02Propositional Logic II CS 3234: Logic and Formal Systems Martin - PowerPoint PPT Presentation

02—Propositional Logic II CS 3234: Logic and Formal Systems Martin Henz August 20, 2009 Generated on Tuesday 12 th January, 2010, 14:38 CS 3234: Logic and Formal Systems 02—Propositional Logic II 1

CS 3234: Logic and Formal Systems 02—Propositional Logic II 2

CS 3234: Logic and Formal Systems 02—Propositional Logic II 3

Propositions Propositions are Declarative Sentences Sentences which one can—in principle—argue as being true or false. CS 3234: Logic and Formal Systems 02—Propositional Logic II 4

Propositions Propositions are Declarative Sentences Sentences which one can—in principle—argue as being true or false. Examples The sum of the numbers 3 and 5 equals 8. 1 Jane reacted violently to Jack’s accusations. 2 Every natural number > 2 is the sum of two prime 3 numbers. CS 3234: Logic and Formal Systems 02—Propositional Logic II 5

Sequents as Logical Arguments A Sequent in English If the train arrives late and there are no taxis at the station then John is late for his meeting. John is not late for his meeting. The train did arrive late. Therefore, there were taxis at the station. CS 3234: Logic and Formal Systems 02—Propositional Logic II 6

Focus on Structure, not Content We are primarily concerned about the structure of arguments in this class, not the validity of statements in a particular domain. CS 3234: Logic and Formal Systems 02—Propositional Logic II 7

Focus on Structure, not Content We are primarily concerned about the structure of arguments in this class, not the validity of statements in a particular domain. We therefore simply abbreviate sentences by letters such as p , q , r , p 1 , p 2 etc. CS 3234: Logic and Formal Systems 02—Propositional Logic II 8

Focus on Structure, not Content We are primarily concerned about the structure of arguments in this class, not the validity of statements in a particular domain. We therefore simply abbreviate sentences by letters such as p , q , r , p 1 , p 2 etc. Instead of English words such as “if...then”, “and”, “not”, it is more convenient to use symbols such as → , ∧ , ¬ . CS 3234: Logic and Formal Systems 02—Propositional Logic II 9

Sequents in Symbolic Notation A Sequent in English If the train arrives late and there are no taxis at the station then John is late for his meeting. John is not late for his meeting. The train did arrive late. Therefore, there were taxis at the station. CS 3234: Logic and Formal Systems 02—Propositional Logic II 10

Sequents in Symbolic Notation A Sequent in English If the train arrives late and there are no taxis at the station then John is late for his meeting. John is not late for his meeting. The train did arrive late. Therefore, there were taxis at the station. The same sequent using letters and symbols p ∧ ¬ q → r , ¬ r , p ⊢ q Remaining task Develop proof rules that allows us to derive such sequents CS 3234: Logic and Formal Systems 02—Propositional Logic II 11

Rules for Conjunction Introduction of Conjunction φ ψ [ ∧ i ] φ ∧ ψ CS 3234: Logic and Formal Systems 02—Propositional Logic II 12

Rules for Conjunction Introduction of Conjunction φ ψ [ ∧ i ] φ ∧ ψ Elimination of Conjunction φ ∧ ψ φ ∧ ψ [ ∧ e 1 ] [ ∧ e 2 ] φ ψ CS 3234: Logic and Formal Systems 02—Propositional Logic II 13

Rules of Double Negation Introduction of Double Negation φ [ ¬¬ i ] ¬¬ φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 14

Rules of Double Negation Introduction of Double Negation φ [ ¬¬ i ] ¬¬ φ Elimination of Double Negation ¬¬ φ [ ¬¬ e ] φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 15

Rules for Eliminating Implication φ φ → ψ [ → e ] ψ φ → ψ ¬ ψ [ MT ] ¬ φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 16

Rule for Introduction of Implication ✄ � φ . . . ψ ✂ ✁ [ → i ] φ → ψ CS 3234: Logic and Formal Systems 02—Propositional Logic II 17

Rules for Introduction of Disjunction φ ψ [ ∨ i i ] [ ∨ i 2 ] φ ∨ ψ φ ∨ ψ CS 3234: Logic and Formal Systems 02—Propositional Logic II 18

Rule for Elimination of Disjunction ✄ � ✄ � φ ψ . . . . φ ∨ ψ . . χ χ ✂ ✁ ✂ ✁ [ ∨ e ] χ CS 3234: Logic and Formal Systems 02—Propositional Logic II 19

Proof of p ∧ ( q ∨ r ) ⊢ ( p ∧ q ) ∨ ( p ∧ r ) 1 p ∧ ( q ∨ r ) premise 2 ∧ e 1 1 p 3 q ∨ r ∧ e 2 1 4 q assumption 5 p ∧ q ∧ i 2,4 6 ( p ∧ q ) ∨ ( p ∧ r ) ∨ i 1 5 7 assumption r 8 p ∧ r ∧ i 2,7 ( p ∧ q ) ∨ ( p ∧ r ) ∨ i 2 8 9 10 ( p ∧ q ) ∨ ( p ∧ r ) ∨ e 3, 4–6, 7–9 CS 3234: Logic and Formal Systems 02—Propositional Logic II 20

A Special Proposition Recall: We are only interested in the truth value of propositions, not the subject matter that they refer to (Martian pizzas or whatever). CS 3234: Logic and Formal Systems 02—Propositional Logic II 21

A Special Proposition Recall: We are only interested in the truth value of propositions, not the subject matter that they refer to (Martian pizzas or whatever). Therefore, all propositions that we all agree must be true are the same! CS 3234: Logic and Formal Systems 02—Propositional Logic II 22

A Special Proposition Recall: We are only interested in the truth value of propositions, not the subject matter that they refer to (Martian pizzas or whatever). Therefore, all propositions that we all agree must be true are the same! Example: p → p CS 3234: Logic and Formal Systems 02—Propositional Logic II 23

A Special Proposition Recall: We are only interested in the truth value of propositions, not the subject matter that they refer to (Martian pizzas or whatever). Therefore, all propositions that we all agree must be true are the same! Example: p → p We denote the proposition that is always true using the symbol ⊤ . CS 3234: Logic and Formal Systems 02—Propositional Logic II 24

Another Special Proposition We denote the proposition that is always true using the symbol ⊤ . CS 3234: Logic and Formal Systems 02—Propositional Logic II 25

Another Special Proposition We denote the proposition that is always true using the symbol ⊤ . Similarly, we denote the proposition that is always false using the symbol ⊥ . CS 3234: Logic and Formal Systems 02—Propositional Logic II 26

Another Special Proposition We denote the proposition that is always true using the symbol ⊤ . Similarly, we denote the proposition that is always false using the symbol ⊥ . Example: p ∧ ¬ p CS 3234: Logic and Formal Systems 02—Propositional Logic II 27

Elimination of Negation ¬ φ φ [ ¬ e ] ⊥ CS 3234: Logic and Formal Systems 02—Propositional Logic II 28

Introduction of Negation ✄ � φ . . . ⊥ ✂ ✁ [ ¬ i ] ¬ φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 29

Elimination of ⊥ ⊥ [ ⊥ e ] φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 30

Basic Rules (conjunction and disjunction) φ ψ φ ∧ ψ φ ∧ ψ [ ∧ i ] [ ∧ e 1 ] [ ∧ e 2 ] φ ∧ ψ φ ψ ✄ � ✄ � φ ψ . . . . φ ∨ ψ . . χ χ φ ψ ✂ ✁ ✂ ✁ [ ∨ i i ] [ ∨ i 2 ] [ ∨ e ] χ φ ∨ ψ φ ∨ ψ CS 3234: Logic and Formal Systems 02—Propositional Logic II 31

Basic Rules (implication) ✄ � φ . . . ψ φ → ψ φ ✂ ✁ [ → i ] [ → e ] φ → ψ ψ CS 3234: Logic and Formal Systems 02—Propositional Logic II 32

Basic Rules (negation) φ . . . ⊥ ¬ φ φ [ ¬ i ] [ ¬ e ] ¬ φ ⊥ CS 3234: Logic and Formal Systems 02—Propositional Logic II 33

Basic Rules ( ⊥ and double negation) ⊥ [ ⊥ e ] φ ¬¬ φ [ ¬¬ e ] φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 34

Some Derived Rules: Introduction of Double Negation φ [ ¬¬ i ] ¬¬ φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 35

Example: Deriving [ ¬¬ i] from [ ¬ i] and [ ¬ e] 1 φ premise ¬ φ 2 assumption 3 ⊥ ¬ e 1,2 4 ¬¬ φ ¬ i 2–3 CS 3234: Logic and Formal Systems 02—Propositional Logic II 36

Some Derived Rules: Modus Tollens φ → ψ ¬ ψ [ MT ] ¬ φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 37

Some Derived Rules: Proof By Contradiction ✄ � ¬ φ . . . ⊥ ✂ ✁ [ PBC ] φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 38

Some Derived Rules: Law of Excluded Middle [ LEM ] φ ∨ ¬ φ CS 3234: Logic and Formal Systems 02—Propositional Logic II 39

Motivation Consider the following theorem. Theorem There exist irrational numbers a and b such that a b is rational. Let us call this theorem χ . CS 3234: Logic and Formal Systems 02—Propositional Logic II 40

Proof Outline for χ Let p be the following proposition: Proposition p √ √ 2 is rational. 2 Then the proof of χ goes like this: ✄ � ✄ � ¬ p p . . . . [ LEM ] . . p ∨ ¬ p χ χ ✂ ✁ ✂ ✁ [ ∨ e ] χ CS 3234: Logic and Formal Systems 02—Propositional Logic II 41