Logic as a Tool Chapter 1: Understanding Propositional Logic 1.3 Logical equivalence Negation normal form of propositional formulae Valentin Goranko Stockholm University September 2016 Goranko
Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A ≡ B , if they obtain the same truth value under any truth valuation (of the variables occurring in them). Goranko
Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A ≡ B , if they obtain the same truth value under any truth valuation (of the variables occurring in them). Examples: ¬ ( p ∧ q ) ≡ ¬ p ∨ ¬ q Goranko
Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A ≡ B , if they obtain the same truth value under any truth valuation (of the variables occurring in them). Examples: ¬ ( p ∧ q ) ≡ ¬ p ∨ ¬ q ¬ ( p ∧ q ) ¬ ∨ ¬ p q p q T T F T T T F T F F T T F T T F F F T T T F F T T F F T T F T F T F F T F F F T F T T F Goranko
Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A ≡ B , if they obtain the same truth value under any truth valuation (of the variables occurring in them). Examples: ¬ ( p ∧ q ) ≡ ¬ p ∨ ¬ q ¬ ( p ∧ q ) ¬ ∨ ¬ p q p q T T F T T T F T F F T T F T T F F F T T T F F T T F F T T F T F T F F T F F F T F T T F p ∧ ( p ∨ q ) ≡ p ∧ p ≡ p Goranko
Logical equivalence of propositional formulae Propositional formulae A and B are logically equivalent, denoted A ≡ B , if they obtain the same truth value under any truth valuation (of the variables occurring in them). Examples: ¬ ( p ∧ q ) ≡ ¬ p ∨ ¬ q ¬ ( p ∧ q ) ¬ ∨ ¬ p q p q T T F T T T F T F F T T F T T F F F T T T F F T T F F T T F T F T F F T F F F T F T T F p ∧ ( p ∨ q ) ≡ p ∧ p ≡ p p q p ∧ ( p ∨ q ) p ∧ p T T T T T T T T T T T F T T T T F T T T F T F F F T T F F F F F F F F F F F F F Goranko
Some basic properties of logical equivalence ◮ Logical equivalence is reducible to logical consequence: A ≡ B iff A | = B and B | = A Goranko
Some basic properties of logical equivalence ◮ Logical equivalence is reducible to logical consequence: A ≡ B iff A | = B and B | = A ◮ Logical equivalence is reducible to logical validity: A ≡ B iff | = A ↔ B . Goranko
Some basic properties of logical equivalence ◮ Logical equivalence is reducible to logical consequence: A ≡ B iff A | = B and B | = A ◮ Logical equivalence is reducible to logical validity: A ≡ B iff | = A ↔ B . ◮ ≡ is an equivalence relation, i.e., reflexive, symmetric, and transitive. Goranko
Some basic properties of logical equivalence ◮ Logical equivalence is reducible to logical consequence: A ≡ B iff A | = B and B | = A ◮ Logical equivalence is reducible to logical validity: A ≡ B iff | = A ↔ B . ◮ ≡ is an equivalence relation, i.e., reflexive, symmetric, and transitive. ◮ Moreover, ≡ is a congruence with respect to the propositional connectives, i.e.: Goranko
Some basic properties of logical equivalence ◮ Logical equivalence is reducible to logical consequence: A ≡ B iff A | = B and B | = A ◮ Logical equivalence is reducible to logical validity: A ≡ B iff | = A ↔ B . ◮ ≡ is an equivalence relation, i.e., reflexive, symmetric, and transitive. ◮ Moreover, ≡ is a congruence with respect to the propositional connectives, i.e.: ⊲ if A ≡ B then ¬ A ≡ ¬ B , and Goranko
Some basic properties of logical equivalence ◮ Logical equivalence is reducible to logical consequence: A ≡ B iff A | = B and B | = A ◮ Logical equivalence is reducible to logical validity: A ≡ B iff | = A ↔ B . ◮ ≡ is an equivalence relation, i.e., reflexive, symmetric, and transitive. ◮ Moreover, ≡ is a congruence with respect to the propositional connectives, i.e.: ⊲ if A ≡ B then ¬ A ≡ ¬ B , and ⊲ if A 1 ≡ B 1 and A 2 ≡ B 2 then ( A 1 • A 2 ) ≡ ( B 1 • B 2 ), where • ∈ {∧ , ∨ , → , ↔} . Goranko
Some basic properties of logical equivalence ◮ Logical equivalence is reducible to logical consequence: A ≡ B iff A | = B and B | = A ◮ Logical equivalence is reducible to logical validity: A ≡ B iff | = A ↔ B . ◮ ≡ is an equivalence relation, i.e., reflexive, symmetric, and transitive. ◮ Moreover, ≡ is a congruence with respect to the propositional connectives, i.e.: ⊲ if A ≡ B then ¬ A ≡ ¬ B , and ⊲ if A 1 ≡ B 1 and A 2 ≡ B 2 then ( A 1 • A 2 ) ≡ ( B 1 • B 2 ), where • ∈ {∧ , ∨ , → , ↔} . Theorem (Equivalent replacement) Let A , B , C be any propositional formulae p be a propositional variable. If A ≡ B then C ( A / p ) ≡ C ( B / p ) , where C ( X / p ) is the result of simultaneous substitution of al occurrences of p by X. Goranko
Some important logical equivalences • Idempotency: p ∧ p ≡ p ; p ∨ p ≡ p . Goranko
Some important logical equivalences • Idempotency: p ∧ p ≡ p ; p ∨ p ≡ p . • Commutativity: p ∧ q ≡ q ∧ p ; p ∨ q ≡ q ∨ p . Goranko
Some important logical equivalences • Idempotency: p ∧ p ≡ p ; p ∨ p ≡ p . • Commutativity: p ∧ q ≡ q ∧ p ; p ∨ q ≡ q ∨ p . • Associativity: ( p ∧ ( q ∧ r )) ≡ (( p ∧ q ) ∧ r ); ( p ∨ ( q ∨ r )) ≡ (( p ∨ q ) ∨ r ) . Note that this property allows us to omit the parentheses in multiple conjunctions and disjunctions. Goranko
Some important logical equivalences • Idempotency: p ∧ p ≡ p ; p ∨ p ≡ p . • Commutativity: p ∧ q ≡ q ∧ p ; p ∨ q ≡ q ∨ p . • Associativity: ( p ∧ ( q ∧ r )) ≡ (( p ∧ q ) ∧ r ); ( p ∨ ( q ∨ r )) ≡ (( p ∨ q ) ∨ r ) . Note that this property allows us to omit the parentheses in multiple conjunctions and disjunctions. • Absorption: p ∧ ( p ∨ q ) ≡ p ; p ∨ ( p ∧ q ) ≡ p . Goranko
Some important logical equivalences • Idempotency: p ∧ p ≡ p ; p ∨ p ≡ p . • Commutativity: p ∧ q ≡ q ∧ p ; p ∨ q ≡ q ∨ p . • Associativity: ( p ∧ ( q ∧ r )) ≡ (( p ∧ q ) ∧ r ); ( p ∨ ( q ∨ r )) ≡ (( p ∨ q ) ∨ r ) . Note that this property allows us to omit the parentheses in multiple conjunctions and disjunctions. • Absorption: p ∧ ( p ∨ q ) ≡ p ; p ∨ ( p ∧ q ) ≡ p . • Distributivity: p ∧ ( q ∨ r ) ≡ ( p ∧ q ) ∨ ( p ∧ r ); p ∨ ( q ∧ r ) ≡ ( p ∨ q ) ∧ ( p ∨ r ) . Goranko
Other useful logical equivalences Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ ⊥ Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ ⊥ • A ∨ ⊤ ≡ Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ ⊥ • A ∨ ⊤ ≡ ⊤ ; Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ ⊥ • A ∨ ⊤ ≡ ⊤ ; A ∨ ⊥ ≡ Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ ⊥ • A ∨ ⊤ ≡ ⊤ ; A ∨ ⊥ ≡ A Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ ⊥ • A ∨ ⊤ ≡ ⊤ ; A ∨ ⊥ ≡ A • ¬ A ≡ A → ⊥ Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ ⊥ • A ∨ ⊤ ≡ ⊤ ; A ∨ ⊥ ≡ A • ¬ A ≡ A → ⊥ • A ↔ B ≡ ( A → B ) ∧ ( B → A ) Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ ⊥ • A ∨ ⊤ ≡ ⊤ ; A ∨ ⊥ ≡ A • ¬ A ≡ A → ⊥ • A ↔ B ≡ ( A → B ) ∧ ( B → A ) • A → B ≡ ¬ A ∨ B Goranko
Other useful logical equivalences • A ∨ ¬ A ≡ ⊤ ; A ∧ ¬ A ≡ ⊥ • A ∧ ⊤ ≡ A ; A ∧ ⊥ ≡ ⊥ • A ∨ ⊤ ≡ ⊤ ; A ∨ ⊥ ≡ A • ¬ A ≡ A → ⊥ • A ↔ B ≡ ( A → B ) ∧ ( B → A ) • A → B ≡ ¬ A ∨ B • A ∨ B ≡ ¬ A → B Goranko
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