Non-linear dynamics as a dialectic logic

Hector Sabelli

Chicago, Illinois, USA

Abstract: Evolutionary science implies a dynamic logic. Non-linear dynamics can be interpreted as a logic of thinking processes that combines the dynamic perspective of dialectic logic, with the mathematical formulation that gives static logic its rigor and its applicability to computation. Process logic postulates (1) Asymmetric and transitive action (becoming) instead of static identity (being); an action is represented by a vector in phase space. (2) The coexistence and mutual implication of opposites (principle of universal contradiction) separated in space, time or respect (principle of local no contradiction), rather than their mutual exclusion (absolute principle of no contradiction); coexisting opposites are operationalized via the phase plane of opposites. (3) The co-creation of complexity, novelty and diversity by the integration of opposites, instead of determined, lineal, transitive implication.

Key words: co-creation, dialectics, dynamics, logic, priority/supremacy, process theory, scientific method, union of opposites.

          Scientific knowledge is (a) grounded on empirical data; (b) formulated clearly, if possible, mathematically (Kant); and (c) empirically testable (Popper [23]).  Method defines science.  Logic is the core of scientific methodology.  When Boole in his “Laws of Thought” formulated logic in mathematical terms, he transformed a philosophical discipline into a science which contributes theoretically to the foundations of mathematics, and serves practically in the design of computer circuits.  However Boole-Russell’s mathematical logic has profound limitations that derive from the static nature of the postulates they adopted from traditional logic, namely: static identity (for all A, A = A); no contradiction (if P is true, then no P is necessarily false); and the third excluded (either P or no P, there is no middle case).  Russell's paradoxes demonstrated the limitations of this static logic.  Assuming that entities are static is at variance with the evolutionary perspective of modern science. As truth is the adaptation of thought to reality, logic should also adopt a process perspective.

Modern mathematical logic no longer postulates a unique set of laws of thought.  In the last decades, mathematicians have developed in abstract multiple formal systems based on different sets of axioms, and an indefinitely larger number can be constructed. Scientific practice remains the ultimate criterion for evaluating the applicability of these various systems to reality, as well as a source for alternative methodologies.  One such is the process approach of Greek physiology (the account of nature).  The term logic itself derives from Heraclitus' logos, the logic of nature and thought.

Dynamic logic:  At the dawn of science, Heraclitus introduced the concepts of becoming and union of opposites as principles that govern reality and hence should govern thought. Similar process views had been developed by Buddhist and Taoist philosophers ("Tao," becoming as the cosmic law, is the Chinese equivalent of logos).  The recognition of evolution in cosmology, biology, and history, has recreated an interest in the process approach. Evolutionary science requires a process logic that deals with action and change, not stable entities; with actual oppositions, not abstract separation of opposites; and with creative processes in nature and thought, not only linearly determined causality and implication.  Quantum mechanics also suggests a departure from traditional logic insofar as it postulates that (1) the universe is made up of quanta of action (Plank constant); (2) particle and wave properties coexist (principle of complementarity) [1]; and (3) interactions create qualitative, non-linear leaps.

Although process philosophies have been also developed by Spencer, Whitehead and Teilhard du Chardin, a process approach to logic is largely limited to the dialectics developed by Hegel.  Dialectic logic, which in our century came to be fostered almost exclusively by Marxist thinkers, had the advantage of recognizing empirical facts essential to scientific understanding which are obscured and denied by mathematical logic, namely, the ever present becoming (so an entity becomes unequal to itself), the existence of coexisting opposites in nature, history, and mind (denied by the principle of no contradiction), and the existence and generation of a multiplicity of alternatives (excluded as third cases).  On the other hand, dialectic logic was not mathematically formulated except in very partial ways, and by a limited number of thinkers [11, 17, 26-28].  Some systems theorists [42] and trialectics [14] have explicitly incorporated dialectic logic. Temporal [25], and fuzzy [16] logic, may also be understood as partial formalizations of dialectic logic under another name.

Logic, neuropsychology and methodology:  Both static and dialectic logic have grown independently from empirical study of cognition. Piaget's proposal for an experimental epistemology" was rejected as "psychologism." Yet it may be reckless to invent norms for thinking disregarding those that nature built in our brain. Because human brain is the highest (known) product of natural evolution, our mind is predisposed to understand reality correctly [22, 24, 41]. This does not mean that every mathematical structure devised by human brain exists apart from it, but rather that the brain itself is part of nature, and therefore all the abstract systems it develops reflect, imperfectly, some objective phenomena. More important, neurophysiological processes should be taken as a model for human-made  systems of logic; structure of the organ of thinking itself reveals important features about the organization of nature [27].

          The human central nervous system shows a fundamental dorso-ventral asymmetry that corresponds to the direction of action, and a rough bilateral symmetry that corresponds to the opposition of eyes, ears, arms, legs and brain hemispheres. In the vertical dimension, the central nervous system is organized hierarchically, with simple processes (temperature regulation, mechanical movement and posture) located at simpler and lower levels (medullary and spinal) that have priority in both the evolution of the species and the function of the individual, while complex processes (cognition, creativity) are integrated at higher and more complex levels (cerebral cortex) that have supremacy of control. Such hierarchical but bidirectional organization of the nature's organ of knowledge organization suggests that also the levels of organization in nature may be best understood as a bidirectional hierarchy in which the simpler levels (physical, chemical) have greater duration and extension (priority), while the more complex (biological, human) have supremacy (priority of the simple and supremacy of the complex [27]). Clinically, we postulate the priority of the biological and the supremacy of the psychological [29], and the priority of the objective and the supremacy of the subjective [6]. Methodologically, we postulate the principle of mathematical priority and psychological supremacy, an approach that we have applied to the interpretation of the electrocardiogram [8,34] and of social processes [7,31]. Here we apply it to logic itself. Logic requires both a mathematical formulation, without which there is no rigor nor practical applicability to computer technology, and a psychosocial analysis of its assumptions and methodological choices. Our methods of observation, and the questions raised by our assumptions, often implicit and unconscious, determine the data we obtain from objective reality. Our current reductionist bias makes us to seek physical principles as explanatory for biological and psychological processes, but discoveries such as Pasteur's cosmic asymmetry from the observation of the asymmetry of biomolecules, indicate that we can also learn about simpler processes by examining their more complex manifestations --a new paradigm in science [27]. Ideally, logic should be mathematical, dialectic and physiological. Mathematical formulation separates process logic from verbal dialectics. Reference to the logic of nature characterizes the process approach, from Heraclitus' physiology to Hegel and Engels' dialectic of nature [10].

Logic, mathematics and the logic of nature: Both patterns of processes and ideas are forms (idea mean form in Greek) embodied in matter. The logic of mathematics governs, not only describes, the forms of nature. The same mathematics must then govern both natural processes and thinking processes. We thus interpret non-linear dynamics as a mathematical formulation of dialectic logic, using empirical cognitive science and process philosophy as guidelines for such an interpretation. A complex mathematical discipline like dynamics has greater chance to illuminate and guide the process of thinking than the relatively simple logic of classes in which Russell and others sought the foundations of mathematics. Gödel [12] demonstrated that arithmetic cannot be reduced to any simpler system of logic. Mathematization needs not imply the reductionistic assumption that all science could be reduced to physics, all physics explained by mathematics, and the entire edifice of mathematics be founded upon simple and universal laws of logic. Such mathematization increased rigor by sacrificing meaning. For instance, negation was reduced to its simplest form, the complement in set theory. Actually mathematics provides models for more complex forms of negation, such as positive vs negative, real vs imaginary numbers in arithmetic, and the inverse in group theory and oppositely directed vectors in algebra. Each of these models captures some facets of the hierarchy of simple and complex forms of negation that obtain in human discourse. It is important not only to find the simplest form of negation, but also its various fundamental forms.

          The entire edifice of mathematics is a formal representation of the logic of thinking, and of the logic of nature. The group of mathematicians who wrote under the collective name of Bourbaki proposed three pillars for mathematics, lattice theory, group theory, and topology. Lattice theory deals with an asymmetric and transitive relation, which process theory takes as a model for action and change; group theory deals with the coexistence of opposites in a set, that process theory interprets as negation; topology deals with continuous and discontinuous change, providing a logical model for creative reasoning. Process logic thus expands Boolean logic by giving a logical interpretation to Bourbaki's three pillars of mathematics.

The three postulates of Process logic:  Assuming that the laws of thought must correspond to the laws of nature, process theory [27] formulates a set of scientific hypotheses regarding natural and logical processes --scientific insofar as they have originated a number of research methods and empirical studies, including the phase plane of opposites [5,7], psychogeometry [9,30], sociodynamics [7,31], the biosociopsychological formulation of medical diagnosis and treatment [32,33], and methods to analyze the influence of neuropsychological processes on cardiac activity [8,34]. Natural processes, including thinking, are action patterns that connect opposites, communicating information and increasing their symmetry; symmetric opposites co-create material structures (complexity) and/or represent disorder. Correspondingly, process postulates an asymmetric identity, the coexistence of opposites, and creative junctions of opposites. Similarly, Xu and Li [42] take evolution, complementary opposition and self-organization as basic assumptions.

1. Dynamic identity: To be is to become. Whereas traditional logic is based on being and the separation of classes, the basic concept of evolutionary science is becoming, that creates unity in diversity, and diversity in unity (Heraclitus). Process logic formulates a dynamic concept of identity in its postulate of universal and asymmetric action. Action has the physical dimensions of energy x time, and it is readily applicable to social and psychological processes. Everything is an action, as the Planck constant has the dimensions of action. Identifying things or formulating a concept are mental actions. Both "to be" and "to assert", both objects and thoughts, are actions, as contrasted with traditional dualism that separates objective reality from the ideal universe of logic. 

          Standard logic neglects actions by reducing all verbs to the copula "to be", and further, interpreting "to be" in a static sense, although it is obviously possible to model mathematically other verbs via logical relations; action verbs may also be modeled by different types of catastrophes [39]. Static identity is defined as reflexive, symmetric and transitive (as the equality relation =), and class belonging and logical implication are modeled by the reflexive, symmetric and transitive relation < which defines lattice theory. Actually < describes becoming: each process is continuous with itself from its beginning to its end, in a unidirectional fashion A_t ® A_t+1, but A_t+1 does not become A_t, and there is a i = j such that A_t+j is not equal to A_t. As there is a fundamental asymmetry in nature discovered in biology by Pasteur [13], demonstrated in atomic [43] and in cosmological processes, and obvious in human processes, including thought [27]. Change is ordered, not random flux, nor a series of independent events, as required for some statistical calculations.

          In traditional logic, qualities are represented by classes. To say that x has the property A is translated as saying that x belongs to the class of entities with property A. This is represented by a point within the circle that represents this class in a Venn diagram. Classification is extremely useful in biology, and has been a major tool of logic at least since Aristotle. Yet, in actual thinking, concepts develop as categories around a prototype, with some members being more central and others more peripheral [19], rather than as mutually exclusive classes. In fact, classes are not mutually exclusive. One entity has many properties, including opposing properties, so it belongs to many classes. A person belongs to many social systems (family, work, club, etc) [35]. Each person's behavior includes opposing actions such as cooperating, competing, fighting and isolating, so these opposite actions do not determine four mutually exclusive classes of persons. Instead, classes can be defined by the intensity with which each of these behaviors are manifested. In clinical studies, we have used the multiaxial phase space of mathematical dynamics to represent the trajectory of behaviors in time [9,30].

          We propose then to represent qualities as axes or dimensions in phase space. As any one entity has multiple properties, the phase space has many dimensions, including the universal dimensions of action (time and energy). The minimum possible observation is a change, that requires at least two points for graphing. An observation is thus represented by a vector or arrow in phase space. Dynamic identity (representing continuity within change) is the trajectory generated by the succession of vectors in such a N-dimensional space. This allows one to represent how a given entity changes its properties in time. Often there is an alternation of opposites, such as the alternation of feet in walking. Two successive steps (right, left, right, left) represents both advance and a partial return to the origin. This is Hegel's notion of the negation of the negation, that Engels considered so important that he made it into a fundamental law of dialectics; likewise Stalin judged it so fundamental that he erased from it (the negation of socialism would then be a partial return to capitalism). 

          Obviously the world is not made of either unchanging objects or ever-changing flux, but there is both permanence and change in every process. Natural and human objects are not permanent and immutable things; they are transient structures which once formed, evolve in continuous interaction with their environment for some relative amount of time, to be  replaced later on by some other equally transient structure. For static logic, change is one thing and permanence is another; they are by necessity separate. Some models of evolution postulate states of relative stability separated by discontinuous revolutionary change; these include punctuated equilibrium theory of biological evolution, Marx, Freud and Piaget's concepts of development in stages, Kühn's idea of "normal science" separated by "paradigm shifts, and trialectics [14] postulate of stable "material manifestation points" separated by "mutations". For process theory, permanence and change coexist at all times. This union of change and stability illustrates a second principle of process logic, the union of opposites.

2. The union of opposites: the principle of universal contradiction and local no-contradiction: Opposition is universal. Every process coexists with its opposite (Heraclitus): harmony and conflict, asymmetry and symmetry, union and separation, positive and negative, male and female. Bunge [3] argued that oppositions are not really universal, there are chairs but there are no "anti-chairs," but chairs are static and artificial objects, not processes, and their form is adapted to the opposite complementary form of the human body. Fundamental physical particles have opposites, e.g. electron and positron, and in fact this symmetry of opposites has served to predict the existence of particles later on discovered. Actions create opposite reactions in physics (Newton) as well as in human affairs. Female and male, parent and child, hormone and receptor, illustrate the fundamental role of opposition at the biological level. The minimum of information is the difference between two possible values, hence the definition of the bit as its unit. Correspondingly, two valued logic plays a fundamental role in formal logic, and process logic starts with dialectic antitheses. Drawing a distinction or asserting a connection between two entities always involves two concepts.

          If opposition is universal in reality, then opposition must be included in logic. In contrast, it is excluded by the principles of no contradiction (nothing is A and no-A) and of the excluded third (either A or no-A). Other formulations of logic dismiss the excluded middle [2] or allow the coexistence of opposites. There are four fundamental ways of conceptualizing their interaction:

(1) Separation of opposites: According to mathematical logic, opposites are mutually exclusive, and more generally, what is different is also separate. One may thus separate abstract logical thought from actual psychological thinking --thought is one thing, and thinking is another-, and, more generally, thought is one thing and reality is another --thought is thought and always remains thought, while reality is reality and always remains reality. Following the axiom of no-contradiction, what is thought is not reality, and what is reality is not thought. A thing cannot be itself and something else.  A cannot co-exist with no-A.  Following the axiom of the excluded middle, something must be either thought or reality; there is no third choice: a thing must be one of the two mutually exclusive opposites (A or not A).  There is no uncertainty, no ambivalence, no contradiction, and no emergence of one process from another. Thinking is a real processes, that emerges from physical processes in human brains. Thoughts modify the way in which people behave. Reality creates thought, and thought creates reality. Reality contains thought in human brains, and forms which correspond to our ideas in objects and processes. Truth is this correspondence between the form of reality and the form of thought.

(2) Harmony of opposites: Harmonic interactions between opposite forces, and cyclic changes in predominance are implicit in mechanics.  Eastern philosophers thought of change as resulting from the harmonious interaction between opposites.  They thus focused on the cyclicity of nature, and advised persons to let themselves flow with the natural flow of nature (Tao). But the stagnation of Eastern civilizations says that harmony alone does not move us along. The idea of harmonic opposites has been taken up by system theorists and holistic philosophers. Trialectics denies that oppositions actually exists in nature, except in balance. Periods of stability are due to the equilibrium of opposites which results from a balanced circulation of energy (trialectics axiom of circulation). Trialectics purports to integrate the timeless permanence of logic and the endless change of dialectics as a logic of cycles. As repetitions, cycles, equilibrium and balance cannot explain creativity, trialectics postulates that situations and things change according to preordained patterns. In trialectics, change is not viewed as the creation of novelty, but as the appearance of what has already been established.

(3) Struggle of opposites: Many Western scientists and philosophers have seen progress as the struggle of opposites. This is the dialectics, a Greek word for dialogue, but here meaning debate as in a legal trial.  Dialectics is a projection of human passions. The idea that progress is promoted by struggle permeates Western civilization. The foundation of modern society resides on the idea of competition between individuals--man is the wolf of man, said Hobbes. Darwin explained the evolution of species by their struggle. Marx explained evolution of society by the struggle of classes; shorn of the Marxist ideological vocabulary, Western political behavior closely mirrored Soviet policy in its dedication to conflict and competition. Even more darkly, racists demagogues, and religious spiritual leaders, have promoted a philosophy of hate that has led to inquisitions and concentration camps.  Even the ideologies of liberation have been conceived in terms of a struggle--of women against male supremacy, of the poor against the rich, of new nations against older empires.  Western tradition has preserved Heraclitus' dictum "War is the Father of all things", but has forgotten the second part of his saying "and Peace is their Mother" --since Heraclitus always spoke of opposites, he could not have omitted it himself.

(4) The union of opposites: The theory of complementary opposition [42] and process theory [27,29] return to Heraclitus/Lao-tzu notion of the union of opposites, where harmony and conflict, objectivity and subjectivity, union and separation, symmetry and asymmetry, and in general all opposites coexist necessarily, albeit in various proportions, and with alternative preponderance of one or the other. Only to the naive a person is either good or bad; even when woman and man are fundamentally different, there are also degrees of maleness and femaleness in all humans. Oppositions are information. Information asserts something and denies its opposite. Omni determinatio is negatio (Spinoza). The mutual implication of opposite concepts was highlighted by Hegel and more recently by Trier and other semanticists [21].

          In traditional logic, truth and false are mutually exclusive. Actually nothing is absolutely true nor absolutely false. Assuming the identity of the logic of nature with the logic of thinking, Hegel named "contradiction" this coexistence of opposites in nature; this terminology created confusion, as it seemed to exclude the logical principle of no contradiction. Some dialecticians denied the existence of contradictions in nature, while asserting them in thought [38]. Others granted that oppositions exist in nature, but required thought to be consistent [45]. Logicians solved the problem in the most straightforward manner, by prohibiting all contradictions, but this seems like throwing out the baby with the bath water. Can we create a consistent logic if we negate the fundamental difference between true and false, or does this simply throw us into the marasma of a verbal dialectic that serves only as rhetoric? Can we do without a principle of no contradiction, when mathematical theorems often are proven by demonstrating the falsity of their opposite?  Already Aristotle, who recognized the inseparability of complementary opposite categories (e.g. content and form, essence and phenomenon, quantity and quality) found a solution in a local formulation of the principle of no-contradiction: opposites cannot coexist at the same time, in the same place and in the same respect. Such a local principle of no-contradiction is compatible with Hegel's principle of contradiction.

          Boole, the founder of mathematical logic, chose the exclusive meaning of "or" ("A or B" means either A or B, but not both) as one of the fundamental logical operations. Later on, because of the complexities introduced by such choice, other logicians choose instead the inclusive sense of "or" ("A or B" means either A or B, or both A and B). Reading "A or no-A" with the exclusive sense of or, asserts the coexistence of A and no-A (dialectic principal of contradiction) but only as separate, in either time, place, or respect (Aristotle's local version of no contradiction).

          One may allow for the coexistence of opposites even within the formalism of two-valued logic. The propositional calculus models negation by means of two-valued truth-functions: the negation of a true proposition is false, and the negation of a false one is true. Truth-tables can be made to include coexisting opposites by bidimensional negation, using ordered pairs <x,y> to represent the assertions made regarding A and no-A. In this manner we create a table of opposites, which allows four values. For instance, if A = "an electron behaves like a particle", then the assertion of A and the negation of no-A to be denoted <1,0>, as well as the assertion of no-A and the denial of A, to be denoted <0,1>, are partially true, because electrons behave like a particle or like a wave (a non-particle) depending on the experimental conditions.  The assertion of both A and no-A, represented as <1,1> is truer than either <1,0> or <0,1>. Likewise objective and subjective are complementary opposites, that can exist in all possible combinations: objective and not subjective (unconscious), subjective but not objective (false), objective and subjective (conscious truth), neither objective nor subjective (absent), and degrees of all of the above. We have then four fundamental cases; true, false, contradictory and irrelevant <0,0>. The traditional principles of no-contradiction and the excluded third are hence replaced by their opposites: a true assertion implies its opposite, a third case, and a contradictory case.

          As strictly speaking, everything is either A and no-A <1,1>, or neither <0,0>, but there are cases that are mainly A, or mainly no-A, we can use fractions between 0 and 1 as values in the ordered pair. Using ordered pairs to represent truth values is not the same as using fractional numbers in a linear scale, such as used in probability and in fuzzy logic. Fuzzy sets are sets whose elements belong to it in different degrees [44]. Fuzzy logic [16,44] attempts to solve paradoxes such as the coexistence of tallness and shortness in the same subject that already puzzled Plato as the result of gradual change. It interprets contradictory cases as midpoints between opposites; in this extreme fuzzy case, 50% A = 50% no-A. Fuzziness is not the same as probability, as the fact that an apple is 50% ripe is not the same as a 50% probability that the apple is ripe. Notwithstanding, both probability and fuzzy logic assume a linear scale in which opposites are inversely related, with absolute values at the extreme poles. However,  in complex configurations, such as in geometric fractals and biological organisms, opposites need not wax and wane together in an inverse fashion. Opposites may vary inversely with each other (as in probabilistic and fuzzy logic); or they may wax and wane together, as for instance closeness can increase both love and conflict in a human relation; or opposites may vary to some extend independently from each other. 

          As actual processes have properties A and no-A in various proportions that vary in time, it is useful to use a graphic representation that allows for expressing differences in quantity, even if only in a purely subjective way. It may be accomplished in the phase plane of opposites (or diamond of opposites), in which the opposites are the orthogonal axes that define the coordinate plane, and processes are represented as vectors.  We have introduced the phase plane of opposites in electrocardiography [34] to study opposing neural inputs through examination of the time series of beat to beat intervals, and in psychodynamics [27] and sociometry [5] for recording the coexistence of opposite feelings, and personal interactions. As in standard mathematical logic, one defines no-A as the complement of A: no-A = U - A, where U represents a local universe of discourse (for instance, the meaningful complement of red is green, not the absolute complement that also includes atoms, rhinoceros and triangles). In contrast to static logic, however, A and no-A are vectors. No-A is the resulting vector of the sum of all vectors except A. Each process interacts with a number of other tendencies, that are in part synergic -at least insofar as they are compatible-, and in part antagonistic -at least insofar as they are not identical. For instance, competitiveness strives for success; any other tendency, such as compassion, duty, or pleasure-seeking, while compatible and coexisting with competitiveness, also represents a competition that at times drives away the person from success; the sum of all these coexisting tendencies creates a vector opposed to success-seeking. Further, slower processes appear to be moving in the opposing direction from the perspective of faster ones [4]. We thus need to consider both the relative intensity, and the angle between vectors, to describe their degree of synergy and of antagonism.    

          The diamond of opposites (figure 1) is determined by the two orthogonal vectors, A and no-A, with their common origin as neither A nor no-A in the bottom vertex, and their junction both A and no-A at the apex. Strictly, all the entities that are within the diamond are either A or no-A or both, while entities which are neither A nor no-A, are in its low vertex. The four quadrants of the phase plane represent four cases defined by iterating negation: the predominance of A <1,0> in the right quadrant, of no-A <0,1> ion the left quadrant, of both <1,1> in the upper quadrant, and the relative absence of either one <0,0> in the lower quadrant. Both the phase plane and the ordered pair representation of negation capture the same fact: that in contrast to simple opposition, in which one excludes the other, complex oppositions (i.e. those in which opposites coexist), are bidimensional. Opposites are not poles of a line, but opposite points in a closed curve --hence at least two dimensions. The empirical use of the diamond of opposites is illustrated in [5,7,30,31,34], and in these proceedings by a two-dimensional study of entropy [36], and in developing the notion of the union of right and left as a political strategy [35]. 

          Coexisting opposites are necessarily similar, and the investigation of this similarity is a fundamental method to gain insight (Antonio Sabelli, see [27]). The similarity of opposites does not deny their difference, nor does the symmetry implicit in their coexistence denies a power asymmetry, such that one predominates at each moment, place and respect --symmetry and asymmetry, as other opposites, coexist. Opposite concepts are asymmetric, as shown by hierarchies of semantic preference (man and woman, captain and sailor, parent and child) [21] which obviously express power relations. A refutation is stronger than an assertion --thus Popper's concept of a scientific knowledge as one that can be falsified. Even in two-valued logic the two cases are asymmetric, as 0 implies 1, and the empty set is contained in every set, but not vice versa. To model negations that are asymmetric, we have developed helicoids (figure 2), partially ordered sets with two classes of elements that alternate with each other in the hierarchical chains [28,37]; they are defined by the relation sublation (dialectic negation) ¯ that is asymmetric (for no A, A ¯ A) and indirectly transitive (if A ¯ B, and B ¯ C, and C ¯ D, then A ¯ D. One can prove that A < C. For instance, if A is a hypothesis, B is an experimental refutation of A, and C refutes B, then C confirms A. In chains of hypotheses, refutations, and new hypotheses or refutations of refutations, etc, the negation of the negation is similar to, and stronger than, the initial case, as in Hegel's dialectic law --in contrast, the negation of the negation is identical to the initial statement in standard logic. This calculus of refutations portrays more realistic the scientific process than Popper's absolute falsification, and provides a definition of confirmation free from the paradoxes of confirmation theory --observing a white shoe confirms that "all ravens are black," because in static mathematical logic "All A are B" is equivalent to "No no-B is A").   

          These models for negation only capture some of its simplest features, and there is no valid reason to believe that complex cases can be reduced to simpler ones. Processes include not one but multiple pairs of opposites. They also manifest periodicities of high dimensionality. In (temporarily) stable processes, states repeat, so for every change there necessarily is an opposite. Here negation can be modeled by group inverse. There are many cases in which there are triads, rather than pairs, of complementary opposites (trifurcation [27]) --processes have 3 aspects, energy, information, and matter; three quarks form a proton; mother, father and child make the nuclear family. Colors offer a particularly interesting model for trifurcations, as the physical continuum of frequencies is divided by the retinal receptors into three primary colors (red, yellow and blue) such that the combination of any two creates a secondary color that is the inverse of the third primary (e.g. blue + yellow = green that is the complementary of red). The three primary and three secondary colors create a lattice, in which black and white represent the least and the last element [27]. Further combinations produce an infinite multiplicity of colors. In many cases of opposition, we can identify an implicit third. A third case arises in many manners; one of them is the union of the opposites.

3. Co-creation -the junction of opposites: Traditional logic deals with lineal deductions, but actual thinking occurs as dialog, including internal dialogues. Scientific investigations generate data by means of appropriate questions to nature --experiments and methodical observations. Logical processes thus generate novelty in the same manner as natural processes. Sexes copulate to procreate a new individual; oppositely charged particles bind to create atoms (matter); the interaction of processes creates, and destroys, novel, complex, diverse, and transiently stable structures of matter. Likewise the interaction of concepts creates new ones. Organization is a co-creation of opposites. Coexisting opposites can create a third case. Opposite processes interact in a non-linear manner thereby co-creating transient patterns and structures --the simplest case is the fold catastrophe [39], in which the non-linear interaction between opposite point attractors creates a distribution of outcomes in a third dimension. Catastrophes and other creative bifurcations involve also an increase in dimensions.           

        In empirical studies of choice [5,7], the coexistence of similarly intense opposite motivations (attraction and repulsion) can produce opposite behavioral outcomes (choice or rejection). Opposite outcomes alternate, that is to say, they coexist (contradiction) but are separated in time (local no contradiction). Representing conflicting motivations in the plane of opposites, behavioral outcomes determine a third dimension, as they are distributed in a folded surface such as in a mathematical catastrophe [39]. Further, the coexistence of intense opposite motivations fosters change and creativity [7]. We view this catastrophe of opposites as the simplest type of creative bifurcation, that generates novelty and complexity.

          The concept of co-creation via the interaction of opposites originates with dialectics but our process theory postulates the bifurcation of opposites, not their synthesis, or their annihilation.

The creative junction of opposite statements is illustrated by the formulation of a hypothesis and its empirical refutation. Hypotheses generate the question tested by observation or experiment; without hypotheses, there are just facts, not significant data. In turn the empirical refutation does not simply eliminate the hypothesis, but serves to develop it, creating new ones [18], illustrating a creative junction. In contrast, confirmations of the hypothesis by observing a case that satisfies it, fail to prove it, and lead to the raven paradox.  In contrast, finding a refutation of the refutation constitutes a confirmation [27,28] as discussed above. 

         Distinctions constitute another example of a creative junction; whenever you find a contradiction, make a distinction, advised scholastic philosophers. A distinction consists in creating a third dimension in the phase space (such as time, space or respect, in Aristotle's principle) that separates the opposites, allowing their coexistence. This separate coexistence of opposites is observed as alternation of opposite values under some experimental conditions [7], as in a catastrophe fold [39].

An algorithm to elicit creativity: Based on the above considerations, we have developed a strategy to analyze processes and to foster creativity. As behavioral creativity is fostered by the coexistence of intense opposite motivations, we propose to develop new perspectives by combining apparently opposite perspectives or hypothesis. For instance, in a companion article [35], we develop a social program as a combination of right and left ideals. As a practical strategy to promote creativity, it seems useful to consider the opposite of each "obvious" truth, and in all dilemmas to search for a third possibility. Instead of a logical principle of the excluded third, a heuristic principle of the implied third. Whereas Mao-tze Tung advised to search for the fundamental contradiction, we examine in each process the multiplicity of oppositions [27], and following Torre [40], we consider that the solution of each problem is facilitated by the simultaneous solution of another.

          Illustrating the creative nature of the union of opposites, process logic results from the co-creative intercourse of dialectic logic (which highlighted change and creativity but lacked scientific rigor and mathematical formulation) and mathematical logic (that provided rigor but focused on things as static beings, and knowledge as a state rather than as process); this is not a unique dialectic synthesis, as a multiplicity of such syntheses are being developed (temporal logic, modal logic, fuzzy logic, trialectics) with or without explicit recognition to the dialectic roots of the endeavor.

Logic of thinking: Static logic studies rational and mainly certain ways of reasoning. In the dynamic logic presented here, there are three different patterns of rational thinking: lineal deduction, opposition and co-creation by taking two different perspectives. These three patterns correspond to the three fundamental types of attractors in mathematical dynamics: linear flow, cyclic recurrences, and divergences and convergences. Logical implication ("if-then") formally resembles the linear, unidirectional flow towards a point attractor. Pathological obsessions and ruminations, and the logical paradoxes illustrate the existence of cyclic logical reasoning, resembling periodic attractors. Paradoxes such as the liar's ("I am lying", if true, implies that it is false, and vice versa) illustrate that even static logic has a dynamic. The existence of cycles in reasoning does not represent a basic flaw in the axioms of logic to be corrected, as it has been done by Zermelo at the expense of intuitiveness. There is a third type of logical relation, that is more important and creative: making distinctions, creative syntheses, formulating hypotheses, making choices, and inferring causes by their effects. This is Pierce's abduction [22], such as in medical diagnoses (Hippocrates' method is based on "symptoms", a term that means clue), in paleontology (print of a cloven hoof allows to conclude that the animal was a ruminant and hence much about its anatomy and physiology, pointed out Cuvier), and in detective stories. Choices depend on external interactions: empirical testing, psychodynamic processes, ideological biases, personal idiosyncrasies, may help us decide between the alternatives. This resembles the influence of external factors in catastrophes and chaotic attractors.

Psychosocial analysis of static logic: Static logic contradicts the three postulates of dynamic logic. It denies becoming in defining identity, the coexistence of opposites in its principle of no contradiction, and the existence and generation of a multiplicity of alternatives are excluded as third cases. The preference for static models is readily understandable. It reflects one of the earliest and most fundamental achievements in psychological development: the recognition of the stable identity of persons and objects within the flux of perceptions. Further, it follows from our natural wish to conserve: those who have, particularly if they have little, want to prevent losses, and nobody wants to age and die. Living depends fundamentally processes of maintenance, repair, and defense against predators.

          Thinking in opposites which are mutually exclusive and irreconcilable probably represents both the simplicity of inexperience, and the experience of conflict. Some primitive languages recognize only two colors, black and white. Simple minded people, and even more the mentally retarded, think in terms of rudimentary oppositions. Persons with borderline personality, a severe character pathology often emerging from childhood abuse, and manifesting high degrees of aggressiveness, think in black and white. Likewise social conflicts lead to think in antagonistic oppositions (cowboys and indians, catholics and heretics, christians and infidels, capitalists and communists), and/or to postulate false harmonies that promote the acceptance of suffering and evil. Black and white thinking promotes conflict and intolerance; it is perhaps the most dangerous characteristic of religious fundamentalism and totalitarian ideologies. Adler, Beck and others have related black and white thinking with neuroticism and tendency to depression. Cognitive therapists thus propose to replace black and white thinking by an awareness of grays. We have proposed instead thinking in color, using the analogy of the 3 primary colors to promote thinking from at least 3 different perspectives (e.g. leftist red, conservative blue, religious yellow; red anger, yellow cowardice, blue depression).

          Two valued logic denies the creation of something new. Recognizing the unpredictability of future events, Aristotle assigned a third value to them, and at the start of the century the Polish logician Lukasiewicz developed a three valued mathematical logic -true, false, indeterminate. Logical, physical, social, and psychological determinism are likely to be related to each other, as they all imply a denial of creative processes. The objective existence of creative processes seems to be validated by cosmological and biological evolution, but regardless of this ultimate metaphysical question, creativity is a subjective experience, so we need to ask ourselves what brings about its denial, religious fatalism that promotes acceptance of the status quo, neurotic avoidance of uncertainty, refusal to accept personal responsibility?  

          Some thinkers have defended the static postulates of logic by claiming that it is a purely mathematical discipline, which has no bases on, or applications to, actual human thinking. However, the term "logic", from Aristotle to Boole, has always meant the normative study of rational thinking. It is acceptable to adopt terms with specific meanings (logic, negation, implication, etc) to name mathematical models for them, but one cannot require that the terms should henceforth be used only in these new "technical" meanings. It is further argued, in defense of static mathematical logic, that a focus on static concepts does not imply a disregard for the ever changing nature of reality, but only a choice to focus on more permanent features. Such claim is disingenuous: why should one pay so much attention to static entities, and disregard entirely change and process, if one will not be committed to the idea that permanent structures are the fundamental form of reality?  Already with Plato, static philosophy was associated with conservative politics. Contrariwise, progressive views on social issues are naturally associated throughout history with evolutionary views of the universe and the species, and with attempts to develop a dynamic logic. In Aristotle, the static conception of logic was associated with a static conception of matter, according to which a motor was required to account for movement --mechanics has dismissed the need for a motor for inertial movement, and evolutionary science and mathematical dynamics imply that also more complex forms of movement are spontaneous.

Process logic, an alternative paradigm:  These methodological issues have practical importance in scientific practice. Static logic prevents us from examining how we can promote change. The sophistic and conflictual interpretation of the union of opposites by marxist dialectics led to a disregard for mathematical logic, quantum mechanics, genetics and psychodynamics. The disregard for Heraclitus dynamic logos in favor of Aristotle's materialistic metaphysics and formal logic has influenced even current scientific endeavors, as illustrated by the neglect of Bohm's quantum theory, of Pasteur's cosmic asymmetry and of Piaget's experimental epistemology by the scientific community, the delay of physics to adopt an evolutionary view of cosmology, and the acceptance of a statistical mechanics formulation of thermodynamics contradictory to evolution. The logic underlying the construction of theories and methods plays a fundamental role in scientific progress.

          In recent years many scientists have come to accept Thomas Kühn's notion of scientific revolution, a shift that occurs when empirical data contradicts the accepted paradigm and hence this is replaced by a new one. As Kühn, also Ichazo [14] adopts a lineal developmental analogy, identifying logic with childhood thinking, dialectics with adolescence and trialectics with mature thinking. The process paradigm has a drastically different view of both scientific and social evolution: static and dynamic logic, materialist and idealist philosophies, rationalism and irrationalism, every opposite viewpoint has existed for much of human history. Contemporary with static logic, the process paradigm was already well developed in the fifth century B.C. in Greece (Heraclitus, Empedocles), India (Buddha), and China (Lao- Tzu). Notwithstanding it was obscured through much of history by alternative paradigms such as materialism and mechanism  on the one hand, and epistemological idealism and spiritualism on the other. The process paradigm reappeared at periods of rapid scientific and social development, such as during the Italian Renaissance (Bruno, Nicholas of Cusa, Pico della Mirandola), in the 19th century with evolutionary theory (Lamarck, Darwin) and process philosophies (Hegel-Marx-Engels' dialectics), and in our times with Bohm's quantum theory, Prigogine's far from equilibrium thermodynamics, cosmological evolutionism, and non-linear dynamics. Thus, the process paradigm has coexisted throughout times with its opposites, static portraits of fundamental reality as unchanging structure, dualisms that separate matter and spirit, determinism that denies creation, and two-valued logic that denies the coexistence of opposites. Likewise, opposite interpretations co-exist throughout history in most fields of scientific endeavor, as illustrated by the idealist Copenhagen version of quantum mechanics which is taught as fact vs Bohm's interpretation which explain the facts equally well while not requiring to postulate counter-intuitive ideas; economic theories supporting capitalist development or socialist revolution during the last one hundred years; gradualist vs punctualist views of evolution; and many others. Changes of predominance from one paradigm to another often appears to result more from changes in the social milieu than from the emergence of specific data --Kühn himself notices a close relationship between scientific and political revolution. Yet different scientific paradigms also coexist in all periods of science rather than representing mutually exclusive systems that replace each other in a linear development. 

          The concept of the union of opposites provides a unique understanding of scientific progress as resulting from the intercourse of multiple disciplines, and of competing hypotheses. To develop an understanding of a scientific question, it is useful to formulate a multiplicity of specific hypotheses, to build evidence for each particular one, as well as to attempt to refute each of them, and to counterpoint each hypothesis with the others. Likewise, the intercourse and competition between coexisting and opposite paradigms fosters the progress of science. The entire history of philosophy is shaped by the contradiction between materialism and spiritualism, between static views and process ones, and between ideologies that support the powerful vs the messianic messages of their challengers. In this article we have developed a dynamic logic through the union of static mathematical logic and dynamic but verbal dialectic logic. In our view, the co-creative junction opposites is a fundamental logical process still largely unexplored.

Acknowledgements: I am thankful to Mrs. Margaret Trobaugh and Mrs. María McCormick for their invaluable support to the Society for the Advancement of Clinical Philosophy.

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