On May 31, 8:36 am, "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
There's much interest in both the foundations of rational development writ large and the particular domains of geometry, mathematics of numbers, mathematics of categories/types/sets as in the establishment of general relations, and of the plainly extra-logical.
There's much to be said of the logicist's tools, plainly inference and deduction, and the notion of most expressive theories with minimal content in that they're foundations for higher-level theorems and increasing abstration from the concrete, as to the concrete.
That the primeval objects of theory embody in so little the utmost of depth and complexity, vis-a-vis the monist or paraconsistent dialetheic of an ur-element or ur-object, the logicist's Ding-an-Sich, with one and a numeric infinity; or a point, and space; or an empty set and the universe: there's a general notion that for their properties to hold, that reversing each implication would yield the same theory. In our physical theories, there is conservation of various properties, for principles of conservation and symmetry as most primitive, here there's a notion of conservation of truth, and as well of declarative expression, that there's a shorter proof that something is true then that it is false, proof via elementary constructs.
Then, for inference and deduction, that given A that A->B and that given B that A->B, our extra-logical theories (or here generally theories where all theories are of some "logical" theory) are generally built on inference and the positive where deduction is built on the contrapositive, say.
Then, it seems the goal of a theory (and here A Theory) is to define A and B as basically the beginning and end, that A <-> Z. For this the defined are a point and all space, one and all, null and universal. Then they are as tools for inference and deduction, of the results from each about the other. In then a general "theory of theories" or of logical theory, generally of their consistency and completeness: the state of research is generally as to theories skewed as to forward inference, instead of deductive reasoning. Forward inference is perfectly valid reasoning, but, when the logical theory starts with both inferential and deductive reasoning as primary and as even bounding each other, and scaling with each other, then it seems the very features of a given theory or any theory (as fundamental) are as to the features of all theories.