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Unformatted text preview: Details of Hunters Informal Proof of Craigs Interpolation Theorem for P Branden Fitelson 02/06/05 Hunters proof of Craigs Interpolation theorem for P is a bit opaque. Heres a more detailed version of his proof, which I sketched in class on Friday. Since this is our first (non-trivial) metatheorem, its worth doing a handout that proves it in some detail. Well see similar kinds of proofs often in the course. Theorem . Let A and B be formulas of P , such that (1) they share at least one propositional symbol in common, and (2) ( P A B . For any two such formulas of P , there exists a formula C (called the P-interpolant of the formulas A and B ) such that (3) ( P A C , (4) ( P C B , and (5) C contains only propositional symbols that occur in both A and B ( i.e. , only propositional symbols shared by A and B ). [Intuitively, if ( P A B (and A and B have some symbols in common!) it is always possible to reason from A to B via a formula C that has no propositional symbols not shared by A and B . This is sometimes called linear reasoning from A to B , since it takes no detours through irrelevant or tangent unshared propositional symbols.] Proof. Case 1 : There are zero propositional symbols occuring in A that do not also occur in B . That is, the set of propositional symbols in A is a subset of those in B . If we let S p A q be the set of propositional symbols occurring in a formula A , then we can express this case as the case in which S p A q S p B q . In this case, just let C A . Then, obviously, (3) ( P A C , since ( P A A . And, since the assumption of the theorem is that ( P A B , we also know that ( P C B . All we need to show is that (5) C contains only proposi- tional symbols that occur in both A and B . But, this follows from the fact that C A , and the assumption of this Case, which is that S p A q S p B q . Hence, S p A q S p C q S p A qX S p B q , which completes Case 1 ....
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This note was uploaded on 08/01/2008 for the course PHIL 140A taught by Professor Fitelson during the Spring '07 term at University of California, Berkeley.
- Spring '07