fom
Posts:
1,030
Registered:
12/4/12
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Re: This is False. 0/0 {x | x ~e x} e {x | x ~e x} A single Principle
to Resolve Several Paradoxes
Posted:
Feb 4, 2013 4:26 PM
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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.
There is a struggle between forms that are purely syntactic
and the fact that well-formedness must convey significance
before interpretation.
>
> A. Naïve 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.
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.
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.
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
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 coherent
>> explanation for the utility of an otherwise incoherent
>> collection of mathematical techniques. If such an
>
> ?ZFC is one of several axiomatic systems proposed to formulate a
> theory of sets without the paradoxes of naive set theory such as
> Russell's paradox.? - Wikipedia
>
>> explanatory role is not forthcoming, such theories
>> at least organize mathematical techniques into a
>> science (in the sense of Aristotle) connected by
>> the argumentation of proofs.
>
> I am not saying to not formalize. (I have personally axiomatized at
> least 5 branches of Computer Science/Logic. Every case of
> incompleteness is handled by a single axiom to distinguish the sets or
> relationships that cannot be characterized in the system. In fact,
> that additional axiom is the only difference between the positive and
> negative sides of a theory e.g. Universal Turing Machine vs. Halting
> Problem in the Theory of Computation.)
>
> I am saying that ZF is a lousy attempt at formalizing and I propose an
> alternate formalization ? a simple addition to Frege?s Naïve Set
> Theory. Just using predicate calculus instead of a specialized
> language to state the axioms makes ZF hard to communicate:
>
> ?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.? - Wikipedia
>
Yes. There is a problem with interpretations.
My own issue lies with the axiom of extension. It can
be eliminated in favor of axioms more consistent with
the historical developments associated with the identity
relation.
Language is topological. The complex of a negation
symbol with the Fregean "the True" and "the False" makes
a formalized language representable as a minimal Hausdorff
topology. This is a semiregular topology.
Not surprisingly, the manipulations used in forcing
involve a topology based on regular open sets which
is also a semiregular topology.
Forcing models manipulate the topological structure
of language just like a coffee cup is made to look
like a donut for classification purposes.
That is before one even gets to separation.
>>> Logic = Set Theory
>>
>> If this is true, it is not the logic of which
>> you are thinking.
>>
>> What you are taking for granted is the structure
>> of logic without a negation symbol. The negation
>> you use in your programming has no reality in the
>> underlying computer architectures.
>
> With negation you have all levels of the Kleene Arithmetic Hierarchy,
> which means any wff that can be expressed. (Each added ~exists adds a
> level.) Without negation you have only Sigma-1 the recursively
> enumerable sets, and the negation (complement) of some included sets
> of natural numbers are not included.
>
> Set Theory, axiomatic Logic used to express sets with wffs that are
> true of its elements, and English all have negation and are
> equivalent. Computer programs, proof in axiomatic Logic and the
> various bases of computing developed during the 1930s (excluding a
> couple of misfires) are Sigma-1, do not allow the complement of every
> set allowed and are equivalent.
>
> Is this what you?re referring to?
No. I really do need to "catch up" with some of the
work computationally-literate mathematicians utilize.
I literally mean considering the nature of logic
without a negation symbol.
My studies on the identity relation have essentially
eliminated logic from the foundations of mathematics.
A negation symbol is like the sign of a determinant.
The sign of a determinant is correlated with the
handedness of a coordinate system. Classical negation
is correlated with geometric reflection through a
line. The algebraic representation for this is
the subdirectly irreducible DeMorgan algebra on
four elements.
Typically, the system of 16 basic Boolean functions
is thought of in relation to a 16 element Boolean
algebra. But, that algebra is simultaneously order
isomorphic with the 16 element DeMorgan algebra
formed as the Cartesian product with the DeMorgan
algebra on four elements.
It is DeMorgan algebra rather than Boolean algebra
which is the foundational form.
At this level, one can actually represent the
structure in a finite projective geometry. The
16 elements corresponding with the truth functions
(as "objects") are the affine points of that
geometry. Negation, DeMorgan conjugation, and
contraposition reflect geometric projectivities
with the involution corresponding to negation
having the line at infinity as its axis.
Thus, negation as a "unary" connective is essentially
the line at infinity.
The exaggeration above concerning the elimination
of logic can now be tempered with various
classical observations from authors such as
Carnap who recognize that the syntactical structure
of logic is very much like a geometric structure.
My studies have simply identified an explicit
form for it.
>
>> More formally, what you are taking for granted
>> is that only 14 of the 16 basic Boolean functions
>> are linearly separable switching functions. The
>> two that are not are logical equivalence (LEQ) and
>> exclusive disjunction (XOR).
>>
>> These particular connectives become problematic
>> when considered in the context of classical quantificational
>> logic because of the relation of identity, definition,
>> and description.
>>
>> The standard account of identity (for example as
>> discussed under "relative identity" at
>> plato.stanford.edu) addresses trivial identity,
>> that is, x=x, and substitutivity.
>>
>> What is not addressed is informative identity,
>> that is, x=y.
>>
>> In classical model theory, however one has
>> determined an object in a model and a name
>> for that object has consequences for the
>> satisfaction map. That is how the classical
>> model theory interprets x=y.
>>
>> In 1971 Tarski directed his attention to the
>> representation of first-order logic in the context
>> of algebraic logic. In those deliberations, he
>> introduced the axiom
>>
>> AxAy(x=y <-> Ez(x=z /\ z=y))
>>
>> In the formulation of these "cylindrical algebras"
>> the formula
>>
>> x=y
>>
>> corresponds geometrically to a 2-dimensional subspace
>> of the domain. In a 3-dimensional domain, this is
>> a hyperspace separating the domain into two regions.
>>
>> This suggests that there is a fundamental geometric
>> reason for LEQ and XOR to not be represented in the
>> underlying propositional logic by linearly separable
>> switching functions.
>>
>>
>>
>>> Logic = NOT AND OR EXISTS simple, easy
>>
>> What happens if I take NOT away?
>
> Interesting question. Assuming you can express without quantifiers
> all recursive sets, since all wffs can be put into prenex normal form,
> you can express the same sets. Neither the quantifiers nor the
> relations need the negation symbol.
>
Well a "full" system of connectives has NAND and NOR. So,
there should be no diminishing of what can be expressed.
My question had been directed at the complexity of determining
a canonical choice between NAND and NOR for use as a negation
operator.
In fact, there are four Boolean functions that negate uniform
arguments. What you call P and Q I call FIX and LET. There
respective negations I call FLIP and DENY. The four Boolean
functions that negate uniform arguments are
NAND, NOR, FLIP, DENY
My solution for the complexity of making a choice was to
recognize that the structure of the projective geometry
could be manipulated to accommodate extensions to the
propositional connectives. In effect, propositional
logic is an incomplete system. The sense of a negation
arises from its relation to quantifiers or operators.
Separately, such quantifiers or operators have the
four-fold structure one characterizes using negation.
Organizing them into a unified system forms the
lattice below, where the NOR connective is an integral
part of the intersection of the component sublattices
and NAND is not.
In regard to the projective geometry, there are 5
points on the line at infinity. The projectivity
corresponding to negation has its center on the line
at infinity. The line without that center corresponds
leaves 4 points, and, that is what I am treating
as a quantificational or operational complex.
Often, mathematicians are interested in invariances.
If you perform DeMorgan conjugations on all 16 Boolean
functions, you will find that
FIX, FLIP, LET, and DENY
are invariant under DeMorgan conjugation.
As I said, I need to catch up on the kinds of
mathematics computationally-literate mathematicians
use. I am almost finished with these particular
geometric concerns.
> (I did go through a period of writing software in search of minimal
> bases (subsets of the 16 binary Boolean functions) for propositional
> calculus.)
>
With what results?
>> There is no real way to post this picture to a newsgroup.
>> It is the ortholattice which is an atomic amalgam of a
>> Boolean lattice with 4 atoms (the usual 16 element lattice
>> associated with basic Boolean functionality) and a
>> Boolean lattice with 3 atoms.
>>
>> ....................................TRU....................................
>> ............................./.../..//\...\.................................
>> ......................../..../.../../....\...\.............................
>> .................../...../..../..../.........\.....\.......................
>> ............../....../...../....../...............\......\.................
>> ........./......./....../......../.....................\.......\...........
>> ....../......./......./........./...........................\........\.....
>> .....IF......NAND.......IMP.....OR.........................ALL........NO...
>> ..../.\.\..../.\.\..../..|.\..././\.\\..................../...\.....././...
>> .../...\./\......\./\....|./..\./..\...\...\...................../.........
>> ../../..\...\.../...\./.\|...../.\..\....\............/....../...\../......
>> .//......\.../\.../....\.|.\../....\.\......\...\......../.................
>> LET.......XOR..FLIP....FIX..LEQ.....DENY........./.../............/\.......
>> .\\....../...\/...\..../.|./..\...././......./...\.......\.................
>> ..\..\../.../...\.../.\./|.....\./../...../...../.............../....\.....
>> ...\.../.\/....../.\/....|.\../.\../.../.../...........\.........\.........
>> ....\././....\././....\..|./...\.\/.//.....................\./.......\.\...
>> .....NIF......AND......NIMP.....NOR........................OTHER......SOME.
>> ......\.......\.......\.........\.........................../......../.....
>> .........\.......\......\........\...................../......./...........
>> ..............\......\.....\......\.............../....../.................
>> ...................\.....\....\....\........./...../.......................
>> ........................\....\...\..\..../.../.............................
>> .............................\...\..\\/../.................................
>> ...................................NTRU....................................
>>
>> The two lattices share TRU, NTRU, OR, and NOR. The structure of
>> the unfamiliar lattice has
>>
>> SOME=EXISTSWITH=Ex
>> OTHER=EXISTSWITHOUT=Ex-
>> ALL=Ax
>> NO=Ax-
>>
>> with
>>
>> ALL=NOT(OTHER)
>> NO=NOT(SOME)
>>
>> on the basis of the order relation alone.
>>
>> This construction, while described specifically
>> for quantificational logic here, actually characterizes
>> the geometric (in the sense of an atomic lattice) structure
>> of any extension to propositional logic with negation.
>> Modal extensions are four-fold interrelating POSSIBLE and
>> NECESSARY. Deontic extensions are four-fold interrelating
>> OBLIGATORY and PERMITTED.
>>
>> For quantificational logic, each variable has this
>> four-fold structure. This corresponds with the indexing
>> of quantifiers found in the cylindrical algebras of
>> Tarski's later work.
>>
>> The negation symbol masks this four-fold structure in
>> the formation rules for formulas.
>>
>> Yes. Logic in the absence of NOT--just like in the
>> hardware of your computer systems--is not easy.
>>
>>> ZF Set Theory = a dozen messy axioms for which people can t even agree
>>> on the specifics ??
>>
>> There are actually an infinity of axioms (damn those
>> first-order logicists) because well-formed formulas are
>> separately generated and present in the axioms of separation.
>
> Yes, but this schema ranges over wffs (rather than sets) so the set
> defined is aleph-0 and in fact recursively enumerable. Better to
> range over sets and pull in that needed aleph-1.
>
How do you mean? Any references? I am always curious why
the Borel hierarchy extends to aleph_1. I am certain your
statement reflects the same "need".
>>> There are a dozen set theories and a dozen interpretations of the most
>>> popular set theory, and 2 or 3 versions of it (with or without Choice,
>>> etc.) none of which decide any of the important questions of set
>>> theory due to exhaustive work (a waste!) by Godel and Cohen.
>>
>> That is an odd thing to say. While I find forcing to be nonsense
>> in set theory (but I reject the axiom of extension as foundational)
>> it is extremely important to recursion theory, is it not? And that
>> has consequences for the practical application in computational
>> contexts, does it not?
>
> You are referring to the advancements in prosthetics made during war.
So, my ignorance is showing, as usual.
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