Symbolic Logic: Propositional Calculus
What the exam tests
Propositional calculus treats propositions as units and connects them with truth-functional operators. The exam tests symbolization, truth tables, tautology/contradiction/contingency, logical equivalence, and natural-deduction rules.
Core rules
- Connectives: negation (¬), conjunction (∧), disjunction (∨), conditional (→), biconditional (↔). Precedence usually ¬, ∧, ∨, →, ↔.
- A tautology is true on every valuation; a contradiction is false on every valuation; a contingency is true on some and false on others.
- Inference rules: modus ponens (P→Q, P ⊢ Q), modus tollens (P→Q, ¬Q ⊢ ¬P), disjunctive syllogism (P∨Q, ¬P ⊢ Q), hypothetical syllogism (P→Q, Q→R ⊢ P→R), simplification, addition, conjunction.
- Equivalences: De Morgan’s laws, double negation, contraposition, distribution, exportation.
Worked example
Symbolize: “If it rains, the match is cancelled. It rains. Therefore the match is cancelled.” Let R = it rains, C = match cancelled. The form is R→C, R ⊢ C (modus ponens). To test it, assume both premises true. If R is true, then for R→C to be true, C must be true; the conditional is false only when R is true and C false. Therefore no valuation makes both premises true and the conclusion false. The argument is valid. A truth table with four rows confirms that the row R = T, C = F is excluded by the second premise.
Traps
Do not confuse modus ponens with affirming the consequent (P→Q, Q ⊢ P), which is invalid. Do not confuse modus tollens with denying the antecedent (P→Q, ¬P ⊢ ¬Q), also invalid. Treat “or” as inclusive unless the question says exclusive. Watch parentheses: P→Q∧R is not the same as (P→Q)∧R. A conditional with a false antecedent is true, which often surprises candidates.