|
|
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, Charlie-Boo wrote:
>
>
>
>
>
>
>
>
>
> > On Feb 4, 4:26 pm, fom <fomJ...@nyms.net> wrote:
> >> On 2/4/2013 8:46 AM, Charlie-Boo wrote:
>
> >>> On Feb 4, 12:25 am, fom <fomJ...@nyms.net> wrote:
> >>>> On 2/3/2013 10:19 PM, Charlie-Boo wrote:
> >>>> <snip>
>
> >>>>>>>> In PROLOG we use lowercase words for TERMS
> >>>>>>>> and uppercase words for VARIABLES
>
> >>>>>>>> ATOMIC PREDICATE
>
> >>>>>>> ATOMIC PREDICATE meaning relation?
>
> >>>>>>> C-B
>
> >>>>>> 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.
>
> >>>> Well-formed formulas are built from the alphabet
> >>>> of a formal language. If the language contains
> >>>> a symbol of negation, then NOT(xex) will be a
> >>>> well-formed 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 well-formedness 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 non-circular 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 non-Euclidean 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
>
> "Non-orthomodular 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!
C-B
> 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 pre-defined 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 Martin-L 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 »
|
|
