Propositional Logic: Soundness of Formal Deduction Alice Gao Lecture 9 CS 245 Logic and Computation Fall 2019 1 / 16
Learning Goals By the end of this lecture, you should be able to deduction and the soundness of formal deduction. contrapositive of the soundness of formal deduction. CS 245 Logic and Computation Fall 2019 2 / 16 ▶ Defjne the soundness of formal deduction. ▶ Prove that a tautological consequence holds using formal ▶ Show that no formal deduction proof exists using the
Tautological Consequence Let Σ be a set of propositional formulas. Let 𝐵 be a propositional formula. Σ ⊨ 𝐵 Several ways of proving a tautological consequence: truth table, direct proof, a proof by contradiction, etc. CS 245 Logic and Computation Fall 2019 3 / 16 ▶ Σ semantically implies 𝐵 . ▶ 𝐵 is a tautological consequence of Σ . ▶ For any truth valuation 𝑢 , if every formula in Σ is true under 𝑢 ( Σ 𝑢 = 1 ), then 𝐵 is also true under 𝑢 ( 𝐵 𝑢 = 1 ).
Formal Deduction Let Σ be a set of propositional formulas. Let 𝐵 be a propositional formula. Σ ⊢ 𝐵 premises in Σ to produce the conclusion 𝐵 . can be checked mechanically. CS 245 Logic and Computation Fall 2019 4 / 16 ▶ Σ formally proves 𝐵 . ▶ There exists a proof which syntactically transforms the ▶ A formal proof is a syntactic manipulation of symbols and it
Tautological Consequence v.s. Formal Deduction Σ ⊨ 𝐵 and Σ ⊢ 𝐵 appear to be similar. Ideally, we would like them to be equivalent. This could mean two properties: 1. If Σ ⊢ 𝐵 , then Σ ⊨ 𝐵 . (Soundness of formal deduction) If there exists a formal proof from Σ to 𝐵 , then Σ tautologically implies 𝐵 . 2. If Σ ⊨ 𝐵 , then Σ ⊢ 𝐵 . (Completeness of formal deduction) If Σ tautologically implies 𝐵 , there exists a formal proof from Σ to 𝐵 . CS 245 Logic and Computation Fall 2019 5 / 16
Soundness and Completeness of Formal Deduction Theorem: Formal Deduction is both sound and complete. Soundness of Formal Deduction means that the conclusion of a proof is always a logical consequence of the premises. That is, If Σ ⊢ 𝛽 , then Σ ⊧ 𝛽 Completeness of Formal Deduction means that all logical consequences in propositional logic are provable in Formal Deduction. That is, If Σ ⊧ 𝛽 , then Σ ⊢ 𝛽 CS 245 Logic and Computation Fall 2019 6 / 16
Other proof systems prove 𝑞 ∨ (¬𝑞) complete. not sound because we can prove 𝑞 ∧ (¬𝑞) which is false. complete because we can prove anything with 𝑞 ∧ (¬𝑞) as an axiom. CS 245 Logic and Computation Fall 2019 7 / 16 ▶ resolution ▶ axiomatic systems ▶ semantic tableaux ▶ intuitionistic logic: sound but not complete. e.g. it cannot ▶ any system plus 𝑞 ∧ (¬𝑞) as an axiom: not sound but
Proving the soundness of formal deduction We will prove this by structural induction on the proof for Σ ⊢ 𝐵 . A proof is a recursive structure. A proof either (Base case) on a proof. (Inductive case) CS 245 Logic and Computation Fall 2019 8 / 16 ▶ derives the conclusion without using any inference rule, or ▶ derives the conclusion by applying a rule of formal deduction
Proof of the soundness of formal deduction Theorem: For a set of propositional formulas Σ and a propositional formula 𝐵 , if Σ ⊢ 𝐵 , then Σ ⊨ 𝐵 . Proof: We prove this by structural induction on the proof for Σ ⊢ 𝐵 . Base case: Assume that there is a proof for Σ ⊢ 𝐵 where 𝐵 ∈ Σ . (To be continued) CS 245 Logic and Computation Fall 2019 9 / 16 Consider a truth valuation such that Σ 𝑢 = 1 . Since 𝐵 ∈ Σ , then 𝐵 𝑢 = 1 . Thus, Σ ⊨ 𝐵 .
Proof of the soundness of formal deduction Induction step: Consider several cases for the last rule applied in the proof of Σ ⊢ 𝐵 . (There is one case for every rule of formal deduction.) with the two premises Σ ⊢ 𝐶 and Σ ⊢ 𝐷 and reaches the conclusion Σ ⊢ 𝐶 ∧ 𝐷 . Let me prove this case for you. (To be continued) CS 245 Logic and Computation Fall 2019 10 / 16 ▶ Assume that the proof of Σ ⊢ 𝐵 applies the rule ∧+
Proof of the soundness of formal deduction Induction step (continued): with the two premises Σ ⊢ 𝐶 and Σ ⊢ (𝐶 → 𝐷) and reaches the conclusion Σ ⊢ 𝐷 . Try proving this case yourself. CS 245 Logic and Computation Fall 2019 11 / 16 ▶ Assume that the proof of Σ ⊢ 𝐵 applies the rule → −
Applications of soundness and completeness if Σ ⊢ ¬𝐵 and Σ ⊢ 𝐵 ∨ 𝐶, then Σ ⊢ 𝐶. where 𝐵 and 𝐶 are well-formed propositional formulas. Prove that this inference rule is sound. That is, prove that if Σ ⊨ ¬𝐵 and Σ ⊨ 𝐵 ∨ 𝐶 , then Σ ⊨ 𝐶 . 2. Show that there does not exist a formal deduction proof for 𝑞 ∨ 𝑟 ⊢ 𝑞 , where 𝑞 and 𝑟 are propositional variables. 3. Prove that (𝐵 → 𝐶) ⊬ (𝐶 → 𝐵) where 𝐵 and 𝐶 are propositional formulas. CS 245 Logic and Computation Fall 2019 12 / 16 1. The following inference rule is called Disjunctive syllogism.
Applications of soundness and completeness The following inference rule is called Disjunctive syllogism. if Σ ⊢ ¬𝐵 and Σ ⊢ 𝐵 ∨ 𝐶, then Σ ⊢ 𝐶. where 𝐵 and 𝐶 are well-formed propositional formulas. Prove that this inference rule is sound. That is, prove that if Σ ⊨ ¬𝐵 and Σ ⊨ 𝐵 ∨ 𝐶 , then Σ ⊨ 𝐶 . CS 245 Logic and Computation Fall 2019 13 / 16
Applications of soundness and completeness Show that there does not exist a formal proof for 𝑞 ∨ 𝑟 ⊢ 𝑞 , where 𝑞 and 𝑟 are propositional variables. CS 245 Logic and Computation Fall 2019 14 / 16
Applications of soundness and completeness Prove that (𝐵 → 𝐶) ⊬ (𝐶 → 𝐵) where 𝐵 and 𝐶 are propositional formulas. Proof: By the contrapositive of the soundness of formal deduction, if (𝐵 → 𝐶) ⊭ (𝐶 → 𝐵) , then (𝐵 → 𝐶) ⊬ (𝐶 → 𝐵) . We need to (𝐵 → 𝐶) ⊬ (𝐶 → 𝐵) . QED CS 245 Logic and Computation Fall 2019 15 / 16 give a counterexample to show that (𝐵 → 𝐶) ⊭ (𝐶 → 𝐵) . Let 𝐵 = 𝑞 and 𝐶 = 𝑟 . Consider the truth valuation where 𝑞 𝑢 = 0 and 𝑟 𝑢 = 1 . By the truth table of → , (𝑞 → 𝑟) 𝑢 = 1 and (𝑟 → 𝑞) 𝑢 = 0 . Therefore, (𝐵 → 𝐶) ⊭ (𝐶 → 𝐵) and
Revisiting the Learning Goals By the end of this lecture, you should be able to deduction and the soundness of formal deduction. contrapositive of the soundness of formal deduction. CS 245 Logic and Computation Fall 2019 16 / 16 ▶ Defjne the soundness of formal deduction. ▶ Prove that a tautological consequence holds using formal ▶ Show that no formal deduction proof exists using the
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