
Re: This is False. 0/0 {x  x ~e x} e {x  x ~e x} A single Principle to Resolve Several Paradoxes
Posted:
Feb 11, 2013 2:07 PM


On Feb 7, 1:51 am, fom <fomJ...@nyms.net> wrote: > On 2/5/2013 9:32 AM, CharlieBoo wrote: > > > > > > > > > > > On Feb 4, 4:26 pm, fom <fomJ...@nyms.net> wrote: > >> On 2/4/2013 8:46 AM, CharlieBoo wrote: > > >>> On Feb 4, 12:25 am, fom <fomJ...@nyms.net> wrote: > >>>> On 2/3/2013 10:19 PM, CharlieBoo wrote: > >>>> <snip> > > >>>>>>>> In PROLOG we use lowercase words for TERMS > >>>>>>>> and uppercase words for VARIABLES > > >>>>>>>> ATOMIC PREDICATE > > >>>>>>> ATOMIC PREDICATE meaning relation? > > >>>>>>> CB > > >>>>>> RELATION > >>>>>> p(a, b, e) > > >>>>> If wffs are built on relations then { x  x ~e x } is not a wff > >>>>> because ~e is not a relation. > > >>>> Wellformed formulas are built from the alphabet > >>>> of a formal language. If the language contains > >>>> a symbol of negation, then NOT(xex) will be a > >>>> wellformed formula. > > >>> You have to define what value a symbol may have  how it is > >>> interpreted in your definition of a wff. You need to complete B > >>> below to see there is no paradox if you are consistent about what a > >>> wff may contain and what values it may equal after substitution > >>> (interpretation) if it contains variables for functions. > > >> First, I was not in a good mood when I posted. So, I may > >> have been too dogmatic. > > >> What you seem to be objecting to is the historical development > >> of a logical calculus along the lines of Brentano and DeMorgan. > > I meant Bolzano here. > > > > > The only objecting in my Set Theory proposal is perhaps objecting to > > the fact that ZF has a dozen messy axioms, a dozen competing > > axiomatizations, a dozen interpretations of the most popular > > Axiomatization, and (Wikipedia), The precise meanings of the terms > > associated with the separation axioms has varied over time. The > > separation axioms have had a convoluted history, with many competing > > meanings for the same term, and many competing terms for the same > > concept. > > > (DeMorgan is an example of why Logic and Set Theory are the same thing > > and should be combined  same as Math and Computer Science etc.) > > How do you see Logic and Set Theory as being the same? > > >> There is a struggle between forms that are purely syntactic > >> and the fact that wellformedness must convey significance > >> before interpretation. > > > But what is the significance of trying to apply axioms to different > > functions than those of original interest? Who uses other than the > > standard interpretation of + * ** ? > > From what I have gathered in reading historical references, various > issues lead to that. > > In Aristotle, the justification for a deductive calculus with > respect to science is epistemic. So, proofs trace a coherent > framework between scientific assertion. And, Aristotelian > necessity admits circular structure in the formation of initial > formulas. But, Aristotle also introduced a notion of substance. > > It is in the fruitless search for "simple" substance that the > history of logic and mathematics reduces itself to more and > more abstract syntactic form. > > Leibniz has no problem following Aristotle and allowing > circularity in axioms. But, when we get to Bolzano, the > search for a noncircular definition of a simple substance > begins to lead to the arguments about undefined language > primitives (different from the symbols needed for transformation > in the calculus). > > By the time we get to DeMorgan, one gets explicit discussion > of the possibility of pure syntax subjected to random > interpretation which, when coherent, we would call a > model. > > Other developments shattered the "meaning" of mathematical > words. The discovery of nonEuclidean geometries and the > admissibility of complex arithmetic and quaternion arithmetic > are two examples. It is this precursor that enables Cantor > to argue for a transfinite arithmetic in the arena of justifying > the differential calculus. > > >>> A. Na ve Set Theory > >>> B. Formal definition of a wff including substitution for variables > >>> (aka interpretation.) > >>> C. Statement that x ~e x is not a relation (aka set or predicate.) > > >>> [The whole idea of interpretations is also not well designed. It is > >>> an example of generalizing the wrong thing, as Productive Sets > >>> generalize the set of true sentences  a fixed aspect of > >>> incompleteness proofs instead of the premises which vary especially > >>> those implicit in the carrying out of the proofs but never stated. > > >> Presuppositions are clearly problematic. > > > Just the omission of indicating they are premises. > > Perhaps. But, if one considers the various perspectives of > people who disagree, the "new" theory of the detractor has its own > presuppositions. It is like a game where the presuppositions > are "hidden variables". > > >> The consequences of many of these landmarks in foundational > >> studies are viewed as definitive epistemic limitations without > >> considering them further. A number of years ago, it was > >> shown that classical propositional logic was not categorical. > > > Who would say such a thing? Propositional calculus is a necessary > > subset of categorical logic. > > Pavicic and Megill. 1999 > > "Nonorthomodular Models for Both > Standard Quantum Logic and Standard > Classical Logic: Repercussions for > Quantum Computers". > > > > > > > > > > >> The apparent discrepancy was identified as a presupposition > >> concerning logical equivalence within the method of proving > >> completeness. > > >> The semantics of "ideal language theory" has been being eroded > >> by the study of pragmatics. But, mathematics has become set > >> in its ways with regard to model theory. > > >>> We don t want to know all functions that satisfy Peano s Axioms. If > >>> it is done right there is only one set of functions that + * ** can > >>> be. What we have lots of variations of is the properties of N that > >>> is what is to be generalized. Saying + is not addition is like > >>> Fortran allowing you to redefine what 1 means. There s no need for > >>> that either. It only muddies the water. > > >> Once again, you are diverting from classical notions of calculi. > > > Isn t this how they do logic now? And do you know of anyone using > > other than the standard interpretation ? > > Mathematics is different from logic. So, for example, one
If you want to know what Logic is, try being a programmer of medical systems and then explain what is special about the set { inpatient , outpatient }.
1. It is a finite set. 2. We don't really decide it but we enumerate it. a. In Logic we decide it and "This is false." is not in it. 3. We have a primitive name for each element. 4. Axioms use the names of the elements. a. E.g. if inpatient then list on inpatient report.
Now change that to { TRUE , FALSE }.
The lesson of the Halting Problem for programmers is that specs may be inconsistent but only after a bit of logic is applied!
CB
> can talk of permutations on a set of objects. The set of > permutations have an arithmetical property with one another. > Thus, they form a system. As a system, they are independent > of the underlying set. Then with some additional terminology, > one speaks of group actions on arbitrary sets, categories of > groups, and group representation theory. > > Usually, that particular system is thought of as a multiplication. But, > there are groups whose intrinsic property is one of > addition. > > In any case, a "naive" programmer would think of this as > beginning with some predefined data types, forming objects > and object methods, and differentiate the object methods > from the arithmetical relations of the predefined data types. > > But, mathematics does not really have predefined data types, > although the drive for foundations has organized mathematics > to now look as if it does. That is a good in many ways. Still, > some things are lost. > > > (A calculus is a cross between a logic and a programming language, and > > they are severely underutilized. Program Synthesis can be easily > > explained as a program calculus, while researchers are clueless as to > > how to address the problem  to the point of making blatantly > > fraudulent claims about MartinL f Type Theory.) > > See, this is how different backgrounds lead to different > things. Some time ago I read a great book on formal > language theory in the sense one would have emphasized in > a computer science curriculum. It would be the kind of > thing someone writing compilers would have to know. I am > assuming that this applies somewhere in what you have written. > > But, I see that it is too much for a quick Wikipedia read. > > I have looked at the lambda calculus a small bit. I hope to > take some time this year to learn enough to understand your > paragraph. But, I also have other interests. I am just > sick of not knowing about so much good work that came out > of intuitionistic logic and constructive mathematics. > > > > > > > > > > >> Part of the reason that one speaks of "number systems" is because > >> of the development of the complex numbers and the quaternions. > >> With arithmetical systems different from the usual arithmetic, > >> mathematicians were confronted with the genesis of model theory > >> and interpretation of calculi. > > >>> Godel/Rosser/Smullyan incompleteness theorems include reference to a > >>> wff being true but not provable. But then it must be true for which > >>> interpretations? . > > >> Yes. But, while there may be a number of ways to introduce > >> Goedel numbering, there is always the method that involves > > > This doesn t have anything to do with the choice of correspondence > > between wff and number. It is the insistence that we use variables > > for functions and indicate what + * and ** represent. > > Ok. The next time I dig around in my logic > texts, I will look at one of the proofs more > closely and think about what you are saying. > > > > > > > > >> prime decompositions. So, what kind of interpretation of > >> arithmetic would alter the configuration of primes and their > >> relation to the number system as a whole? > > >>> That is left out and opens up the question, why > >>> the standard interpretation works and which ones work? But we really > >>> don t care about using other functions for + * ** in the first place! > >>> Certainly not in the middle of an incompleteness theorem.] > > >>>>> We don t need ZF  at all. All we need is Na ve Set Theory, a > >>>>> complete formal definition of wff and recognition that x ~e x is not a > >>>>> relation due to diagonalization on sets. > > >>>> The reason for ZF and other inquiries into the > >>>> foundations of mathematics has to do with a > > ... > > read more »

