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notes_12_02_08

notes_12_02_08 - Applied Logic Lecture Notes 1 Generalized...

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Unformatted text preview: Applied Logic Lecture Notes 12-02-08 1 Generalized Refutation Trees Up until this point we were only concerned with validity and provability without hypotheses. We saw that both of these notions were closely related to the existence of consistent finished subtrees of the refutation tree of a proposition. To accomodate hypotheses, we need to extend this notion. When dealing with finitely many hypotheses, there is a simple way to do this. Using Modus Ponens finitely often, it is easy to check that σ 1 , , σ n satisfies α ⇔ satisfies ( σ n → → ( σ 1 → α ) ) So σ 1 , , σ n satisfies α iff the refutation of the proposition ( σ n → → ( σ 1 → α ) ) has no consis- tent subtree iff ⊢ ( σ n → → ( σ 1 → α ) ) in the sense of tableau proofs. To handle infinitely many hypotheses, we need to extend our definition of refutation tree. Definition. Let Σ = { σ 1 , σ 2 , } be a (finite or infinite) set of propositions. The Σ-refu- tation tree of α is the tree with the following properties. • The (unlabelled) root has 1+ | Σ | children, 1 labelled α F , σ 1 T , σ 2 T , , from left to right. • If a node is labelled ( ¬ β ) T then its unique child is labelled β F . If a node is labelled ( ¬ β ) F then its unique child is labelled β T . • If a node is labelled ( β ∨ γ ) T then its two children are labelled β T and γ T , respec- tively. If a node is labelled ( β ∨ γ ) F then its two children are labelled β F and γ F , respectively. • If a node is labelled ( β ∧ γ ) T then its two children are labelled β T and γ T , respec- tively. If a node is labelled ( β ∧ γ ) F then its two children are labelled β F and γ F , respectively. • If a node is labelled ( β → γ ) T then its two children are labelled β F and γ T , respec- tively. If a node is labelled ( β → γ ) F then its two children are labelled β T and γ F , respectively....
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