Propositional : 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 ▶ Define 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 퐴 be a propositional formula.

Σ ⊨ 퐴

▶ Σ semantically implies 퐴. ▶ 퐴 is a tautological consequence of Σ. ▶ For any valuation 푡, if every formula in Σ is true under 푡 (Σ푡 = 1), then 퐴 is also true under 푡 (퐴푡 = 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 퐴 be a propositional formula.

Σ ⊢ 퐴

▶ Σ formally proves 퐴. ▶ There exists a proof which syntactically transforms the premises in Σ to produce the conclusion 퐴. ▶ 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 ⊨ 퐴 and Σ ⊢ 퐴 appear to be similar. Ideally, we would like them to be equivalent. This could mean two properties: 1. If Σ ⊢ 퐴, then Σ ⊨ 퐴. (Soundness of formal deduction) If there exists a formal proof from Σ to 퐴, then Σ tautologically implies 퐴. 2. If Σ ⊨ 퐴, then Σ ⊢ 퐴. ( of formal deduction) If Σ tautologically implies 퐴, there exists a formal proof from Σ to 퐴.

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 of the premises. That is,

If Σ ⊢ 훼, then Σ ⊧ 훼

Completeness of Formal Deduction means that all logical consequences in propositional logic are provable in Formal Deduction. That is,

If Σ ⊧ 훼, then Σ ⊢ 훼

CS 245 Logic and Computation Fall 2019 6 / 16 Other proof systems

▶ ▶ axiomatic systems ▶ semantic tableaux ▶ intuitionistic logic: sound but not complete. e.g. it cannot prove 푝 ∨ (¬푝) ▶ any system plus 푝 ∧ (¬푝) as an axiom: not sound but complete. not sound because we can prove 푝 ∧ (¬푝) which is false. complete because we can prove anything with 푝 ∧ (¬푝) 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 ⊢ 퐴.

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 on 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 퐴, if Σ ⊢ 퐴, then Σ ⊨ 퐴.

Proof: We prove this by structural induction on the proof for Σ ⊢ 퐴. Base case: Assume that there is a proof for Σ ⊢ 퐴 where 퐴 ∈ Σ. Consider a truth valuation such that Σ푡 = 1. Since 퐴 ∈ Σ, then 퐴푡 = 1. Thus, Σ ⊨ 퐴.

(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 ⊢ 퐴. (There is one case for every rule of formal deduction.) ▶ Assume that the proof of Σ ⊢ 퐴 applies the rule ∧+ with the two premises Σ ⊢ 퐵 and Σ ⊢ 퐶 and reaches the conclusion Σ ⊢ 퐵 ∧ 퐶.

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 ⊢ 퐴 applies the rule → − with the two premises Σ ⊢ 퐵 and Σ ⊢ (퐵 → 퐶) and reaches the conclusion Σ ⊢ 퐶.

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 .

if Σ ⊢ ¬퐴 and Σ ⊢ 퐴 ∨ 퐵, then Σ ⊢ 퐵.

where 퐴 and 퐵 are well-formed propositional formulas. Prove that this inference rule is sound. That is, prove that if Σ ⊨ ¬퐴 and Σ ⊨ 퐴 ∨ 퐵, then Σ ⊨ 퐵. 2. Show that there does not exist a formal deduction proof for 푝 ∨ 푞 ⊢ 푝, where 푝 and 푞 are propositional variables. 3. Prove that (퐴 → 퐵) ⊬ (퐵 → 퐴) where 퐴 and 퐵 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 ⊢ ¬퐴 and Σ ⊢ 퐴 ∨ 퐵, then Σ ⊢ 퐵.

where 퐴 and 퐵 are well-formed propositional formulas. Prove that this inference rule is sound. That is, prove that if Σ ⊨ ¬퐴 and Σ ⊨ 퐴 ∨ 퐵, then Σ ⊨ 퐵.

CS 245 Logic and Computation Fall 2019 13 / 16 Applications of soundness and completeness

Show that there does not exist a formal proof for 푝 ∨ 푞 ⊢ 푝, where 푝 and 푞 are propositional variables.

CS 245 Logic and Computation Fall 2019 14 / 16 Applications of soundness and completeness

Prove that (퐴 → 퐵) ⊬ (퐵 → 퐴) where 퐴 and 퐵 are propositional formulas.

Proof: By the contrapositive of the soundness of formal deduction, if (퐴 → 퐵) ⊭ (퐵 → 퐴), then (퐴 → 퐵) ⊬ (퐵 → 퐴). We need to give a counterexample to show that (퐴 → 퐵) ⊭ (퐵 → 퐴). Let 퐴 = 푝 and 퐵 = 푞. Consider the truth valuation where 푝푡 = 0 and 푞푡 = 1. By the truth table of →, (푝 → 푞)푡 = 1 and (푞 → 푝)푡 = 0. Therefore, (퐴 → 퐵) ⊭ (퐵 → 퐴) and (퐴 → 퐵) ⊬ (퐵 → 퐴). 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 ▶ Define 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