Propositional Logic: Tautological Consequence and Translations - PowerPoint PPT Presentation

Propositional Logic: Tautological Consequence and Translations Alice Gao Lecture 6 CS 245 Logic and Computation Fall 2019 Alice Gao 1 / 25

Outline Learning goals Satisfaction of a Set of Formulas Tautological Consequence Proving/Disproving a Tautological Consequence Subtleties of a Tautological Consequence Translations between English and Propositional Logic Revisiting the learning goals CS 245 Logic and Computation Fall 2019 Alice Gao 2 / 25

Learning goals propositional formula. Alice Gao Fall 2019 CS 245 Logic and Computation propositional formulas are not tautologically equivalent. multiple propositional formulas and prove that the 3 / 25 By the end of this lecture, you should be able to tables. using the defjnition of tautological consequence, and/or truth tautological consequence. ▶ Determine if a set of formulas is satisfjable. ▶ Defjne tautological consequence. Explain subtleties of ▶ Prove that a tautological consequence holds/does not hold by ▶ Translate an English sentence with no logical ambiguity into a ▶ Translate an English sentence with logical ambiguity into

Logical Deduction and Tautological Consequence tautological consequence. CS 245 Logic and Computation Fall 2019 Alice Gao 4 / 25 ▶ Logic is the science of reasoning. ▶ The process of logical deduction is formalized by the notion of ▶ Can we deduce a conclusion based on a set of premises?

Outline Learning goals Satisfaction of a Set of Formulas Tautological Consequence Proving/Disproving a Tautological Consequence Subtleties of a Tautological Consequence Translations between English and Propositional Logic Revisiting the learning goals CS 245 Logic and Computation Fall 2019 Alice Gao 5 / 25

Satisfying a Set of Formulas Let Σ denote any set of formulas. 0, otherwise Defjnition (Satisfjability) Σ is satisfjable if and only if there is some truth valuation 𝑢 such CS 245 Logic and Computation Fall 2019 Alice Gao 6 / 25 Σ 𝑢 = {1, if for each 𝐶 ∈ Σ, 𝐶 𝑢 = 1 , What does Σ 𝑢 = 0 mean? that Σ 𝑢 = 1 . When Σ 𝑢 = 1 , 𝑢 is said to satisfy Σ .

CQ Is Sigma Satisfjable? CS 245 Logic and Computation Fall 2019 Alice Gao 7 / 25

Outline Learning goals Satisfaction of a Set of Formulas Tautological Consequence Proving/Disproving a Tautological Consequence Subtleties of a Tautological Consequence Translations between English and Propositional Logic Revisiting the learning goals CS 245 Logic and Computation Fall 2019 Alice Gao 8 / 25

Tautological Consequence Defjnition (Tautological Consequence) Suppose Σ ⊆ 𝐺𝑝𝑠𝑛(𝑀 𝑞 ) and 𝐵 ∈ 𝐺𝑝𝑠𝑛(𝑀 𝑞 ) . 𝐵 is a tautological consequence of Σ (that is, of the formulas in Σ ), written as Σ ⊨ 𝐵 , if and only if CS 245 Logic and Computation Fall 2019 Alice Gao 9 / 25 for any truth valuation 𝑢 , Σ 𝑢 = 1 implies 𝐵 𝑢 = 1 .

Tautological Equivalence 𝐵 and 𝐶 are (tautologically) equivalent if and only if 𝐵 ⊨ ⊨ 𝐶 holds. 𝐵 ⊨ ⊨ 𝐶 denotes 𝐵 ⊨ 𝐶 and 𝐶 ⊨ 𝐵 . CS 245 Logic and Computation Fall 2019 Alice Gao 10 / 25

Outline Learning goals Satisfaction of a Set of Formulas Tautological Consequence Proving/Disproving a Tautological Consequence Subtleties of a Tautological Consequence Translations between English and Propositional Logic Revisiting the learning goals CS 245 Logic and Computation Fall 2019 Alice Gao 11 / 25

CQ: Prove a tautological consequence CQ: Consider the tautological consequence Σ ⊨ 𝐵 . To prove that the tautological consequence holds, we need to consider CS 245 Logic and Computation Fall 2019 Alice Gao 12 / 25 (A) Every truth valuation 𝑢 under which Σ 𝑢 = 1 . (B) Every truth valuation 𝑢 under which Σ 𝑢 = 0 . (C) One truth valuation 𝑢 under which Σ 𝑢 = 1 . (D) One truth valuation 𝑢 under which Σ 𝑢 = 0 .

CQ: Disprove a tautological consequence CQ: Consider the tautological consequence Σ ⊨ 𝐵 . To prove that the tautological consequence does NOT hold, we need to consider CS 245 Logic and Computation Fall 2019 Alice Gao 13 / 25 (A) Every truth valuation 𝑢 under which Σ 𝑢 = 1 and 𝐵 𝑢 = 1 . (B) Every truth valuation 𝑢 under which Σ 𝑢 = 1 and 𝐵 𝑢 = 0 . (C) One truth valuation 𝑢 under which Σ 𝑢 = 1 and 𝐵 𝑢 = 1 . (D) One truth valuation 𝑢 under which Σ 𝑢 = 1 and 𝐵 𝑢 = 0 .

CQ: Proving/disproving a tautological consequence 0 1 0 1 0 1 0 0 1 1 1 0 1 0 1 CS 245 Logic and Computation Fall 2019 Alice Gao 1 1 using a truth table (¬(𝑞 ∧ 𝑟)) CQ: Let Σ = {¬(𝑞 ∧ 𝑟), 𝑞 → 𝑟} , 𝑦 = ¬𝑞 , and 𝑧 = 𝑞 ↔ 𝑟 . Based on the truth table, which of the following statements is true? A) Σ ⊨ 𝑦 and Σ ⊨ 𝑧 . B) Σ ⊨ 𝑦 and Σ ⊭ 𝑧 . C) Σ ⊭ 𝑦 and Σ ⊨ 𝑧 . D) Σ ⊭ 𝑦 and Σ ⊭ 𝑧 . 𝑞 𝑟 (𝑞 → 𝑟) 0 (¬𝑞) (𝑞 ↔ 𝑟) 0 0 1 1 1 1 14 / 25

Prove a tautological consequence using the defjnition Exercise. Show that {(¬(𝑞 ∧ 𝑟)), (𝑞 → 𝑟)} ⊨ (¬𝑞) . CS 245 Logic and Computation Fall 2019 Alice Gao 15 / 25

Disprove a tautological consequence using the defjnition Exercise. Show that {(¬(𝑞 ∧ 𝑟)), (𝑞 → 𝑟)} ⊭ (𝑞 ↔ 𝑟) . CS 245 Logic and Computation Fall 2019 Alice Gao 16 / 25

Disproving propositional logical consequence A student is trying to prove that {(𝐵 → 𝐶)} ⊭ (𝐶 → 𝐵) where 𝐵 and 𝐶 are well-formed predicate formulas. The student starts the proof by writing down the following sentence. Is the above sentence true (a valid claim)? (A) Yes, it is true. (B) No, it is false. (C) There is not enough information to tell. CS 245 Logic and Computation Fall 2019 Alice Gao 17 / 25 There exists a truth valuation 𝑢 such that 𝐶 𝑢 = 1 and 𝐵 𝑢 = 0 .

Outline Learning goals Satisfaction of a Set of Formulas Tautological Consequence Proving/Disproving a Tautological Consequence Subtleties of a Tautological Consequence Translations between English and Propositional Logic Revisiting the learning goals CS 245 Logic and Computation Fall 2019 Alice Gao 18 / 25

Subtleties of a Tautological Consequence Does the tautological consequence hold under each of the following conditions? 1. Σ is the empty set. 2. Σ is not satisfjable. 3. 𝐵 is a tautology. 4. 𝐵 is a contradiction. CS 245 Logic and Computation Fall 2019 Alice Gao 19 / 25 Consider the tautological consequence Σ ⊨ 𝐵 .

CQ Subtleties of a Tautological Consequence CS 245 Logic and Computation Fall 2019 Alice Gao 20 / 25

Outline Learning goals Satisfaction of a Set of Formulas Tautological Consequence Proving/Disproving a Tautological Consequence Subtleties of a Tautological Consequence Translations between English and Propositional Logic Revisiting the learning goals CS 245 Logic and Computation Fall 2019 Alice Gao 21 / 25

English sentences with no logical ambiguity Translate the following sentences to propositional formulas. 1. Nadhi eats a fruit if the fruit is an apple. Nadhi eats a fruit only if the fruit is an apple. 2. Soo-Jin will eat an apple or an orange but not both. 3. If it is sunny tomorrow, then I will play golf, provided that I am relaxed. CS 245 Logic and Computation Fall 2019 Alice Gao 22 / 25

English sentences with logical ambiguity Give multiple translations of the following sentences into propositional logic. 1. Sidney will carry an umbrella unless it is sunny. 2. Pigs can fmy and the grass is red or the sky is blue. CS 245 Logic and Computation Fall 2019 Alice Gao 23 / 25

Translations: A reference page 𝑟 ; 𝑞 although 𝑟 for 𝑟 ; 𝑟 is necessary for 𝑞 suffjcient for 𝑟 CS 245 Logic and Computation Fall 2019 Alice Gao 24 / 25 ▶ ¬𝑞 : 𝑞 does not hold; 𝑞 is false; it is not the case that 𝑞 ▶ 𝑞 ∧ 𝑟 : 𝑞 but 𝑟 ; not only 𝑞 but 𝑟 ; 𝑞 while 𝑟 ; 𝑞 despite 𝑟 ; 𝑞 yet ▶ 𝑞 ∨ 𝑟 : 𝑞 or 𝑟 or both; 𝑞 and/or 𝑟 ; ▶ 𝑞 → 𝑟 : 𝑞 implies 𝑟 ; 𝑟 if 𝑞 ; 𝑞 only if 𝑟 ; 𝑟 when 𝑞 ; 𝑞 is suffjcient ▶ 𝑞 ↔ 𝑟 : 𝑞 is equivalent to 𝑟 ; 𝑞 exactly if 𝑟 ; 𝑞 is necessary and

Revisiting the learning goals propositional formula. Alice Gao Fall 2019 CS 245 Logic and Computation propositional formulas are not tautologically equivalent. multiple propositional formulas and prove that the 25 / 25 By the end of this lecture, you should be able to tables. using the defjnition of tautological consequence, and/or truth tautological consequence. ▶ Determine if a set of formulas is satisfjable. ▶ Defjne tautological consequence. Explain subtleties of ▶ Prove that a tautological consequence holds/does not hold by ▶ Translate an English sentence with no logical ambiguity into a ▶ Translate an English sentence with logical ambiguity into