On 26/05/2013 3:52 AM, Zuhair wrote: > On May 26, 11:03 am, Nam Nguyen <namducngu...@shaw.ca> wrote: >> On 26/05/2013 12:52 AM, Zuhair wrote: >> >>> Frege wanted to reduce mathematics to Logic by extending predicates by >>> objects in a general manner (i.e. every predicate has an object >>> extending it). >> >> [...] >> >>> Now the above process will recursively form typed formulas, and typed >>> predicates. >> >> Note your "process" and "recursively". >> >> >> >> >> >> >> >> >> >> >> >>> As if we are playing MUSIC with formulas. >> >>> Now we stipulate the extensional formation rule: >> >>> If Pi is a typed predicate symbol then ePi is a term. >> >>> The idea behind extensions is to code formulas into objects and thus >>> reduce the predicate hierarchy into an almost dichotomous one, that of >>> objects and predicates holding of objects, thus enabling Rule 6. >> >>> What makes matters enjoying is that the above is a purely logically >>> motivated theory, I don't see any clear mathematical concepts involved >>> here, we are simply forming formulas in a stepwise manner and even the >>> extensional motivation is to ease handling of those formulas. >>> A purely logical talk. >> >> Not so. "Recursive process" is a non-logical concept. >> >> Certainly far from being "a purely logical talk". > > Recursion is applied in first order logic formation of formulas,
Such application isn't purely logical. Finiteness might be a purely logical concept but recursion isn't: it requires a _non-logical_ concept (that of the natural numbers).
> and all agrees that first order logic is about logic,
That doesn't mean much and is an obscured way to differentiate between what is of "purely logical" to what isn't.
> similarly here > although recursion is used yet still we are speaking about logic, > formation of formulas in the above manner is purely logically > motivated.
"Purely logically motivated" isn't the same as "purely logical".
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