(1) An Observation That Does Not Directly Yield a General Logic-system and How an Interpretation of the Data Yields a Physical Law That Does Yield a General Logic-system and a Consequence Operator. (2) Observations That Will Falsify the GID-model.
(1) Recall that natural laws are supposed to be human inventions or descriptions that depict natural-system behavioral regularities. Further, they are used to logically predict behavior. An accepted natural law can be specified in the sense that it applies only to certain entities. In this example, it is shown that one can accept basic observations that do not form a general logic-system or, how applying logical arguments (human mental processes), the data can be shown to represent a natural law - the box-law - with a corresponding general logic-system. Also recall, that general logic-systems and consequence operators are equivalent when viewed operationally, where general logic-systems tend to display additional refinements.
A box with the following properties was found in a cave in New Mexico and later examined in a laboratory setting. Three different sized round inputs are located at one end of the box and an apparent output at the other end. Also found were 120 balls in three sizes - small, medium, and large - which are denoted in what follows by the names a, b, c, respectively. These balls were found in 10 collections of 12 each. The 12 are collected into the sets {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}. We let Joe conduct 10 experiments using these found items. Joe inserted into the input each of these 7 sets of balls, one set at a time. Joe took no more than 5 seconds to insert a single ball or any of the collections of different sized balls. Immediately after Joe inserted a ball or simultaneously two or three balls of the same sizes as the input holes, the holes would close while the machine did its work. That is, Joe could not insert more than ball of the same size even if Joe had attempted to do so. Joes waited until the inputs holes opened and to be on the safe side and extra 30-minutes and then retrieved the balls that had come from the output and made an observation as to the sizes of the balls that emerged. In each case, the observations Joe made after waiting 30-minutes are the same Joe would have made at the moment the insert holes opened. The assumption, although it could be wrong, is that the processes inside the box require less than 30-minutes to complete. Joe repeated this experiment 10 times and recorded the exact same results. For the 10 repetitions, the actual order in which the 7 sets of balls were inserted was obtained in a rather "random" way. The apparent processes occurring inside the box led to the following results, where the ball name(s) on the left represent the input and the ball names on the right the observed output.
(If the S(1) relation between all of the subsets of ball sizes and other subsets of ball sizes were to satisfy the requirements for application of a general logic-system, then certain features would need to be maintained. Consider the behavior-signatures for steps (1), (2). These are, respectively, (1)' {(a,a),(a,b)}, (2)' {(b,b),(b,c)}. If there were a general logic-system, that yields the combined behavior, then for the single set {a} the general logic-system would yield {a,b,c} and if the hypothesis is removed as often required the result is {b,c}. In either case, this contradicts observation (1). If no other experiments were possible after those made by Joe, say the box caught fire and was incinerated, and these observations are accepted as a complete statement of the facts, then these observations would falsify the notion that a GGU-modeled natural law exists that yields each of the 8 observations. But, in this case, further experiments were possible.S(1)(1) {a} => {a,b}
(2) {b} => {b,c}
(3) {c} => {c,a}
(4) {a,b} => {a,b}
(5) {b,c} => {a,b,c}
(6) {a,c} => {a,b,c}
(7) {a,b,c} => {a,b,c}
(8) no balls => no balls
I gathered the same collections of balls and I tried to duplicate the exact actions that produced the Joe data. Again, the order in which the collections were inserted was "randomized." Further, I did 50 experiments and I noticed that if A was any collection of balls, then A was used, at least, once as the first inserted. This is what I obtained.
(In what follows, please notice that human logical processes are applied.) Why have I changed some input and all output symbols? Well, as a first assumption for the box-law, I did not assume that at each step that the balls inserted also emerged. This is the reason for the prime notation. Next, I assume that the same ball cannot be in two different place at the same time. In 5 cases, there appear to be "other" balls that emerge and that differ in some characteristic manner from the inserted objects, balls that are not as yet inserted. For example, suppose I'm "holding" all of these sets of balls. Then when {a} is inserted the other balls with names a, b and c, still remain to be inserted. They have not been used. Is it not self-evident that the medium sized ball not marked with the Joe observed "b" name that emerged, although it appears to be exactly equal to the "b"s that are still being held for insertion, has the different characterization that I'm not "holding it," so to speak. So, this is why I have named it "b_1". The informal S(1) general logic-system-contradiction for the step (4) Joe observation is now corrected using this "logical argument." However, there was still a problem with my observations for the {a,b},{b,c}, {a,c} and {a,b,c} insertions. Technically, if there is no difference between inserting, in a randomly ordered manner, the {a} and, when the hole opens, the {b} as single balls insertions and the combination {a,b}, then the {a,b} insertion should yield {a',b',b_1,c_1} and similar results for {b,c}, {a,c} and {a,b,c}. But this did not happen. So, I postulated that there are box-processes (part of a box-law for all such boxes) that when {a} and {b} are inserted together in there proper holes then the box processes bind a and b together in such a way that they do not act like single objects but act as a coupled pair. As a coupled-collection, they yield at the output only the same size balls {a',b'}. So, I needed to use for this pair a new symbol, say, a|b. This yields {a|b} => {a',b'}. By the same argument, the b and c types are coupled in another fashion denoted by b||c, and this coupling yields (5). In like manner, one obtains steps (6), (7).S(2)(1) {a} => {a',b_1}
(2) {b} => {b', c_1}
(3) {c} => {c', a_1}
(4) {a|b} => {a',b'}
(5) {b||c} => {a_2,b',c'}
(6) {a|||c} => {a',b_2,c'}
(7) {a|b|c} => {a',b',c'}
(8) no balls => no balls.
Different general logic-systems can generate the same consequence operator. Further, unless a special symbol and special relation is used and since a consequence operator always includes the hypotheses, the operator cannot differentiate between a hypothesis and the important fact that a hypothesis is not affected by a natural-process. However, in general, this is not the case for general logic-systems, where this important null-effect can be represented by a "prime" notation. If it is known that the object has either not been replaced or altered, then usually the "prime" notation means that exact same object appears after the process is applied. Thus far, however, the prime notation does not indicate this. A determination would need to be made by additional experimentation. Let's see what the actual formal general logic-system looks like when generated from system S(2).
When a general logic-system is used to generate a consequence operator C with values C(X) for a specific set of hypotheses X, it is required that the hypotheses X also be a subset of the conclusions C(X). This fact need not be included when a general logic-system is being considered unless one wishes to include the notion of the null-effects. The general logic-system S(3) may include null-effects, the prime notation, but this is yet to be determined. In this case, when the general logic-system algorithm is applied, the observed results will be those where the hypotheses is removed from the deduced conclusions. The non-extraneous information being generated by the general logic-system is how the box-processes alter behavior or produce null-effects. Technically, how one either generates a general logic-system that corresponds to alternations in natural-system behavior or how one employs one to determine natural-system behavior requires some additional analysis. The one being considered here is rather easy to apply due to the presence of the prime notation. However, in general, a second operator called "intelligent finite choice" needs to be applied in order to eliminate all extraneous deductions.S(3)(1) (a,a'),(a,b_1)
(2) (b,b'), (b,c_1)
(3) (c,c'), (c,a_1)
(4) (a|b,a'), (a|b, b')
(5) (b||c,a_2), (b||c,b'),(b||c,c')
(6) (a|||c,a'), (a|||c,b_2), (a|||c,c')
(7) (a|b|c,a'), (a|b|c,c'), (a|b|c,c')
(8) No set of axioms present.
Does this formal general logic-system generate the observations based upon the regularity notions I have introduced and argued for? The symbols a|b, b||c, a|||c, a|b|c denote the specific mentioned balls coupled with special processes. Pure general logic-system deduction from hypotheses would include as deductions the notation a||b, b||c, a|||c, a|b|c that represents simultaneous insertion, but these are not part of the actual final observed physical events and are considered as extraneous. Using these procedures, the following is how one would interpret S(3).
S(4)(1) From {a}, the set {a',b_1} is only produced.
(2) From {b}, the set {b',c_1} is only produced.
(3) From {c}, the set {c', a_1} is only produced.
(4) From {a|b}, the set {a',b'} is only produced.
(5) From {b||c}, the set {a_2,b',c'} is only produced.
(6) From {a|||c}, the set {a,b_2,c'} is only produced.
(7) From {a|b|c}, the set {a,b,c'} is produced.
(8) For no balls inserted, not balls are deduced.
The letter names should be removed and replaced with the terms, small, medium or large in order to get the actual box-law behavior in terms of the small, medium and large characteristics. The symbols |, ||, ||| now denote that you must insert the objects as a set in their appropriate holes rather simultaneously. This is the additional, obvious, requirement indicated by these extra symbols and adds to the physical description. This physical language interpretation is logically correct. But, to obtain the complete box-law as a general logic-system an additional analysis was necessary.
Although to obtain logically the physical description, it was not necessary, further experimentation was done in order to determine whether the prime notation is properly applied. The prime notation is actually used when a result is a null-effect. That is, that processes do not alter or replace the object. This assumption was tested. I gathered the same small, medium, and large ball collections using the emerged balls and put a small black-dot on each ball to be inserted. Thus, I used the same letter to denote the same size ball with its block-dot as indicating that the ball that emerged was the same ball as the one inserted. (Unfortunately, Joe could not do this.) The experiment was repeated and this time when the same size balls emerged appropriate ones did carry the additional identifying black-dot. This verified my assumption, under the additional non-cloning hypothesis. This additional fact can now be added to the physical descriptions. Of course, for the other balls that emerged, we don't know how long the "creation" of these new balls will continue.
So, this general logic-system does generate the same results as observed, gives us further insight into the box-law processes. Now there is a box-law that has been intelligently designed and can be used as a representation for my form of the displayed Joe observations. For this box-law to be a true physical law, the same results must be obtained for every such box, if more are found. The significance of this box-law is that it can now be included within a science-community's scientific theory, where one might need the generation of medium sized balls from the repeated use of the same size small ball.
But, why didn't the rules that Joe follows for observation include these "obvious" alterations? Why couldn't Joe show that there is an intelligently designed box-law that produces these results? Well, the reason is that Joe is actually a machine that has been programmed to perform the insertion operation and only observe the size of the balls emerging and then record the information. Is it possible that there are "intelligent" human beings that would accept the Joe observations without any further investigation? Well, this is almost the case. Although, thus far, all empirical data accepted by science-communities has been shown to be intelligent designed via GID, many science-communities do not accept all the analysis that shows this. They do admit that, at present, they have Joe type observations that have not yet yielded a "comprehensible natural law" that predicates the behavior completely. But, they also claim, "We, will find it - some day." Hence, although in these actual cases, the observations would, if not further "explained" via other GGU-model processes, technically falsify the GGU-model, these science-communities believe that they can intelligently design a natural law, like I have done, that will replicate the observations. By definition, this belief means that the natural laws and the actual natural-system behaviors they control are intelligently designed via the GID interpretation of the GGU-model. This fact cannot eliminate although it can be ignored.
This example does not give some of the actual difficulties associated with applying general logic-systems to physical behavior. The most significant is that general logic-system constructions are not unique and their applications usually must be restricted to particular scenarios. However, from a "logic" viewpoint, obtaining extraneous results not part of an actual observation is very common. An intelligent individual when predicting the behavior of a physical entity may use thousands of mathematical entities within a derivation that only gives a relative position statement at some specific "time." The individual intelligently selects only the results that apply to the scenario being investigated.
(2) Suppose that one uses the following general logic-systems to predict experimental events and the events occur. (A) = {(a,b,c), (d,e)}, B = {(a,b,d)}. Suppose that when both experiments are combined, it is observed that for every member of the language L = {a,b,c,d,e}, the results of the combined experiments is the set-theoretic union of the A and B results. As shown in [2], these combined results cannot be produced by a general logic-system. This would falsify, in the sense of Popper, the GID-model hypothesis that every form of natural-system behavior is controlled by a general logic-system. Note: If one assumes that any such combined events must be humanly comprehensible, then one can argue that the combining of such experiments is not possible unless the hypotheses are altered. But, this example still stands since its purpose is to show that there can be defined scenarios that are not humanly comprehensible in the sense of general logic-systems.
[1] Herrmann, Robert A. 2001. "Standard and hyperfinite unifications for physical theories," http://arxiv/abs/physics/0105012
[2] Herrmann, Robert A. 2002. Science Declares Our Universe IS Intelligently Designed, Xulon Press, Longwood, FL