The Wondrous Intelligent Design for Probabilistic Behavior

Robert A. Herrmann

(3 Dec. 1998; Last slight revision 27 AUG 2020)

Abstract:

In this article, it is shown specifically that physical system chance events as represented by theory predicted (a priori) probabilistic statements used in such realms as modern particle physics, biology, among others, are only random relative to the restricted language of the theory that predicts such behavior. It is shown that all such "chance" physical events are related one to another by a remarkably designed, systematic and wondrous collection of equations that model how physical laws specifically yield such physical events. A second result shows that all probabilistic "chance" behavior is produced by actions of an higher-intelligence as modeled by well-defined ultralogics. These results show specifically that processes that cannot be differentiated from those that mirror the behavior of an infinitely powerful mind control the fundamental underlying behavior associated with all physical systems that internally comprise our universe. From a Scriptural viewpoint, these results imply the existence of a complex yet specific design for the behavior of the most fundamental of physical world entities and this reveals signatures for God that represent His continual presence and control over all aspects of the physical universe.Before I start this article, I must make an important disclosure. All the Herrmann models mentioned and the results they predict utilize a few common everyday human experiences that can be verified within a laboratory setting. Except for the use of Principle (1) and the most basic mathematical axioms used throughout all of science, NO additional mathematical or physical axioms or presuppositions are required for the construction of these models. This must always be kept in mind, for there will arise the myth that the results have somehow or other been included within the axioms or presuppositions. This myth is entirely and utterly false.

[A large amount of useful material is presented in this article and a rather easily understood model is discussed in the body of this paper and it is still viable. However, as mentioned below, a Herrmann (2001) highly complex mathematical disclosure verifies other aspects of probabilistic behavior that the secular-science community considered as "impossible" to achieve. These aspects are briefly discussed at the end of this article and they formally appear in the appendix.]

From Nobel Laureate Louis de Broglie, comes the statement:

. . . the structure of the material universe has something in common with the laws that govern the workings of the human mind. (March, 1963, p. 143)In an interesting series of lectures that will be discussed more fully, Nobel Laureate Richard Feynman tells his audience

I am describinghowNature works. (1985, p. 10)These two quotations present a rather obvious fact that should come as no surprise. What Feynman and all other materialistic scientists do during their lectures, within their textbooks and journal papers is to

describein words, diagrams, computer images, and the like, physical laws that they claim will lead to a physical event or a change in a physical system. For humanly comprehensible "descriptions," as broadly defined, the following is fully discussed in Herrmann (1994) and is denoted asOne of the most fundamental deductive procedures that models the development of a physical system is, itself, modeled by the "consequence" operator. Such operators are discussed in more detail later in this article and many interesting properties appear in Herrmann (1987).Principle (1).How nature combines physical laws to produce the moment-to-moment "evolution" (i.e. development or changes in the behavior) of a physical system is modeled (i.e. mirrored, mimicked, imitated) by certain aspects of human mental activity.In the appendix, Principle (1) is illustrated, where electromagnetic radiation is being emitted by a collection of atomic structures as the collection travels further and further into an ever-increasing gravitational potential. There appear to be 12 physical laws being combined using 9 human deductive processes. It is shown that even to determine the correct structure to which this physical law applies and to produce the stated outcome requires additionally some type of "physical law" that also mimics human mental procedures. When all of this has been accomplished and this illustrated physical law is applied to a collection of excited hydrogen atoms, then the results that are observed are but "images" that display the changes in the "color" of the emitted photons.

There are displayed no actual descriptions for physical laws anywhere within "nature" itself. The physical laws and regions of application are constructed by means of human mental processes. Indeed, the usual machines used to "verify," by various modes of measurement various physical laws, do not exist in "nature" until they are built by application of human thought processes. If physical laws exist, at all, they must be hidden as actual objects from either direct or indirect human observation. Possibly this is what the following, attributed to Hermann Weyl, is attempting to convey in the form of a question.

Is it conceivable that immaterial factors having the nature of images, ideas, 'building plans' also intervene in the evolution of the world as a whole?Principle (1), although clearly restrictive, completely materialistic and secular in character, is verified, relative to deductive science, by the largest amount of empirical evidence that could ever exist since whenever a scientist predicts physical system behavior from a collection of hypotheses deductive human mental activity is applied. This even includes the application of principles used to gather and analyze evidence that would tend to verify a scientific hypothesis. Thus, the secular scientist is using modes of "description" to tell the world "how" nature "works," as well as describing experiments and machines to verify the claimed "how."

No one has ever observed physical "laws" as some type of objects within nature itself, even those claimed "laws" describing "the how" of the "unobserved" world of particle physics. Hence, a

cautiouschoice for secular science would be to admit that there is "something" going on within nature, called "physical" laws or "natural" laws that forces objects to behave in a predicted manner. And, whatever these "things" are they are but being "modeled" by human modes of "description" and logical deduction. These last two sentences depict the actual character of what science terms as "physical laws." [This notion as to what constitutes "physical laws" is not necessary in the "Solution to the General Grand Unification Problem . . . " (Herrmann 1994) for the physical laws are replaced by the much more general idea - the developmental paradigm. This allows the actual solution method to be apply to infinitely many different "universes" not just the one in which we dwell.]During his 1998 American tour, Stephen Hawking stated while visiting the President at the White House that in a very few years scientists will be able to "describe" all the physical laws that govern the behavior of every physical system that exists within the universe and use these laws to predict correctly such behavior. Once again, Hawking emphasizes the basic philosophy of science that drives the secular scientist. It is immaterial whether or not the Hawking philosophy is correct for the driving force behind

basic researchis that the human brain, let us call it the mind instead, is, at the least, capable of describing and comprehending all of the so-called physical laws that govern the workings of our universe. This does not alter the above remarks as to what the phrase "physical law or process" actually signifies.Let's call this Hawking philosophy theStrong Principle (2).I point out that during recent history strong principle (2) was not accepted by all scientists. For example, Nobel Laureate Max Planck wrote that:

Nature does not allow herself to be exhaustively expressed in human thought. (1932, p. 2)Of course, Max Planck is one of the founders of "Quantum Mechanics." Since strong principle (2) is not yet established, it's better to drop the "strong" from the Hawking Strong Principle (2) and consider it as but a partial statement. Thus, we modify (2) to include the word "probably," and accept that some of the present day humanly constructed theories are "probably" correct predictors for objective reality. [Philosophically, the theories themselves should be considered as probable in character (Cohen & Nagel, 1934, p. 393).] When theories are considered, they will be tacitly assumed to be among the ones that are "probably" correct predictors.

In a course on Quantum Information and Computers given at Cal. Tech. by John Preskill, we are told in the textbook he wrote for the course that:

. . . fundamentally the universe is quantum mechanical. . . . For example, clicks registered in a detector that monitors a radioactive source are described by atruly randomPoisson process. In contrast, there is no place for true randomness in deterministic classical dynamics (although of course a complex (chaotic) classical system can exhibit behavior that is in practice indistinguishable from random). (Preskill, 1997, p. 4)

Principle (3)is the notion thatfundamental physical system behavior is probabilistic in character.

[A]Clearly, Preskill requires his students to take a specific stance in the philosophy of science, a philosophic stance that cannot be scientifically verified in any manner. Although there are deterministic mathematical descriptions for behavior that cannot be distinguished from a pure statistical distribution that it is claim is produced by the "pure" random behavior of a physical system, an individual must not accept these deterministic models as reality.I repeat that according to Preskill and his quantum view, you must not accept these deterministic models as mirroring reality although there is no scientific method that can distinguish such deterministic design for physical system behavior from the claimed non-design displayed by pure random behavior. Of course, we do have the very great mystery why there is, indeed, a Poisson mathematical design being displayed when a large number of clicks are being considered. But relative to the behavior of each individual click, we are to believe that there is no possible intelligent relationship between the occurrence of one click and the occurrence of the very next click. It is this apparent "non-relation" between successive clicks that appears to make them "random" in character.

First, no one can truly force you to accept this philosophy. You can study chaos theory and other mathematical areas and accept that there are deterministic and highly designed physical world processes being modeled by the mathematics and these physical world processes only give the appearance of random behavior. Of course, I suppose that you would not voice your opinion while a student in such courses at Cal. Tech. However, more importantly, does the basic idea of "random" imply lack of design? Is the apparent random character of quantum mechanics the true underlying feature that describes physical system behavior at all levels of human comprehension?

The particle physicists claim that all physical system behavior is fundamentally controlled by the "invisible" and individually undetectable entities that abound within the realm of quantum physics and, especially, within a nonempty physical background called the "vacuum." The major evidence that such entities might exist in objective reality is that various theories predict that under certain hypotheses gross matter, that

canbe detected by a machine or a human sensor, will behave in a certain way. This is calledindirect evidence. But the entities being described need only be objects within ananalog model, a collection of fictitious entities that are used solely to aid the human mind to comprehend and describe processes that predict an experimental outcome although the entities and processes need not exist in objective reality.Many times the subatomic entities start out as imaginary. In the original paper Einstein wrote where he gives his model for the photoelectric effect, he called the photon [a Planck energy element] an "imaginary" particle-like entity. But, Richard Feynman in his book "QED: The Strange Theory of Light and Matter" (1985) [QED = quantum electrodynamics] insists that light particles exist in objective reality. However, in this same book in order to explain partial reflection Feynman states relative to the direction of an arrow that will measure a probability:

To determine the direction of each arrow, let's imagine that we have a stopwatch that can time a photon as it moves. This imaginary stopwatch has a single hand that turns around very, very rapidly. When a photon leaves a source, we start the stopwatch. As long as the photon moves, the stopwatch hand turns . . . ; when the photon ends up at the photomultiplier, we stop the watch. The hand ends up pointing in a certain direction. This is the direction we will draw the arrow. (Feynman, 1985, p. 27.)I point out that not only is the stopwatch imaginary, at least at the present time, but the vectors, the arrows, also do not exist in objective reality. Such descriptions form but an analog model for such behavior and nothing more than that. Indeed, some physicists who describe QED properties use such phrases as "a photon is absorbed" or "emitted by an electron" without ever considering any description as to how an "electron" absorbs or emits anything. An assumed process such as the directly unobserved interaction between a photon and an electron is an essential part of this QED theory. But, not withstanding these obvious logical faults, I will consider as another principle of modern science,

Of course, principle (4) immediately implies that principle (1) is empirical fact. Also note that, from a theoretical point of view, a theoryPrinciple (4),the principle of indirect verificationas fundamental to modern research. This principle states that if assuming the existence of undetectable entities or processes yields correct predictions for the behavior of gross matter, then these entities or processes should be assumed to exist in objective reality.Tis often consider better than another theory ifTuses few hypotheses.Assuming principle (2) and (4), human beings describe laws of nature as part of human theories and predict correctly from these theories physical system behavior, even if all such predictions are probabilistic in character. Consequently, the following should hold. Let G be a set of hypotheses for a scientific theory. The facts are that G need not contain all of the physical laws that lead to a prediction. The set G need only contain a description for the physical entities to which these physical laws are applied. Then the human brain takes G and performs upon G some sort of mental process that yields a prediction P. This prediction is then verified in the laboratory or, similar to the Big Bang cosmology, is

assumedto have occurred sometime in the past. These, at present, unknown human mental processes are indicated by theturnstilesymbol |--. If G does not contain all of the applicable physical laws, then these laws become part of the human mental process symbolized by |--. In either case, this simple sequence of events is symbolized by

G |-- P. . . . . . . . . . [1] In all that comes next, the fundamental hypotheses will include the previously mentioned four principles. These partially materialistic principles are re-stated as follows:

(1) How nature combines physical laws to produce the moment-to-moment "evolution" (i.e. development or changes in the behavior) of a physical system is modeled or mirrored by certain aspects of human mental activity.(2) Some of the present day and yet to be discovered humanly constructed theories are "probably" correct predictors for objective reality. When theories are considered, they will be tacitly assumed to be among the ones that are "probably" correct predictors.

(3) Predictions of correct fundamental physical system behavior will always require the behavior to be associated with a statistical statement that implies that physical system behavior originates from behavior that is always probabilistic in character.

(4) Although physical entities and processes described by means of a language using "physical" terms may not be observed directly, they will be considered as existing in objective reality if their continued use yields correct predictions.

It is self-evident that according to principle (2) that [1] displays an absolute fundamental process that occurs within the human brain. Expression [1] corresponds to principle (1) in that it displays a fundamental relationship between human mental processes and physical system behavior. Expression [1] and the next expression [2] can also be related to the

informational contentof a theory and Preskill claims that information is "physical" (Preskill, 1997, p. 4) in character. Thus Preskill and many others, I included, claim that there exists "something" within the physical world that can not be observed either directly or indirectly called information. However, it isspecificinformation, rather than Shannon information, that corresponds to a physical law in the sense that it is modeled by "language" and "descriptions."Mathematicians study patterns associated with human mental activity in various ways. There are general properties associated with these mental patterns and these mental properties can be modeled by means of a mathematical object called a

consequenceoperator C rather than the turnstile |--. (Consequence operators correspond to a general logic-system that gives exact rules for "deduction.") The turnstile relation defines aconsequence operatoras follows: for each P such that G |-- P

P ∈ C(G) and if P ∈ C(G), then G |-- P. . . . . . . .[2] Expressions [2] display, in the simplest possible manner, the possible existence of a physical process that is being mirrored by the properties of operator C. Since the consequence operator properties are fundamental to all physical science theory and principle (2) states that some of the probably correct humanly generated theories are known and these theories predict the behavior of physical systems within the universe, then, using principle (4), we should accept that it's absolutely true in objective reality that the behavior of operator C is mirroring the most fundamental of all physical processes.

One of the major characteristics of a modern scientific community is that it is exceptional myopic when it deals with scientific theory. A scientist learns and then concentrates upon a very complex often mathematically based theory and predicts verified results. But, usually, the scientist cannot step back and observe the actual essence of what has been accomplished through such efforts. What is actually happening in nature itself is usually hidden among thousands of pages of symbolic representations. This also happens when mathematicians investigate patterns represented by consequence operators as models and, using principle (1), how such patterns mirror physical processes.

A consequence operator is defined on all of the subsets of a language L. The language L can be very broad in character and can not only include the ordinary strings of symbols one associates with a language such as those strings displayed in this article but can include audio and visual impressions as well. Indeed, as mentioned, it can include all representations for all modes of human sensory perception via the notion of virtual reality. You need not start with a turnstile operator |-- as your basic operator. It has been customary to include in the set G not only a specific source event (the input) that leads to a specific outcome but all of the assumed physical laws that the human mind would need to predict the probability that the outcome will occur. For example, the physical source event for radioactive decay is the presence of a "decaying" radioactive material. It should be possible to apply certain general physical laws that explain such decay and, of course, applications of human mental processes, and predict the probability, with respect to time intervals, of the occurrence of the detector "clicks."

Then as another example, using the Feynman described theory for partial reflection, the actual source is a "photon generator" and the detectors are photomultipliers. As Feynman states it on page 17 of the above mentioned book, QED should predict that for a glass surface and a supply of photons of a fixed frequency, say red light, on the average 4% of the photons out of 100 photons are reflected by the surface at a specific angle and be counted by a specially placed photomultiplier. The input or

trialsare the 100 photons emitted from the source and the output events are the 4 photons detected by the photomultiplier. This customary approach can be continued. However, as mentioned, since the entire collection of all physical laws could be included in G, then the only differences would be in the source event information. Thus, we need only consider G, for the radioactive case, as a single description for the radioactive material, and, for the photon case, a description for the photon source. All the physical laws can be consider as what actually generates the consequence operator relation, a binary relation that has the set G as a first coordinate and the set of all P as the second coordinate.With respect to these event predictions, the essence of the concept of random is that the occurrence of one of these events has no influence upon the occurrence of the "next event." (Olkin, et al. 1994, p. 7.) The basic problem is what does the phrase "no influence" mean. One of the aspects of this new material is that the phrase "no influence" does not mean no design. Further, it will be shown that the phrase "no influence" relative to predictions produced by any physical theory means

[B] no influence that can be described using the theory's restricted language (Theory Randomness), with the exception of trivial identity styled relations. But, although principle (3) is used by most secular scientists to mean "no influence" by means of any physical law or process, which is the concept calledabsolute randomness, we show that there is no such scientific concept as absolute randomness. Indeed, absolute randomness is a false statement. We show that the myopia of the scientific community has prevented it from "seeing," so to speak, the actual wondrous design and the actual basic influences that must be present before any so-called random events can occur.In explaining the following new research findings, certain philosophical statements made by Feynman are upheld by the secular-science community. Relative to theoretic constructs, Feynman writes:

The . . . reason that you might not understand what I am telling you is, while I am describinghowNature works, you won't understandwhyNature works that way. But, you see, nobody understand that. (1985, p. 10)There is a reason why "Nature" behaves this way, but Feynman and other secular scientists simply will not accept the reason. From my experience, another aspect of the Feynman philosophy is correct with the exception of, at the least, one notable research discovery.

Finally, there is the possibility: after I tell you something, you just can't believe it. You can't accept it. A little screen comes down and you don't listen anymore. I'm going to describe to you how Nature is -- and if you don't like it, that's going to get in the way of your understanding it. It's a problem that physicists have learned to deal with: They've learned to realize whether they like a theory or they don't like a theory isnotthe essential question. Rather, it is whether or not the theory gives predictions that agree with experiment. It is not a question of whether a theory is philosophical delightful, or easy to understand, or perfectly reasonable . . . . (1985, p. 10)The notable exception is a theory that produces

theories of everything(Herrmann, 1994) and that solves the General Grand Unification problem. This falsifiable theory uses processes that satisfy, in general, principle (1) and, since it predicts the behavior of all of the physical systems that exist within our universe, it should, by principle (4), be accepted by all of the scientific community. In general, the model shows, with respect to principle (4), that our universe was created by "infinitely" powerful processes. The model is called the GGU-model. It uses processes and concepts associated with a mathematical entity called the Nonstandard Physical World that also, according to principle (4), should be accepted as existing in objective reality. But, the GGU-model can beinterpretedScripturally since the processes have signatures that mirror the behavior of an infinitely powerful mind. Hence, it is not philosophically "delightful" to the secular scientific community. So, such a Feynman statement would need to be restricted so as not to include the GGU-model.Thus, the fact that the secular scientific community presently rejects the GGU-model is a counterexample to the last sentence in the above Feynman quotation. Since the GGU-model does depend upon the existence of a "background" or "substratum" universe that is the domain for the universe creating operators, then, counter to principle (4) and the Feynman statement, this model is further rejected based upon the philosophical stance that no such undetectable substratum exists. But, the basic entities within QED are also directly undetectable. Thus, in modern science, postulating away the existence of such a background universe is not sufficient in order to eliminate the conclusion that physical systems within our universe are, indeed, controlled by processes that mirror exceptionally remarkable and wondrous mental processes.

As a final re-enforcement to the above four principles, where I have added remarks between the [ and ], Feynman states:

I am not going to explain how the photons actually "decide" when to bounce back or go through: this is not known . . . . I will only show you how to calculate the correct [Indeed, as perfect as one wishes]probabilitythat light will be reflected from the glass of a given thickness, because that's the only thing physicists know how to do! What we do to get the correct answer tothisproblem is analogous to the things we have to do to get the answers toevery otherproblem explained by quantum electrodynamics. (1985, p. 24)The theory [QED] describes

allof the phenomena of the physical world except the gravitational effects . . . and radioactive phenomena [nuclear physics] . . . Most phenomena we are familiar with involve suchtremendousnumbers of electrons that it's hard for our poor minds to follow that complexity. [But, in theory, a computer, which follows the specific logical rules of the propositional logic, should be able to make such calculations.] (1985, p. 7-8)The following discussion and results, although applicable to various physical scenarios, are being restricted to the realm of "particle physics." For probability models, where the probability does not vary between events, probability statements can be re-expressed in terms of the number n of describable

trialsand describable physical eventsAthat can occur during these specific trials. Since all of these theories use mathematical procedures to predict a probability p, the basic requirement for a probability statement is that the trials form arandomly produced sequenceunder a given set of conditions. If during the n trials, where n is a large number of trials, m eventsAoccur, then the eventAoccurs with a probability p approximated by m/n. The approximation becomes more certain as the number of trials increases "without limit." With the exception that one might claim, as Preskill has done, that certain events seem to follow a mathematical distribution, the "randomly produced sequence" statement cannot be established with certainty within a laboratory setting. Also as pointed out in the dictionary by James and James (1968, p. 285 p. 307),such definitions may have either logical or empirical difficulties.Of course, any description for the source that produces the trials, the trials themselves and a description for the events require a specific scientific language. Note that some trial descriptions are solely relative to "time." One calculates that over certain time intervals the probability that the detector will click is such and such. On the other hand, you might have a collection of different events that can be described in the appropriate language. For example, suppose that out of 100 photons the probability that they will be "reflected" at the measured angle is 0.04 (i.e. event

Aoccurs) and the probability that they will be "scattered" or not reflected at the measured angle is 0.96 (i.e. eventAdoes not occur). Each trial n and each event produced by a trial must correspond to some sort of "counting label" or else you would have no such probability statement.Since mental activity is used to predict probability notions, the n trials can be modeled by means of n very simple consequence operators, one for each of the counting labels and the n possible outcomes, of which m are the

Aevents. Let G be a fixed description for the source such as "A photon from S." Let each b ∈ B, where B contains more than one member, correspond to a description for the event outcome produced by each of the n trials. There needs to be n applications of a mentally produced statement such as "G yields a member of B." In the case ofAevents, m of these trials correspond to a fixed b ∈ B that describes theAevents. Using the concept called "a power set map," the following set of statements model the essence of this probabilistic scenario, where if b ∈ B, then there is an i such that 1 ≤ i ≤ n and b = a_{i}.

a _{1}∈ H_{1}({G)}, . . . , a_{n}∈ H_{n}({G}), . . . . . . . . . [3]where the counting label is represented by the n applications of the operator H (as denoted by the H

_{1}, . . . ,H_{n}symbols) as H is associated with the labeled a_{1}, . . . , a_{n}events.To indicate the intuitive ordering of any sequence of events, the set T of Kleene styled "tick" marks, with a spacing symbol, is used (Kleene, 1967, p. 202) as they might be metamathematically abbreviated by symbols for the non-zero natural numbers. Using the philosophy of science concept of simplicity, assume that our language L = {G} U B U T. (That is, an object is a member of L if and only if it is a member of {G}, which is just the G , or a member of B or a member of T. This is the definition of the

unionoperation U.) In 1987, it was discovered that there exists a set of consequence operators

Ç = {C _{i}=C({a_{i}},{G}) | 1 ≤ i ≤ n}, . . . . . . . . . . . [4]that satisfies the requirements of [3] and is minimal with respect to L and [3] (the Occam Razor requirement). (Herrmann, 1987, p. 2, Definition 2.4 (i). Note: This 1987 paper, unfortunately, contains numerous topographical printers errors.) In order to maintain the "no order" requirement and, hence, prediction that

[C] no two or more of the trial events b should be in a recognizable order (an aspect of theory randomness), principle (1) requires that any two or more members of Ç in expression [4], when combined, be a simple type of consequence operator such that the corresponding event outputs of this combination maintain this requirement. The appropriate combination that is absolutely necessary is called theunion consequence operatorand is denoted by C' = C_{1}V C_{2}V · · · V C_{n}(i.e. C'(X) = C_{1}(X) ∪ · · · ∪ C_{n}(X)). The union consequence operator does not usually exist. However, in the case of the set of operators in Ç such an operator defined on L does exist and is minimal in the sense of it being the weakest consequence operator (mental activity) that is stronger than each C_{i}and, hence, satisfies the Occam Razor requirement for models (Herrmann, 1987, p. 5). These consequence operators are the most simplistic entities one can use to model the actual rational processes that yield the probabilistic predictions and theory requirements.There is another minimal mental-like operator that would be applied after each C

_{i}. In human communication, one of the most significant processes is the selection from a vast collection of words and phrases a particular collection that faithfully describes an event. This operator is called thefinite human choiceoperator. This operator mirrors the physical process known as therealismoperator and might be considered as a slight generalization of the natural selection operator used throughout secular evolutionary theories. However, this operator is usually considered as but an integral part of each C_{i}. This operator, denoted by R, is restricted to members of Ç as they are applied to {G} and such operators as C'. Define R(C({G}) = C({G}) - {G}, where - means to remove the G from the deduced results. This yields for each (trial) R(C_{i}({G})) = {a_{i}} and R(C'({G}) = B.A source G

_{1}can have numerous associated but different B_{j}, 1 ≤ j ≤ k,Adescribed events probabilistically predicted by a specific theory. For example, simply consider the source as sunlight. Again these probabilistic predictions correspond to a set of consequence operators Ç_{1}each defined on a language L_{1}= {G_{1}} ∪ B_{1}∪ · · · ∪ B_{k}∪ T. Again these consequence operators must satisfy the union operator requirement. The simplest set of such consequence operators that contains Ç_{1}and satisfies the union operator property forms one of the most wondrous and beautifully designed of all mathematical objects: it is a "complete distributive lattice" of finitary consequence operators (Herrmann, 1987, p. 5). (Of course, such mathematical terms may be foreign to you.) In the appendix zip file, is a short PDF file that shows that if {G_{1}} is replaced by L, then even more remarkably this structure is a complete Boolean algebra. [For reference purposes, the set is H = {C(X,{G_{1}}) | X ⊂ L_{1}}, where if G_{1}∈ Y, then C(X,{G_{1}})(Y) = Y ∪ X; if G_{1}∉ Y, then C(X,{G_{1}})(Y) = Y.]Usually, the probability statement is considered to be more accurate only if you increase the number of trials n without limit (i. e. n -> infinity). In order to avoid the philosophical difficulties one might have with the "infinity" concept, one can use the "potential" infinite notion. In this case, one can describe this process by saying, "Pick any natural number you wish, then the necessary number of trials for an accurate prediction is greater than your choice." The conclusion that the set of all such consequence operators has this remarkable mathematical property is not dependent upon the number of trials.

Such sets of consequence operators have a much more startling property, however. Any set of consequence operators such as those in [4] and that satisfies the requirement that C' is a consequence operator are related one to another. In the appendix is a new nontrivial result that shows that for any nonempty set of consequence operators {C

_{i}| 1 ≤ i ≤ n, 1 < n}, each of which is defined on a language L and for which C' is a consequence operator defined on L, it follows that

C _{1}(C_{2}V · · · V C_{n}) = C_{2}(C_{1}V C_{3}V · · · V C_{n) = · · ·}= C _{(n-1)}(C_{1}V · · · V C_{(n-2)}V C_{n}) = C_{n}(C_{1}V C_{2}V · · · V C_{(n-1)}). . . . . . . . . [5]Although, there may not be a relation between individual events a

_{i}that can be described in terms of the restricted language used for the predictive physical theory, there is the relation [5] between each of the individual operators described above for a probabilistic model and that are needed to predict each a_{i}. Using principle (1), the consequence operators C_{i}represent the combined theory dependent physical laws that yield the events as predicted probabilistically. Hence, the physical laws modeled by the C_{i}are individually related by the n(n - 1)/2 equations that appear in [5].Expression [5] is what one might describe as being in a symmetric form. But, actually, there are many more such equations since the operation V is commutative (not dependent upon which of two operators one places on the right or left of the V) and, hence, any permutation of the operators between the ( and ) will also lead to many more such equations. Consequently, the events a

_{i}that are produced by the necessary consequence operators C_{i}are not absolutely random in character when the processes are described by the theory of consequence operators. This also indicates that the actual concept of "randomness" should be considered as only relative to the original theory restricted language that predicts the probabilistic statement.It is self-evident from paragraph

[A], the theory randomness statement[B], and statement[C]relative to recognizable order that such "randomness" need not be considered as a hypothesis within such probabilistic theories as QED. Indeed, theory randomness is a statement about and, hence, exterior to such theories and as such is a statement within the employed philosophy of science. For this reason, expression [5] is falsifiable. For if it can be demonstrated that[C]is false, then this will falsify the expression [5].

ConclusionsThe GGU-model (Herrmann, 1994) shows that it's rational to assume that an external intelligent action, an ultralogic, as constructed by an intelligent agent and an external object that behaves like a very compact collection of symbols that represents specific information, an ultraword, exist. Moreover, they can be used as the underlying entities that produce and sustain not only our universe in its development but can be used to create and sustain other "universes" as well. [The now complete GGU-model is a substantial refinement but these original results are still viable.] The particle physics community tends to accept the original Hawking principle, especially since their contention is that it's the probabilistic rules and processes of quantum physics that will be shown, shortly, to govern completely the four fundamental interactions produced by what are often called the electromagnetic, strong, weak, and gravitational "forces."

Due to the often-stated requirement that probabilities be considered as related to potentially infinitely many trials, many would consider the n that appears in display [5] to be potentially infinite in character. Notwithstanding this last possible requirement, since the n mental-like consequence operators that appear in [4] and [5] must apply to every predictive quantum physical scenario in the vicinity of every spacetime location throughout our universe, as assumed by particle physics, then the fundamental behavior that governs our universe can be described, in general, in the exact same terms as those used to describe the GGU-model conclusions if there exists an ultralogic that is the underlying control. If such is the case, then processes that mirror the behavior of an infinitely powerful mind control all physical system behavior for all of the physical systems that comprise our universe, both internally and externally.

The research result Theorem 2, in the appendix, gives a very significant signature for God's ever present control over all physical system behavior. It is a remarkable design by such an higher-intelligence. Relative to quantum physical behavior of photons and partial reflection as QED predicates the probability that an event will occur, Feynman writes:

I am not going to explain how the photons actually "decide" whether to bounce back or go through; that is not known. (Probably the question has no meaning.) (1985, p. 24)Feynman uses at this point in his lecture the terms "bounce back" and "go through" for what he later terms as new photons (emitted from electrons) and that reach the detector or photons that are "scattered" by electrons. The probability that an observed event will occur depends upon thecombinedprobability that a set of individual events will occur. Since physical systems are also controlled, in general, by ultrawords, it would be the ultranatural events and ultrawords that force the individual probabilities to combine in the appropriate manner that would be described as quantum physical "physical law." Consequently, this combined probability depends upon the probability that a describable individual eventEwill occur.The above Feynman statement may, indeed, be correct for the language of QED and quantum mechanics in general, but it is a false statement relative to the theory of ultralogics and the physical-like behavior of ultralogics. Relative to theory predicted probabilistic behavior, Theorem 2, in the appendix, shows explicitly that for any sequence of relative frequencies m/n that converges to a predicted probability p that an individual event will occur, whether the convergence be fast or slow. Moreover, there is a physical-like ultralogic P

_{p}that does, indeed, force the event to occur or not to occur in such a manner that the relative frequency sequence is duplicated exactly. Except for satisfying the basic consequence operator properties for what are called internal sets, it is significant, that the ultralogic P_{p}that forces a specific physical system to comply with such a probabilistic pattern isnotrelated to any known human deductive process. It is also significant that the P_{p}is a member of the nonstandard extension *Hof the set H.Preskill's (1997, p. 4) quoted remark is about a probability distribution, the Poisson distribution. The facts are that all such distributions that are obtained from a frequency (mass, density) function follow patterns dictated by an ultralogic generated by a finite set of ultralogics P

_{p}(i). Relative to Theorem 2, the only difference is that the "source" statement G may be more complex in that it describes a particular physical scenario and the "event" statementEmay also be more complex. Under the view that equivalent descriptions represented by G andEcontain all the necessary information that depicts physical system behavior for a particular physical scenario, then a frequency function f(x) is used to obtain the probability that the scenario will occur. A product consequence operator (an ultralogic) is generated by finitely many of the chosen P_{p}that appear in Theorem 2, and yields or sustains the appropriate sequence of events that satisfies the required distribution.The probability convergence property used in Theorem 2 can be dropped and the result still holds. This yields the following highly significant result.

For specifically formed types of physical events as they are defined in Theorem 2, these two levels of higher-intelligent design also follow without the sequence actually converging.As seen in the proof of Theorem 2, human modes of deductive thought will not be able to predict the individual occurrences of such behavior that is rationally designed and produced by an higher-intelligence.Using these event sequence ultralogics, one would conclude that the so-called "unregulated or random" behavior that is often associated with quantum mechanics, biology and other disciplines and the statistical behavior of entities within our universe is actually one of the most powerful signatures that an infinitely powerful higher-intelligence is sustaining these aspects of physical system behavior.

Many times the Scriptures present a linguistic model for God's attributes, "And God said, . . ." A major aspect of the linguistic attributes of human beings is modeled by logical operators such as the consequence operator. Hence, from a Scriptural viewpoint, the above startling results coupled with a theological interpretation for the GGU-model demonstrate God's continual presence and control over all aspects of the physical universe.

AppendixAn illustration, both mathematically established theorems and the Boolean algebra result can be found in this pdf file.

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Mathematics Department, U. S. Naval Academy, 572 Holloway Rd., Annapolis, MD 21402-5002