fom
Posts:
1,968
Registered:
12/4/12


Re: This is False. 0/0 {x  x ~e x} e {x  x ~e x} A single Principle to Resolve Several Paradoxes
Posted:
Feb 7, 2013 1:51 AM


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 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 MartinLöf 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 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 Na ve 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 Sigma1 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 Sigma1, 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 computationallyliterate 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 firstorder 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 2dimensional subspace >>>> of the domain. In a 3dimensional 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. > > I mean AND and OR only. > >> My question had been directed at the complexity of determining >> a canonical choice between NAND and NOR for use as a negation >> operator. > > Only a program and a few minutes of processing can answer that. First > question is the spec: what do you want to check for in the finite > world of propositional calculus?
It is much more complex than that. :)
I have 4096 axioms that look like
AND (IF,AND) = AND AND (IF,NIMP) = NIMP AND (IF,XOR) = NIMP AND (IF,IMP) = LEQ
Under DeMorgan transformation, each axiom transforms into an axiom. So,
AND (IF,NIMP) = NIMP
transforms into
OR (NIMP,IF) = IF
This is why I no longer think about (or believe) Boolean algebra as a foundation for mathematics.
Other transformations (i.e., negation) do not have that invariance.
> > BTW Why does NOT and EXISTS create a Kleene Arithmetic Hierarchy level > and not AND and OR? I recently realized the answer. I mean from a > high level properties of the connectives point of view? P(NOT) ^ > P(EXISTS) ^ ~P(AND) ^ ~P(OR).
Well, I do not really grasp your notation here.
By definition, the Kleene hierarchy is based on quantifier complexity, is it not? ALL is just the DeMorgan conjugate of EXISTS (in the sense of "negate the arguments and negate the connective" applied to a single argument operator  ALL=NOT(EXISTS(NOT))) so that NOT and EXISTS should generate the hierarchy as a matter of definition.
> >> 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 >> fourfold 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 computationallyliterate 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? > > Unfortunately I must say that history does not record what happened > next. But given the spec and technical design I can whip up a PHP > script. You want the subsets of the 16 BBF (binary Boolean functions) > which can express all 16 and have no subset that can? Then what would > the technical design be? For each BBF . . . ? >
I cannot imagine a need for that information, although both memory and imagination fail me far too often in life.
What my question really was directed toward was whether or not you discovered anything interesting or surprising.
So, for example, I always believed that only NAND and NOR negated a uniform argument just because I did not immediately think of the semantics of the propositional connectives as itself being meaningful only in the context of a full system of representation (fancy way of saying that all sixteen basic Boolean functions are presupposed even if a minimal set of connectives is used).
In the context of all sixteen functions, I realized there were two more.
So, do you remember anything notable?
>>>> 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 fourfold interrelating POSSIBLE and >>>> NECESSARY. Deontic extensions are fourfold interrelating >>>> OBLIGATORY and PERMITTED. >> >>>> For quantificational logic, each variable has this >>>> fourfold structure. This corresponds with the indexing >>>> of quantifiers found in the cylindrical algebras of >>>> Tarski's later work. >> >>>> The negation symbol masks this fourfold structure in >>>> the formation rules for formulas. >> >>>> Yes. Logic in the absence of NOTjust like in the >>>> hardware of your computer systemsis 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 >>>> firstorder logicists) because wellformed 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 aleph0 and in fact recursively enumerable. Better to >>> range over sets and pull in that needed aleph1. >> >> 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". > > I am merely thinking that the reason that ZF is said to not be > finitely axiomatizable is simply because there are only aleph0 > expressions and at least aleph1 sets (Cantor/Godel.) And the axiom > schemes that are blamed are not the ones that actually address or > cause this. The schema would have to range over sets not expessions. > No references. >
I see.
In my imagination, they do.
What I mean by this is that the separation and replacement schemas merely ensure that one cannot use grammatical methods to create subsets that would conflict with our basic sense of what a subset ought to be. If we can use language to delineate some heap as a wellconstrued collection for which identity is meaningful, it certainly ought to exist in a theory of sets as objects.
And, thank you. You seem to have directed me to where I should be looking. If one opens a book on descriptive set theory, everything is aleph_1 without explanation. Now it makes sense. The consideration is prior and from sources not specifically associated with the theory.
>>>>> 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. > > But it is a good idea to get that fact out of the way early. > > All the best, > > CB >

