
Re: This is False. 0/0 {x  x ~e x} e {x  x ~e x} A single Principle to Resolve Several Paradoxes
Posted:
Feb 11, 2013 1:12 PM


On Feb 6, 10:59 pm, fom <fomJ...@nyms.net> wrote: > On 2/5/2013 10:01 AM, CharlieBoo wrote: > > > On Feb 4, 12:25 am, fom <fomJ...@nyms.net> wrote: > >> On 2/3/2013 10:19 PM, CharlieBoo wrote: > >> <snip> > > >> ....................................TRU.................................... > >> ............................./.../..//\...\................................. > >> ......................../..../.../../....\...\............................. > >> .................../...../..../..../.........\.....\....................... > >> ............../....../...../....../...............\......\................. > >> ........./......./....../......../.....................\.......\........... > >> ....../......./......./........./...........................\........\..... > >> .....IF......NAND.......IMP.....OR.........................ALL........NO... > >> ..../.\.\..../.\.\..../...\..././\.\\..................../...\.....././... > >> .../...\./\......\./\...../..\./..\...\...\...................../......... > >> ../../..\...\.../...\./.\...../.\..\....\............/....../...\../...... > >> .//......\.../\.../....\..\../....\.\......\...\......../................. > >> LET.......XOR..FLIP....FIX..LEQ.....DENY........./.../............/\....... > >> .\\....../...\/...\..../../..\...././......./...\.......\................. > >> ..\..\../.../...\.../.\./.....\./../...../...../.............../....\..... > >> ...\.../.\/....../.\/.....\../.\../.../.../...........\.........\......... > >> ....\././....\././....\.../...\.\/.//.....................\./.......\.\... > >> .....NIF......AND......NIMP.....NOR........................OTHER......SOME. > >> ......\.......\.......\.........\.........................../......../..... > >> .........\.......\......\........\...................../......./........... > >> ..............\......\.....\......\.............../....../................. > >> ...................\.....\....\....\........./...../....................... > >> ........................\....\...\..\..../.../............................. > >> .............................\...\..\\/../................................. > >> ...................................NTRU.................................... > > > Is there a graphic of this online? Do you know of a smaller > > representation? Do you believe one exists? There is so much symmetry > > it suggests this is not the most efficient representation. > > > How in general do you know each link? > > > Of course the most efficient representation is a . . . written in > > a . . . language. > > I am working on an alphabet. Since my current understanding > of the functional behavior of truth functions consists of > 4096 equational axioms (16^3) the logical alphabet I am > developing is not tiny. At present, I have completed descriptions > for the 96 letters. The next level of complexity will involve > working out the details for approximately 40,000 geometric relations > between names....
WADR if you have to figure out umpteen things, then that is not a very good axiomatization. OTOH if this is just legwork and you plan to see the pattern in what you did to create a small set of rules, then all the better  but do you still need so many?
On July 2010 FOM "18 Word Proof" proves that some of the theorems of the Theory of Computation are axioms of Incompleteness in Logic. (I recently added that some of the theorems of Program Synthesis are axioms to prove the theorems of Theory of Computation.) But we don't have to list all of those theorems that may be axioms! As long as we list the ones used in our finite discussion. (The FOM thread lists about 10.)
CB
> I cannot make the above diagram simpler except to direct you to > do google searches for ortholattices, orthologics, orthomodular > lattices, orthomodular logic, or quantum logic. You will find > discussions of axioms and representations. But this is algebraic > logic. I am working on how to identify its objects in a way that > they may be viewed truth functionally. First, find the objects. > > Here is the excerpt I promised to explain the diagram above > relative to the text where I found it. Sorry about the undue > length. > > First of all, the easiest representation > is a Greechie diagram. These diagrams > are used to represent ortholattices and > orthomodular lattices that can be > understood as amalgams of Boolean > lattices. The points of a Greechie > diagram corresponds with the atoms > of the lattice and the directionality > of the connecting lines in the > diagram make a rigid, angular change > to indicate the connection point > for atomic amalgams. > > So, the given ortholattice takes the > form > > O > .\ > ..O > ...\ > ....OOOO > > although this Greechie diagram > swaps the righttoleft orientation > of the given ortholattice in order > to minimize the spacefilling points > I have added with the hope of > preserving graphics in newsgroup > readers. > > The next simplest presentation is > the orthogonality diagram, > > ..............OTHER > .............* > .........../. > ........../.. > ........./... > ......../.... > ......./......NOR...........NIF > .SOME.*** > .............\.........../ > ...............\......./.. > .................\.../.... > ...................X...... > ................./...\.... > .............../.......\.. > ............./...........\ > .............** > .........NIMP................AND > > But, the explanation of this > diagram lies with the actual notion > of an orthologic. The following > example is transcribed from "Orthomodular > Lattices" by Beran > > ==================== > > As an illustration of a general guiding > principle, consider the following experiment > in which a single electron, say e, is confined > to move along the xaxis through a small slit > in a screen. After passing through the > slit, its ycoordinate, say q_y, and its > ycomponent of momentum, say p_y, can be > measured by two measuring devices. > > This experiment can be used to define two > physical operations E_1 and E_2. > > There is one symbol common to both > operations. > > Symbol: > r > > Description: > e is not present > > There are three additional symbols > for operation E_1 > > Symbol: > a > > Description: > e is present and q_y > 1 > > Symbol: > b > > Description: > e is present and 1/2 < q_y < 1 > > Symbol: > c > > Description: > e is present and q_y < 1/2 > > There are two additional symbols > for operation E_2 > > Symbol: > s > > Description: > e is present and p_y > 1 > > Symbol: > t > > Description: > e is present and p_y <= 1 > > It is customary to identify each > operation E_i with its set of > outcomes, i.e., write > > E_1={a,b,c,r} and E_2={r,s,t} > > We now need to define a manual > and the relation of orthogonality > associated with the manual. > > Let E_i, ieI be nonvoid sets and let > M={E_i:ieI}. Let TOP denote the union > of the sets E_i. Given xeTOP and yeTOP, > we write > > x__y > > and call the > elements x and y orthogonal when > there exist E_j, jeI such that > {x,y}cE_j and when (x=y) > > A subset D of TOP is called an event > of M if and only if there exists > E_k, keI, such that DcE_k. > > The set of all events will be > denoted E(M). A subset NcTOP is > said to be orthogonal if and only > if x__y for all (x=y) of N. > > Under these conventions we shall > say that M is a manual if and only > if the following two conditions > are satisfied: > > 1. if E_i,E_j are in M and E_icE_j > then E_i=E_j > > 2. if E_i,E_j are in M and if N is > an orthogonal set such that > Nc(E_iuE_j), then there exists > E_keM with NcE_k > > Next, for KcTOP, define > > K^__={ aeTOP: AkeK a__k} > > Returning to construction of the > example, let M={E_1,E_2} be the manual > consisting of the operations > E_1, E_2. The orthogonality relation > __ defined by M is shown in > the figure > > ..............s > .............* > .........../. > ........../.. > ........./... > ......../.... > ......./......r.............c > ....t.*** > .............\.........../ > ...............\......./.. > .................\.../.... > ...................X...... > ................./...\.... > .............../.......\.. > ............./...........\ > .............** > ............a................b > > Let us now carry out the construction of > the orthologic affiliated with the manual > M={E_1,E_2}. > > According to the general procedure, we > obtain the orthologic from the following > list of events where we for brevity write > abcrst for {a,b,c,r,s,t}, etc. > > D=BOT=null > > D^__=TOP=abcrst > > D^__^__=BOT > > D=a > > D^__=bcr > > D^__^__=a > > D=b > > D^__=acr > > D^__^__=b > > D=c > > D^__=abr > > D^__^__=c > > D=r > > D^__=abcst > > D^__^__=r > > D=ab > > D^__=rc > > D^__^__=ab > > D=ac > > D^__=br > > D^__^__=ac > > D=ar > > D^__=bc > > D^__^__=ar > > D=bc > > D^__=ar > > D^__^__=bc > > D=br > > D^__=ac > > D^__^__=br > > D=cr > > D^__=ab > > D^__^__=cr > > D=abc > > D^__=r > > D^__^__=abcst > > D=abr > > D^__=c > > D^__^__=abr > > D=acr > > D^__=b > > D^__^__=acr > > D=bcr > > D^__=a > > D^__^__=bcr > > D=abcr > > D^__=BOT > > D^__^__=TOP > > D=s > > D^__=rt > > D^__^__=s > > D=t > > D^__=rs > > D^__^__=t > > D=rs > > D^__=t > > D^__^__=rs > > D=rt > > D^__=s > > D^__^__=rt > > D=st > > D^__=r > > D^__^__=abcst > > D=rst > > D^__=BOT > > D^__^__=TOP > > Hence, as a preliminary remark it may be > said that the base set of the orthologic > has exactly twenty elements. By means of > the D^__^__ the construction of the > orthologic is now elementary. The > resulting diagram represents an orthomodular > lattice which can be obtained as an > atomic amalgam of the Boolean algebras > 2^4 and 2^3. > > ....................................TOP.................................... > ............................./.../..//\..\................................. > ......................../..../.../../....\...\............................. > .................../...../..../..../.........\.....\....................... > ............../....../...../....../...............\......\................. > ........./......./....../......../.....................\.......\........... > ....../......./......./........./...........................\........\..... > .....bcr......acr.......abr.....abcst.......................rt.........rs... > ..../.\.\..../.\.\..../...\..././\.\\..................../...\.....././... > .../...\./\......\./\...../..\./..\...\...\...................../......... > ../../..\...\.../...\./.\...../.\..\....\............/....../...\../...... > .//......\.../\.../....\..\../....\.\......\...\......../................. > .ab........ac...ar......bc..br......cr.........../.../............/\....... > .\\....../...\/...\..../../..\...././......./...\.......\................. > ..\..\../.../...\.../.\./.....\./../...../...../.............../....\..... > ...\.../.\/....../.\/.....\../.\../.../.../...........\.........\......... > ....\././....\././....\.../...\.\/.//.....................\./.......\.\... > ......a........b.........c.......r..........................s..........t... > ......\........\........\.........\........................./......../..... > .........\........\.......\.......\..................../......./........... > ..............\......\......\......\............../....../................. > ...................\.....\....\....\........./...../....................... > ........................\....\...\..\..../.../............................. > .............................\...\..\\/../................................. > ....................................BOT.................................... > > ======================================= > > This explanation is a lot, I know. But, it > is what it is. > > As for my own researches, the notion of > semantics for a complete logical connective > such as NAND or NOR depends on the fixed > representation of a truth table. > > In considering the nature of truth table > semantics, I divorced my understanding of > a truthfunction from its representation. > > Given that there are six possible fixed > representations for logical equivaelence > (vectors containing 2 T's and 2 F's), the > "odd man out" position for NAND (its False > position) identifies the locus of constant > T values in a truth table. Similarly, the > "odd man out" for NOR (its True position) > identifies the locus of constant F values. > > Relative to this pattern matching, consider > the assignments > > There is one symbol common to both > operations. > > Symbol: > r > > Description: > TFTF > > There are three additional symbols > for operation E_1 > > Symbol: > a > > Description: > FFTT > > Symbol: > b > > Description: > FTFT > > Symbol: > c > > Description: > FTTF > > There are two additional symbols > for operation E_2 > > Symbol: > s > > Description: > TTFF > > Symbol: > t > > Description: > TFFT > > Without going into any more details, > this should suffice to explain why my > thoughts on the nature of logic and its > geometric foundation are somewhat > different from most.

