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hiz - Some Remarks and Extra-Credit Exercises Concerning...

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Some Remarks and Extra-Credit Exercises Concerning the Deductive System of Hi˙z Branden Fitelson 03/04/05 Hi˙z’s system (H) for propositional logic consists of the following axiom and inference rule schemata for P: Axiom schemata for H : (HA1) A B A (HA2) A B B Inference Rule schemata for H : (HR1) From H A B and H B C , infer H A C . (HR2) From H A B C and H A B , infer H A C . (HR3) From H A B and H A B , infer H A . Note : the premises of these rules must all be theorems of H! This is different than MP in PS! Some interesting facts about H, and some extra-credit exercises concerning H : 1. H is weakly semantically sound . For all formulae A of P, if H A , then P A . Prove this! 2. Question: Is H strongly semantically sound ? That is, are there sets of formulae Γ and formulae A of P such that Γ H A , but Γ P A ? Settle this question with a proof! 3. H is weakly semantically complete . For all formulae A of P, if P A , then H A . Note: One cannot use Henkin’s method to prove this! This proof is worth a serious amount of extra credit! 1 You may assume (3) for the other exercises below. [Note: (1) and (3) imply: H A P A ( PS A ).] 4. H is not strongly semantically complete . That is, there are sets of formulae Γ and formulae A of P such that Γ P A , but Γ H A . Why? One way to see why is to note that: 5. Modus Ponens is not (in at least one sense) an inference rule schemata of H . That is, there exist formulae A and B of P such that A B, A H B . Exercise: prove this! [Hint: choose a

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hiz - Some Remarks and Extra-Credit Exercises Concerning...

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