Predicate Logic: Logical Consequence

Alice Gao

Lecture 14

CS 245 Logic and Computation Fall 2019 1 / 12 Outline

The Learning Goals

Definition of Logical Consequence

Proving/Disproving a Logical Consequence

Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 2 / 12 Learning goals

By the end of this lecture, you should be able to: ▶ Define logical consequence for predicate logic. ▶ Prove that a logical consequence holds. ▶ Prove that a logical consequence does not hold.

CS 245 Logic and Computation Fall 2019 3 / 12 Definition of Logical Consequence

Define the symbols. ▶ Σ is a set of predicate formulas. ▶ 퐴 is a predicate formula.

Σ ⊨ 퐴 (Σ logically implies 퐴) (퐴 is a logical consequence of Σ) iff for every valuation 푣, if Σ푣 = 1, then 퐴푣 = 1.

CS 245 Logic and Computation Fall 2019 4 / 12 Prove a logical consequence

Consider the logical consequence Σ ⊨ 퐴. To prove that the logical consequence holds, we need to consider

(A) Every valuation 푣 such that Σ푣 = 1. (B) Every valuation 푣 such that Σ푣 = 0. (C) One valuation 푣 such that Σ푣 = 1. (D) One valuation 푣 such that Σ푣 = 0.

CS 245 Logic and Computation Fall 2019 5 / 12 Disprove a logical consequence

Consider the logical consequence Σ ⊨ 퐴. To prove that the logical consequence does NOT hold, we need to consider

(A) Every valuation 푣 such that Σ푣 = 1 and 퐴푣 = 1. (B) Every valuation 푣 such that Σ푣 = 1 and 퐴푣 = 0. (C) One valuation 푣 such that Σ푣 = 1 and 퐴푣 = 1. (D) One valuation 푣 such that Σ푣 = 1 and 퐴푣 = 0.

CS 245 Logic and Computation Fall 2019 6 / 12 Example: Prove/Disprove a logical consequence

Consider the logical consequence below.

∀푥 ¬퐴(푥) ⊨ ¬(∃푥 퐴(푥)).

If the logical consequence holds, prove it. If it does not hold, provide a counterexample.

CS 245 Logic and Computation Fall 2019 7 / 12 Example: Prove/Disprove a logical consequence

Consider the logical consequence below.

∀푥(퐴(푥) → 퐵(푥)) ⊨ ∀푥퐴(푥) → ∀푥퐵(푥)

If the logical consequence holds, prove it. If it does not hold, provide a counterexample.

CS 245 Logic and Computation Fall 2019 8 / 12 Example: Prove/Disprove a logical consequence

Consider the logical consequence below.

∀푥퐴(푥) → ∀푥퐵(푥) ⊨ ∀푥(퐴(푥) → 퐵(푥))

If the logical consequence holds, prove it. If it does not hold, provide a counterexample.

CS 245 Logic and Computation Fall 2019 9 / 12 Example: Prove/Disprove the logical consequence

Prove the following

∃푥(퐴(푥) ∧ 퐵(푥)) ⊨ ∃푥퐴(푥) ∧ ∃푥퐵(푥).

CS 245 Logic and Computation Fall 2019 10 / 12 Example: Prove/Disprove the logical consequence

Prove the following

∃푥퐴(푥) ∧ ∃푥퐵(푥) ⊭ ∃푥(퐴(푥) ∧ 퐵(푥)).

CS 245 Logic and Computation Fall 2019 11 / 12 Revisiting the learning goals

By the end of this lecture, you should be able to: ▶ Define logical consequence for predicate logic. ▶ Prove that a logical consequence holds. ▶ Prove that a logical consequence does not hold.

CS 245 Logic and Computation Fall 2019 12 / 12