Symbolic Logic: Propositional Calculus and Truth Tables
Propositional calculus represents whole propositions by variables and connects them with truth-functional operators. The exam tests symbolisation, truth tables, tautology, contradiction, contingency, equivalence and validity.
Core rules
- Connectives: not, and, or, if-then, if and only if. Material implication P implies Q is false only when P is true and Q is false.
- A tautology is true on every row of its truth table; a contradiction is false on every row; a contingency varies.
- An argument is valid if there is no row in which all premises are true and the conclusion false. A conditional is valid as a theorem when its corresponding conditional is a tautology.
- Precedence normally runs: negation, conjunction, disjunction, conditional, biconditional; use brackets when unsure.
Worked example
Test modus ponens: P implies Q; P; therefore Q. The truth table has four rows for P and Q. The only row where P implies Q is true and P is true is the row T, T; there Q is true. There is no counterexample row, so the form is valid. Its corresponding conditional, ((P implies Q) and P) implies Q, is a tautology.
Traps
Do not read if P then Q as P causes Q or as Q if P reversed. The converse Q implies P is not equivalent to P implies Q. Do not confuse inclusive or with exclusive or. In truth tables, one counterexample row is enough to prove invalidity.