notes_12_02_08 - Applied Logic Lecture Notes 12-02-08 1...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 1 , , n satisfies satisfies ( n ( 1 ) ) So 1 , , n satisfies iff the refutation of the proposition ( n ( 1 ) ) has no consis- tent subtree iff ( n ( 1 ) ) in the sense of tableau proofs. To handle infinitely many hypotheses, we need to extend our definition of refutation tree. Definition. Let = { 1 , 2 , } be a (finite or infinite) set of propositions. The -refu- tation tree of is the tree with the following properties. The (unlabelled) root has 1+ | | children, 1 labelled F , 1 T , 2 T , , from left to right. If a node is labelled ( ) T then its unique child is labelled F . If a node is labelled ( ) F then its unique child is labelled T . If a node is labelled ( ) T then its two children are labelled T and T , respec- tively. If a node is labelled ( ) F then its two children are labelled F and F , respectively. If a node is labelled ( ) T then its two children are labelled T and T , respec- tively. If a node is labelled ( ) F then its two children are labelled F and F , respectively. If a node is labelled ( ) T then its two children are labelled F and T , respec- tively. If a node is labelled ( ) F then its two children are labelled T and F , respectively....
View Full Document

Page1 / 4

notes_12_02_08 - Applied Logic Lecture Notes 12-02-08 1...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online