Propositional Logic: Soundness of Formal Deduction Alice Gao - - PowerPoint PPT Presentation

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

▶ Defjne the soundness of formal deduction. ▶ Prove that a tautological consequence holds using formal

deduction and the soundness of formal deduction.

▶ Show that no formal deduction proof exists using the

contrapositive of the soundness of formal deduction.

CS 245 Logic and Computation Fall 2019 2 / 16

Tautological Consequence

Let Σ be a set of propositional formulas. Let 𝐵 be a propositional formula. Σ ⊨ 𝐵

▶ 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

(Σ𝑢 = 1), then 𝐵 is also true under 𝑢 (𝐵𝑢 = 1). 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

Formal Deduction

Let Σ be a set of propositional formulas. Let 𝐵 be a propositional formula. Σ ⊢ 𝐵

▶ Σ formally proves 𝐵. ▶ There exists a proof which syntactically transforms the

premises in Σ to produce the conclusion 𝐵.

▶ A formal proof is a syntactic manipulation of symbols and it

can be checked mechanically.

CS 245 Logic and Computation Fall 2019 4 / 16

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

▶ resolution ▶ axiomatic systems ▶ semantic tableaux ▶ intuitionistic logic: sound but not complete. e.g. it cannot

prove 𝑞 ∨ (¬𝑞)

▶ any system plus 𝑞 ∧ (¬𝑞) as an axiom: not sound but

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

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

▶ derives the conclusion without using any inference rule, or

(Base case)

▶ derives the conclusion by applying a rule of formal deduction

• n a proof. (Inductive case)

CS 245 Logic and Computation Fall 2019 8 / 16

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. Consider a truth valuation such that Σ𝑢 = 1. Since 𝐵 ∈ Σ, then 𝐵𝑢 = 1. Thus, Σ ⊨ 𝐵. (To be continued)

CS 245 Logic and Computation Fall 2019 9 / 16

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.)

▶ Assume that the proof of Σ ⊢ 𝐵 applies the rule ∧+

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

Proof of the soundness of formal deduction

Induction step (continued):

▶ Assume that the proof of Σ ⊢ 𝐵 applies the rule → −

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

Applications of soundness and completeness

• 1. 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.

• 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

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 give a counterexample to show that (𝐵 → 𝐶) ⊭ (𝐶 → 𝐵). Let 𝐵 = 𝑞 and 𝐶 = 𝑟. Consider the truth valuation where 𝑞𝑢 = 0 and 𝑟𝑢 = 1. By the truth table of →, (𝑞 → 𝑟)𝑢 = 1 and (𝑟 → 𝑞)𝑢 = 0. Therefore, (𝐵 → 𝐶) ⊭ (𝐶 → 𝐵) and (𝐵 → 𝐶) ⊬ (𝐶 → 𝐵). QED

CS 245 Logic and Computation Fall 2019 15 / 16

Revisiting the Learning Goals

By the end of this lecture, you should be able to

▶ Defjne the soundness of formal deduction. ▶ Prove that a tautological consequence holds using formal

deduction and the soundness of formal deduction.

▶ Show that no formal deduction proof exists using the

contrapositive of the soundness of formal deduction.

CS 245 Logic and Computation Fall 2019 16 / 16