
Re: This is False. 0/0 {x  x ~e x} e {x  x ~e x} A single Principle to Resolve Several Paradoxes
Posted:
Feb 9, 2013 7:19 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?
Both are concerned with mappings to {true,false}. A propositional calculus proposition is 0place. A set is 1place. A relation is any number of places. (A relation is a set  of tuples.)
So you have the same rules of inference: Double Negative, DeMorgan etc. apply to propositions and sets.
To prove incompleteness, Godel had to generalize wffs as expressing propositions to expressing sets when the wff has a free variable.
CB
> >> 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 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 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

