(p.113) Only concepts or properties so introduced may form part of a consistent and useful theory of empirical nature. The procedure may be comparable to Whitehead's principle of “extensive abstraction” [69].

  4.5.3. The complement: what we do not assert

  Above we introduced the set by selecting those elements ofwhich satisfy b A. This is enough to extract the property Ωb once we access the class collected by satisfying a A. This says something about the objects exemplifying the property Ωa, but we now ask what is the set of elements {X} satisfying

  what is the negation of a property introduced by collecting elements found to pass a test devised for this purpose?

  This is asking about the character of the negation of a statement (here taken to be the occurrence of a physical property). A proposition may be negated in several ways; ie the test for this may be constituted differently. In particular there is no set defined by writing

  The problem here is the range of applications of the quantifier ∀; to which universe do we have to apply the test Ωa? Its proper use is in the formula (1), which fixes the range of entities that may exemplify a property Ωb. The result (2) collects those elements of which fail to satisfy the test for Ωb.

  Our fundamental domain for the investigation must be . We may always encounter many entities which fail the test for belonging to this set. We may find:

  • The test is not applicable to the concept at all: “Red” cannot be applied to pressure, velocity hopes, or love. Any formal theoretical analysis is valid only in a limited domain of total experience.

  • The concept is not unambiguously definable, thus no crucial test is possible: Absolute velocity, number of clouds, and human motivations are not quantifiable in reproducible manner.

  • The entity may be subjected to the test, and either the result becomes ambiguous, ie it is not reproducible, or it depends on the observer's own situation: Some variable may be random or like velocity depend on the observer's own situation.

  After fixing A we may encounter a new entity Y:

  which definitely is found to obey

  (p.114) In this case we have learned something about the empirical reality, ie the complement property has been instantiated and the first element of a new class has been found. This may be used to extend our concepts to describe a broader range of experience. If this behavior is validated, the set may be constructed and we can extend the discussion to the class 𝒜 Thus the physical description of empirical reality can be extended to encompass ever larger domains.

  This model of concept formation hinges on the logical concept of negation. Negating the sentence:

  "I have found a red object"

  (3)

  may be either of

  "I have not found a red object"

  "I have found an object that is not red"

  Which one holds depends on the task set as a judge of the correctness of the statement. This may be found to make (3) true, but to deny its truth, we need to know more. Thus negation of an assigned test implies an extra investigation of the intended concept. Only when this is concluded can we use both a statement and its negation. In making a statement, we need to know what its validity serves to exclude.

  In formal proofs of mathematics, the conclusion proved is usually clearly stated, but it does not seem to be customary to tell what the proof claims not to be a possible state of affairs. Thus the standard diagonal proof of the cardinality of the reals only proves that all the decimal numbers cannot be numbered. To claim that this implies that their cardinality is “larger” than that of the integers implies a genuine extension of the concept of size of a set. The action is allowed as long as it may be used with impunity. We do, however, note that the existence and construction of the continuum is far from philosophically opaque [70]. The same holds even more with respect to those reals that complement the algebraic numbers.

  4.5.4. Is there a total world?

  Let us approach the ambitious task to describe everything there is. We proceed by catching one entity after another. To these we apply a class of tests {Ωk}. The entities investigated can then be assigned to sets k according to the outcomes. These sets may then be analyzed further with respect to additional tests to make each one uniquely identifiable. Let us assume that this is done by the classification

  (p.115) When the index set C becomes large enough (infinite) we hope that the full set 𝒲 will describe the whole world. This is taken to signify

  If a physical property has been found to be possible, it must be logically possible for it not to be realized.

  In addition we may require that the properties tested for by {Ωk} are mutually exclusive

  (4)

  The properties are the case or not the case independently of each other.

  We now take the set

  If this collects all information accessible about the empirical universe, it constitutes a description of all possible worlds. It includes all elements with W and excludes all elements for which W. When n and k independently go over all values possible we get a full description of the set 𝒲.

  The procedure fails if the description does verifiably not have a property necessary for the consistency of the world to be constructed. Remember that I am here still talking about properties as something which we know a procedure to test for. These conditions constitute the elements for a classical description of a “possible world”. It remains to be seen if our empirical knowledge about reality allows us the luxury of such a description.

  4.5.5. Classical approach

  The set of “all possible worlds” is a complete description in terms of all possible facts. The totality of facts determines what is the case and what is not the case. Each item can be the case or not the case while everything else remains the same. These words in Wittgenstein's Tractatus lay the foundations for “possible world semantics”. But even totally classically this program cannot be realized. An initial effort to reduce the facts to “sense data” has failed to lead to the desired goal. The facts affecting our orientation in reality do not form a denumerable sequence. We want to talk about many various entities, and it is not clear how to extract a complete set of independent “facts”, and it is obvious that they cannot be accommodated into a denumerable set. When all is said and done, usually something needs to be added. This requires complementing concepts without obvious limitations. On the other hand, even if such a set is built, it is not denumerable and thus quantifying over it is dubious to say the least. This is our version of the set‐theoretic statement: “There cannot be a set of all sets.”

  Even if all purely scientific issues have been brought into the frame of a theoretical description (which they have not), we want to discourse over hopes, beliefs, and love; all requiring totally new multitudes of concepts. There is (p.116) no reason to believe that such a process will converge. The various ranges of topics will require their own nonoverlapping sets of meanings. This may be what Wittgenstein acknowledges in his concept of “language games”.

  4.5.6. Quantum approach

  The ultimate theoretical description of scientific reality must rest on the conceptual framework of quantum theory. In this it is obscure what part is ontology and what is formalism. So far it has not been possible to derive features central to human life except within the technology of natural science. It may even be contended that it will never be possible. I may propose that what exists in science is manifestations of a universal substance called “energy”. Where there is no energy above the empty vacuum, there are no processes to be observed or described by scientists. The modern energy concept emerges as the successor to the classical concept of substance.

  An occurrence of energy offers an entity subject to scientific investigation. Such entities possess certain properties which allow us to recognize their occurrences in various contexts. Thus, e.g. the electron has a definite charge, spin, and rest mass. These variables are not appearing randomly in the empirical material, but take ranges of possible values: the spectrum of elementary particles emerges. These are the primary properties of possible excitations of energy.

  The accidental secondary properties of quantum entities are given by the quantum state imposed in an experiment by the initial preparation. In quantum theory it is customary to assign the term “system” to the entities assumed given. The quantum state determines all outcomes of experiments directed at the system under investigation. The peculiar feature of the quantum description of reality is, however, that it can make only probabilistic predictions. This happens after the investigator has decided which measurement he is to perform, and intending different observational procedures he gets different probability predictions. The concept of complementarity governs which procedures are possible and which are excluded. The investigator can relate his information to a potential outcome in reality only after having decided what to measure and calculated the corresponding probabilities. Then these quantum probabilities can be related to features of reality; the reality of the associated quantum state remains shadowy at best.

  The quantum description of reality is extremely well tested. It agrees with observations in all cases where tests have been possible. Where it produces numbers, these emerge with unprecedented accuracy. But it supplies a weird description of reality. All occurrences can only be interpreted as stochastic events, obeying a highly nonclassical probability calculus. The state as a property of quantum entities does not in itself allow direct observation; only its manifestations constrain possible measurement outcomes.

  The quantum description of a state cannot be complete in the sense introduced in the previous section. The two‐level state (p.117)

  assigns the probabilities

  (5)

  to an observable . In the same situation, the state

  assign to its possible observed outcomes the probabilities

  (6)

  The answer we obtain depends on the question we pose. The same state may entail different propositions in different interrogations.

  Any experiment determining probabilities must have access to an assembly of identically prepared basic systems (in principle an infinite number). Measuring these we may try to verify (5) or (6). Each observation utilizes one system, but each one can only contribute to verifying one choice or the other, both cannot be verified from the same subassembly of systems. The two variables cannot be tested on the same sample.

  In the situation described above, we have considered two observables (physical properties) assignable to the same system. However, we may not assign these probabilities independently. The quantum description needed to realize one set of probabilities cannot be changed without affecting the other one. Still, to give a complete description of the state of the system, the “state of affairs”, we need both, and in fact even more. There is no complete description in terms of independent properties. Thus we cannot give a complete description of the actual world in terms of lists of facts.

  Actually, quantum theory is even more intricate. We will have to include pairs of simple entities into the description of empirical reality. These pairs have correlates which are independent of the spatial separation of the constituents. The quantum correlations are based on the probabilistic interpretation of quantum states. These are conditioned by the amount of information imposed on the pair during the process of preparation. Thus the state we have to assign to the systems will depend on what conditions we assign to the process of preparation. If we learn more about the system here and now, we have to instantaneously adjust the expectations over the whole universe. This appears to be some queer action at a distance, but it is a natural consequence of the essential probabilistic interpretation of the formalism. This only predicts probabilities, its interpretation rests on assigning distributions to ensembles of (classical) measurements. When these are calculated, one must condition them on all information previously extracted from the system by anybody anywhere. The quantum probabilities are not determined by the knowledge possessed by (p.118) any particular individual; they are an absolute measure of the leeway left when all possible experimental information has been used to condition the observation. In practice all this information may not be available to any observer, in which case quantum ensembles need to be introduced. They require the use of density matrices, which, however, does not affect the general argument.

  Nonlocal objects can have nonlocal properties. If one constituent in a delocal‐ ized system is acted upon, the nonlocal relation may be changed. As an example I take my wife and me; we are married even if she is away in Las Vegas. If she divorces me there, I am instantaneously divorced even before the mail brought me the official papers confirming the act.

  One may object that such relations exist only within the social network of human relations. It is easy to see that this is not true: The stick A in Nevada is longer than the stick B here. If someone breaks the stick A where it is, stick B instantaneously becomes the longer one. Thus even classical two‐party relations may be affected nonlocally. This holds even more so if we deal with probabilistic relations. If I know the length of stick A, but I need to guess the length of stick B, it is a priori longer by 50% probability. When I learn here and now that it was manufactured just like A, I will know its length wherever it resides.

  As a pair of entities make up a correlated quantum system, we need to represent its state by tensor products of states referring to the individual constituents. Assume we have prepared the highly correlated state of systems a and b

  such a state is called “entangled”. This is a collective property if looked at from the viewpoint of the constituents; if we consider the pair as the basic quantum system, the entanglement is a property of this emerging new totality.

  We test for the property A, of system a, independent of the value of B in system b, we find that they occur with equal probability

  Thus: having no information about the observed value in b, we must assign equal likelihood to the outcomes at system a.

  On the other hand, if the value of B has been measured and found to be “+”, we have to assign different probabilities to the potential outcomes at a

  Thus knowledge gained at b constrains what can be obtained at a even if no one there knows the outcome at b. Thus information gained imposes its outcome on further measurements; the initial preparation has acquired a new constraint.

  This situation is sometimes erroneously described as a jump of the state of system a induced by the observation at b. That this is incorrect is most clearly seen from the fact that the correlation so described is independent of the actual (p.119) order of the two observations. Thus no information can be transmitted through this bipartite system. This is comforting because of the threat of conflict with relativity. The relative state of motion between the two constituting systems does not influence the observed correlation.

  Finally we have to inquire whether the quantum description can be interpreted as a possible world formalism. We have to remember that the basic unit of a system has predicates defined by the quantum state. This already excludes a full description in terms of independent properties or facts. We can measure an arbitrary component of the spin, but only when knowing the values of the spins in three directions can we assign a state to the system. However, we cannot know all these components either in preparation or observation. The outcomes of these observations cannot then be taken as independent.

  It becomes worse, however. Starting from individual basic entities, we know that we can prepare quantum‐correlated states which relate to the pairs as constituting complete systems. They offer many novel quantum systems which have to be assigned common properties by correlations. Affecting one of the constituents, we will significantly change the probability assignments for the other component. In addition, there is a range of possible actions, all leading to different results. The process of “teleportation” is an application of this possibility.

  We find that the quantum description of empirical reality is not of the form assumed in a possible world view of reality. It is not possible to identify a basic set of facts which fully and independently fix the state of the world. The fact that the observer can select what to measure, i.e. define a “context”, makes it impossible to describe even the basic units classically. In addition, we may posit combined entities which constitute states on their own right. This process seems to have no end: We can imagine a multitude of incompatible observers; there are more than two noncommuting variables. We have discussed pairs, but multiparty entanglement is possible and challenging. Here the paradoxical nonclassicality of quantum theory enters in full force.

  If quantum theory describes the world of reality, this is not a “possible world”.

 

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