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 6, 2013 10:59 PM


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....
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.

