Predicate Logic: Soundness and Completeness of Formal Deduction - PowerPoint PPT Presentation

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Predicate Logic: Soundness and Completeness of Formal Deduction Alice Gao Lecture 17 CS 245 Logic and Computation Fall 2019 1 / 9 Outline The Learning Goals The Soundness of Formal Deduction Revisiting the Learning Goals CS 245 Logic and

Predicate Logic: Soundness and Completeness of Formal Deduction Alice Gao Lecture 17 CS 245 Logic and Computation Fall 2019 1 / 9
Outline The Learning Goals The Soundness of Formal Deduction Revisiting the Learning Goals CS 245 Logic and Computation Fall 2019 2 / 9
Learning goals By the end of this lecture, you should be able to: and completeness theorems. consequence using the soundness and completeness theorems. CS 245 Logic and Computation Fall 2019 3 / 9 ▶ Defjne soundness and completeness. ▶ Prove that an inference rule is sound or not sound. ▶ Prove that a logical consequence holds using the soundness ▶ Show that no natural deduction proof exists for a logical
The Soundness of Formal Deduction Theorem Formal Deduction for Predicate Logic is sound. Proof Sketch. Since Formal Deduction for Propositional Logic is sound, it suffjces to prove the soundness of the ∀− , ∃+ , ∀+ and ∃− rules. CS 245 Logic and Computation Fall 2019 4 / 9
The soundness of ∀− Theorem The ∀− inference rule is sound. That is, if Σ ⊨ ∀𝑦 𝐵(𝑦) , then Σ ⊨ 𝐵(𝑢) where 𝑢 is a term. CS 245 Logic and Computation Fall 2019 5 / 9
The soundness of ∃+ Theorem The ∃+ inference rule is sound. That is, if Σ ⊨ 𝐵(𝑢) , then Σ ⊨ ∃𝑦 𝐵(𝑦) where 𝑢 is a term. CS 245 Logic and Computation Fall 2019 6 / 9
The soundness of ∃− Theorem The ∃− inference rule is sound. That is, if Σ, 𝐵(𝑣) ⊨ 𝐶 , 𝑣 not occurring in Σ or 𝐶 , then Σ, ∃𝑦 𝐵(𝑦) ⊨ 𝐶 . CS 245 Logic and Computation Fall 2019 7 / 9
The soundness of ∀+ Theorem The ∀+ inference rule is sound. That is, if Σ ⊨ 𝐵(𝑣) , 𝑣 not occurring in Σ , then Σ ⊨ ∀𝑦 𝐵(𝑦) . CS 245 Logic and Computation Fall 2019 8 / 9
Revisiting the learning goals By the end of this lecture, you should be able to: and completeness theorems. entailment using the soundness and completeness theorems. CS 245 Logic and Computation Fall 2019 9 / 9 ▶ Defjne soundness and completeness. ▶ Prove that an inference rule is sound or not sound. ▶ Prove that a semantic entailment holds using the soundness ▶ Show that no natural deduction proof exists for a semantic

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