Logic as a Tool Chapter 1: Understanding Propositional Logic 1.2 - - PowerPoint PPT Presentation

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Logic as a Tool Chapter 1: Understanding Propositional Logic 1.2 Propositional logical consequence Logically correct inferences

Valentin Goranko Stockholm University September 2016

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Propositional logical consequence

A propositional formula C is a logical consequence from the propositional formulae A1, . . . , An, denoted A1, . . . , An | = C, if C is true whenever all A1, . . . , An are true, i.e., every assignment of truth-values to the variables occurring in A1, . . . , An, C which renders the formulae A1, . . . , An true, renders the formula C true, too. If A1, . . . , An | = C, we also say that C follows logically from A1, . . . , An, and that A1, . . . , An logically imply C. Logical consequence is reducible to validity: A1, . . . , An | = C iff A1 ∧ . . . ∧ An | = C iff | = (A1 ∧ . . . ∧ An) → C.

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Propositional logical consequence is reducible to validity

Proposition

For any propositional formulae A1, . . . , An, B, the following are equivalent:

• 1. A1, . . . , An |

= B

• 2. A1 ∧ . . . ∧ An |

= B

• 3. |

= (A1 ∧ . . . ∧ An) → B

• 4. |

= A1 → (. . . → (An → B) . . .)

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Testing propositional consequence with truth tables Example 1

p, p → q

?

| = q p q p p → q q T T T T T T F T F F F T F T T F F F T F Yes.

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Testing propositional consequence with truth tables Example 2

p → q

?

| = q → p p q p → q q → p T T T T T F F T F T T F F F ... ... No.

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Testing propositional consequence with truth tables Example 3

p → r, q → r

?

| = (p ∨ q) → r p q r p → r q → r p ∨ q (p ∨ q) → r T T T T T T T T T F F F T F T F T T T T T T F F F T T F F T T T T T T F T F T F T F F F T T T F T F F F T T F T Yes.

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Sound rules of propositional inference

A rule of propositional inference (for short, inference rule) is a scheme: P1, . . . , Pn C , where P1, . . . , Pn, C are propositional formulae. The formulae P1, . . . , Pn are called premises of the inference rule, and C is its conclusion. An inference rule is (logically) sound if its conclusion logically follows from the premises. A propositional inference is an instance of a rule, where propositions are uniformly replaced by the propositional variables. A propositional inference is (logically) correct if it is an instance of a sound inference rule.

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Propositional inference: example 1

Consider the propositional inference: Alexis is singing. If Alexis is singing, then Alexis is happy. Alexis is happy. It is obtained from the following rule, called Modus Ponens: p, p → q q This rule is sound, therefore, the inference is logically correct.

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Propositional inference: example 2

Now consider the propositional inference: 2 plus 2 equals 4. If 5 is greater than 3, then 2 plus 2 equals 4. 5 is greater than 3. It is based on the rule p, q → p q which is not sound. Therefore, the inference is not logically correct.