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12/4/12


An attempt at a finite consistency thesis
Posted:
Jan 24, 2014 5:51 AM


In December 2012 I began this series of analyses with the link,
http://mathforum.org/kb/plaintext.jspa?messageID=7935660
which arranges the names for 16 logical connectives as the points for an affine plane within a projective plane. The geometric relations account for the basic relations of negation, De Morgan conjugation, and a form of contraposition. The three operations can be expressed as a commutative diagram.
A second link established the functional behavior by combining ideas from Church and Birkhoff:
http://mathforum.org/kb/plaintext.jspa?messageID=7933608
The semantic structure is provided by two constructions. First, the incidence matrix for a trivial affine geometry corresponds with the incidence matrix for a tetrhedron  the simplex for 3dimensional space. Since affine geometries are associated with projective geometries, the intended use of the affine geometry to establish a semantic correspondence with the biconditional connective (LEQ) also permits relation to a complete connective (NOR).
The orginal first step is in the link:
http://mathforum.org/kb/plaintext.jspa?messageID=7933617
Upon further consideration, I realized that both of the possible planes in that construction should be used. The LEQ selection is bound with the NOR extension. The second possible plane is semantically extraneous. It carries, however, the XOR deselection and the OR extension for the deselected geometry.
Upon full interpretation, the 8 connectives of this construction will be configured as labels for the Vamos matroid,
http://en.wikipedia.org/wiki/Vamos_matroid
The particulars of the matroid specification segregate a single quadruple
{ LEQ, XOR, NOR, OR }
in the sense that this is the only quadruple containing two pairs of negations admitted into the matroid basis.
Conceptually, this obfuscates any implicit fixed truth table representations, making the system amenable to the notion that connective meaning is discerned from language structure through logical analysis.
The second construction uses a De Bruijn graph to serialize the combinatorial relations of the 16 points from the geometry.
The specific graph used can be seen in the link,
http://en.wikipedia.org/wiki/File:De_bruijn_graphfor_binary_sequence_of_order_4.svg
The canonical enumeration for the 16 truth functions begins with LEQ and ends with NOR. It is given in the link,
http://mathforum.org/kb/plaintext.jspa?messageID=7933628
This link contains fixed representations intended to provide a reader with some familiar sense of matters. The truth table for NOR is the concatenation of the last three columns. Irregardless of any given fixed representation, the first two components of a truth table are named
FIX LET
to coincide with the interpretation of the canonical enumeration. Any given system of truth tables can be put into compliance with this labeling by virtue of the geometric relations discussed earlier.
To the extent that there is, in fact, a fixed representation, it corresponds with the sequential interpretation of a De Bruijn sequence as described in
http://en.wikipedia.org/wiki/De_Bruijn_sequence#Example
Hence, there is no semantic presupposition in the assignments.
Before moving away from De Bruijn graphs, it would be prudent to observe that the metamathematical use of ordinal numbers as algebraic indexes is coordinated with the combinatorics of power set operations for De Bruijn graphs on 2 symbols. One can find an explanation for this in the link,
http://en.wikipedia.org/wiki/De_Bruijn_graph
This correspondence is to be considered in conjunction with the mathematization of quantifiers posted recently in the link:
http://mathforum.org/kb/plaintext.jspa?messageID=9370921
That link treats the ordinality of Peano succession in contrast with the incidence structure of the natural numbers based upon aliquot parts and unique prime decompostion.
In addition, there will be a type of incidence structure for with representatives for every cardinality 2^n. This incidence structure is a Steiner Quadruple System. It will play a significant role.
The point here is that the the combinatorics associated with power set operations arise from mathematical structures related to geometry and topology rather than from naive set theory. In addition, the role of enumeration corresponds with traversals of partition lattices. These lattices are nonBoolean. So, Boolean combinatorics can be accounted for without Boolean semantics.
The review of past work is almost finished.
As noted, the functionality of the names had been axiomatically stipulated following ideas from Church and Birkhoff. However, additional work is required in order to represent the compositional structure associated with the Fregean development of logic.
The system must be extended by a single symbol. This is accomplished by restricting attention to the axioms for which NOR is the principal connective and introducing a applicative interpretation in the sense of Shonfinkel and Curry. There are no combinators here. It is a simple applicative system base on the NOR connective.
However, because of the NAND connective, the applicative system can be extended to a simple magma. The system is extended by a symbol for this product, and, in the magma a connective that forms a product with itself has the value of its truthfunctional negation.
This work is presented in the link:
http://mathforum.org/kb/plaintext.jspa?messageID=7935661
The extended system supports the Fregean notion of "truth objects" because it admits language elements to be organized as a minimal Hausdorff topology. The first attempt at that explanation is given in the link:
http://mathforum.org/kb/plaintext.jspa?messageID=7933648
But, the significance is better expressed in the analysis of the TarskiGivant axioms in the link:
http://mathforum.org/kb/plaintext.jspa?messageID=9337125
The two key properties of minimal Hausdorff topologies are feeble compactness and the embeddability of semiregular topologies. The topologies used in Booleanvalued forcing, for example, are semiregular. And, feeble compactness accounts for a small countable "surplus". This seems to correspond with the sense of a set of measure zero that prevents acceptance of Freiling's axiom of symmetry in set theory,
http://en.wikipedia.org/wiki/Freiling%27s_axiom_of_symmetry
This completes the review of past work needed for this post. The material moving forward will attempt to coordinate certain arithmetical structures with statements made previously in such a fashion that the 17 symbols discussed above will be related to the arithmetical forms.
The post mentioned above describing the mathematicization of quantifiers involves an infinitary assumption of an infinity of primes. The next step is to offer a finite arithmetical characterization in terms of a system of Diophantine equations. There are two things significant with this system.
First, there are 14 equations. In topology, Kuratowski's 14 set problem is well known. But, few people associate topology with semantics despite the work of Tarski and McKinsey. What is also little known is that of the 16 basic Boolean functions that comprise the semantics of truth tables, 14 are threshold functions. Only LEQ and XOR fail to correspond with linearly separable systems of equations in threshold logic.
Moreover, LEQ is significant toward recognizing that Boolean algebras are not faithful models of the syntactic structure of the propositional calculus. This is discussed in the paper by Pavicic and Megill,
http://arxiv.org/pdf/quantph/9906101v3.pdf
To understand the significance of the expression of 14 elements in these systems, one must realize that the free Boolean lattice on 2 generaors which is usually associated with truthfunctional connectivity is orderisomorphic with the lattice representaton for the Cartesian product of the 4element subdirectly irreducible De Morgan algebra with itself. When one applies De Morgan conjugation to all 16 truth functions, there are 4 which are invariant. These are the 4 which are different from LEQ and XOR in the formation of truth table semantics. So, the "constancy" of logical constants derives from the De Morgan invariance.
It turns out that a system of 14 symbols forms 4 nonisomorphic Steiner Quadruple Systems. The thesis here is that it is appropriate to reduce these systems according to this relation to the incidence structures.
This is further supported by the fact that the 16 element affine geometry has 20 lines. The labeling for those lines in the original post, reproduced here,
http://mathforum.org/kb/plaintext.jspa?messageID=7935660
preserves the geometric relationships in the dual plane. The only difference is that the symbol NOT exchanges with the "all false" NTRU in the point labeling. One may consider the NOT to correspond with "absurdity" in standard treatments that use absurdity as an expression by which to define negation.
Now, the free Boolean lattice on 2 generators is also orderisomophic to a Boolean subblock in the lattice N_1 x 2 on page 433 (pdf pg 7) in the paper,
http://www2.latech.edu/~greechie/2008%20Coverings%20of%20%5BMOn%5D%20and%20minimal%20OMLs.pdf
If you examine N_1, you can see the relation of 6 interconnected elments organized into 3 orthocomplemented pairs and 2 additional orthocomplemented elments. This should be compared with the semantic relations discussed earlier. When this lattice is put into a Cartesian product with the 2element De Morgan algebra, it forms a 20element ortholattice having a Boolean subblock with 4 atoms and 16 elements. It is this subblock that is orderisomorphic with the free Boolean algebra on 2 generators.
All of the 14 element systems are taken to relate to these structures by way of the 4 nonisomorphic SQS(14)'s in relation to the 4 invariant truth functions that ground the semantic interpretations.
The arithemtical system upon which the finitary description of prime numbers rests is to be found in the Diophantine equations discovered by Jones,
http://en.wikipedia.org/wiki/Formula_for_primes#Formula_based_on_a_system_of_Diophantine_equations
This system is to be associated with the affine geometry on 16 points by virtue of its system of 26 variables.
Without concern for a specific choice, one variable is taken as a distinguished element. Thus, the ground for this construction is a pointed set. It is significant that the distributivity laws do not hold for the category of pointed sets. In the paper discussed above in which Boolean algebras are shown to not be faithful to the syntactic structure of the logic, it had been upon the failure of the distributivity laws that that conclusion had been reached.
In addition, there is the relationship of a pointed directed complete partial order to the structure of arithmetic in terms of a base element and a fixed point under orderpreserving maps into itself,
http://en.wikipedia.org/wiki/Directed_complete_partial_order#Properties
This is reminiscent of the Dedekind formulation of arithmetic.
To obtain 16 objects that may be organized into the elements of a finite affine geometry as in the first post of December 2012, the use of order 5 pandiagonal squares is employed.
There are exactly 16 such squares that are self similar. Mitsumi Suzuki has compiled those squares here:
http://mathforum.org/te/exchange/hosted/suzuki/MagicSquare.5x5.selfsim.html
Our objective, however, is to obtain a system with 17 squares.
Now, with respect to a De Morgan transformation, the 5 elements on the line at infinity are not fixed. Only one point on the line at infinity is fixed because it is the center of the collineation. In addition, a De Morgan transformation exchanges TRU and NTRU.
This duality is captured in the base 5 representation of the GraecoLatin pandiagonal magic square described in the link,
http://www.grogono.com/magic/5x5.php
With respect to the involution described above, let two copies of the center of the collineation be the center of the pandiagonal square labeled '33'. Observe that in the secondary diagonal, digits are exchanged when reflected across the main diagonal.
The scheme used in labeling the points at infinity had been the term logic quantifiers. These are mere schemes reflecting any extension of the logic by operators. Any extension, in the context of exclusion negation, has De Morgan conjugations.
In the case of mathematical logic, let ALL correspond to Ax and let SOME correspond to Ex. Then they form a De Morgan conjugate pair. Similarly, let NO correspond to Ax~ and let SOME correspond to ~Ex. This forms another pair.
Situating these pairs on the secondary diagonal to reflect their exchange in a De Morgan transformation will populate this pandiagonal square.
Now, the 17 desired objects have been obtained.
All of these constructions have been directed at formulating the truth table for the complete connective NOR.
Associated with the construction of a truth table is the combinatorial complexity of its formation.
There are two symbols from which one must be the preferred symbol to represent truth with respect to transformations in a deductive calculus.
There are two symbols which will correspond to propositional variables. They can have any order.
And, there are four rows which may also be in any order.
The possibilities correspond to a 96 element group,
Z/2 x S_2 x S_4
Based on a group representation capable of representing both a 3 symbol group and a 6 symbol group, a 96 element block design had been formulated. By virtue of the canonical enumeration, the block design has a lexicographical ordering. By construction, it is a (96,20,4,4) strongly regular graph.
But, the lexicographic ordering permits it to form a labeling for the 96 element Horton graph with automorphism group,
Z/2 x S_2 x S_4
There may be other such graphs. But, the principle being expressed here is that the combinatorial complexity must be accounted for. The 96 element Horton graph with a 96 element automorphism group expresses a minimal representation of this complexity. But, there may be others.
The links,
http://mathforum.org/kb/plaintext.jspa?messageID=8324328
http://mathforum.org/kb/plaintext.jspa?messageID=8334585
http://mathforum.org/kb/plaintext.jspa?messageID=8334586
describe the construction of the block design and its relation to combinatorial structures associated with the Mathieu groups.
The link,
http://mathforum.org/kb/plaintext.jspa?messageID=9335214
contains a labeling of the Horton graph on 96 elements.
The particular combinatorial structure used in the links above is the Miracle octad Generator and is related to the 24dimensional binary Golay code.
Recalling that the presumed fundamental invariant here is a 4element collection, the order of S_4 is 24. Moreover, when Curtis invented the Miracle Octad Generator, he formulated his construction to support the representation of a 21 element projective geometry.
The "constancy" of logical constants should be associated with the uniqueness of the Mathieu groups up to isomorphism. The Mathieu groups are in the family of 3transposition groups. So, too, are the finite symmetric groups.
From the standpoint of prooftheoretic semantics, the question of grounding the constancy of logical constants is an issue. Evidence for this can be found in the SEP link,
http://plato.stanford.edu/entries/prooftheoreticsemantics/#StrChaLogCon
where the duality of introduction and elimination rules is discussed.
The account given in this posting deals with a general duality across the spectrum of issues that include, for example, the association of paradoxes with a specific equation for the mathematization of quantifiers.
Where the account in the link above perceives duality with a preference for one class of rule over another, the construction here gives preference to the line elements because that is where the order relations coincide.
That the modern compositional forms of logic do, indeed, introduce problems concerning logical constants is discussed in the SEP link:
http://plato.stanford.edu/entries/logicalconstants/
For the most part, the construction has met its goals. There are some additional details, however.
First, in the link,
http://www.uwyo.edu/moorhouse/pub/planes16/
one can find projective planes of order 16 having 17 points on their lines at infinity. On lines 11 and 12 you can find planes with orbits that include an orbit of order 96.
Finite geometries relate to one another. Many incidence structures are recursive. So, in some sense, the construction described here is a "basepoint" with respect to larger incidence structures.
In fact, it appears that there may be an entire system of partial structures relating to one another. If one subtracts 5 from the number of elements in the free orthomodular lattice on 2 generators the remainder is 91.
Now, that lattice has 96 elements and its lattice polynomials correspond to the labeling described above. But, the 91point projective planes of order 9 seem to relate to one another as do the subdirectly irreducible De Morgan algebras.
To see why the projective planes of order 9 might be of interest here, consider again the system of Diophantine equations discussed earlier. According to the Wikipedia article, the system with 14 equations and 26 variables can be replaced by a system having only 9 variables. This is related to Matiyasevich's theorem. It is better described in the link:
http://www.scholarpedia.org/article/Matiyasevich_theorem
Now, that link expresses the fact that the 9 variable solution corresponds to positive values. That will be the focus here. However, when values are arbitrary integers, the number is 11.
Consider the 11dimensional lattice in the link,
http://www.math.rwthaachen.de/~Gabriele.Nebe/LATTICES/ANABASIC_11.html
I would be misrepresenting myself if I were to "interpret" that lattice in the context of this construction. However, the numbers 13 and 7 are significant with respect to the 91 point projective planes about to be discussed. The number 96 corresponds with the work above. I know that my own investigations have led to tangential references to the Kirkman points
http://mathworld.wolfram.com/KirkmanPoints.html
because of the Pascal configuration,
http://en.wikipedia.org/wiki/Pascal%27s_theorem
However, what certainly does correspond to the work above is the kissing number of 24. And, if I recall correctly, there is a Mathieu group M_11. But, I do not know if it is related to this lattice.
Now, turning to the projective planes of order 9, the failure of distributivity reappears in the mathematics. Of the 4 planes, only one is based upon a Galois field. The other 3 are nearfields based on a system called miniquaternions.
In the link,
https://groups.google.com/forum/#!original/sci.logic/uttWPezDLpk/FWg3Ob8C1AEJ
there is another construction based upon matroids. This time it is an 11elment matroid formed with the 15 elments among the line names  that is, excluding the NOT (absurdity)  less the 4 names corresponding to the truth functions that are invariant with respect to De Morgan conjugation.
This matroid again express a structure of order 14. And, from this structure, the miniquaternions are developed.
The 11 elements of the matroid are segregated into a group of 8 augmented by LEQ, XOR, and TRU. That group of 8 has a significant role in the link
http://mathforum.org/kb/plaintext.jspa?messageID=9347270
In particular, these 8 connectives are associated with the threshold logic of pixelation through edge intensity diagrams. The link,
http://patentimages.storage.googleapis.com/EP0853293B1/00620001.png
indicates the simple form of directionality involved. And, when the 8 connectives are correlated with the remaining 8, octonion geometry is expressed. The octonion geometry is discussed in the link,
http://mathforum.org/kb/plaintext.jspa?messageID=9334936
But, in relation to the 14 element constructions and the 4 element De Morgan algebra involutions, the origin of the quaternionic forms lies with the 91 point projective geometries.
On this account, it is also conjectured that the order of these projective planes is expressed algebraically by the Heegner numbers,
http://en.wikipedia.org/wiki/Heegner_number
To conclude these remarks, let me observe that I have spent the last year working on constructions based on selfsimilarity and based on eliminating Boolean logic. To the extent that a bivalent logic applies in mathematics, it applies in the sense of a Suszko reduction to support the presuppositions of a deductive calculus.
I have worked hard to distinguish nominal, metamathematical uses of natural numbers from ordinal and cardinal concepts.
My formal systems are based upon notions of fixed points and my fundamental relations generate tree structures.
With the elimination of a Boolean logic, Goedel's incompleteness is better expressed in terms of Diophantine unsolvability as follows from the work of Matiyasevich
It should be noted that Tarski's semantics is consistent with a constructive approach to mathematics. It is only the mistaken understanding that truth tables ground the recursive definition of formal languages. The axiomatization of truthfunctionality eliminates that interpretation. The axioms provide the necessary mathematical form for the recursive construction directly.
Well, I cannot be the one who decides whether or not the arguments I have given over the last year can be construed as justifying a finite consisency argument. It is, however, the best that I can manage toward that end.
I'm just a guy who swings a sledgehammer.
:)



