THE
GENESIS OF ONE AND ALL
The
void, call it darkness or emptiness, by definition does
not exist, for in that transition toward becoming a
concept there must be something which for just that
instant can conceive of nothing: its own absence (of
something). Now, since there is but something, this is
all there is: a universe of all and nothing, one juxtaposed
the other. Yet this ‘all and nothing ‘ is an indivisible
element, for its universe has but one, itself, for member. But ‘one’ is meaningless unless there also
is a next or other one, for void has been accounted
for and offers no measure for one.
And then again, one as ‘only one’ would be an
unreasonable proposition: why not another? Clearly if there can be one, so can be another,
and a next one, and so on without bound other than by
‘an all’ that will admit up to an infinite number of
such members.
The appearance
of the second member (and a third etc.) heralds another
concept: the possible aggregate of the two (or more)
extant members in which discrete individuality is submerged
and their uniting quality (predicate) defines the class
of their composition.
In this
section we will discuss certain properties that are
clearly adjoined with, but not part of formal logic,
ostensibly because logic in order to be self-consistent
does not appear to require them.
Yet, this extension does more than complement
logic, as will be learned, it forges a remarkable connection
to the physical world and thus should provide a different
viewpoint for understanding the latter.
Class
theory does not provide a satisfactory reconciliation
of the contradiction embedded in the precept ‘the universal
class includes void and all’. It would appear self-evident that ‘void’ excludes the concept ‘all’
(but not vice versa: ‘all’ can admit ‘void’).
Nevertheless, ‘void’ is unquestionably contained
in the null or empty class, which in turn is included
with ‘all’ members of the universal class in the latter.
How can this enigma of class theory be resolved?
We will
start by conceiving a class that includes elements for
propositions that are negated in a second class with
otherwise corresponding elements; we can say each class
contains a sub class that is the complement of the other. The axioms of conventional logic ensure that
classes admitting exclusive propositions cannot coexist:
two mutually contradictory statements are not admitted
to be true simultaneously. In other words, they can neither be conjoined
nor disjoined in a super class, not even in the universal
class, except paradoxically: the null class is deemed
to include all incompatible properties and, hence, must
have an as yet undefined mechanism for making these
contradictions transparent in all classes of which the
null class is also part (the aggregate of members of
a class do not equate to the class itself). It would appear that the universal class has
two mutually exclusive attributes: it cannot include
contradictions that are yet included in the null class
which in turn is part of the universal class! The enigma
is clearly profound and appears to implicate conventional
logic as incomplete.
TO BE AND NOT TO
BE, THAT IS THE ANSWER
In daily
life objects will undergo state transformations with
the passing of time. Propositions that evolve over time are complex
comprising both temporal elements and ‘real’ elements,
the latter signifying a domain possessing spatial extension,
i.e. ‘not temporal’. Of course, complex propositions will exhibit
real components at any one instant, which, therefore,
can be subject of a logic statement.
In contrast, logic's universal class refers to
a domain of physical and abstract objects, and their
respective states that can be conceived all at any one
instant. Time does not play an integral role in the
relations of logic propositions.
Complex propositions are not exclusively ‘real’;
hence, their temporal components are outside the realm
of conventional logic. The universal class, while in some way subordinate,
is not synonymous with the universe of space-time in
which we exist. The
reason for this subordinate role of the universal class
to the universe of space-time is simply that, unlike
in the universal class, a proposition may initially
be false and then become true in our space-time universe,
and similarly two contradictory propositions may both
be true, albeit at different times, i.e. not at one
instant of time. This
difference between the universal class and the space-time
universe raises three further issues:
·
Temporal relations between a proposition's initial
state and any successor states may not be expressed
by conventional formal logic.
In other words, propositions that will admit
both real spatial and temporal relations, cannot necessarily
be presented in a planar, two-dimensional diagram (i.e.
Euler circles and Venn diagrams are inherently unsuitable).
Consequently, complex propositions will require
an extension of the form and discourse of formal logic
(and beyond sequential analysis).
Such extension must permit that relations be
formulated that cannot be realised in ‘flatland’- a
cartoon only makes sense because we appeal to the external
knowledge from our life in the space-time universe.
(Complex propositions including abstractions,
e.g. a music performance will express real components
as spatial state functions of a temporal relation; in
contrast, a music composition, a body of text and scripts
contains temporal elements, which are subject to spatial
relations. Other
apparently complex propositions can be expressed in
conventional logic, such as ‘diligent students will
succeed’, however, any temporal relation is effectively
removed if the future tense indicates a preordained
outcome: students who are diligent succeed).
·
Relativity theory shows that two observers in
different observation (reference) frames may disagree
on the physical state of the same real object (while
its state transformations remain wholly consistent with
the laws of Physics and with respect to those of all
other objects in each frame).
Since, however subtly different, no one shares
the same frame, it follows that no two observers will
discern the same subclass of the space-time universe.
While this effect has many important implications,
for the purpose of the present discussion it will suffice
to note that since members of the (conventional) universal
class are also members in the space-time universe, the
individual ‘logician’ in the class of above observers,
will encounter at least some subclass of members, their
predicates and relations of propositions that will be
unique and hence not part of the universe of the other
observer. In
other words, the state of an assertion cannot be considered
absolute (in the sense of conventional logic) but is
relative to both real and temporal conditions in the
space-time universe that cannot be expressed by conventional
logic.
·
If the class members and their predicates are
real components of logic that are connected by relations
in a spatial domain, then this domain must necessarily
extend into the quantum world since the components can
ultimately be elemental indivisible units.
By the same token, the temporal domain, into
which we will seek to extend logic, should not be assumed
confined to the conventional interpretation of space-time,
but must be part of the same quantum world, since temporal
relations can be formulated that are elemental in nature.
Relations
in a temporal domain represent a definite departure
from the conventional logic of states implicit in the
relations between elements of propositions that are
either true or false.
Yet, science and our experience of its manifestation
accept time as an intrinsic parameter (even though only
the former are generally founded in logic, while time
is not granted any such profound basis).
In order to illustrate the intimate connection
that the temporal domain has with logic, we will next
examine some examples of propositions, involving relations
between real elements, and show that these relations
are more differentiated than can be relegated to mere
semantics and than conventional logic appears able to
express.
Consider
the following assertions: ‘travel on foot is possible’
(is true), and ‘air travel is possible’ (is true); the
alternate state for both statements being not true,
i.e. ‘impossible’. The first statement may refer to a proposition
that relates the condition of the weather, the traveller
or of the terrain, to whether or not this makes travel
on foot feasible, and is hence either true or false,
one state being the negation of the other. Of course, this same statement could also be
a generalisation like the second assertion.
Yet, the latter was clearly conjectural before
the year 1783, when a manned hot air balloon first took
to the skies, (as the first was conjectural before our
human ancestors descended from the safety of the boreal
canopy), an unproven assertion that can neither be called
true nor untrue, i.e. lacking substantiation of either
state and which, hence, should properly be considered
vacant or empty. Furthermore,
unlike the absolute true or false states, the empty
state exists only in ‘anticipation’ of a temporal transition
to either the true or false state. In other words the above assertions are either
true or empty (or in the case of some different proposition,
false or empty, as opposed to true or false), depending
on whether or not they have been proved so (true or
false). Note
that unlike true and false states, which are absolute
and timeless, the empty condition's complement ‘not
empty’, would imply a domain in which both true and
false states are valid.
However, other than when applied to the universe
of the ‘all and null’ class, if the context of the assertion
concerns our macroscopic world, then the proper alternate
to ‘empty’ must be either true or false, and hence ‘not
empty’ is indefinite and would not be an admissible
state. (however, feasible and thus admissible in Quantum
Physics: 'not empty' would require time reversal in
the macroscopic world to gain foreknowledge that a certain
true or false state will enter into a superposition
of true and false).
Let us
examine the nature of the state transition that occurs
between successive classes. For instance, what is the relation between
a proposition, which is false in the preceding class
and finding the same proposition to be true in a succeeding
class? If not
for this one proposition with opposing states in successive
classes, the latter would be completely conjoined and
a diagrammatic representation of the two classes in
turn would find them coincident e.g. in the universal
class. Yet attempt
to depict both simultaneously and we find the proposition's
truth and its denial dissolve into the null class (note
the latter cannot be represented diagrammatically, even
though it is included in every class): evidently our
successive classes are not altogether on the same plane
of presentation- they do not exist in the same universe
(of classes), i.e. they are not conjoined and neither
can be disjoined since this would imply a super class
containing contradictory propositions.
In other words between the two states there exists
a temporal condition in which the state of the proposition
is in transition and indeterminate, both true and false,
and we might consider the null class a non-spatial conjunction
between the two successive classes in which exclusive
states sum up to null.
When
examining a proposition's state transition to or from
the empty class and the class in which it has a true
(or false) state, as distinct from that between the
true and false states, we will note that the temporal
condition of our proposition is neither true nor false;
the latter conditions being states only valid in the
preceding or succeeding class; in other words in this
transition the state of the proposition is empty.
In our
extended logic system we find that two successive classes
are adjoined by means of the null or empty class in
which a proposition is both true and false or neither
true nor false! We
must conclude that the null and empty classes must be
differentiated in our extended logic, each possessing
a distinct nature from their synonymy in conventional
logic, implying that the (null and empty) classes intersect
only in, hence not outside, the latter's spatial domain.
The consequence of the null and empty class not
being synonymous yet intersecting, implies that both
feature distinct and separate extensions to conventional
logic in a temporal domain in which relations can be
defined that produce respectively a superposition and
absence of states. Neither of these conditions can be readily understood in terms of
conventional time- but draw a curious parallel with
quantum theory: the temporal domain of the extended
logic lacks directional bias (time is not an ‘arrow’)
and extends in more than one ‘present’.
The state
of a (initially empty) proposition may be asserted in
ignorance of the existence of a proof that establishes
its proper state (and that also disavows the ‘not empty’
condition). Accordingly,
an empty proposition's temporal relation with its proper
state is relative to the state that was initially asserted.
Temporal relations may be removed altogether
from an assertion, if they can be considered disjoined
with any potential proof of a true or a false state,
i.e. by an act of faith at the expense of abandoning
all connection, even the relative nature of this connection,
to any unique proof of the assertion's state (by confining
the assertion to conventional logic which is absolute). However, if we should put our faith in an assertion's
state ultimately being proved, while making that assertion
before such a proof exists, then we have in effect acknowledged
the existence of an implicit temporal relation to either
state.
An example
of a proposition that subsequent to having been false,
proved empty is the proclamation ‘all slaves are free’
(once the abolition of slavery is proclaimed its assertion
refers to a class without members). Yet conventional interpretation will hold that a class exists whether
or not it has any members and would consider the proposition
a generalisation that is either true or false.
Nevertheless, the example possesses the same
duality of also being empty once we deem the concept
slavery irrevocably incompatible with being human(e)!
IN
MY TIME
We
have shown that in the extended logic system a statement
and its repudiation can both be true.
For instance a class exists whose members include
the conditions ‘to be’ and ‘not to be’ as true statements. Yet, it is clear that both conditions cannot coexist in (the same) objective
reality. We
are led to conclude that both conditions can only be
true if time has a dimensional quality (i.e. it is not
a scalar) contributing members to a class representing
‘to be’ and, representing ‘not to be’, an another
class that does not share any members with the former. The latter will not be some negative state
of being which would have an effect in objective reality,
but will be imaginary because ‘not to be’ means the
true absence of ‘being’. In other words, the extended logic system is
synonymous with a universe of space-time whose continuum
is defined by real and imaginary spatial and temporal
dimensions. Hence,
what we have called objective reality is actually subject
to the constituent components of space-time that lie
outside our physical reality; hence, the latter appears
to be relative and subjective. Accordingly, the temporal components can be said to determine the
reality that we experience as much as being a consequence
of our reality.
ADJUNCTIONS
The
ambivalent nature of the extended logic also finds an
echo in experiments involving quantum states of a system
of correlated twin particles (photons). In such experiments the results obtained by
an observer of one particle are instantaneously affected
depending on whether or not a second observer who is
non-local to the first, has made an observation (measurement)
of a quantum state of the particle's correlated twin. (However, the instantaneous action will only
become evident after correlating the two otherwise random
results- i.e. after the fact: Bell's theorem shows that
the uncertainty relation in quantum measurements is
preserved even when the choice of measurement is made
after separation of the two particles, and cannot be
communicated in time to explain the preservation of
the correlation, i.e. will not exceed the velocity of
light). These parallels may be less surprising if we
remember that propositions consist of elements, class
members and their predicates linked by relations that
quantify their state as either true or false- the above
discussion demonstrates that these will need to be expanded.
The dimensional
quality of time is also discussed in the context of
expanded vector
theory that has been introduced
herein. The
expansion of both vector theory and of class theory
presented above provides a basis for removing the cause
of incompatibility between Relativity and the Quantum
theory of physics: neither of these theories excludes
the possibility of time having dimensionality and allows
that reality is subjective - i.e. cause and effect are
impositions from our macroscopic reality, yet both pre-empt
the concept axiomatically. The vectorial nature of time, the imaginary
aspect of spatial and temporal vectors in the quantum
domain and their relation to the macroscopic ‘arrow
of time’ is further discussed in the main presentation
on extended space-time geometry.