28 Logic as a Tool 1.3 Logical equivalence: negation normal form of propositional formulae 1.3.1 Logically equivalent propositional formulae Definition 16 The propositional formulae A and B are logically equivalent, denoted A ≡ B , if for every assignment of truth values to the variables occurring in them they obtain the same truth values Being a little imprecise (you’ll see why), we can say that A and B are logically equivalent if they have the same truth tables For example, p ơp ã Every tautology is equivalent to • Every contradiction is equivalent to ⊥ For example, p ơp ã ơơp p: a double negation of a proposition is equivalent to the proposition itself ã ơ(p q ) (ơp ¬q ) and ¬(p ∨ q ) ≡ (¬p ∧ ¬q ) These are known as De Morgan’s laws Let us check the first, using simplified truth tables: p T T F F q T F T F ¬ F T T T (p T T F F ∧ T F F F q) T F T F (¬ F F T T p T T F F ∨ F T T T ¬ F T F T q) T F T F • (p ∧ (p ∨ q )) ≡ (p ∧ p) There is a small problem here: formally, the truth tables of these formulae are not the same, as the first contains two variables (p and q ) while the second contains only p However, we can always consider that q occurs vacuously in the second formula and include it in its truth table: p T T F F q T F T F (p T T F F ∧ T T F F (p T T F F ∨ T T T F q )) T F T F (p T T F F ∧ T T F F p) T T F F Checking logical equivalence can be streamlined, just like checking logical validity and consequence, by systematic search for a falsifying assignment, as follows In order to check if A ≡ B we try to construct a truth assignment to the variables occurring in A and B which renders one of them true while the other false If such an assignment exists, the formulae are not logically equivalent; otherwise, they are For example, let us check the second De Morgan’s law: ¬(p ∨ q ) ≡ (¬p ∧ ¬q ) There are two possibilities for an assignment to falsify that equivalence: (i) ¬(p ∨ q ) : T and (¬p ∧ ¬q ) : F Then p ∨ q : F hence p : F, and q : F, but then (¬p ∧ ¬q ) : T: a contradiction