B
3. A ∨ C ∴ C From (1) and (2), Modus Tollens gives ~A. From (3) and ~A, Disjunctive Syllogism gives C. The direct proof is complete in two steps.

Traps that cost marks

Rules of inference may not be applied to part of a line - from A ⊃ (B • C) and A you get B • C, never B alone. Do not confuse affirming the consequent with Modus Ponens. In CP and IP, indent the assumption and discharge it before the final line; a conclusion resting on an undischarged assumption is invalid.