Logic in Action Chapter 9: Proofs http://www.logicinaction.org/ ( - PowerPoint PPT Presentation

Natural Deduction for Propositional Logic Proving the axioms (2) The axiom ( ϕ → ( ψ → χ )) → (( ϕ → ψ ) → ( ϕ → χ )) can be proved from modus ponens and deduction : ϕ → ( ψ → χ ) 1 2 ϕ → ψ 3 ϕ 4 ψ modus ponens 3,2 5 ψ → χ modus ponens 3,1 6 χ modus ponens 4,5 7 ϕ → χ deduction 3-6 ( ϕ → ψ ) → ( ϕ → χ ) 8 deduction 2-7 ( http://www.logicinaction.org/ ) 8 / 24

Natural Deduction for Propositional Logic Proving the axioms (2) The axiom ( ϕ → ( ψ → χ )) → (( ϕ → ψ ) → ( ϕ → χ )) can be proved from modus ponens and deduction : ϕ → ( ψ → χ ) 1 2 ϕ → ψ 3 ϕ 4 ψ modus ponens 3,2 5 ψ → χ modus ponens 3,1 6 χ modus ponens 4,5 7 ϕ → χ deduction 3-6 ( ϕ → ψ ) → ( ϕ → χ ) 8 deduction 2-7 9 ( ϕ → ( ψ → χ )) → (( ϕ → ψ ) → ( ϕ → χ )) deduction 1-8 ( http://www.logicinaction.org/ ) 8 / 24

Natural Deduction for Propositional Logic We need more The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) cannot be proved from modus ponens and deduction . ( http://www.logicinaction.org/ ) 9 / 24

Natural Deduction for Propositional Logic We need more The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) cannot be proved from modus ponens and deduction . We need a way to deal with negations. ( http://www.logicinaction.org/ ) 9 / 24

Natural Deduction for Propositional Logic The refutation rule Suppose you want to prove ϕ . ( http://www.logicinaction.org/ ) 10 / 24

Natural Deduction for Propositional Logic The refutation rule Suppose you want to prove ϕ . Assume ¬ ϕ . ¬ ϕ ( http://www.logicinaction.org/ ) 10 / 24

Natural Deduction for Propositional Logic The refutation rule Suppose you want to prove ϕ . Assume ¬ ϕ . If after further steps ¬ ϕ . . . ( http://www.logicinaction.org/ ) 10 / 24

Natural Deduction for Propositional Logic The refutation rule Suppose you want to prove ϕ . Assume ¬ ϕ . If after further steps you can prove a contradiction ⊥ , ¬ ϕ . . . ⊥ ( http://www.logicinaction.org/ ) 10 / 24

Natural Deduction for Propositional Logic The refutation rule Suppose you want to prove ϕ . Assume ¬ ϕ . If after further steps you can prove a contradiction ⊥ , then ¬ ϕ cannot be true ¬ ϕ . . . ⊥ ( http://www.logicinaction.org/ ) 10 / 24

Natural Deduction for Propositional Logic The refutation rule Suppose you want to prove ϕ . Assume ¬ ϕ . If after further steps you can prove a contradiction ⊥ , then ¬ ϕ cannot be true so you actually have ϕ . ¬ ϕ . . . ⊥ refutation ϕ ( http://www.logicinaction.org/ ) 10 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ¬ ϕ → ¬ ψ 1 ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ¬ ϕ → ¬ ψ 1 2 ψ ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ¬ ϕ → ¬ ψ 1 2 ψ 3 ¬ ϕ ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ¬ ϕ → ¬ ψ 1 2 ψ 3 ¬ ϕ ¬ ψ 4 modus ponens 3,1 ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ¬ ϕ → ¬ ψ 1 2 ψ 3 ¬ ϕ ¬ ψ 4 modus ponens 3,1 5 ⊥ modus ponens 2,4 ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ¬ ϕ → ¬ ψ 1 2 ψ 3 ¬ ϕ ¬ ψ 4 modus ponens 3,1 5 ⊥ modus ponens 2,4 6 ϕ refutation 3-5 ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ¬ ϕ → ¬ ψ 1 2 ψ 3 ¬ ϕ ¬ ψ 4 modus ponens 3,1 5 ⊥ modus ponens 2,4 6 ϕ refutation 3-5 7 ψ → ϕ deduction 2-6 ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ¬ ϕ → ¬ ψ 1 2 ψ 3 ¬ ϕ ¬ ψ 4 modus ponens 3,1 5 ⊥ modus ponens 2,4 6 ϕ refutation 3-5 7 ψ → ϕ deduction 2-6 ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) 8 deduction 1-7 ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic Proving the axioms (3) The axiom ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) can be proved from modus ponens , deduction and refutation : ¬ ϕ → ¬ ψ 1 2 ψ 3 ¬ ϕ ¬ ψ 4 modus ponens 3,1 5 ⊥ modus ponens 2,4 6 ϕ refutation 3-5 7 ψ → ϕ deduction 2-6 ( ¬ ϕ → ¬ ψ ) → ( ψ → ϕ ) 8 deduction 1-7 For step 5, note that ¬ ψ can be seen as an abbreviation of ψ → ⊥ . ( http://www.logicinaction.org/ ) 11 / 24

Natural Deduction for Propositional Logic So . . . ϕ , ϕ → ψ modus ponens ψ ϕ ¬ ϕ . . . . . . ⊥ ψ deduction refutation ϕ → ψ ϕ ( http://www.logicinaction.org/ ) 12 / 24

Natural Deduction for Propositional Logic So . . . ϕ , ϕ → ψ modus ponens ψ ϕ ¬ ϕ . . . . . . ⊥ ψ deduction refutation ϕ → ψ ϕ The modus ponens , deduction and refutation rules are a complete system for propositional logic. ( http://www.logicinaction.org/ ) 12 / 24

Natural Deduction for Propositional Logic To facilitate things . . . ( http://www.logicinaction.org/ ) 13 / 24

Natural Deduction for Propositional Logic To facilitate things . . . Natural deduction introduces rules to manipulate all the connectives in an easy way. ( http://www.logicinaction.org/ ) 13 / 24

Natural Deduction for Propositional Logic For implication → ( http://www.logicinaction.org/ ) 14 / 24

Natural Deduction for Propositional Logic For implication → ϕ , ϕ → ψ modus ponens ψ ( http://www.logicinaction.org/ ) 14 / 24

Natural Deduction for Propositional Logic For implication → ϕ , ϕ → ψ modus ponens ψ E → ( http://www.logicinaction.org/ ) 14 / 24

Natural Deduction for Propositional Logic For implication → ϕ . . ϕ , ϕ → ψ . modus ponens ψ ψ deduction ϕ → ψ E → ( http://www.logicinaction.org/ ) 14 / 24

Natural Deduction for Propositional Logic For implication → ϕ . . ϕ , ϕ → ψ . modus ponens ψ ψ deduction ϕ → ψ E → I → ( http://www.logicinaction.org/ ) 14 / 24

Natural Deduction for Propositional Logic For negation ¬ ( http://www.logicinaction.org/ ) 15 / 24

Natural Deduction for Propositional Logic For negation ¬ ¬ ϕ , ϕ ⊥ ( http://www.logicinaction.org/ ) 15 / 24

Natural Deduction for Propositional Logic For negation ¬ ¬ ϕ , ϕ ⊥ E ¬ ( http://www.logicinaction.org/ ) 15 / 24

Natural Deduction for Propositional Logic For negation ¬ ¬ ϕ . . ¬ ϕ , ϕ . ⊥ ⊥ refutation ϕ E ¬ ( http://www.logicinaction.org/ ) 15 / 24

Natural Deduction for Propositional Logic For negation ¬ ¬ ϕ . . ¬ ϕ , ϕ . ⊥ ⊥ refutation ϕ E ¬ I ¬ ( http://www.logicinaction.org/ ) 15 / 24

Natural Deduction for Propositional Logic For conjunction ∧ ( http://www.logicinaction.org/ ) 16 / 24

Natural Deduction for Propositional Logic For conjunction ∧ ϕ ∧ ψ ϕ ϕ ∧ ψ ψ ( http://www.logicinaction.org/ ) 16 / 24

Natural Deduction for Propositional Logic For conjunction ∧ ϕ ∧ ψ ϕ ϕ ∧ ψ ψ E ∧ ( http://www.logicinaction.org/ ) 16 / 24

Natural Deduction for Propositional Logic For conjunction ∧ ϕ ∧ ψ ϕ ϕ , ψ ϕ ∧ ψ ϕ ∧ ψ ψ E ∧ ( http://www.logicinaction.org/ ) 16 / 24

Natural Deduction for Propositional Logic For conjunction ∧ ϕ ∧ ψ ϕ ϕ , ψ ϕ ∧ ψ ϕ ∧ ψ ψ E ∧ I ∧ ( http://www.logicinaction.org/ ) 16 / 24

Natural Deduction for Propositional Logic For disjunction ∨ ( http://www.logicinaction.org/ ) 17 / 24

Natural Deduction for Propositional Logic For disjunction ∨ ϕ ψ . . ϕ ∨ ψ , . , . . . χ χ χ ( http://www.logicinaction.org/ ) 17 / 24

Natural Deduction for Propositional Logic For disjunction ∨ ϕ ψ . . ϕ ∨ ψ , . , . . . χ χ χ E ∨ ( http://www.logicinaction.org/ ) 17 / 24

Natural Deduction for Propositional Logic For disjunction ∨ ϕ ϕ ψ . . ϕ ∨ ψ ϕ ∨ ψ , . , . . . χ χ ψ χ ϕ ∨ ψ E ∨ ( http://www.logicinaction.org/ ) 17 / 24

Natural Deduction for Propositional Logic For disjunction ∨ ϕ ϕ ψ . . ϕ ∨ ψ ϕ ∨ ψ , . , . . . χ χ ψ χ ϕ ∨ ψ E ∨ I ∨ ( http://www.logicinaction.org/ ) 17 / 24

Natural Deduction for Predicate Logic For predicate logic In order to present introduction and elimination rules for both ∀ and ∃ , we need to recall two notions. ( http://www.logicinaction.org/ ) 18 / 24

Natural Deduction for Predicate Logic For predicate logic In order to present introduction and elimination rules for both ∀ and ∃ , we need to recall two notions. Bounded variable . ( http://www.logicinaction.org/ ) 18 / 24

Natural Deduction for Predicate Logic For predicate logic In order to present introduction and elimination rules for both ∀ and ∃ , we need to recall two notions. Bounded variable . Substitution of a variable for a term in a formula . ( http://www.logicinaction.org/ ) 18 / 24

Natural Deduction for Predicate Logic Bounded variable ( http://www.logicinaction.org/ ) 19 / 24

Natural Deduction for Predicate Logic Bounded variable Scope of a quantifier. In a formula of the form ∀ xϕ ( ∃ xϕ ), the subformula ϕ is said to be the scope of the quantifier ∀ ( ∃ ). ( http://www.logicinaction.org/ ) 19 / 24

Natural Deduction for Predicate Logic Bounded variable Scope of a quantifier. In a formula of the form ∀ xϕ ( ∃ xϕ ), the subformula ϕ is said to be the scope of the quantifier ∀ ( ∃ ). Binding a variable. In a formula of the form ∀ xϕ ( ∃ xϕ ), the quantifier ∀ ( ∃ ) binds any occurrence of x in ϕ that is not bounded by another quantifier inside ϕ . ( http://www.logicinaction.org/ ) 19 / 24

Natural Deduction for Predicate Logic Bounded variable Scope of a quantifier. In a formula of the form ∀ xϕ ( ∃ xϕ ), the subformula ϕ is said to be the scope of the quantifier ∀ ( ∃ ). Binding a variable. In a formula of the form ∀ xϕ ( ∃ xϕ ), the quantifier ∀ ( ∃ ) binds any occurrence of x in ϕ that is not bounded by another quantifier inside ϕ . Bounded variable . An occurrence of a variable x is bounded in a formula ϕ if there is a quantifier in ϕ that binds it. ( http://www.logicinaction.org/ ) 19 / 24

Natural Deduction for Predicate Logic Substitution (1) ( http://www.logicinaction.org/ ) 20 / 24

Natural Deduction for Predicate Logic Substitution (1) Substitution inside a term. Replacing the occurrences of the variable y for the term t inside the term s produces the term denoted by ( s ) y t ( http://www.logicinaction.org/ ) 20 / 24

Natural Deduction for Predicate Logic Substitution (1) Substitution inside a term. Replacing the occurrences of the variable y for the term t inside the term s produces the term denoted by ( s ) y t Formally, ( c ) y For a constant : t := c 8 ( x ) y t := x for x different from y < For a variable : ( y ) y t := t : ( http://www.logicinaction.org/ ) 20 / 24

Natural Deduction for Predicate Logic Substitution (1) Substitution inside a term. Replacing the occurrences of the variable y for the term t inside the term s produces the term denoted by ( s ) y t Formally, ( c ) y For a constant : t := c 8 ( x ) y t := x for x different from y < For a variable : ( y ) y t := t : Examples: ( a ) x c := a ( x ) y a := x ( z ) z y := y ( http://www.logicinaction.org/ ) 20 / 24

Natural Deduction for Predicate Logic Substitution (2) ( http://www.logicinaction.org/ ) 21 / 24

Natural Deduction for Predicate Logic Substitution (2) Substitution inside a formula. Replacing the free occurrences of the variable y for the term t inside the formula ϕ produces the formula denoted by ( ϕ ) y t ( http://www.logicinaction.org/ ) 21 / 24

Natural Deduction for Predicate Logic Substitution (2) Substitution inside a formula. Replacing the free occurrences of the variable y for the term t inside the formula ϕ produces the formula denoted by ( ϕ ) y t Formally, ( P t 1 · · · t n ) y t := P ( t 1 ) y t · · · ( t n ) y 8 ( ∀ xϕ ) y t := ∀ x ( ϕ ) y t < t ( ¬ ϕ ) y t := ¬ ( ϕ ) y ( ∀ yϕ ) y t := ∀ yϕ t : ( ϕ ∧ ψ ) y t := ( ϕ ) y t ∧ ( ψ ) y t ( ϕ ∨ ψ ) y t := ( ϕ ) y t ∨ ( ψ ) y t 8 ( ∃ xϕ ) y t := ∃ x ( ϕ ) y ( ϕ → ψ ) y t := ( ϕ ) y t → ( ψ ) y < t t ( ∃ yϕ ) y t := ∃ yϕ ( ϕ ↔ ψ ) y t := ( ϕ ) y t ↔ ( ψ ) y : t ( http://www.logicinaction.org/ ) 21 / 24

Natural Deduction for Predicate Logic For the universal quantifier ∀ ( http://www.logicinaction.org/ ) 22 / 24

Natural Deduction for Predicate Logic For the universal quantifier ∀ ∀ x ϕ ( ϕ ) x t provided that no variable in t occurs bounded in ϕ ( http://www.logicinaction.org/ ) 22 / 24

Natural Deduction for Predicate Logic For the universal quantifier ∀ ∀ x ϕ ( ϕ ) x t provided that no variable in t occurs bounded in ϕ E ∀ ( http://www.logicinaction.org/ ) 22 / 24