Propositional Logic: Formal Deduction Alice Gao Lecture 7 CS 245 - - PowerPoint PPT Presentation

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Jun 15, 2023 304 likes •471 views

Propositional Logic: Formal Deduction Alice Gao Lecture 7 CS 245 Logic and Computation Fall 2019 1 / 15 Outline Learning goals Motivation for formal deduction Rules of formal deduction Revisiting the Learning Goals CS 245 Logic and

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Propositional Logic: Formal Deduction

Alice Gao

Lecture 7

CS 245 Logic and Computation Fall 2019 1 / 15

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Outline

Learning goals Motivation for formal deduction Rules of formal deduction Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 2 / 15

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Outline

Learning goals Motivation for formal deduction Rules of formal deduction Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 3 / 15

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Learning goals

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

▶ Describe rules of inference for natural deduction. ▶ Prove that a conclusion follows from a set of premises using

rules of formal deduction.

CS 245 Logic and Computation Fall 2019 4 / 15

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Outline

Learning goals Motivation for formal deduction Rules of formal deduction Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 5 / 15

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Why study formal deduction?

▶ Want to prove that a conclusion can be deduced from a set of

premises.

▶ Want to generate a proof that can be checked mechanically.

CS 245 Logic and Computation Fall 2019 6 / 15

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Formal Deducibility

Let the relation of formal deducibility be denoted by Σ ⊢ 𝐵, which means that 𝐵 is formally deducible (or provable) from Σ. Comments:

▶ Σ is a set of formulas, which are the premises. ▶ 𝐵 is a formula, which is the conclusion. ▶ Formal deducibility is concerned with the syntactic structure

f formulas.

CS 245 Logic and Computation Fall 2019 7 / 15

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Outline

Learning goals Motivation for formal deduction Rules of formal deduction Revisiting the Learning Goals

CS 245 Logic and Computation Fall 2019 8 / 15

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Rules of Formal Deduction

▶ Refmexivity (Ref):

𝐵 ⊢ 𝐵.

▶ Addition of premises (+):

if Σ ⊢ 𝐵, then Σ, Σ′ ⊢ 𝐵.

▶ (∈):

if 𝐵 ∈ Σ, then Σ ⊢ 𝐵.

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Conjunction Rules

And introduction (∧+) if Σ ⊢ 𝐵, Σ ⊢ 𝐶, then Σ ⊢ 𝐵 ∧ 𝐶. And elimination (∧−) if Σ ⊢ 𝐵 ∧ 𝐶, then Σ ⊢ 𝐵. if Σ ⊢ 𝐵 ∧ 𝐶, then Σ ⊢ 𝐶.

CS 245 Logic and Computation Fall 2019 10 / 15

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Disjunction Rules

Or introduction (∨+) if Σ ⊢ 𝐵, then Σ ⊢ 𝐵 ∨ 𝐶. if Σ ⊢ 𝐶, then Σ ⊢ 𝐵 ∨ 𝐶. Or elimination (∨−) if Σ, 𝐵 ⊢ 𝐷, Σ, 𝐶 ⊢ 𝐷, then Σ, 𝐵 ∨ 𝐶 ⊢ 𝐷.

CS 245 Logic and Computation Fall 2019 11 / 15

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Negation Rules

Negation introduction (¬+) if Σ, 𝐵 ⊢ 𝐶, Σ, 𝐵 ⊢ ¬𝐶, then Σ ⊢ ¬𝐵. Negation elimination (¬−) if Σ, ¬𝐵 ⊢ 𝐶, Σ, ¬𝐵 ⊢ ¬𝐶, then Σ ⊢ 𝐵.

CS 245 Logic and Computation Fall 2019 12 / 15

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Implication Rules

Implication introduction (→ +) if Σ, 𝐵 ⊢ 𝐶, then Σ ⊢ 𝐵 → 𝐶. Implication elimination (→ −) if Σ ⊢ 𝐵, Σ ⊢ 𝐵 → 𝐶, then Σ ⊢ 𝐶.

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Equivalence Rules

Equivalence introduction (↔ +) if Σ, 𝐵 ⊢ 𝐶, Σ, 𝐶 ⊢ 𝐵, then Σ ⊢ 𝐵 ↔ 𝐶. Equivalence elimination (↔ −) if Σ ⊢ 𝐵, Σ ⊢ 𝐵 ↔ 𝐶, then Σ ⊢ 𝐶. if Σ ⊢ 𝐶, Σ ⊢ 𝐵 ↔ 𝐶, then Σ ⊢ 𝐵.

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Revisting the Learning Goals

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

▶ Describe rules of inference for natural deduction. ▶ Prove that a conclusion follows from a set of premises using

rules of formal deduction.

CS 245 Logic and Computation Fall 2019 15 / 15