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

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

View Full Document Right Arrow Icon
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) p A  B q  A (HA2) p A  B q  B Inference Rule schemata for H : (HR1) From $ H A  B and $ H B  C , infer $ H A  C . (HR2) From $ H A  p B  C q 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 ô ( P A ( ô $ 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
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 2

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

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