fom
Posts:
1,037
Registered:
12/4/12
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Re: Matheology § 210
Posted:
Feb 7, 2013 9:41 AM
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On 2/7/2013 1:04 AM, WM wrote:
> Now try to learn to apply the basics.
And, again, you are to stupid to understand my
version of the basics.
In Quine or Carnap you will find discussion
of components for truth table semantics. With
regard to the representation of logical
equivalence, there are six components that
are admissible.
TFFT
FTTF
TTFF
FFTT
TFTF
FTFT
What follows is an explanation of how
those six symbols generate the structural
relations between the elements of logical
discourse. This is how your precious quantifiers
are realized...
Hopefully, the diagrams will display properly
in your browser. Sadly, you are even
too stupid to understand the slit experiments
of your own profession.
================================
Another poster asked for clarification with regard to
the lattice,
....................................TRU....................................
............................./.../..//\..\.................................
......................../..../.../../....\...\.............................
.................../...../..../..../.........\.....\.......................
............../....../...../....../...............\......\.................
........./......./....../......../.....................\.......\...........
....../......./......./........./...........................\........\.....
.....IF......NAND.......IMP.....OR.........................ALL........NO...
..../.\.\..../.\.\..../..|.\..././\.\\..................../...\.....././...
.../...\./\......\./\....|./..\./..\...\...\...................../.........
../../..\...\.../...\./.\|...../.\..\....\............/....../...\../......
.//......\.../\.../....\.|.\../....\.\......\...\......../.................
LET.......XOR..FLIP....FIX..LEQ.....DENY........./.../............/\.......
.\\....../...\/...\..../.|./..\...././......./...\.......\.................
..\..\../.../...\.../.\./|.....\./../...../...../.............../....\.....
...\.../.\/....../.\/....|.\../.\../.../.../...........\.........\.........
....\././....\././....\..|./...\.\/.//.....................\./.......\.\...
.....NIF......AND......NIMP.....NOR........................OTHER......SOME.
......\.......\.......\.........\.........................../......../.....
.........\.......\......\........\...................../......./...........
..............\......\.....\......\.............../....../.................
...................\.....\....\....\........./...../.......................
........................\....\...\..\..../.../.............................
.............................\...\..\\/../.................................
....................................NTRU...................................
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
...\
....O--O--O--O
although this Greechie diagram
swaps the right-to-left orientation
of the given ortholattice in order
to minimize the space-filling 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 x-axis through a small slit
in a screen. After passing through the
slit, its y-coordinate, say q_y, and its
y-component 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 truth-function 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.
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