On May 31, 6:36 pm, "Julio Di Egidio" <ju...@diegidio.name> wrote: > "fom" <fomJ...@nyms.net> wrote in message > > news:UY-dnUnFksIIUDrMnZ2dnUVZ_g-dnZ2d@giganews.com...> On 5/30/2013 11:20 AM, Julio Di Egidio wrote: > > <snipped> > > > is there a criterion > > for deciding whether either of the complete connectives should > > be viewed as the canonical complete connective? That is, is > > there an asymmetry in the dyslexia we call truth-functional > > logic? > > Why should there be a canonical one? Isn't indeed self-referentiality > (circularity) the essential character of the (any) purely logical system? > (And, I am intrigued, why would you call such system dyslexic?) > > > Well, my ideas are actually motivated by considering > > set theory. So, arithmetic, for me, is inside of a > > theory of classes... > > Kant says that space and time are given prior to experience, in fact as a > pre-requisite for the possibility of any experience of the world. Along > similar lines, counting and distinguishing may very well be a-priory > faculties, and, at least as far as I can presently see, that's the > confusion: the use of logic or mathematics to describe what is properly > pre-logical or pre-mathematical. > > Julio
This is interesting. But I don't think that a complex system like for example second order arithmetic would be of the sort of entities you are speaking about, you might be right about some the meta-logic symbols like the natural indices for example, also about the rules of logic themselves (the logical axioms) those might be provably necessary to any kind of thought, yes, but what follows from those is purely analytic (in Frege's sense).
We are not trying to characterize those pre-logic or pre-mathematical concepts by using logic or mathematics as you think, we are just trying to know from where we start and what inference system is necessary and from those what we can build further. As has been shown here it seems that the major bulk of 'traditional' mathematics is traceable to logic, more specifically into a kind of predicative extensional logic, i.e. most of what we think mathematics is about like for example 'intuitions' about spacial relations and temporal successor like those concerned with geometry and arithmetic are actually not really required for having those systems, both geometry and arithmetic and most of traditional math. can follow from systems that are purely logically motivated, i.e. motivated by defining "General" non-contradictory reasoning that is not specific to any particular realm of thought that they can be applicable to, this is really a subtle issue, to see that mathematics can follow from such subtle logical concepts that are by far way weaker than what commonly thought of mathematics to be about, is really an interesting thing. Mathematics turns to be just an offshoot of logic, an analytic machinery rather than derived by any particular intuition about structure, constructs or succession, this is really interesting to see that mathematics is so weakly based.