Logic As The Language Of Innate Order In The Universe
Jeremy Horne, Ph.D.
15 Copper Hill Court
Durham, North Carolina 27713
(919) 402-9470
Our logic has assumed paramount importance as the foundation of modern computer science and much of artificial intelligence. Binary logic usually is the first and most common logic students encounter, but they rarely encounter meaning beyond it being a mechanical convenience for analyzing mathematical relationships and attempting to analyze ordinary language arguments. Yet, exciting mathematical, neurophysiological, and psychological work recently done in binary logic suggests a connection between our consciousness and the cosmos.
This essay is conjectural in some parts and cross-disciplinary. It does not purport to be a deep analysis of competing ideas, but I wanted to tie together some crucial observations to create enough of a focal point for questioning present conventions, re-directing pedagogy, and proffering groundwork for a philosophy of binary logic and consciousness.
The first section describes logical aggregation and its importance. Section Two briefly examines three examples suggesting that the ease of logical thinking depends upon ordering of operators. Cases found in human learning theory and Boolean neural networks suggest that each operator has a unique level of complexity. In Section Three, I propose a method for finding a natural order of operators that more closely fits the way in which humans think, and Section Four advances a procedure to analyze a seemingly unordered phenomena. The fifth section describes the philosophy upon which this proposed research scheme is predicated. Binary logic's syntax displays a semantics of order in the universe, as biophysical and cosmological research indicate. The syntax, itself, may be a semantic expressed by a deeper structure. Section Six suggests a direction in which research should proceed to understand how the source of our being may be communicating to us.
Prioritization and its importance
In parenthesis-free expressions, such as p & q v r, the truth value of (p & q) v r is different than p & (q v r). Using commonly accepted notation and by convention, the priority of operators is =, =>, v, &, and ~ in descending order of scope, or precedence. (Note that these symbols aren't proper, because of their having to be ASCII characters. The "=" is equivalence, "=>" containment, and "v" or.) That is, ~ affects only the adjacent variable, & affects only the variables on either side, v affects the & expression inside the parentheses and the first variable outside, and so forth. So, p = q => r v s & ~t would be grouped p = (q => (r v (s & ~t))), and p v ~q = r => s would be (p v ~q) = (r => s), with = affecting every variable, and ~ affecting only one (e.g.: Stoll, 60; Copi, 219; Massey, 34-62; Rosser, 19-23). The same occurs with arithmetic operators, as 9 + 5 x 4 + 3 would be [9 + (5 x 4)] + 3. It is generally recognized that the prioritization of these relational operators in logic is patterned after mathematical ordering, conjunction being analogous to multiplication, disjunction resembling addition, and so forth. Which operator has a greater scope than another is determined merely by convention (Church 1992, p. 79-80; Exner 1959, p. 38- 40; Rosser 1978, p. 19; Margaris 1967, p. 26; Copi 1979, p. 219).
As to the values that the variables may assume, there are 16 relationships generated from the four ways (00, 01,10, 11) the two elements in a basic linear order may be permuted. Each of these relationships may be seen as a way we describe how we know that the first element is related to the second. For example, the value of p as 0011 (the "0" traditionally regarded as "false," and 1 as "true") can be related to the q value of 0101 as 0111 (0 or 0 = 0, 0 or 1 = 1, etc.), because the relationship is "or." In standard truth table form:
p | q | p or q |
---|---|---|
0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 1 |
Prioritizing enters as a problem in parsing a sequence of variables connected by operators in a parenthesis-free expression. There are two cases. In the first case, the components of the idea are already known and determine the grouping. The second case is significant in this paper, where groupings of ideas are not known, but we group according to the convention described above. Logic texts, such those by Copi and Rosser will discuss the convention, but the reader if left wondering about how the problem of ungrouped expressions arises in the first place.
The convention serves convenient purposes, not the least of which is to preserve consistency in logical computation. For mathematics, it is easier from a visual perspective (because of the more closely spaced x and y) to multiply x times y first in xy+p, rather than separate the x from the y and add y to p. Even while maintaining the visual preference for doing the xy calculation first, the xy just as easily could have meant "x plus y" and the plus "x times y," the order now being addition first and multiplication second. Signs have not always meant the same thing throughout the ages, nor have they always been used. The Bakhshali (Indian) system, used "+" for "negative." In European mathematics, the plus and minus appeared at the end of the 15th century (Cajori 1993, p. 77).
Aggregation did not assume much importance until the end of the 15th century, when Pacioli in his Summa found a need to compute roots in polynomial expressions (Cajori 1993, p. 385). Subsequent treatments of roots depended upon appropriate aggregation techniques. However, notation usually arose after the concepts were formulated and had the purpose of punctuation, or separating the symbols that represented concepts. Roman numerals ultimately were replaced by the Arabic system we see today. Long division by Roman numerals demonstrates the superiority of the present method.
A literature search has failed to yield a convincing philosophical rationale for the current is prioritization convention. Quite to the contrary, as indicated above, standard logic texts refer to the ordering as a convention. Notations represent concepts, and the question at hand is why one operation should precede another. What we are looking for, of course, is an ordering of operators based on concepts rather than simple arbitrary agreement. Truth value or computational results obviously depend upon how operators are prioritized, but the importance of prioritization is greater than obtaining consistent calculation. Logical operators do not have the same degree of complexity, and the operators may be hierarchically ordered according to what constitutes complexity, as the following three briefly discussed examples indicate. The essence of the complexity may carry through to the result of the computation. Consequences of different aggregations Piaget wrote that the building block of a child's ideas of movement and speed is an awareness of serial order. A simple order is linear and "requires only a simple perceptual situation." (Piaget 1958, p. 36) Through adulthood, people's cognitive abilities depend upon apprehension of operational complexity. Piaget and Inhelder demonstrated that children learn in logical stages, i.e., "... memory is a function of operational developments)." (Piaget and Inhelder 1973, p.160) For example, a randomly selected five year old child will do conjunctive operations before disjunctive ones when given a task depending upon the conservation of length. The meaning of the addition (or) function is apprehended more easily than the multiplication (and) one, suggesting that each operator represents a level of intellectual complexity that affects the ability to memorize. More recent investigations confirm the same type of phenomenon in adults. (Taylor 1987, passim.) While Piaget's and Inhelder's research methodology may be slighted, the general thrust of a decade or more of their work points to differences in operational complexity.
Research indicates that learning in Boolean neural nets depends upon the arrangement of operators. A network is designed to search for its own structure to solve a problem by accepting data and attempting to discover the rule governing the relationships among items in the data. The rule must be recognized with the least number of errors. Operators, as logic gates, are serial with no feedback or back propagation. Researchers seek to discover the ordering of operators based upon ascending energy levels required for a successful discovery to occur. Energy is defined as "the discrepancy between the correct result of the operation and the one obtained from the circuit averaged over the number of examples Ne shown to the level required by that gate. Only when the result is zero can the correct gating be identified by the system. It is apparent that if the network is configured differently with the ordering of operations changed, the energy levels will differ as well. While the methods and schema for Boolean neural network computation are quite complex, the results indicate that there is an optimum configuration of operators the net uses to learn a task.
Patarnello's work on performance energy of neural nets corresponding to configurations of input bits as a way of ordering operators is supported by Martland's work classifying network behavior corresponding to truth table elements. The complexity of operators in Boolean neural nets has not been studied extensively, but it seems that the density of truth outputs varies with the type of connective involved, thus suggesting a basis for classification (Martland 1989, p. 222-234). In the same fashion, Kauffman has shown that when specific functions are forward fed into an element (propositional schema) specific patterns of function orders emerge. There are attraction points found within autonomous random Boolean network (BN) state-space (Kauffman 1993, Chapter 5). With respect to ordering in Boolean complexity, it can be demonstrated that numerous random couplings of operations result in patterns resembling those of cellular automatons (Wuensche 1993, passim). If logical operators are randomly coupled to produce patterns, it would be interesting to see what patterns emerge if the operators were coupled according to an empirically determined scheme.
Discovering a natural aggregation
A method exists for determining how the concepts of operations are prioritized according to the way we think. Piaget and Inhelder built a foundation to show that operators have differing degrees of intellectual complexity. While it is beyond the scope of this paper to explain the details, suffice it to say that an approach exists to constrain the operator to a specific learning parameter (color, size, speed, and so forth) and make their experiments mainly phenomenological, rather than having the testing rely upon words. One form of such an experiment I have created builds upon a Miller's Analogies-style of presenting information but pictorially represents the meaning of each operator, with the subject being asked to identify the meaning. Complexity is registered as a function of rapidity and accuracy of response. Each parameter may result in a different order or priority. For example, recognizing an operational concept (such as implication) may be more difficult with weights than with sizes. The priority given the operator would depend upon the time taken to recognize the operation and the accuracy of recognition. Several orderings may emerge from these experiments. For an artificially intelligent device, a task involving so many parameters may have that number of processors operating in parallel, each one configured to an ordering.
Applying empirically derived ordering
How can one use an empirically derived ordering? The significance of ordering may be seen by comparing the computational outcomes of several different orderings of a bit stream. For example, take a sequence like 0 v 0 => 0 & 1 v 0. Normally, this would be (0 v 0) => ((0 v 1) & 0)), the final result being 1. If the new grouping were ((0 v (0 => 0)) v 1) & 0, the result would be 0. Without knowing how the ideas are structured, what are we to do? What if the source meant: (0 v 0) => (0 v (1 & 0))? For ordinary language, the problem is somewhat simplified when the person starts the sentence with "if," for we know often this is the main operator. However, what do we do with the second part? Does the source want us to use & or v as the main operator? This does not mean that there is a universal way of parsing ordinary language utterances expressed in a parenthesis-free manner, but it does illustrate a problem of parsing a bit stream from what could be an intelligent source. As mentioned above, logic texts often raise the issue of unaggregated expressions but do not say when, where, or why we would encounter them. However, one can think of the binary digitizing of any phenomenon without pattern (seemingly non-repeating decimals, like sqrt(2) and pi, cellular automatons, electroencephalograms, or even data from the esoteric former Search for Extraterrestrial Life (SETI) project. one can think of the binary digitizing of any phenomenon (seemingly non-repeating decimals, like the square root of 2 and pi, cellular automatons, electroencephalograms, or even data from the esoteric former Search for Extraterrestrial Life (SETI) project.
How could an empirically derived prioritization of operators be used to derive meaning Many ways exist for using different aggregation schemes to extract patterns from an ungrouped bit stream?. The following example procedure is vastly oversimplified, possibly flawed technically, quite abstract, and merely suggests a general direction in which research might proceed in observing emerging order from such an ungrouped bit stream.
Taking 0s and 1s to represent values (as opposed to operators):
Identify an experimentally-derived prioritization scheme, such as v, =, => (using the commonly accepted notation), where v has the greatest scope of operation, = the next, and => the least. (Other analyses may use different operators and orderings.) The present example might be the ordering found for persons doing logical operations involving sizes. That is, persons might find that it is easier to do logical operations involving size with = than with v and =>.
Divide the bit stream into n+1 bit segments, where n is the number of operators in the prioritization scheme. With three operators, each segment would be four bits long. For 01100001110011001010..., this would look like 0110-0001-1100-1100-1010... .
Aggregate each segment according to the prioritization scheme by inserting the operators in between the bits. In the above example, in the first segment of 0110, the grouping would be (0 ( 1) ( (1 ( 0). The next would be (0 = 0) v (0 => 1), and so forth.
Evaluate the segment. Evaluate (1 => 0) first, to get 0. Next is (0 = 1), with 0 also being the result. Finally, 0 v 0 is 0.
Do the rest of the four bit segments the same way, the next being 0001, the aggregation being (0 = 0) v (0 => 1), and the value being 1.
Concatenate the results, 010 up to a length to be determined experimentally.
"Stack" the lengths on top of each other, such as:
010111011110001001
110001101010110001
110011000110001110
010100110001110000
Observe the patterns and compare to those generated by cellular automatons, or electroencephalograms, as suggested by Wuensche. (Wuensche 1993, p.11) These patterns, however, would be generated from logical operations based upon an empirically derived ordering of operators.
Philosophy of aggregation
Why would a natural ordering scheme tell us more about how we think? Why should this research tell us about innate order? Part of the answer lies in re-thinking how we view logic. Wuensche, Kauffman and others have demonstrated that patterns emerge from variously randomly coupled operators, each operator seeming to have a different degree of complexity. Two arguments may be advanced for regularity in the random coupling, each resting upon a view of causality. Either the patterns originate from something within the entity itself, or something outside the entity imparts that which is needed for the entity to generate the pattern. Modern philosophers refer to the first view as autopoiesis, or the theory of self-organization. How do systems self-organize? What "propels" organization? Artificial life and automatons are two types of apparent self-organization that have current interest. Adherents of the second view of causality would argue that an external agent is responsible for the entity; entities don't arrange themselves. Patterns don't simply "happen." The first assumes independence, the latter interconnectedness. Not denying the first for now, I will focus on making a case for the second, which, in turn may help clarify the discussion of autonomy and the nature of the complexity.
Operators and their ordering(s) are a reflection of complexity, as illustrated by the three examples above, and its structure in human thinking, complexity, and the universe itself, that logic represents. According to Piaget,
There exist outline structures which are precursors of logical structures,... It is not inconceivable that a general theory of structures will...be worked out, which will permit the comparative analysis of structures characterizing the outline structures to the logical structures characteristic of the higher stages of development. The use of the logical calculus in the description of neural networks on the one hand, and in cybernetic models on the other, shows that such a programme is not out of the question. (emphasis included) (Piaget 1958, p. 48).
Other researchers hold that propositional logic reflects an order innate in the universe and human thinking. The arrangement in the universe is according to "pregeometry as the calculus of propositions," such that "...a machinery for the combination of yes-no or true-false elements does not have to be invented. It already exists (Misner et al. 1973, p. 1209)." Everything is reducible literally to the primordial - first ordering. The binary structure may be a very natural expression of the way the universe exists in a fundamental and profound way. That is, the logic is a discovery more than a creation.
Our universe began from a singularity, or an undifferentiated phenomenon. Hesiod in the Theogeny and Lucretius in The Nature of Things spoke of a chaos, or unordered condition, prior to the beginning of our universe. It may be compared to Peirce's state of doubt, a feeling of undifferentiated or uniform energy. Bound up with the singularity was process; potential changed to kinetic, manifested by movement. Out of this "condensed chaos" came what we have in our dimension. We see this emergence of being into our dimension today at both the infinitesimal and the infinite ends of the spectrum of our discernible world. At the infinitesimal end of existence, it has been found that there is a pressure exerted by elements within a space deemed to be a vacuum. In this space of "zero point energy" are particle density fluctuations as photons enter and exit this discernible vacuum space with no known reason (Science "The Subtle Pull..." 1997, p.58 ). At the infinite end of the spectrum, Stephen Hawking's latest research indicates that microscopic black holes "..eat one kind of particle and emit another" (Science "Visions" 1997, p. 476). While p articles entering the event horizon may be "flattened" with their primordial constituents scattered over the boundary layer, and ultimately the lost information may be ejected from the black hole, this does not say anything of the force inside the black hole (Susskind 1997, p. 52).
The dialectic between the discernible (what we see) and the unformed (bound up within the singularity) is the first and most basic of processes. Out of this process came and still does come, from that which exists in terms of what is not apparent, or what we do know in terms of what we don't. Order was born as the object of this dialectic process, allowing us to discern existence through existents (our world around us through the things in it). This order is expressed by the language of logic.
What is the nature of this binary logic? Minimally required for order is a set of two elements, and each operators establishes a relationship between these two elements. In binary logic, the two elements normally are semantically regarded as "false" and "true" and "true" (often symbolized by 0 and 1, respectively). What we say is true or false depends upon our knowledge. Logic displays a structure of existents that comprises "natural" semantics, a structure standing as an ontology of knowing. (An example of such an ontological system may be found in James K. Feibleman's Ontology.) A thing must exist in order for us to know it, knowing being a way of accounting for an assertion. Setting aside criterion of how we determine whether something is true or not, the primary existents for what I call an "epistemological logic" are two: that which is known, or measured, and that which is not known This is not the same as equating the unknown with "false." Actually, "false" would fall into the category of "known," for one knows in order to state something to be false. Likewise, true also is in the category of known. For the other existent, unknown, something can be true but unknown and exist, such as the actual number of stars in the universe. (While asserting a truth presumes its existence, existence of something does not mean that one has knowledge of its nature. Hence, my use of the binary logic is not as a truth-functional calculus of propositions (the former), but as a structure displaying epistemological relationships (the latter).
Symbolically, we can say 0 represents the unknown (undifferentiated, etc.), and 1 represents known; 0 is prior and 1 subsequent. The unknown becomes the known. Another way of expressing this is that 0 contains 1, for the universe of the unknown is larger than and precedes the known.. Containment is the subject of deductive logic.
As a quantum semantics, 0 means a wave, and 1 the collapse of the wave function, or unity, where the observer apprehends the particle density fluctuation as information bounded by space-time. As a maximal expression of information, this would be regarded as an entropy, that which has emerged from chaos, or energy that has been expended (dissipated) to reveal information. Once the information has been expressed, it isn't expressed again from the same source in that space-time. In particle physics, the one is an absolute unit, a dimensionless number representing the speed of light, Planck's constant divided by 2 pi, and the gravitational constant (Hameroff and Penrose 1996, p. 520). Wave function collapse apparently is a constituent of our consciousness, i.e., those 0s and 1s may very well represent our thought processes.
This wave function collapse to the value one is seen in the cytoskeleton, or microtubules, of neurons. Tubulin subunits make up the microtubule and are dimers (a bipolar entity that can assume either a positive or negative state), and these act as binary computational structures (Rasmussen et al. 1990, p. 428-449). When polarization occurs in the gigahertz range (10^9 to 10^11 Hz) (Frohlich 1975, p.1412) among groups of these dimers, the neuron assumes a shape that seems to modulate the neural pulse (Hameroff and Penrose 1994, p. 517-518). However, the phenomenon seems to have cosmological correlates.
About 100 (10^11 Hz) to 1000 GHz most clearly shows the uniformity of cosmic background radiation (CBR at 2.73 K +/-.01 with a 95% degree of confidence) (Smoot 1995, p. 5), the same as black body radiation and about the same value as the natural logarithm e (2.718). Frohlich's upper boundary of 10^11 is the lower boundary of CBR, or the unit measure of 1 mm. More than being simply "true" in a semantics table, 1 very well may signify a resonance with CBR and what gave rise to it. As Penrose said: "...there should vibrational effects within active cells which would resonate with microwave electromagnetic radiation, at 10^11 Hz, as a result of biological quantum coherence phenomenon" (Penrose 1994, p. 352). If the 10^11 frequency is what "activates" consciousness, more support is given the view that the universe is, itself, conscious. Binary logic is the language describing this consciousness. What is the mechanism of the language, and, more importantly, its meaning?
Sixteen operators with four sets of relationships between placeholders for two entities (p and q) spatio- temporally relate the unknown to the known and the wave function (symbolized as 0) to its collapse (symbolized as 1). The conditional, so often the focal point of "paradoxes of material implication," consistently and faithfully describes the spatio-temporal nature of deduction, or the structure and processes of closed systems. One should note that the often used proper subset symbol (p < q , with p=/= q) for the traditional material implication "horseshoe" is incorrect, since it is the improper subset symbol, >= denoting deduction, that says that the first set can contain the second either totally or partially. With this correct use, there is intuitive, as well as logical sense of material implication. An element contains itself (0 >= 0, 1 >= 1). In popular logical parlance, the relationship is true, or 1. Obviously, 0 >= 1 is the case, and that leaves 1 >= 0 being false, or 0, hence completing the truth table for >= (described before as "=>") . What is known consists of a smaller universe than the universe of the unknown. Quantitatively, the universe of the unknown contains what is known. A particle (evidence of an instance, or "collapse") is a constituent of the wave. (Or, in Kant's view, the appearance, or instance, is bounded by the reality of the whole. Kant 1963, p. 185-186 and passim.)
Where do we go from here?
To bring the philosophical speculation and theoretical system constructs into the tangible domain, it would be useful to demonstrate empirically that the links exist among the 1011 frequency, the binary logic, and consciousness. While the technology currently may not be available, the following offers one possible route of exploration for such a test.
Technology is such that simultaneous Positron Emission Tomography (PET), functional Magnetic Resonance Imaging (fMRI), and electroencephalogram (EEG) measurements can be taken (and mapped onto each other) of an individual doing a mental task, such as learning the meaning of a logical operator (Science "New Dynamic Duo" 1997, p.1423). That is, the EEG can measure various structures exhibiting mental activity. With each logical operator, there should be two frequency ranges: that of the neural pulse (1-40 Hz) and the high GHz frequency that causes the microtubule to assume the shape that modulates the wave to produce the EEG matching the brain activity associated with processing a particular logical function. A confirmation that this approach has merit would be to re-introduce the measured electrical signals back into the brain structures to induce the subject to perform the mental task and possibly to report other thoughts that may be embedded in that code. While on the surface it may be that only the thought of an operator would emerge, there may be associated thoughts "grabbed" from other areas of the brain to create a more complete idea of what thoughts are associated with the random bit stream For example, see Newman for how stimulating one area of the brain induces activity in other areas (Newman 1993, p. 267, 270-271). In principle, the object would be to correlate the EEG with the logical operator or series of logical operations done by the subject. Would it be farfetched to suggest that an extended "truth table" of 0s and 1s might pictorialize the EEG wave form or that the 0s and 1s could be mapped to the EEG?
A similar approach of correlating 0 and 1 patterns to EEGs may exist for the random concatenation of operations done by Kauffman and Wuensche. Wuenche suggests that his basins of attraction diagrams resulting from a random concatenation of logical operations densities may even indicate an "...electroencephalogram (EEG) measure of the mean excitatory states of a path of neurons in the brain" (Wuensche 1993, p. 11).
A third area of investigation would be to correlate the patterns of conformational collapse on the surface of the microtubule to EEGs and the patterns exhibited in the work by Wuensche. If the display of the conformational collapse does correspond to the 1 in binary operations, and the 1 is symbolic of quantum collapse, then, this might bring us closer to showing that binary logic is the language of at least one form of consciousness.
Summary
I have presented the issue of aggregating logical operators in parenthesis-free expressions and discussed the importance of finding a method based on how we think. Three studies suggest that it is the complexity of the operators that determines the priority of operations in a parenthesis-free expression. If a string of 0s and 1s representing absences or pieces of information is generated by the complexity represented by operators, it would not be unreasonable to analyze such a bit stream using a prioritization that more closely resembles human thinking. Discovering such a natural order or orders is predicated upon a philosophy that is being borne out by emerging research in biophysics and cosmology. It gives new reason to an old logic.
Our logical thought processes, as expressed by 16 operators,
are ordered according to a type of
intellectual complexity. These processes are mappable to brain
structures, and the frequency against which
cosmic background radiation is measured drives these brain
structures, thus making logic as a language of
innate order in consciousness. Consciousness as we know it is
immanent in the universe.
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Copyright, Jeremy Horne, Ph.D., All Rights Reserved
This article originally appeared in Informatica,
Reprinted from issue no. 4, December 1997,
issued by the
Josef Stefan Institute, Ljubljana, Slovenia
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