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Logic as a Tool Chapter 1: Understanding Propositional Logic 1.3 - - PowerPoint PPT Presentation
Logic as a Tool Chapter 1: Understanding Propositional Logic 1.3 - - PowerPoint PPT Presentation
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
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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 A1 ≡ B1 and A2 ≡ B2 then (A1 • A2) ≡ (B1 • B2), 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.
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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).
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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
- A ∨ B ≡ ¬(¬A ∧ ¬B);
A ∧ B ≡ ¬(¬A ∨ ¬B)
- A → B ≡ ¬B → ¬A
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Important equivalences for negations of propositional formulae
- ¬¬A ≡ A,
- ¬(A ∧ B) ≡ ¬A ∨ ¬B,
- ¬(A ∨ B) ≡ ¬A ∧ ¬B,
- ¬(A → B) ≡ A ∧ ¬B,
- ¬(A ↔ B)
≡ ¬((A → B) ∧ (B → A)) ≡ ¬(A → B) ∨ ¬(B → A) ≡ (A ∧ ¬B) ∨ (B ∧ ¬A). Using these equivalences, all occurrences of negations in any propositional formula can be driven inwards, so eventually they only
- ccur in from of propositional variables.
Then the formula is transformed to a negation normal form.
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