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

Propositional Logic: Completeness of Formal Deduction Alice Gao Lecture 10 CS 245 Logic and Computation Fall 2019 1 / 37

Learning Goals By the end of this lecture, you should be able to their defjnitions. theorem. CS 245 Logic and Computation Fall 2019 2 / 37 ▶ Defjne the completeness of formal deduction. ▶ Defjne consistency and satisfjability. ▶ Prove properties of consistent and satisfjable sets based on ▶ Reproduce the key steps of the proof of the completeness

The Soundness of Formal Deduction Theorems 1 and 2 are equivalent. Theorem 1 (Soundness of Formal Deduction) If Σ ⊢ 𝐵 , then Σ ⊨ 𝐵 . Theorem 2 If Σ is satisfjable, then Σ is consistent. CS 245 Logic and Computation Fall 2019 3 / 37

The Completeness of Formal Deduction Theorems 3 and 4 are equivalent. Theorem 3 (Completeness of Formal Deduction) If Σ ⊨ 𝐵 , then Σ ⊢ 𝐵 . Theorem 4 If Σ is consistent, then Σ is satisfjable. CS 245 Logic and Computation Fall 2019 4 / 37

Outline Learning Goals Defjnitions of Satisfjability and Consistency Two Proofs of Completeness of FD Proof of Completeness of FD using the Stronger Defjnition of Maximal Consistency Proof of Completeness of FD using the Weaker Defjnition of Maximal Consistency Revisiting the Learning Goals CS 245 Logic and Computation Fall 2019 5 / 37

Σ is satisfjable Defjnition 5 Σ is satisfjable if there exists a truth valuation 𝑢 such that Note that this is a semantic notion. CS 245 Logic and Computation Fall 2019 6 / 37 for every 𝐵 ∈ Σ , 𝐵 𝑢 = 1 .

Σ is consistent Intuitively, Σ is consistent if it doesn’t prove a contradiction. Two equivalent defjnitions: 1. There exists a formula 𝐵 , Σ ⊬ 𝐵 . ∃𝐵 (Σ ⊬ 𝐵). ∀𝐵 (Σ ⊢ 𝐵 → Σ ⊬ ¬𝐵). Note that consistency is a syntactical notion. Let’s prove that these two defjnitions are equivalent. CS 245 Logic and Computation Fall 2019 7 / 37 2. For every formula 𝐵 , if Σ ⊢ 𝐵 , then Σ ⊬ (¬𝐵) .

Σ is consistent - two equivalent defjnitions Theorem 6 Def 2 implies def 1. Proof. Assume that for every formula 𝐵 , if Σ ⊢ 𝐵 , then Σ ⊬ (¬𝐵) . We need to fjnd a formula 𝐵 such that Σ ⊬ 𝐵 . CS 245 Logic and Computation Fall 2019 8 / 37

Σ is consistent - two equivalent defjnitions Theorem 7 Negation of def 2 implies negation of def 1. Proof. Assume that there exists a formula 𝐵 such that Σ ⊢ 𝐵 and Σ ⊢ (¬𝐵) . We need to prove that for every formula A, Σ ⊢ 𝐵 . CS 245 Logic and Computation Fall 2019 9 / 37

Sketch of the Proof of The Completeness of Formal Deduction Theorem 8 If Σ is consistent implies Σ is satisfjable, then Σ ⊨ 𝐵 implies Σ ⊢ 𝐵 . Proof Sketch. Assume that Σ ⊨ 𝐵 . If Σ ⊨ 𝐵 , then we can prove that Σ ∪ {¬𝐵} is not satisfjable. (Part of assignment 4) By our assumption, if Σ ∪ {¬𝐵} is not satisfjable, then Σ ∪ {¬𝐵} is inconsistent. If Σ ∪ {¬𝐵} is inconsistent, then Σ ⊢ 𝐵 . (Let’s prove this part.) CS 245 Logic and Computation Fall 2019 10 / 37

Properties of a Consistent Set — Direction 1 Theorem 9 If Σ ∪ {¬𝐵} is inconsistent, then Σ ⊢ 𝐵 . Proof. Similarly, we can prove that “if Σ ∪ {𝐵} is inconsistent, then Σ ⊢ (¬𝐵) .” CS 245 Logic and Computation Fall 2019 11 / 37

Exercise: Properties of a Consistent Set — Direction 2 Theorem 10 If Σ ⊢ 𝐵 , then Σ ∪ {¬𝐵} is inconsistent. Proof. Similarly, we can prove that “if Σ ⊢ (¬𝐵) , then Σ ∪ {𝐵} is inconsistent.” CS 245 Logic and Computation Fall 2019 12 / 37

Outline Learning Goals Defjnitions of Satisfjability and Consistency Two Proofs of Completeness of FD Proof of Completeness of FD using the Stronger Defjnition of Maximal Consistency Proof of Completeness of FD using the Weaker Defjnition of Maximal Consistency Revisiting the Learning Goals CS 245 Logic and Computation Fall 2019 13 / 37

Two Proofs of the Completeness of Formal Deduction We will present two versions of the proofs of the completeness of Fall 2019 CS 245 Logic and Computation the proofs require difgerent defjnitions of maximal consistency. 2. Because of the defjnitions of the truth valuation 𝑢 , maximally consistent set Σ ∗ . 1. The proofs defjne the truth valuation 𝑢 based on the These two versions are almost identical except for two key points. formal deduction. 14 / 37 ▶ Proof 1 defjnes 𝑞 𝑢 = 1 ifg 𝑞 ∈ Σ ∗ . ▶ Proof 2 defjnes 𝑞 𝑢 = 1 ifg Σ ∗ ⊢ 𝑞 . ▶ Proof 1 requires the maximally consistent set Σ ∗ to satisfy 𝐵 ∈ Σ ∗ or (¬𝐵) ∈ Σ ∗ for every formula 𝐵 . ▶ Proof 2 requires the maximally consistent set Σ ∗ to satisfy Σ ∗ ⊢ 𝐵 or Σ ∗ ⊢ (¬𝐵) for every formula 𝐵 .

Two Defjnitions of Maximal Consistency The two proofs require two difgerent defjnitions of a maximally Fall 2019 CS 245 Logic and Computation Given a consistent Σ , Σ is maximally consistent if and only if 2. Weaker defjnition given in Assignment 5 This defjnition is re-stated on slide 18. Given a consistent Σ , Σ is maximally consistent if and only if 1. Stronger defjnition given in the Lu Zhongwan textbook the second defjnition. 15 / 37 consistent set. The fjrst defjnition is stronger than and implies ▶ For every formula 𝐵 , if 𝐵 ∉ Σ , then Σ ∪ {𝐵} is inconsistent. ▶ For every formula 𝐵 , 𝐵 ∈ Σ or (¬𝐵) ∈ Σ but not both. ▶ For every formula 𝐵 , if Σ ⊬ 𝐵 , then Σ ∪ {𝐵} is inconsistent. ▶ For every formula 𝐵 , Σ ⊢ 𝐵 or Σ ⊢ (¬𝐵) but not both. This defjnition is re-stated on slide 29.

Outline Learning Goals Defjnitions of Satisfjability and Consistency Two Proofs of Completeness of FD Proof of Completeness of FD using the Stronger Defjnition of Maximal Consistency Proof of Completeness of FD using the Weaker Defjnition of Maximal Consistency Revisiting the Learning Goals CS 245 Logic and Computation Fall 2019 16 / 37

Every Consistent Set is Satisfjable To fjnish the proof of the completeness theorem, which says “if Σ is consistent, then Σ satisfjable.” Proof Sketch. Assume that Σ is consistent. We need to fjnd a truth valuation 𝑢 Extend Σ to some maximally consistent set Σ ∗ . Let 𝑢 be a truth valuation such that for every propositional variable 𝑞 , Σ is satisfjed by 𝑢 . CS 245 Logic and Computation Fall 2019 17 / 37 it remains to prove theorem 4, such that 𝐵 𝑢 = 1 for every formula 𝐵 ∈ Σ . 𝑞 𝑢 = 1 if and only if 𝑞 ∈ Σ ∗ . For every 𝐵 ∈ Σ , 𝐵 ∈ Σ ∗ . We can prove that 𝐵 𝑢 = 1 . Therefore,

Defjnitions of a Maximally Consistent Set (Stronger Version) A key step in proving theorem 4 is to construct a maximally consistent set that includes Σ . First, let’s look at the defjnition of a maximally consistent set. This defjnition is given in the Lu Zhongwan textbook and it is stronger than the defjnition on slide 29. CS 245 Logic and Computation Fall 2019 18 / 37 Given a consistent Σ , Σ is maximally consistent if and only if ▶ For every formula 𝐵 , if 𝐵 ∉ Σ , then Σ ∪ {𝐵} is inconsistent. ▶ For every formula 𝐵 , 𝐵 ∈ Σ or (¬𝐵) ∈ Σ but not both.

Extending Σ to a Maximally Consistent Set Σ ∗ ⎨ Fall 2019 CS 245 Logic and Computation (We can prove this by induction on 𝑜 .) Σ 𝑜 , otherwise ⎩ Let Σ be a consistent set of formulas. { { ⎧ 𝐵 1 , 𝐵 2 , 𝐵 3 , … following sequence. Arbitrarily enumerate all the well-formed formulas using the 19 / 37 We extend Σ to a maximally consistent set Σ ∗ as follows. Construct an infjnite sequence of sets Σ 𝑜 as follows. Σ 0 = Σ Σ 𝑜+1 = {Σ 𝑜 ∪ {𝐵 𝑜+1 }, if Σ 𝑜 ∪ {𝐵 𝑜+1 } is consistent Observe that Σ 𝑜 ⊆ Σ 𝑜+1 and Σ 𝑜 is consistent.

Extending to Maximal Consistency (continued) 𝑜∈ℕ Σ 𝑜 . CS 245 Logic and Computation Fall 2019 20 / 37 Defjne Σ ∗ = ⋃ Think of Σ ∗ as the largest possible set that ▶ contains Σ , and ▶ is consistent. We will now prove that Σ ∗ is maximally consistent.

CS 245 Logic and Computation Fall 2019 21 / 37 Σ ∗ is maximally consistent First, we prove that Σ ∗ is consistent. Next, we prove that Σ ∗ is maximally consistent.

A Maximally Consistent Set Proves Its Elements Note that direction 2 of this lemma does not hold for the weaker defjnitions of maximal consistency given in assignment 5. Lemma 11 (Lemma 5.3.2 in Lu Zhongwan) Suppose Σ is maximally consistent. Then, 𝐵 ∈ Σ ifg Σ ⊢ 𝐵 . Proof. Direction 1: Assume 𝐵 ∈ Σ . Then, Σ ⊢ 𝐵 by (∈) . Direction 2: Assume Σ ⊢ 𝐵 . Towards a contradiction, assume that 𝐵 ∉ Σ . Since Σ is maximally consistent, Σ ∪ {𝐵} is inconsistent. Then, Σ ⊢ (¬𝐵) and Σ is inconsistent, contradicting the maximal consistency of Σ . Hence, 𝐵 ∈ Σ . CS 245 Logic and Computation Fall 2019 22 / 37

Satisfying a Maximally Consistent Set Lemma 12 Let 𝑢 be a truth valuation such that Then, for every well-formed propositional formula 𝐵 , Proof. By induction on the structure of 𝐵 . (Continued..) CS 245 Logic and Computation Fall 2019 23 / 37 Let Σ ∗ be a maximally consistent set. 𝑞 𝑢 = 1 if and only if 𝑞 ∈ Σ ∗ for every propositional variable 𝑞 . 𝐵 𝑢 = 1 if and only if 𝐵 ∈ Σ ∗ .