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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

SLIDE 1

Predicate Logic: Soundness and Completeness of Formal Deduction

Alice Gao

Lecture 17

CS 245 Logic and Computation Fall 2019 1 / 9

SLIDE 2

Outline

The Learning Goals The Soundness of Formal Deduction Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 2 / 9

SLIDE 3

Learning goals

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

▶ Defjne soundness and completeness. ▶ Prove that an inference rule is sound or not sound. ▶ Prove that a logical consequence holds using the soundness

and completeness theorems.

▶ Show that no natural deduction proof exists for a logical

consequence using the soundness and completeness theorems.

CS 245 Logic and Computation Fall 2019 3 / 9

SLIDE 4

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

SLIDE 5

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

SLIDE 6

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

SLIDE 7

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

SLIDE 8

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

SLIDE 9

Revisiting the learning goals

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

▶ Defjne soundness and completeness. ▶ Prove that an inference rule is sound or not sound. ▶ Prove that a semantic entailment holds using the soundness

and completeness theorems.

▶ Show that no natural deduction proof exists for a semantic

entailment using the soundness and completeness theorems.

CS 245 Logic and Computation Fall 2019 9 / 9