User talk:Lord Farin/Backup/Definition:Abbreviation of WFFs of PropCalc

Why can't we replace $$\left({\mathbf{A} \and \left({\mathbf{B} \and \mathbf{C}}\right)}\right)$$ with $$\left({\mathbf{A} \and \mathbf{B} \and \mathbf{C}}\right)$$? Isn't that essentially what the rule of association states? Either way, the statement is only true if all three are true. --Cynic (talk) 16:57, 6 July 2009 (UTC)

It's not that it's invalid as such, it's just that in order to be able to prove other stuff rigorously, you've got to be able to state that you can always get back to a unique WFF from a given abbreviated WFF. Despite both being WFFs, $$\left({\mathbf{A} \and \left({\mathbf{B} \and \mathbf{C}}\right)}\right)$$ and $$\left({\left({\mathbf{A} \and \mathbf{B}}\right) \and \mathbf{C}}\right)$$ are not technically the same WFF, but both could be obtained by allowing abbreviation from both directions. I suppose the trick is to ensure you build the WFFs up in the correct direction for being able to neatly unbracket them.

So, by restricting the rules on how you're "allowed" (in this context) to abbreviate, you reach the situation where any formula in PropCalc can be specified uniquely. This sort of thing becomes more important when you're proving theorems about uniqueness of representations and Godelisations and things. Not quite sure where this is going yet, but it's in the same general direction (made a few false starts so far, this may be another one). --Matt Westwood 18:46, 6 July 2009 (UTC)