User talk:Lord Farin/Backup/Definition:Classical Propositional Calculus

I am crashing into the boundaries of the possibilities the current state of PW allows me to go, since the axioms for Definition:Natural Deduction take, in some sense, three paths at once, using the notations for proofs with: However it feels like the distinction between axioms and rules of inference is blurred when writing things like $(p\vdash q), p\vdash q$ (modus ponens) which is actually a rule of inference, and not an axiom as such (though presented as were it one).
 * $\vdash$
 * vertical lines with annotation (convenient on paper, not so with TeX)
 * tableaus

There are LaTeX packages out there which allow for writing proof trees, but I have never liked how hard it is to read them when you didn't write them. I will proceed by incorporating the tableau proof method. In due time I will add the formalised interpretation of the sequent calculus (the one using $\vdash$). Sorry for any incoherence, this monologue took one hour to develop. --Lord_Farin 09:48, 16 June 2012 (EDT)

It just occurred to me that we probably want to reserve $\vdash$ for 'there exists a proof'; I also recall that the sequent calculus uses $\to$ instead. --Lord_Farin 09:50, 16 June 2012 (EDT)


 * If there's stuff you're running into problems with, leave them (perhaps use the "Help" template) and move on. I find this helps when I just need a break from a concept, and then suddenly everything falls into place.


 * Alternative techniques for demonstrating a proof (Tableau, truth table, natural deduction) are going to be difficult to resolve in a satisfactory way. My gut feeling is: the formal language approach has been taken as far as it need to with PropCalc, as the relevant and important theorem have been proved. The many detailed results which have been proved by natural deduction and truth table are probably also okay as they are - their validity has been demonstrated by the fact that the formal language approach has justified the natural deduction approach.


 * The hard work comes with PredCalc because it all becomes an order of magnitude more complicated. --prime mover 09:57, 16 June 2012 (EDT)


 * Thanks for the advice. I will move on reformatting the pages that exist, then at an appropriate moment get back to creating new stuff. --Lord_Farin 10:29, 16 June 2012 (EDT)