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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

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Propositional Logic: Tautological Consequence and Translations

Alice Gao

Lecture 6

CS 245 Logic and Computation Fall 2019 Alice Gao 1 / 25

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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

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Learning goals

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

▶ Determine if a set of formulas is satisfjable. ▶ Defjne tautological consequence. Explain subtleties of

tautological consequence.

▶ Prove that a tautological consequence holds/does not hold by

using the defjnition of tautological consequence, and/or truth tables.

▶ Translate an English sentence with no logical ambiguity into a

propositional formula.

▶ Translate an English sentence with logical ambiguity into

multiple propositional formulas and prove that the propositional formulas are not tautologically equivalent.

CS 245 Logic and Computation Fall 2019 Alice Gao 3 / 25

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Logical Deduction and Tautological Consequence

▶ Logic is the science of reasoning. ▶ The process of logical deduction is formalized by the notion of

tautological consequence.

▶ Can we deduce a conclusion based on a set of premises?

CS 245 Logic and Computation Fall 2019 Alice Gao 4 / 25

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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

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Satisfying a Set of Formulas

Let Σ denote any set of formulas. Σ𝑢 = {1, if for each 𝐶 ∈ Σ, 𝐶𝑢 = 1, 0, otherwise What does Σ𝑢 = 0 mean?

Defjnition (Satisfjability)

Σ is satisfjable if and only if there is some truth valuation 𝑢 such that Σ𝑢 = 1. When Σ𝑢 = 1, 𝑢 is said to satisfy Σ.

CS 245 Logic and Computation Fall 2019 Alice Gao 6 / 25

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CQ Is Sigma Satisfjable?

CS 245 Logic and Computation Fall 2019 Alice Gao 7 / 25

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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

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Tautological Consequence

Defjnition (Tautological Consequence)

Suppose Σ ⊆ 𝐺𝑝𝑠𝑛(𝑀𝑞) and 𝐵 ∈ 𝐺𝑝𝑠𝑛(𝑀𝑞). 𝐵 is a tautological consequence of Σ (that is, of the formulas in Σ), written as Σ ⊨ 𝐵, if and only if for any truth valuation 𝑢, Σ𝑢 = 1 implies 𝐵𝑢 = 1.

CS 245 Logic and Computation Fall 2019 Alice Gao 9 / 25

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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

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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

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CQ: Prove a tautological consequence

CQ: Consider the tautological consequence Σ ⊨ 𝐵. To prove that the tautological consequence holds, we need to consider (A) Every truth valuation 𝑢 under which Σ𝑢 = 1. (B) Every truth valuation 𝑢 under which Σ𝑢 = 0. (C) One truth valuation 𝑢 under which Σ𝑢 = 1. (D) One truth valuation 𝑢 under which Σ𝑢 = 0.

CS 245 Logic and Computation Fall 2019 Alice Gao 12 / 25

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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 (A) Every truth valuation 𝑢 under which Σ𝑢 = 1 and 𝐵𝑢 = 1. (B) Every truth valuation 𝑢 under which Σ𝑢 = 1 and 𝐵𝑢 = 0. (C) One truth valuation 𝑢 under which Σ𝑢 = 1 and 𝐵𝑢 = 1. (D) One truth valuation 𝑢 under which Σ𝑢 = 1 and 𝐵𝑢 = 0.

CS 245 Logic and Computation Fall 2019 Alice Gao 13 / 25

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CQ: Proving/disproving a tautological consequence using a truth table

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)) (𝑞 → 𝑟) (¬𝑞) (𝑞 ↔ 𝑟) 1 1 1 1 1 1 1 1 1 1 1 1 1 1

CS 245 Logic and Computation Fall 2019 Alice Gao 14 / 25

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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

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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

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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. There exists a truth valuation 𝑢 such that 𝐶𝑢 = 1 and 𝐵𝑢 = 0. 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

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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

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Subtleties of a Tautological Consequence

Consider the tautological consequence Σ ⊨ 𝐵. 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

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CQ Subtleties of a Tautological Consequence

CS 245 Logic and Computation Fall 2019 Alice Gao 20 / 25

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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

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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

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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

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Translations: A reference page

▶ ¬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

𝑟; 𝑞 although 𝑟

▶ 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

for 𝑟; 𝑟 is necessary for 𝑞

▶ 𝑞 ↔ 𝑟: 𝑞 is equivalent to 𝑟; 𝑞 exactly if 𝑟; 𝑞 is necessary and

suffjcient for 𝑟

CS 245 Logic and Computation Fall 2019 Alice Gao 24 / 25

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Revisiting the learning goals

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

▶ Determine if a set of formulas is satisfjable. ▶ Defjne tautological consequence. Explain subtleties of

tautological consequence.

▶ Prove that a tautological consequence holds/does not hold by

using the defjnition of tautological consequence, and/or truth tables.

▶ Translate an English sentence with no logical ambiguity into a

propositional formula.

▶ Translate an English sentence with logical ambiguity into

multiple propositional formulas and prove that the propositional formulas are not tautologically equivalent.

CS 245 Logic and Computation Fall 2019 Alice Gao 25 / 25