Propositional Logic: Completeness of Formal Deduction

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 ▶ Define the completeness of formal deduction. ▶ Define consistency and satisfiability. ▶ Prove properties of consistent and satisfiable sets based on their definitions. ▶ Reproduce the key steps of the proof of the completeness theorem.

CS 245 Logic and Computation Fall 2019 2 / 37 The Soundness of Formal Deduction

Theorems 1 and 2 are equivalent.

Theorem 1 (Soundness of Formal Deduction) If Σ ⊢ 퐴, then Σ ⊨ 퐴. Theorem 2 If Σ is satisfiable, 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 ⊨ 퐴, then Σ ⊢ 퐴. Theorem 4 If Σ is consistent, then Σ is satisfiable.

CS 245 Logic and Computation Fall 2019 4 / 37 Outline

Learning Goals

Definitions of Satisfiability and Consistency

Two Proofs of Completeness of FD

Proof of Completeness of FD using the Stronger Definition of Maximal Consistency

Proof of Completeness of FD using the Weaker Definition of Maximal Consistency

Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 5 / 37 Σ is satisfiable

Definition 5 Σ is satisfiable if there exists a truth valuation 푡 such that for every 퐴 ∈ Σ, 퐴푡 = 1.

Note that this is a semantic notion.

CS 245 Logic and Computation Fall 2019 6 / 37 Σ is consistent

Intuitively, Σ is consistent if it doesn’t prove a contradiction.

Two equivalent definitions: 1. There exists a formula 퐴, Σ ⊬ 퐴. ∃퐴 (Σ ⊬ 퐴). 2. For every formula 퐴, if Σ ⊢ 퐴, then Σ ⊬ (¬퐴). ∀퐴 (Σ ⊢ 퐴 → Σ ⊬ ¬퐴).

Note that consistency is a syntactical notion. Let’s prove that these two definitions are equivalent.

CS 245 Logic and Computation Fall 2019 7 / 37 Σ is consistent - two equivalent definitions

Theorem 6 Def 2 implies def 1. Proof. Assume that for every formula 퐴, if Σ ⊢ 퐴, then Σ ⊬ (¬퐴). We need to find a formula 퐴 such that Σ ⊬ 퐴.

CS 245 Logic and Computation Fall 2019 8 / 37 Σ is consistent - two equivalent definitions

Theorem 7 Negation of def 2 implies negation of def 1. Proof. Assume that there exists a formula 퐴 such that Σ ⊢ 퐴 and Σ ⊢ (¬퐴). We need to prove that for every formula A, Σ ⊢ 퐴.

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 satisfiable, then Σ ⊨ 퐴 implies Σ ⊢ 퐴. Proof Sketch. Assume that Σ ⊨ 퐴.

If Σ ⊨ 퐴, then we can prove that Σ ∪ {¬퐴} is not satisfiable. (Part of assignment 4)

By our assumption, if Σ ∪ {¬퐴} is not satisfiable, then Σ ∪ {¬퐴} is inconsistent.

If Σ ∪ {¬퐴} is inconsistent, then Σ ⊢ 퐴. (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 ∪ {¬퐴} is inconsistent, then Σ ⊢ 퐴. Proof.

Similarly, we can prove that “if Σ ∪ {퐴} is inconsistent, then Σ ⊢ (¬퐴).”

CS 245 Logic and Computation Fall 2019 11 / 37 Exercise: Properties of a Consistent Set — Direction 2

Theorem 10 If Σ ⊢ 퐴, then Σ ∪ {¬퐴} is inconsistent. Proof.

Similarly, we can prove that “if Σ ⊢ (¬퐴), then Σ ∪ {퐴} is inconsistent.”

CS 245 Logic and Computation Fall 2019 12 / 37 Outline

Learning Goals

Definitions of Satisfiability and Consistency

Two Proofs of Completeness of FD

Proof of Completeness of FD using the Stronger Definition of Maximal Consistency

Proof of Completeness of FD using the Weaker Definition 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 formal deduction. These two versions are almost identical except for two key points. 1. The proofs define the truth valuation 푡 based on the maximally consistent set Σ∗.

▶ Proof 1 defines 푝푡 = 1 iff 푝 ∈ Σ∗. ▶ Proof 2 defines 푝푡 = 1 iff Σ∗ ⊢ 푝. 2. Because of the definitions of the truth valuation 푡, the proofs require different definitions of maximal consistency.

▶ Proof 1 requires the maximally consistent set Σ∗ to satisfy 퐴 ∈ Σ∗ or (¬퐴) ∈ Σ∗ for every formula 퐴. ▶ Proof 2 requires the maximally consistent set Σ∗ to satisfy Σ∗ ⊢ 퐴 or Σ∗ ⊢ (¬퐴) for every formula 퐴.

CS 245 Logic and Computation Fall 2019 14 / 37 Two Definitions of Maximal Consistency

The two proofs require two different definitions of a maximally consistent set. The first definition is stronger than and implies the second definition. 1. Stronger definition given in the Lu Zhongwan textbook Given a consistent Σ, Σ is maximally consistent if and only if ▶ For every formula 퐴, if 퐴 ∉ Σ, then Σ ∪ {퐴} is inconsistent. ▶ For every formula 퐴, 퐴 ∈ Σ or (¬퐴) ∈ Σ but not both. This definition is re-stated on slide 18. 2. Weaker definition given in Assignment 5 Given a consistent Σ, Σ is maximally consistent if and only if ▶ For every formula 퐴, if Σ ⊬ 퐴, then Σ ∪ {퐴} is inconsistent. ▶ For every formula 퐴, Σ ⊢ 퐴 or Σ ⊢ (¬퐴) but not both. This definition is re-stated on slide 29.

CS 245 Logic and Computation Fall 2019 15 / 37 Outline

Learning Goals

Definitions of Satisfiability and Consistency

Two Proofs of Completeness of FD

Proof of Completeness of FD using the Stronger Definition of Maximal Consistency

Proof of Completeness of FD using the Weaker Definition of Maximal Consistency

Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 16 / 37 Every Consistent Set is Satisfiable

To finish the proof of the completeness theorem, it remains to prove theorem 4, which says “if Σ is consistent, then Σ satisfiable.” Proof Sketch. Assume that Σ is consistent. We need to find a truth valuation 푡 such that 퐴푡 = 1 for every formula 퐴 ∈ Σ.

Extend Σ to some maximally consistent set Σ∗. Let 푡 be a truth valuation such that for every propositional variable 푝, 푝푡 = 1 if and only if 푝 ∈ Σ∗.

For every 퐴 ∈ Σ, 퐴 ∈ Σ∗. We can prove that 퐴푡 = 1. Therefore, Σ is satisfied by 푡.

CS 245 Logic and Computation Fall 2019 17 / 37 Definitions 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 definition of a maximally consistent set.

Given a consistent Σ, Σ is maximally consistent if and only if ▶ For every formula 퐴, if 퐴 ∉ Σ, then Σ ∪ {퐴} is inconsistent. ▶ For every formula 퐴, 퐴 ∈ Σ or (¬퐴) ∈ Σ but not both.

This definition is given in the Lu Zhongwan textbook and it is stronger than the definition on slide 29.

CS 245 Logic and Computation Fall 2019 18 / 37 Extending Σ to a Maximally Consistent Set Σ∗

Let Σ be a consistent set of formulas. We extend Σ to a maximally consistent set Σ∗ as follows.

Arbitrarily enumerate all the well-formed formulas using the following sequence.

퐴1, 퐴2, 퐴3,…

Construct an infinite sequence of sets Σ푛 as follows.

⎧Σ = Σ { 0 ⎨ Σ푛 ∪ {퐴푛+1}, if Σ푛 ∪ {퐴푛+1} is consistent {Σ푛+1 = { ⎩ Σ푛, otherwise

Observe that Σ푛 ⊆ Σ푛+1 and Σ푛 is consistent. (We can prove this by induction on 푛.)

CS 245 Logic and Computation Fall 2019 19 / 37 Extending to Maximal Consistency (continued)

∗ Define Σ = ⋃ Σ푛. 푛∈ℕ

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 20 / 37 Σ∗ is maximally consistent

First, we prove that Σ∗ is consistent.

Next, we prove that Σ∗ is maximally consistent.

CS 245 Logic and Computation Fall 2019 21 / 37 A Maximally Consistent Set Proves Its Elements

Note that direction 2 of this lemma does not hold for the weaker definitions of maximal consistency given in assignment 5. Lemma 11 (Lemma 5.3.2 in Lu Zhongwan) Suppose Σ is maximally consistent. Then, 퐴 ∈ Σ iff Σ ⊢ 퐴. Proof. Direction 1: Assume 퐴 ∈ Σ. Then, Σ ⊢ 퐴 by (∈). Direction 2: Assume Σ ⊢ 퐴. Towards a contradiction, assume that 퐴 ∉ Σ. Since Σ is maximally consistent, Σ ∪ {퐴} is inconsistent. Then, Σ ⊢ (¬퐴) and Σ is inconsistent, contradicting the maximal consistency of Σ. Hence, 퐴 ∈ Σ.

CS 245 Logic and Computation Fall 2019 22 / 37 Satisfying a Maximally Consistent Set

Lemma 12 Let Σ∗ be a maximally consistent set. Let 푡 be a truth valuation such that 푝푡 = 1 if and only if 푝 ∈ Σ∗ for every propositional variable 푝.

Then, for every well-formed propositional formula 퐴, 퐴푡 = 1 if and only if 퐴 ∈ Σ∗. Proof. By induction on the structure of 퐴. (Continued..)

CS 245 Logic and Computation Fall 2019 23 / 37 Base case and Inductive case 1

▶ Base case: 퐴 is a propositional variable 푝. 푝 ∈ Σ∗ iff 푝푡 = 1 by the definition of 푡.

▶ Inductive case 1: 퐴 = ¬퐵. Induction hypothesis: 퐵푡 = 1 iff 퐵 ∈ Σ∗. We need to show that (¬퐵)푡 = 1 iff ¬퐵 ∈ Σ∗.

CS 245 Logic and Computation Fall 2019 24 / 37 Inductive case 2

▶ Inductive case 2: 퐴 = 퐵 ∧ 퐶. Induction hypotheses: 퐵푡 = 1 iff 퐵 ∈ Σ∗. 퐶푡 = 1 iff 퐶 ∈ Σ∗. We need to show that (퐵 ∧ 퐶)푡 = 1 iff 퐵 ∧ 퐶 ∈ Σ∗. Direction 1:

Direction 2:

CS 245 Logic and Computation Fall 2019 25 / 37 Inductive cases 3, 4, and 5

▶ Inductive case 3: 퐴 = 퐵 ∨ 퐶. Induction hypotheses: 퐵푡 = 1 iff 퐵 ∈ Σ∗. 퐶푡 = 1 iff 퐶 ∈ Σ∗. We can show that if 퐵 ∨ 퐶 ∈ Σ∗ iff 퐵 ∈ Σ∗ or 퐶 ∈ Σ∗.

▶ Inductive case 4: 퐴 = 퐵 → 퐶. Induction hypotheses: 퐵푡 = 1 iff 퐵 ∈ Σ∗. 퐶푡 = 1 iff 퐶 ∈ Σ∗. We can show that 퐵 → 퐶 ∈ Σ∗ iff 퐵 ∈ Σ∗ implies 퐶 ∈ Σ∗.

▶ Inductive case 5: 퐴 = 퐵 ↔ 퐶. Induction hypotheses: 퐵푡 = 1 iff 퐵 ∈ Σ∗. 퐶푡 = 1 iff 퐶 ∈ Σ∗. We can show that 퐵 ↔ 퐶 ∈ Σ∗ iff (퐵 ∈ Σ∗ iff 퐶 ∈ Σ∗).

CS 245 Logic and Computation Fall 2019 26 / 37 Outline

Learning Goals

Definitions of Satisfiability and Consistency

Two Proofs of Completeness of FD

Proof of Completeness of FD using the Stronger Definition of Maximal Consistency

Proof of Completeness of FD using the Weaker Definition of Maximal Consistency

Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 27 / 37 Every Consistent Set is Satisfiable

To finish the proof of the completeness theorem, it remains to prove theorem 4, which says “if Σ is consistent, then Σ satisfiable.” Proof Sketch. Assume that Σ is consistent. We need to find a truth valuation 푡 such that 퐴푡 = 1 for every formula 퐴 ∈ Σ.

Extend Σ to some maximally consistent set Σ∗. Let 푡 be a truth valuation such that for every propositional variable 푝, 푝푡 = 1 if and only if Σ∗ ⊢ 푝.

For every 퐴 ∈ Σ, 퐴 ∈ Σ∗. We can prove that 퐴푡 = 1. Therefore, Σ is satisfied by 푡.

CS 245 Logic and Computation Fall 2019 28 / 37 Definitions of a Maximally Consistent Set (Weaker Version)

A key step in proving theorem 4 is to construct a maximally consistent set that includes Σ.

Let’s look at the definition of a maximally consistent set.

Given a consistent Σ, Σ is maximally consistent if and only if ▶ For every formula 퐴, if Σ ⊬ 퐴, then Σ ∪ {퐴} is inconsistent. ▶ For every formula 퐴, Σ ⊢ 퐴 or Σ ⊢ (¬퐴) but not both.

This definition is given in Assignment 5and it is weaker than the definition on slide 18.

CS 245 Logic and Computation Fall 2019 29 / 37 Extending Σ to a Maximally Consistent Set Σ∗

Let Σ be a consistent set of formulas. We extend Σ to a maximally consistent set Σ∗ as follows.

Arbitrarily enumerate all the well-formed formulas using the following sequence.

퐴1, 퐴2, 퐴3,…

Construct an infinite sequence of sets Σ푛 as follows.

⎧Σ = Σ { 0 ⎨ Σ푛 ∪ {퐴푛+1}, if Σ푛 ∪ {퐴푛+1} is consistent {Σ푛+1 = { ⎩ Σ푛, otherwise

Observe that Σ푛 ⊆ Σ푛+1 and Σ푛 is consistent. (We can prove this by induction on 푛.)

CS 245 Logic and Computation Fall 2019 30 / 37 Extending to Maximal Consistency (continued)

∗ Define Σ = ⋃ Σ푛. 푛∈ℕ

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 31 / 37 Σ∗ is maximally consistent

First, we prove that Σ∗ is consistent.

Next, we prove that Σ∗ is maximally consistent.

CS 245 Logic and Computation Fall 2019 32 / 37 Satisfying a Maximally Consistent Set

Lemma 13 Let Σ∗ be a maximally consistent set. Let 푡 be a truth valuation such that 푝푡 = 1 if and only if Σ∗ ⊢ 푝 for every propositional variable 푝.

Then, for every well-formed propositional formula 퐴, 퐴푡 = 1 if and only if Σ∗ ⊢ 퐴. Proof. By induction on the structure of 퐴. (Continued..)

CS 245 Logic and Computation Fall 2019 33 / 37 Base case and Inductive case 1

▶ Base case: 퐴 is a propositional variable 푝. Σ∗ ⊢ 푝 iff 푝푡 = 1 by the definition of 푡.

▶ Inductive case 1: 퐴 = ¬퐵. Induction hypothesis: 퐵푡 = 1 iff Σ∗ ⊢ 퐵. We need to show that (¬퐵)푡 = 1 iff Σ∗ ⊢ (¬퐵).

CS 245 Logic and Computation Fall 2019 34 / 37 Inductive case 2

▶ Inductive case 2: 퐴 = 퐵 ∧ 퐶. Induction hypotheses: 퐵푡 = 1 iff Σ∗ ⊢ 퐵. 퐶푡 = 1 iff Σ∗ ⊢ 퐶. We need to show that (퐵 ∧ 퐶)푡 = 1 iff Σ∗ ⊢ 퐵 ∧ 퐶. Direction 1:

Direction 2:

CS 245 Logic and Computation Fall 2019 35 / 37 Inductive cases 3, 4, and 5

▶ Inductive case 3: 퐴 = 퐵 ∨ 퐶. Induction hypotheses: 퐵푡 = 1 iff Σ∗ ⊢ 퐵. 퐶푡 = 1 iff Σ∗ ⊢ 퐶. We can show that Σ∗ ⊢ 퐵 ∨ 퐶 iff Σ∗ ⊢ 퐵 or Σ∗ ⊢ 퐶.

▶ Inductive case 4: 퐴 = 퐵 → 퐶. Induction hypotheses: 퐵푡 = 1 iff Σ∗ ⊢ 퐵. 퐶푡 = 1 iff Σ∗ ⊢ 퐶. We can show that Σ∗ ⊢ (퐵 → 퐶) iff Σ∗ ⊢ 퐵 implies Σ∗ ⊢ 퐶.

▶ Inductive case 5: 퐴 = 퐵 ↔ 퐶. Induction hypotheses: 퐵푡 = 1 iff Σ∗ ⊢ 퐵. 퐶푡 = 1 iff Σ∗ ⊢ 퐶. We can show that Σ∗ ⊢ (퐵 ↔ 퐶) iff (Σ∗ ⊢ 퐵 iff Σ∗ ⊢ 퐶).

CS 245 Logic and Computation Fall 2019 36 / 37 Revisiting the Learning Goals

By the end of this lecture, you should be able to ▶ Define the completeness of formal deduction. ▶ Define consistency and satisfiability. ▶ Prove properties of consistent and satisfiable sets based on their definitions. ▶ Reproduce the key steps of the proof of the completeness theorem.

CS 245 Logic and Computation Fall 2019 37 / 37