Catuskoti: A logical treatment to its qualitative properties and
implementation on quantum double slit experiment.
Introduction
The conventional logic consists with two logic states which the truth assignment follows only true or
false. On the other, hand the event and its negation or the inverse. But in eastern method of reasoning,
the logic consists of four states, the catuskoti (chatur + koti) four corners. They are as follows,
Being, non being, (both) being and non being, neither being and non being, such is the
method that the wise should always use with regard to identity and all other
(Aurvedha)
All is so or not just so.
Both just so and not just so
Neither just so nor not just so
This is the graded teaching of Buddha
(Trans Ruegg p3)
The most important characteristic of the catuskoti is that it contains eight arrays and cannot be plot in a
conventional coordinate space.
If we use conventional logic symbols for an event p the basic, logic appears as,
i.
ii.
iii.
iv.
P
~P
P and ~ P
Neither P nor ~ P
Negations follow as,
i.
ii.
iii.
iv.
~P
(~P)
(both P and ~P)
(neither p nor ~ P)
More roughly speaking the third logic state is analogues to having an event, which
shares both its and its inverse property. In addition, the fourth represent the non-existing even
space, which is discussed later in more detail. In addition, the most important applications occur
in the first four states so it is most interesting of all.
Assumptions.
1. Events are defined only if they are observable.
2. Until it is observed the event is not defined and only an event space exists. Under
these circumstances the third logic states governs.
3. Events are distinguish from their properties
Definitions,
Event space.
The event space is where contains all events in their unobserved state. Observer dependent
function is mapped from event space E
Observer dependent function (ODF).
The function, which defined as O,
:
i.e. The function represents the tendency of observing a particular event, which the event space is at E.
=
I.e. the sub script denotes that the ODF tends to observe the event
The set S
The set S denote the set of all events, which mapped from the event space E from the function O.
Truth assignment
The function maps from S to {T, F}
: ,
property of an event.
Any event contains unlimited number of properties defining from the ODF. Property of an event
,
=
Where define as the resultant of each property. Such as,
=
Where, is the binding operator of the resultant.
Axioms
1. Let be any element of S, then the element exists iff there exist a mapping from S to {T, F}.
: , . ".
2.
Any event space is observable iff there exit a mapping from S to {T, F}.
: , #$% &
Property of an event '(
Event is a collection of infinite number of observable properties. Then,
=
Where is the property function and is the resultant notation.
Then any event can be written in a form such that to present the observed property rest
of its observable predefined properties. Then the notation for observer dependent function is given by,
=
If it is distinct from other observation of an event ,
,
Then can be simply written to be,
=
Which implies that have three different configuration,
1. ,
2. ,
3. ,
From 1 Can be define as the compliment of the event . Then if and 1 for , , &.
Then 2 & can be define as,
3 = 1
That is,
2 =
Then,
P 2
But,
6 2
For all
,
Now let consider an event defined as,
P
And
P
If this happens, the two basic axioms are violated and the ODF does not exist. Such negation is
defined the external negation and the previous negation is known as internal negation. This implies
does not exists. That is,
O: #$% &
Remark on
The resultant property is different from the sum of the property that is when the properties get
to gather to perform a resultant may be any of its sub properties will not observable.
For example number 2 have no all properties of number 1 though it can be made by two 1 s.
Negations.
From above discussion, two negations were observed. One was defined as internal negation 2, and the
external negation, , the fourth state of the catuskoti is defining under the occurrence of the external
negation. That is if to avoid abstractness in this the situation can be roughly taken into account as,
" "=>?@"&&
But
"&
If
"
Basic assumptions and properties of Observer dependent function A(
The function represent that the observation is done to observe . That is is predefined. This
is only
B: & ,
Consider = and 1 = , where and are observable events and let * be any operations.
Then,
1 1
This is of the form of uncertainty principle.
And also the observer dependency differ from function to function as,
=
And
= D #$% &$=" D &
This is of the form of the relativity.
For any truth assignment with respect to and
B , = B [1 ]
But
B , B [ 1 ]
This is obvious since , and * are three distinct events, which are interrelated but give some other
distinct properties.
Using the notion of property, let
6 =
And
6, = ,
Also according to above assumption
6 , = ,
Since events are different, ODF implies that,
6 6, 6 ,
That is,
, ,
Let another event D & is define such that,
, = , D
This resembles the importance of a resultant property than sum of properties in collected events. In
addition, the properties of event depend on the operator *.
Implementation to the quantum double slit experiment.
Let define an event space for electrons to be G. Then the ODF is define for any event of an electron
such that,
: G & H@I &
01
Now let is defined to be an electron was found to be passing slit 1 and be the compliment of it that
is the electron was found to be passing the slit 2 so that the properties are marked as,
6 =
02
6, = ,
Then the resultant property which observing the diffraction pattern is,
1 G = 6 ,
03
1 G G 1 G
04
6 , 6 6,
05
6 , = 6 6, 6D
06
And by above model,
So
But
Where, P {} is the set of properties that cause the diffraction pattern.
Really, what was assumed is that the wave function to be the event space of an electron.
Moreover, the slit was scattered with single electron (the worst case). That is the reason to take as the
conjugate of . Now observing the event and separately during the observation pull the logic state
to form G 1 G which merely end up with an observation with no diffraction pattern. What was
done to observe it, was just observing not the separate incidents but the resultant1 G .
The paradox regarding the experiment can be also described using the expression 06. ,
does not imply that the particle has gone through both slits. Yet this argument of the particle have to
come from both slit is visible in this expression. Instead of that, the third part makes a critical change to
the entire expression since the resultant is very much differ from sum of properties. Therefore, the
electron had never come from slit 1 nor from slit 2 in an awareness of an observer. Because to observe
the diffraction pattern the logical state was pulled to the resultant but not to a single event. In other
words the observation was applied not to the event space before the incident but after the incident.
The assumption, to define the event space in this situation, was that it has to be the wave
function of the particle. This still does not violate fundamental facts since according to basic quantum
mechanics it is still possible to define probability of the position that a particle can be observed is,
G = G G
