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

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 Computation Fall 2019 2 / 15

Outline Learning goals Motivation for formal deduction Rules of formal deduction Revisiting the Learning Goals CS 245 Logic and Computation Fall 2019 3 / 15

Learning goals By the end of this lecture, you should be able to rules of formal deduction. CS 245 Logic and Computation Fall 2019 4 / 15 ▶ Describe rules of inference for natural deduction. ▶ Prove that a conclusion follows from a set of premises using

Outline Learning goals Motivation for formal deduction Rules of formal deduction Revisiting the Learning Goals CS 245 Logic and Computation Fall 2019 5 / 15

Why study formal deduction? premises. CS 245 Logic and Computation Fall 2019 6 / 15 ▶ Want to prove that a conclusion can be deduced from a set of ▶ Want to generate a proof that can be checked mechanically.

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: of formulas. CS 245 Logic and Computation Fall 2019 7 / 15 ▶ 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

Outline Learning goals Motivation for formal deduction Rules of formal deduction Revisiting the Learning Goals CS 245 Logic and Computation Fall 2019 8 / 15

Rules of Formal Deduction 𝐵 ⊢ 𝐵. if Σ ⊢ 𝐵, if 𝐵 ∈ Σ, then Σ ⊢ 𝐵. CS 245 Logic and Computation Fall 2019 9 / 15 ▶ Refmexivity (Ref): ▶ Addition of premises (+): then Σ, Σ ′ ⊢ 𝐵. ▶ ( ∈ ):

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

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

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

Implication Rules Implication introduction ( → + ) if Σ, 𝐵 ⊢ 𝐶, then Σ ⊢ 𝐵 → 𝐶. Implication elimination ( → − ) if Σ ⊢ 𝐵, Σ ⊢ 𝐵 → 𝐶, then Σ ⊢ 𝐶. CS 245 Logic and Computation Fall 2019 13 / 15

Equivalence Rules Σ ⊢ 𝐵 ↔ 𝐶, Fall 2019 CS 245 Logic and Computation then Σ ⊢ 𝐵. Σ ⊢ 𝐵 ↔ 𝐶, if Σ ⊢ 𝐶, then Σ ⊢ 𝐶. if Σ ⊢ 𝐵, Equivalence introduction ( ↔ − ) Equivalence elimination then Σ ⊢ 𝐵 ↔ 𝐶. Σ, 𝐶 ⊢ 𝐵, if Σ, 𝐵 ⊢ 𝐶, ( ↔ + ) 14 / 15

Revisting the Learning Goals By the end of this lecture, you should be able to rules of formal deduction. CS 245 Logic and Computation Fall 2019 15 / 15 ▶ Describe rules of inference for natural deduction. ▶ Prove that a conclusion follows from a set of premises using