On May 25, 4:40 pm, Zuhair <zaljo...@gmail.com> wrote: > On May 24, 10:56 pm, Graham Cooper <grahamcoop...@gmail.com> wrote: > > > On May 22, 8:17 am, "LudovicoVan" <ju...@diegidio.name> wrote: > > > > "Zuhair" <zaljo...@gmail.com> wrote in message > > > > > The investigation of behaviors that are formalizable in consistent > > > > formal systems extending logic. > > > > I'd think that only captures the interests of formal logic. > > > > > Logic is the study of Tautology. > > > > As I have it, logic is the study of self-contradiction, itself informally > > > defined after the notion of mutual incompatibility of predicates. > > > If we examine a Table of Tautologies.. > > >www.tinyurl.com/SetTheory2 > > > So we can make Induction a Tautology in the form A^B->C > > > TAUTOLOGIES > > ----------- > > A B C TYPE > > a a->c c Modus Ponens > > d->e e->f d->f Transitivity > > !(!d) TRUE d Double Negation > > P(0) P(n)->P(S(n)) P(n) Induction Formula > > No induction formula is not a tautology. > > > > > LOGIC might be defined as the FORMAL *USE* of TAUTOLOGIES. > > When making an informal definition I like some general words, I > defined logic as: > > The study of Tautology. > > This includes having a system that can capture any tautology, and also > includes tautologistic implication from a set of non tautologies which > is what you call > the formal use of tautologies. So the word "study" informally covers > both of those aspects. >
Right, but the goal of logic is to be mechanistic, so taking logic as a discipline for granted it's a practical field, a utility, much like calling the Institute Of Sport where athletes go to 'study'.
> > > > and MATHEMATICS the set of FORMALIZ*ABLE* Studies. > > > for instance, it would impossible to have LOGIC (or mathematics) > > without MODUS PONENS. > > > FACT1 & FACT1 IMPLIES FACT2 > > ergo FACT2 > > > So LOGIC appears to use a PROVEN-THEORY, an EXTENSION of TAUTOLOGIES > > In the strict formal sense we are using the word logic here, we can > say it is confined to those systems that merely capture all > tautologies. Those systems can have nine or 5 axioms and together with > Modus Ponens they can capture all tautologies. Also this will cover > tautologistic implication of non tautologies. And for various reasons > I tend to also include "equality" among Logic. So if you are working > with first order languages then logic will be confined to tautologies > and tautologistic implication and also have the axioms of identity > theory added to it. This identity theory is an extension of > tautologyies in your sense it is both consistent and complete and I > accept it as a part of logic although identity axioms by themselves > are not tautologies. So in reality identity is mathematical but I'll > joint it to logic sense it is too simple actually. > > Now all systems obtained by adding sentences that are not tautologies > as axioms (aka non logical axioms) to the axioms that can capture all > tautologies (aka logical axioms) and axioms of identity theory, are to > be considered as mathematical systems because simply what is > formalizable in those is not logics in the strict sense it involves > something external to logic, this thing is mathematics! > > So I'm looking at a spectrum here at the bottom of which are the > logical theories then mathematical and those will branch to > specialized mathematics that is of use to other disciplines of human > knowledge (applied mathematics) or to fantasy mathematics (non applied > end mathematics). > > You can view this as a TREE. The roots are logic the Stem and the > branches are Mathematics and the fruits are applied mathematics and > the leafs are non applied too particular mathematics (like the > solutions to some fantasy questions)
Sheer poetry Zuhair! misc.writing added for a reason!
> > This Tree has much more leafs than fruits, therefore I tend to say > that mathematics is a kind of Rationalization, in simple words > Logicism. > > Zuhair
Herc -- TM-SIZE MAX-1s OUTPUT ------------------------- BB(2) 6 BB(3) 38 BB(4) 3,932,964 BB(5) 1.7 x 10^352 BB(6) 1.9 x 10^4933 ... BB(199) COOPERS NUMBER BB(200) UNIVERSAL TURING MACHINE SIZE includes PorkyPig Jnr's Number = CN+1