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Proof rules for negation #11
Via email, I said:
Richard Zach said:
So your suggestion is to have the current-botI as new-¬E, and have botE and ¬I as they currently are. Then the new system would have no botI rule, and you still need TND as a rule (eg to derive LEM) so that doesn't then fit in.
A nicer alternative would maybe to keep bot rules as they are and replace TND with
(That's then the proof system used in Halbach's logic manual. But a consequence of that change is that proving LEM from it is a massive task! So perhaps that's not so useful, although it's more pleasing on the eye, as it were).
That is the solution I favor. It's hat prawitz calls the classical absurdity rule; I propose we call it Indirect proof (IP). TND would become a derived rule. (Tricky to derive, but not so massive! Or, not more massive than DeM).
Instead of \bot E, I think we should call it ECQ (ex contradictione quodlibet). What do you think?
In the end, the I/E rules make up minimal logic, add ECQ for intuitionistic logic, add IP for classical. (Not that we'd have to discuss that, but it would make it easier to tie in our system to a discussion of minimal and intuitionistic logic in a future chapter or another course.)
referenced this issue
Oct 1, 2017
This is great; thanks for making these changes! One nitpick: now that \bot-Introduction has been renamed to \neg-Elimination, wouldn't it make sense to switch the order of the lines in the citation? That is, as with the other elimination rules, the premise containing the eliminated connective should be cited first.
Would it make sense to start converting proofs in the solutions manual to use the new rules?
Ah, good point, yes. [fixed it!] So ~E and _|_I are actually different rules then. So I'll have to go through it and change all the \ri's to \ne's which is probably cleaner anyway. (Are we supposed to be sticklers about the order of the justifications? The solution to the second problem of 15.A suggests yes, but I didn't see a statement explicitly anywhere. The proof checker doesn't care.)
If you're volunteering to go through the solutions manual, sure! I fear we might have to actually move some exercises around -- with TND it was not too big of a deal to prove 15.C.7, but this is in the chapter on basic rules, and it'll be a lot harder with IP -- so probably move that into chapter 16 which needs examples of TND/LEM anyway. 15.C.12 will be easier with IP (just the second subproof), but we'll need a few more examples of IP in use/as exercises. I also added a few exercises already. Must keep track somewhere so we can make a list of what's changed for the next edition. [see #18 ]