UNIVERSAL REALITY
UNIVERSAL REALITY
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.
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)!
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.
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 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.