The word "logic" is used in common parlance no less than in formal treatises. And yet that word is one of the most dangerous and misleading tools of communication, because there is not one system of logic. There are many, and there are profound differences among them.
This discovery occurred to this writer in the 70s while doing research on economic theory, and specifically when his need to understand "economic logic" became imperative. The deeper the search, the clearer became the realization of the existence of that many systems of logic—as well as the need to add one or more systems to the list. But, if logic is supposed to provide the "rules of correct reasoning" and these rules change from system to system, what type of guidance does Logos/Logic provide?
The answer is no longer simple. It becomes threefold. First, one must indeed recognize the existence of that many systems of logic. Second, one must recognize differences and similarities among those systems. Third, one must learn not so much to choose among those systems and adopt the one that is best suited to one's needs as, when communicating across systems, to translate each and all verbal and written expressions from one system into another.
These operations are less complex than they might appear beforehand, and they are less complex because all systems of logic share basic common characteristics. Each one of them is built on a set of equivalence relations; if properly developed and applied, each system has an internal impregnable consistency; and, in their consistency and sufficiency, those systems are all equivalent to each other. Just like languages: see, Noam Chomsky.
These are the basic issues treated in this paper. Interspersed in the analysis is the attempt to show how systems of logic influence culture—from economics to physics, from religion to mathematics, from art to politics.
The short range purpose of the paper is to demonstrate that all discussions across systems of logic are fruitless. The long range purpose is to foster tolerance of all systems of logic—not only through the demonstration that all systems are indeed equivalent in their consistency and sufficiency, but especially through the presentation of a synthesis which includes the essential elements of all previous systems. This synthesis is the outline of a new system of logic that might be called Relational Logic.
Most of what follows is bound to be well known to expert logicians. It seems to be totally new to most other people. Expert logicians also know the characteristics of equivalence relations. Yet even they might be surprised at how well does the principle of equivalence serve to unify the manifold fields of logic.
1. THE VALUE OF THE EQUIVALENCE RELATION
If the equivalence relation has indeed such a far reaching and fundamental value, it might be useful to inquire about the reasons for this status. The equivalence is a relation among three terms (A, B, C). There are two forms to express the equivalence relation: A ≡
B ≡ C; or, alternatively, A ↔ B ↔ C. The first reason for the extraordinary importance of the equivalence relation is that it gives us the opportunity to triplecheck the basis of our reasoning. The second reason undoubtedly can be found in the development of precise rules concerning its application: Each term of the equivalence must be reflexive, symmetric, and transitive, cf. Allen (1970, pp. 43547, 74852) or Suppes (1957, pp. 21320). Thus we obtain not only three, but nine checks on our reasoning. If any of those conditions is not respected, our reasoning is not valid. To say the least, it requires more or different support.
A term is reflexive when it remains identical to itself all through the discussion—and, ideally at least throughout an entire discipline let alone across disciplines. A term is symmetric if each term is a mirror image of the other two terms. A term is transitive if it allows us to observe the same reality from three different points of view.
A personal experience will perhaps succinctly relate the importance of the equivalence relation in no uncertain terms. The writer was in the new Philadelphia airport. At landing, he went straight for his car. In the hall, he glanced at one exit (A) and doubted that that was the exit to the parking lot. He went to the opposite symmetric exit (B). The two exits were so identical that he could have forever gone in circles and never been offered the certainty of where the parking lot was. The solution was to actually step out of any one of the two doors, thus creating a third term of reference (C) and ascertain whether he needed Exit A or Exit B. This case does not establish the equivalence of Exit A to Exit B to Parking Lot; it only verifies the validity of the methodology to ascertain the truth. Technically, if A was the exit, B was not the exit needed to reach C, the parking lot. As proof of its extraordinary importance, this method is profoundly used by mathematicians as well as by philosophers.
A third reason why the tool of equivalence is so important, perhaps, is that "the world" itself seems to be built on a set of equivalence relations. The triangle, after all, is an equivalence; and Buckminster Fuller (see, 1975, esp. p. 4) has provided incisive insights and convincing applications of this tantalizing possibility. But this is not the place to explore such an issue at any depth. When this paper was submitted to Buckminster Fuller (1979) for his examination, he found it “to be extremely interesting." In a similar vein, incidentally, did Moses Abramovitz, then editor of the Journal of Economic Literature, react. Equally encouraging were a few other readers to whom this paper has been shown over time.
In any case, if the above reasoning is correct, we have the basis for carrying the analysis forward. And, going forward, we shall eventually lay the foundation to a new system of logic: Relational Logic.
2. MANY SYSTEMS OF LOGIC
In the following paragraphs we shall review the essential elements of the following systems: (1) Primordial Logic; (2) Classical Logic; (3) Rational Logic; and, (4) Dialectic Logic. We shall find that Eastern Logic (5) occupies a place of its own in the history of the mind. Thereafter, we shall observe some of the subsets of these major systems, namely, Conventional Logic, Positive Logic, Deductive or Syllogistic Logic, Inductive Logic, Instrumental Logic, and Economic Logic. We shall conclude with the bare outline of Relational Logic.
Two warnings are in order at the outset. First, we shall have to neglect a great many details that actually put flesh and blood on the basic structure of these many systems. Second, the order in which these systems are presented is dictated by ease of exposition. Any inclination to see in this order the possible existence of a temporal succession, or, worse, an order of importance, should be strenuously resisted.
If these two warnings are firmly kept in mind, only a little prompt might occasionally be necessary to make the reader aware that all systems share common characteristics. Provided they are indeed complete systems, they provide sufficient assistance to the conscientious thinker to make him, or her, reach a comprehensive understanding of the surrounding world—the world of ideas, no less than the world of things. But how can an unbiased observer realize that this feat is in fact accomplished, while in the presence of that many systems of logic? The observer simply needs to realize that certain elements in the various systems undergo not always subtle shifts in position. Specifically, the observer needs to realize that while the shifts in position—or modes of usage—lead the adherents to each system to emphasize or deemphasize certain elements; the shift itself does not make those elements disappear.
2.1 Primordial Logic
There is obviously no written record on the system of logic used by our early ancestors, with the CroMagnon man of 40,000 years ago as the most famous and best known of our ancestors—and, most likely, much—much— earlier men and women still. But that system must be reconstructed, if we truly want to understand not only the development of thought but especially the function which every system of logic performs. More importantly perhaps, today we have all the tools needed to reconstruct this fundamental first link in our habits of reasoning. Of course, the reader has to overcome a potential bias. The reader has to be open to consider the possibility that, as Herbert L Calhoun (2015) points out, “while the content of thinking may have changed over various epochs, ecologies and cultures, the process of thinking itself has not.”
To insist on one point, in the following paragraphs we shall be concerned with the broad outline of this system of thought, which might be called Primordial Logic, and neglect the many details of this reconstruction—especially because we shall have to adhere strictly to an intellectual distinction that does not exist in reality, the distinction between logic and epistemology. That continuum has to be broken here. Eventually, much more will be written to catch the manifold nuances of the necessary encounter with our past.
The question is: How did it all start? What are the beginnings of knowledge, our conscious understanding of the world? Let us try to recapture that miraculous moment in which a CroMagnon man is exiting the Lascaux cave—and faces the moon rising and the salmon jumping on the horizon. How did he learn to distinguish between the two flickering signs?
Many years ago, while he was analyzing the structure of economic theory and finding principles of logic to be essential to the task, a friend and editor of much of this writer’s work, David S. Wise, brought to his attention a book by Alexander Marshack titled The Roots of Civilization (1972). Fairly soon after devouring Marshack’s book, this writer jotted down a sketch of the use that CroMagnon man must have made of three basic principles of logic and traced their development over time. Apart from necessary distinctions, most philosophers as well as mathematicians agree, an agreement of extraordinary importance in itself, that there are three fundamental principles of logic: the principle of identity, the principle of noncontradiction, and the principle of equivalence. Our CroMagnon man made use of all three. Here is that sketch, only slightly altered over the years, which will eventually have to be validated, or its validity denied, by many scholars in many fields.
2.1.1 The Use of Three Principles of Logic by CroMagnon Man
Alexander Marshack announced a fantastic hypothesis in his book. He had discovered scratches— i.e., generally points but at times also lines—on bone upon bone and stone upon stone that are preserved in archeological and anthropological museums the world over. He announced the hypothesis that with those scratches our early ancestors were actually inscribing "notations."
One point represented something! Scratches were neither idle doodles nor an esthetic arabesque. Their simplicity and repetitiveness implied purposefulness. What was this something? Let us unpack the question. Alexander Marshack has been acclaimed as singlehandedly revolutionizing the field of Paleolithic art research. Let us stress, art research. In fact, it is fully granted that he allowed scholars to see portable art objects and cave paintings “with fresh eyes, to ask new questions, and to understand their technology and production far more precisely” (Bahn, 2009). However, Marshack himself concentrated his attention on a deeper penetration of the meaning of those scratches. He suggested that those notations record the early steps of our knowledge of astronomy and calendar time. And he has been chided for going too far (Robinson, 1992).
The suggestion here is that neither Marshack nor other scholars have gone far enough in their understanding of those notations. The inspired idea that scratches made by CroMagnon man on stones and bones were notations is a discovery of enormous importance indeed. To repeat, the question is: A CroMagnon man is exiting the Lascaux cave while facing what we today know as the moon rising and the salmon jumping on the horizon. How did he learn to formulate a complete vision of each image? How did he learn to distinguish between the two images—and the “billions and billions” of other images on his horizon?
Once he stopped relying on memory, the first decision of this human being was to grab a stone and start recording observations about any specific object at any one time. Let us pause on this incredibly important first step in the history of humanity. In plain words, with that single act he made use of all three fundamental principles of logic at once: the principle of equivalence, the principle of identity, and the principle of noncontradiction. Let us see how.
Undoubtedly unaware of all the implications of what he or she was doing, by placing a mark on a stone our ancestor established this set of mental relationships:
This notation ≡ One digit ≡ Specific information about… the flicker
Clearly, our ancestor made full use of what is today known as the relationship of equivalence. There are three “equivalent” terms in that construction: (1) the scratch on the stone, which corresponded to (2) the number “one,” eventually to be followed by a second and a third scratch, and (3) these two entities stood for something else in the mind of the researcher: What was that? Evidently, whatever information was of interest to the researcher at the moment. Let us say, shape of flicker, length of flicker, duration of flicker. Clearly, all this mental activity was preceded by the development of the number system, which must have started with counting fingers of the hand. Why else are numbers called digits?
What stood inside the above equivalence is a fullfledged application of the Principle (or Axiom) of Identity. This notation represents facts concerning this flicker—and this flicker alone. Geometrically, it is useful to represent the Principle of Identity through a single point. Thus:
Our ancestors did not need any other mental apparatus to proceed with their discovery of the world. Duplicating the use of the Principle of Identity over and over again, our ancestors eventually discovered that they needed another stone to represent facts concerning different flickers. Let us see why.
At first, our ancestors must have made notations about both flickers at once; needless to say, our attention here is concentrated on the two flickers of our imagination. Our ancestors were faced with thousands of stimuli all at once. What is suggested here is a “glorious” simplification that will have to be appropriately broken down into more and more finite details in order to be validated or its validity denied. Returning to this terrain from the point of view of gnoseology and epistemology, we shall be able to retrace more of the steps that led our early ancestors from the observation of “flickers” to naming the objects they were observing. The conclusion is this. Soon after he put one notation down, and it does not make much difference to assume that, likely, it was after eons of trials and errors, one simple fact became apparent. Our ancestor was observing two different things.
Simply by taking a few steps away from the cave, our early ancestor must have experienced that the two flickers on his horizon represented two different entities: One was near, the other was far away; the behavior of one event was completely different from the behavior of the other. Soon (?), a careful study of who knows how many objective recordings intellectually, convincingly confirmed that the two flickers in the sky actually belonged to two different phenomena, two different aspects of reality: Our ancestor was compelled to realize he was observing two distinct objects.
Therefore, he needed two stones to record his further observations. Let us call them Stone A and Stone B. Only when they were satisfied that they had observed each flicker from every possible point of view, they named one of them “moon” and the other “salmon.” That was the ultimate inestimable gift by the principle of identity to humans. Our CroMagnon man—or woman—was still using the principle of identity, but with a twist.
Let us think this through. If he had not made the distinction between the two Stones and two Events—and respected it religiously—our CroMagnon man would have soon become utterly confused. Does this notation belong to the object near me or the one far away? This operation appears simple and obvious today, and it might have occurred in one single spurt. But its importance does not need to be underestimated. Indeed, the full importance of the discovery and application of the Principle of Identity can be better appreciated considering that there are mental disciplines upon mental disciplines, and intellectual discussions upon intellectual discussions, which still do not make scrupulous use of this essential distinction. Economics, above all, still insists on making use—as R.W. Goldsmith (l955, p. 69n) calculated—of any one of the possible 100,000 definitions of Saving at any one moment in its analysis. For some of the theoretical effects of this practice upon economic science, see Gorga (2002 and 2009, esp. pp. 69137). It is as if our ancestors had had one hundred thousand tablets representing the object far away, which they eventually called the moon, and each person had insisted on inscribing different notations—one’s own perceptions—on any one of those tablets at random. The disrespect of the dictates of the principle of identity does indeed necessarily lead to incredible misconceptions of the objective reality. If our ancestors had insisted on the prevalence of their personal perceptions, we might have ended up with eventually, necessarily, giving 100,000 different names to the one moon that is in the sky—with consequent 100,000 different descriptions (and analyses) of that one phenomenon in the sky. Our early ancestors did not do that.
Our ancestors avoided that confusion. They instinctively (?) respected the dictates of the Principle of Identity. Using modern symbols, they made sure that: (Stone) A = (Stone) A; (Stone) B = (Stone) B...; (Stone) Z = (Stone) Z.
In fact, as the Greeks were eventually to point out, the Principle of Identity carries implicitly with it, unavoidably, the Principle of NonContradiction: A is not B. In classical notation,
A ≠ B
Pace Hegel and some Hegelians, Stone A is not Stone B; the moon is not the salmon. The moon is always identified by Stone A, and notations belonging to the moon are never placed on Stone B that identifies the salmon.
As Boland (l979, p. 503) points out, "Aristotle was probably the first to systemize the principles of [classical] logic"; but, he continues, "most of them were common knowledge at this time." To Aristotle belongs the pride of paternity. With the Greeks we meet recorded history, and a system of logic that is universally identified as Classical Logic.
2.2 Classical Logic
Aristotle enunciated three fundamental principles of Classical Logic. He proceeded in this fashion. He put the Principle of Identity at the foundation of an elaborate construction. The moon must be the moon throughout our entire discussion: A = A. We cannot change the definition of our terms in the middle of the discussion—or, abstracting even further, the "truth" remains the truth throughout. But how do we know that the moon is actually the moon, or the truth remains the truth?
The simplest test is that the moon is not the salmon; the truth is not falsity: A ≠ notA. The Principle of NonContradiction was formally born.
Geometrically, it can be said that Aristotle (and many other thinkers) began to look not only at single points, but also at the area surrounding individual points. In order to make this area visible, we shall draw a circle around the point. Thus:
which can formally be read: "The circle is not the point."
But what was the area between the point and the circle? Aristotle, with classic Greek clarity, did not tolerate "grey" areas. He formulated the third principle of Classical Logic to take care of this issue, the Principle of Excluded Middle. Nothing can be the moon and the salmon at the same time; or, abstracting the issues even further, a statement cannot be true and false at the same time. As Boland (op. cit., p. 504n) puts it, "statements that cannot be true or false, or can be something else, are prohibited." Aristotle simply denied the existence of "grey" areas. He declared them inadmissible in any conversation or logical formal reasoning.
Appropriately compressing the issues, namely making all grey areas between the point and the circle of Fig. 2 disappear, Classical Logic can be represented through a sphere. In this fashion:
We thus obtain an immediate image of the "wholeness" of Greek thought. And it is this wholeness that explains not only the great clarity and firmness of that thought, but also its enormous successes especially in philosophy and physics, sculpture and the theater—arts and sciences which must be presented in "tutto tondo." And yet, it was perhaps the selfassurance that existed at the foundation of Greek thought that enabled it to avoid any serious confrontation with economic theory and other topics that require the study of messy, "grey" areas.
And yet the reality of economics could not be avoided altogether. Rather than through hard theory, it was faced with the softer tools of economic policy, which was treated in such a realistic fashion that it endured, basically unaltered, for two thousand years. Aristotle enunciated the doctrine of economic justice, a doctrine that, fully accepted by Saint Thomas Aquinas, was silenced by Adam Smith. That system of thought has been revived and completed in the form of the theory of economic justice by Gorga (1999 and 2014).
Before leaving the field, it behooves us to inquire: Have we gathered all that we could from the vast field of Classical Logic? What does the shift from Primordial Logic to Classical Logic imply? Evidently, the shift implies a momentous stepping stone in the history of mankind. It almost seems as if men and women had no more concrete items—such as the salmon and the moon—to discover, to name, and to analyze. The finite world of physical products of the universe was almost nearly all catalogued. By the time we meet the Greeks, man had named nearly all objects within his range of observation. What was there left to do?
The world of abstraction, the world created by all our moral and intellectual capabilities was all there to be explored.
2.2.1 Classical Logic and the Principle of Equivalence: Deductive or Syllogistic Logic
What happened to the Principle of Equivalence, which as we have seen played such an important part in Primordial Logic? Logically enough, it had two places to go, and it went there. On the one hand, it went (1) underground—where it can still be found today. Strangely enough, it is well known to both (mirabile dictu) philosophers and mathematicians. And it is much used by both. But where is it openly located? Where the Greeks left it: (2) in the development of syllogisms, of course! More generally still, the Greeks used the principle of equivalence to build syllogisms and led us to the stratosphere of abstraction.
The principle of equivalence became a tool of analysis of such abstract entities as propositions. A syllogism is the equivalence of three propositions: If the major premise and the minor premise are true, the logical conclusion is also true. All men are mortal. Socrates is a man. Socrates is mortal.
Do not laugh. For two thousand and more years, our culture—what our intellect produced—was unified by extended, detailed exploration of the applications of this methodology. All advancements in philosophy, theology, and science were sustained by this methodology. Even the mathematical model of the economic system developed by Keynes (1936, p. 63) is a syllogism:
If, Income = Consumption + Investment
And Saving = Income – Consumption
Therefore, Saving = Investment
From such a simple set of starting propositions, the “figures” of the syllogism, through a variegated set of permutations and combinations, over the years became more and more complicated (see, e.g., Van Vleck, 2014)—not unlike econometric techniques today. All certainties in science up to the 15th or 16th century were derived from the validity of the syllogism. The applications of syllogistic logic culminated in the “realism” of Saint Thomas Aquinas, and from there it degenerated in the formalistic casuistry of the late Middle Ages and the Renaissance.
Formalistic casuistry of the syllogism led to complicated formulas that at times yielded contradictory results—just like the results of econometric analysis today (see, e.g., McCloskey, 1983). Logic, and especially syllogistic logic, could no longer be relied upon as a guide to thought.
Left alone without theoretical guidance, the mind was thrust in a state of unsustainable allencompassing doubt. Even theology, to remain living and vital, was confined to mysticism, especially the mysticism of Saint John of the Cross and Saint Teresa of Avila. We shall see how philosophy gradually devoured itself from within; for the time being, let us draw an important distinction in the existence of mysticism. Mysticism is mysticism. Yet, apart from such forms of mysticism as Kabbalah and Sufism, there is a fundamental difference between the mysticism of the East and the mysticism of the West. The mysticism of the East is immersed into the sublimity of emptiness; the mysticism of the West is immersed into the sublimity of the fullness of Jesus Christ.
At the end of the Renaissance, everyone was left on his own—just like we are all, again, left on our own today. No guidance is accepted or acceptable from logic or philosophy, let alone theology. A mind as acute as that of Lorenzo il Magnifico could issue such degenerate instructions as “He who wants to be happy, be happy. Tomorrow is uncertain.” The mind could not tolerate an abject state of doubt.
René Descartes came to the rescue.
2.3 Rational Logic
The Greek civilization collapsed. Was it because the sphere was actually empty? Thereafter, its heirs, the Roman and the Byzantine empires, also collapsed. And the Renaissance gradually became not at all sure of itself; indeed, it became so unsure of itself that its heirs were led to believe that they could settle theological disputations through religious wars. The certainty of the Greeks had disappeared. The Age of Uncertainty was in the saddle.
What was that changed in the classical system of logic? The Principle of Identity and the Principle of NonContradiction were preserved and much strengthened over the centuries. But the soft Principle of Excluded Middle was replaced by the Principle of Indifference. Was the collapse of past civilization which created the need for a change in the ruling system of logic, or was it the other way around, or a combination of the two? Thinking “firmly” and “securely” is the foundation of life and civilization.
No one seems to know when or by whom the Principle of Indifference was formulated. But it was wellknown in the Sixteenth and Seventeenth Century. In fact, it was formally incorporated into a new system of logic by Descartes, which is denoted as Rational Logic. Descartes reached this new synthesis formally, but not explicitly. One needs to operate a considerable "translation" and interpretation of his words in order to see how those principles are "clearly and distinctly" (Descartes, l938 [l637], p. 17) organized to form the system of Rational Logic.
Descartes reduced to three "the great number of precepts of which Logic is composed" (ibid.): "The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt" (ibid.). (Observing Fig. 4 below, it becomes apparent that one single "point" corresponds to this "precept"; and, as seen in Fig. 1, one point is also the simplest representation of the Principle of Identity.)
"The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution" (ibid.). Observing Fig. 4 below, it becomes apparent that Descartes had simply in mind points B, C, D on any line; and the validity of such points—while they undergo examination—does not need to be determined a priori. One has to be "indifferent" to their validity. Hence this second precept formally incorporates the Principle of Indifference into Rational Logic. For confirmation, it is sufficient to quote the following passage: "If some of the matters... should offend at first sight, because I... seem indifferent about giving proof of them, I request a patient and attentive reading of the whole... for it appears to me that the reasonings are so mutually connected... that, as the last are demonstrated by the first which are their causes, the first are in their turn demonstrated by the last which are their effects" (op. cit., p. 60). For the discourse not to be too cumbersome, not every point needs to be proved at once.
"The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence" (op. cit., p. 17). Observing Fig. 4 below, it becomes apparent that with this third precept Descartes was observing an entity which might—for short—be called notA. This entity can not only be properly defined as "Infinity." It can also be said to incorporate the essence of the Principle of NonContradiction; e.g. Infinity is the negation of the concreteness of A and Z.
Rational Logic is best represented geometrically by a line; specifically, an infinite line. Thus:
This representation of Rational Logic is not arbitrary. Descartes himself, in the next breath, stated: "...I thought is best for my purpose to consider these propositions in the most general form possible, without referring them to any objects in particular.... Perceiving further, that in order to understand these relations I should sometimes have to consider them one by one, and sometimes only in the aggregate, I thought that, in order the better consider them individually, I should view them as subsisting between straight lines...“ (op. cit., p. l8).
This representation of Rational Logic also explains why Descartes further specified that his method called for "enumerations so complete, and reviews so general that I might be assured that nothing was omitted" (op. cit., p. 17). This maxim is not a fourth precept but an implicit conjunction of the three principles of logic mentioned above. In his summary, he simply spoke of “the three preceding maxims" (op. cit., p. 24).
Rational Logic is, of course, the dominant mode of thinking in the West. And if there were any doubt as to the direct influence exercised by Rational Logic in our culture, it would suffice to consider its impact on the development of economic thought: One can simply open any economics text and observe how much of its analysis is conducted with the assistance of straight lines. Yes, our entire Western culture is imbued with Rational Logic. Indeed, it is upon this system of logic that the entire philosophical structure of Rationalism is built.
2.3.1 Rationalism
While Rational Logic might not be very familiar, everyone knows about Rationalism. René Descartes based his entire system of thought on the wellknown proposition, I think, therefore I am. He did not only create Rationalism; he reinforced his rational thinking processes with the explicit, uninterrupted help of mathematics and geometry.
The union of Rationalism with mathematics and geometry raised high hopes that the Age of Uncertainty would soon come to an end. And people have lived under that impression for a few centuries now. All our certitudes are “rational” certitudes. The world of the intellect is beautiful and perfect.
But what is the reality? Where is beauty and perfection in everyday life? Where is beauty and perfection for most people? It is the examination of these realities that is making us trace our steps back again. If we do that, a new world opens up to observation. We soon discover that the most enduring result of the exalted union of Reason with mathematics and geometry was a total surprise. It gave birth to an unexpected child: materialism. The reason is clear. Only material things can be counted, can be measured, and can be represented geometrically.
And then materialism became absolutist. It did not tolerate the existence of anything that cannot be measured. Absolute scientism is older and deeper than the scientism identified by Hayek (1955). It denies the existence of anything spiritual. As a consequence of this encroachment of scientism, intellectually and rationally—as distinguished from the practical perception of billions of human beings—first, the soul went, and with the soul also went religion. Indeed, among the illuminati and the cognoscenti, any conception of God was also excluded from Nature first and from the nature of men and women thereafter. The strangest phenomenon of all is the ongoing battle to reduce the mind to a set of material—chemical and electric—nodes and switch operations. Off has gone the unity of body, mind, and soul. The human reality has been reduced to a set of stick figures that only think: no feelings, no spirit. Martians, indeed. But also extreme forms of individualism—“I” think, therefore “I” am. An individualism that degenerates in solipsism, whose forerunner was a lovable narcissism.
Rationalism cannot endure. We are living in the throes of its demise. As John Lukacs, among many others, has authoritatively pointed out, we are At the End of an Age (2002), the Age of Rationalism. The examination of fundamental questions is throwing us back into the world of uncertainty. Much uncertainty is in the saddle. We are back where we were in the late Renaissance. The analytic disaggregating tools of reason need to be fused with integrative and synthesizing tools of analysis; this is the work of the future.
Before we do that, we need to ask: Is Rational Logic the only mode of thought today? Far from it. Just under the surface, and prevailing in Communist or Fascist regimes, is another system of logic: Dialectic Logic.
2.4 Dialectic Logic
Dialectic Logic's fatherhood is credited to Hegel, but it was already known to Aristotle—and indeed was implicit in Primordial Logic. The principles of Dialectic Logic are exactly the same as those of Rational Logic, but they change direction. Instead of the two extreme points of the line proceeding toward infinity, they converge toward the middle, a point which Hegel called "process." Thus:
When Marx acquired this system of logic from Hegel, he saw in the new shape of things the Class Struggle and all that. While some people put their ear to the ground and hear the rumble of the Class Struggle, others cannot even pick up a faint tremor. Who is right? Who is wrong? Are there other positions in between? These are substantive issues that cannot be addressed in this paper. But the methodological question is clearcut. Once one's thought is being guided by Dialectic Logic, the Class Struggle becomes a selfevident phenomenon. The poor fight against the rich. Those whose thought is guided by other systems of logic are likely to be insensitive to it. Incidentally, notice also that, once the third point—or "infinity" of Rational Logic—was eliminated (or, better, shifted position and turned inward), it was very consistent of Marx to fight against all religions. Allowing the mind to escape toward the infinite, instead of concentrating its attention on the affairs of this earth, religion became “The opium of the people.” And since the extremes always have points in common, some of the fundamental positions are shared by capitalism as well. Fight, fight, fight is the order of the day. Indeed, competition, competition is for the Alpha man; coordination and cooperation is for the weak.
Dialectic Logic’s second major innovation was the destruction of the principle of noncontradiction. We are thus plunged back into the preSocratic tradition of sophists and rhetoricians that Aristotle had put to rest. After point A and Z have fought with each other and are transformed into P, the distinction of A from Z disappears. They are both transformed into P. The destruction of the principle of noncontradiction has opened the mind to the consideration of infinite possible transformations (cf. Brandom, 2014).
Yet, the destruction of the principle of nomcontradiction has also done much damage to the clarity of the discourse. Now everything can be contradicted and the speaker is authorized to become dictatorial. Giovanni Gentile, the friend of Fascists, was explicit about this. He said: “He who possesses the truth has the right to repudiate the affirmation of those who contradict him” (quoted in Holmes, 1937, p. 61). Who is the judge of the one “who possesses the truth”? Luimême, he himself, of course. Brute force is then the last resort to resolve disagreements.
Avicenna got it right: "Anyone who denies the law of noncontradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned."
The miasma emanating from the type of discourse engendered by the negation of the validity of the principle of noncontradiction is asphyxial; it leads to suffocation. The only retort must be this. Granted that the principle of noncontradiction is not necessarily valid in a dynamic analysis of the longrun, its denial at the static moment of the initiation—or ending—of the discourse is utter nonsense and leads only to confusion of mind. A tiny example might suffice: True, over time something ugly can be transformed into something beautiful, to deny this possibility is to deny life; yet, before that final moment the ugly cannot enter the container called beauty, it cannot be called beautiful. At any one moment, anything is either beautiful or ugly. The beautiful is the antagonist of the ugly; it is not the ugly; it cannot be confused with the ugly.
Where is the ultimate flaw in the attempted destruction of the validity of the principle of noncontradiction? The flaw is hidden in the assumption that A and Z necessarily fight with each other in order to reach point P. As we will see, in Eastern Logic the transformation of A and Z into P is generally—very creatively and imaginatively—conceived as an entirely peaceful process.
2.5 Eastern Logic
Generally, no one speaks of Eastern Logic. But there could hardly not be one such system of logic. The hint of its existence included in the following paragraphs can hardly be considered satisfactory, especially because this writer does not know of any formal treatment of the issue—whether through a Western or an Eastern source. A glance at Tibetan logic (Brinkman, 2014) suggested by Bob Arnold, an expert in the field, reveals not a system of logic but an application of syllogistic logic.
In any case, Eastern Logic can be represented geometrically through the wellknown YinYang symbol:
An analysis of this symbol and cursory acquaintance with Eastern literature reveals that Eastern Logic is perhaps the available synthesis of all other systems of logic observed above. One can recognize in Fig. 6 all symbols belonging to each one of the systems of thought mentioned so far: the point, the circle, and of course the sphere. What is a circle, if not a flat mental image of a sphere?
The line is also in Eastern Logic. Snap a circle, and you obtain a line. In reverse, going back one step or two you can build a circle out of a line, by bending the line. Whether you extend the line point by point or you reduce it to a point, and whether or not you blow up the point into a sphere, the line easily leads the mind to infinity. Smale transformations help.
Only one point might perhaps be emphasized here. Yin and Yang can actually appear to be revolving around each other and even fighting each other. Is this the major reason why China accepted the Marxist ideology? Are the differences between Dialectic Logic (Fig. 5) and Eastern Logic (Fig. 6) sufficient to explain the differences in the behavior of Stalin and Mao? Are these differences sufficient to provide clues for the different approaches to industrialization and "modernization" between Russia and China?
Generally, however, the transformation of Yin into Yang, like the transformation of night into day, is assumed to be totally peaceful. Pace Hegel, if we had the capacity of Eskimos to distinguish snow from snow, we would not be deceived by our inability to see Dawn as neither Night nor Day. We would call it a “littlelessnightbutstillnight” or a “littlemoredaybutnotyetday” or much more imaginative words. Does Dawn fight the Night?
3. PROPOSITIONS AND SYSTEMS OF THOUGHT
Logic cannot be separated from the theory of knowledge. It is the integration of the two fields that eventually will give us a reliable, complete method of analysis. For the time being, faced by the reliance of both Rational Logic and Eastern Logic on individual propositions it is important to realize that there is no way of determining the validity of propositions by themselves; only systems of thought determine the validity of individual propositions. Quick proof. Do parallel lines meet? In Euclidean geometry, they do not meet; in modern Imaginary nonEuclidean geometry, they do meet.
Generally unaware of this fundamental weakness of syllogistic logic, in the West as well as the East we are still going on basing our beliefs on—and arguing the validity of—single propositions. No wonder we are having such a difficult time clearing our minds and communicating with each other. No wonder the Greeks and the Buddhists (at the same time) have both thrown us into the stratosphere of abstraction, and left us there.
Difficulties are compounded by the personal attachment to modes of thought that do not form complete systems of thought. These are more clearly understood if grouped together under the heading of nonsystems of logic. We can rationalize, we can argue the validity of any proposition; and we like to cow our opposition by maintaining that our position is logical (implying the position of our opponents is illogical).
4. NONSYSTEMS OF LOGIC
As we have seen so far, a system of logic includes all fundamental principles of logic that are necessary to understand reality. The following subsets use the word “logic” but they are not systems of logic. Let us see some of their particular limitations.
4.1 Conventional Logic
Believers in Conventional Logic maintain that there is not such thing as the truth. For them, it is all matter of "conventions." If they stopped here, they would not build any "system" of logic. But they shift position, and accept other systems of logic. It is useless therefore to insist on their nonsystem—except to notice that they who sincerely believe in their original position give up the struggle for understanding before they enter the ring. What is Thought supposed to do, if not being constantly engaged in the search for Truth? (The key words are purposely capitalized). Why carry an empty shell in our brain? Why behold a Convention at all? Is it because other people hold it? Is it to feel superior or, at least, more cynical than other people? As interpreted by Peter J. Bearse: “Conventional Logic is not a system of logic, because it accepts illogical statements that make no sense.”
4.2 Positive Logic
Positive Logic has apparently taken its first steps from Primordial Logic. Geometrically, Positive Logic can be represented in this fashion:
in which each point represents an individual observation.
Econometrics and the appearance of the computer are two of the latest developments which keep much economics—and many economists—locked in within the realm of Positive Logic. Just like Conventional Logic, however, Positive Logic is not a "system" of logic. There is no relation between points. Just the facts, Ma’am; just the facts. What are its characteristics? There are two fundamental characteristics. It is enough to observe Fig. 7 to realize that there is not a rule to help its practitioners reach a decision as to when it is possible to stop making observations and start analyzing whatever information has been collected so far. Positive Logic, denying the existence of ideas, is locked in within the realm of "facts.” As Boland (op. cit., pp. 507, 5ll) points out, "Contrary to the hopes of the inductivists, even though one can distinguish between positive and normative statements, there is no inductive logic that will guarantee the sufficiency of any finite set of singular statements. There is no type of argument that will validly proceed from assumptions that are singular to conclusions that are general statements.... one cannot directly solve the problem of induction." Let us put it this way: In the midst of a serious monetary crisis, much effort is spent analyzing inexhaustible reams of financial data and we are still proceeding without the definition of money. Economics textbooks tell us the functions of money; they do not tell us what money is.
Perhaps it is not entirely unfair, then, to conclude that pure Positivists are still at the level of the first observations taken by primordial man. They have a long way to go. Of course, “pure" Positivists (or, as we shall see, pure Deductivists or Inductivists) are stereotypes that —if encountered in reality—would be more akin to caricatures than to truly thinking people.
4.3 Deductive or Syllogistic Logic
Deductive Logic is locked in within the realm of ideas. Ideas are generally expressed through simple propositions; it is not generally realized that single words also contain ideas. The “big” idea, of course, is the idea of Being. Even when combined with the complementary idea of notBeing (especially notBeing of the East), discussion has exhausted the intellectual resources of many a philosopher for 2,500 years, but not too much enlightenment has been gained.
Perhaps all to the good, contemporary thinkers who are imprisoned in this nonsystem of logic, Deductive or Syllogistic Logic, can be counted on the fingers of a hand in the West.
Do these statements and those concerning the limitations of Positive and Inductive Logic that follow contradict the assertion made at the beginning of the paper that all systems of logic are equivalent to each other in their consistency and sufficiency? Not really. First of all, it is questionable whether all claims of expressing logical propositions are planted indeed in systems of logic. But, more fundamentally, all men are locked in within limitations. Those who prefer to be locked in within the confines of facts and those who prefer to remain locked in within the confines of ideas, observed from inside their "systems" of thought, hardly suffer from any limitation. After all, both facts and ideas are infinite. And the (supposedly) mutually exclusive observation of facts or ideas has its own hidden potential which bursts forth as soon as one considers the work of Leontief (1976) or Boulding (1956) for instance.
What are the limitations of Deductive Logic? Deductive Logic has forgotten how systems of logic ever developed. It was from the observation of factual evidence, and the abstraction from that evidence of successive sets of ideas that eventually helped give names to “things.” Progress occurred when common properties (ideas) were found to bind individual facts together. Thus one operated a synthesis of Inductive and Deductive Logic.
4.4 Inductive Logic
What has been said in relation to Positive Logic and Deductive Logic largely exhausts what needs to be said in relation to Inductive Logic. The abstraction of ideas from factual evidence is essential to the construction of our understanding of the world.
And yet, there is no such thing as an absolute "inductive" or experimental proof of the truthfulness or validity of any statement.
In brief, the old dichotomy between Inductive and Deductive Logic is a false one. The two approaches are complementary. Only then are they both productive; otherwise, they are both sterile.
4.5 Instrumental Logic
The reader who has read Boland's paper very carefully and who now perhaps has a better acquaintance with Eastern Logic will automatically discover the great similarities between the latter and Instrumental Logic. But one characteristic of Eastern Logic needs to be made explicit: its mysticism.
How does Eastern Logic reach its conclusions? The answer might be put this simply: "The ways to knowledge are many and arcane. One needs to assiduously study (contemplate) the evidence. Do not judge me on how do I reach certain conclusions. I do not know. Rather, judge me on the validity/truthfulness of those conclusions." Instrumentalism might thus be called modern Western mysticism. Both Instrumentalism and Eastern Logic aim at communication, predictability, and control (of at least a temporary attention from their audience and disciples). Both rely on luck. But their luck is often the result of hard thought.
Essentially, Milton Friedman, who with his Essays in Positive Economics (1953) was the major exponent of Instrumentalism, could have used Descartes’ words: "For myself, I have never fancied my mind to be in any respect more perfect than those of the generality; on the contrary, I have often wished that I were equal to some others in promptitude of thought, or in clearness and distinctness of imagination, or in fullness and readiness of memory.... I will not hesitate, however, to avow the belief that it has been my singular good fortune to have very early in life fallen in with certain tracks which have conducted me to considerations and maxims, of which I have formed a Method that gives me the means, as I think, of gradually augmenting my knowledge.... My present design, then, is not to teach the Method which each ought to follow for the right conduct of his Reason, but solely to describe the way in which I have endeavoured to conduct my own" (op. cit., pp. 5, 6).
There is a very good reason for singling out Milton Friedman in this paper. Milton Friedman is the last of the great economists who so believed in the need for a method that he created his own tools: Instrumentalism. All other economists have succumbed to the numbing dictum of Paul Feyerabend, who declared himself Against Method: Outline of an Anarchistic Theory of Knowledge (1975). So today everything goes. The result is that economics, in the famous dictum of Jacob Viner, has become "what economists do.” Everyone is free at last. Everyone is for himself. The penalty is that followers are free to follow no one.
4.6 And then there Is Economic Logic, the Logic of Balancing Contradictions
…And then there is economic logic. As Keynes specified in the preface to the General Theory, "...if orthodox economics is at fault, the error is to be found not in the superstructure, which has been erected with great care for logical consistency, but in a lack of clearness and of generality in the premisses." If one applies the principles of formal logic to the inner structure of economic analysis, one discovers that not one of its major component terms respects fundamental principles of logic (Gorga, 2002 and 2009, Chs. 415). As already noted, this is what R. W. Goldsmith, a professor of economics at Yale, tells us in a footnote in the second of his three volumes titled A Study of Saving in the United States (1955, p. 69n). Examining the "specific (operational) definitions of saving," he found that ". . . the number of theoretically possible variant definitions of saving as change in earned net worth is as high as 2^{5} x 5^{5} or 100,000." Investment is never defined—except as being “equal" to investment. Consumption, in modern theory, does not mean physical destruction of wealth; it means expenditure of money; yet, not all expenditures of money are counted as consumption; only expenditures to buy consumer goods are arbitrarily counted as consumption. Not one of the fundamental building blocks of economic theory respects the dictates of the principle of identity: A term must clearly mean one thing and one thing only.
If principles of logic are not applied, what is the logic of economics, then?
Analysis demonstrates that the consistency of economics that was claimed by Keynes and, implicitly or explicitly, by most economists is not an external consistency measured against principles of logic, but an internal consistency. History proves that the logic of economics is the logic of balancing contradictions. As soon as a contradiction is discovered, great effort is undertaken to repair the flaw; yet, since the effort does not go to the root of meeting the challenges of formal logic, the attempt is destined to fail. That is the reason why mainstream economics has been in a state of crisis since the publication of Keynes' General Theory. This is commonly recognized. What is not generally admitted is that the crisis is older still. It goes all the way back to Adam Smith. That is why economics has regularly been subjected to such major upheavals ever since 1776 as recorded in the books of the history of economics. From classical economics we were led to neoclassical economics to the marginalist revolution to the economics of Keynes to Keynesian economics to postKeynesian economics to monetarism to real business cycle theory to behaviorism—let alone Marxist economics or Austrian economics or Georgist economics or Kelsonian economics; indeed, let alone such (major) splinter programs of research within each major school of economic thought as labor economics, industrial economics, feminists economics, and so on and so forth.
5. THE ISSUE OF CONSISTENCY
It should be apparent now that discussions across systems of logic are destined to be fruitless. The external manifestations of these systems are so many and so varied (even neglecting the myriad of details) that one can hardly become thoroughly familiar with any of them through a lifetime of study. And without complete familiarity that is available only to insider cognoscenti, action and reactions based on external criticism can be irritating, and are certainly pointless. Dialogue begins to become possible only when any two interlocutors are within the same system of logic. Otherwise, an optimistic assumption relies on three conditions: First, that the interlocutors are aware of the existence of different systems of logic—thus avoiding the pitfall of believing that there is such a thing as only one system and using the word "logic" indiscriminately; second, that each interlocutor reasons exclusively from within one system of logic; third, that an attempt is made to “translate” meanings from one to the other system.
The situation begins to appear a little less despairing as soon as one observes the internal structure of those systems. Then it is possible to agree, first of all, that all systems of logic are consistent. The simplest proof of the validity of this position is not so much to suggest the tautology that if they are not consistent they are not systems of logic, but especially to advance the proposition that all fully constructed systems of logic are built upon at least one or a set of equivalence relations. They contain at least three basic propositions, and these propositions are equivalent to each other.
Let us observe this allimportant phenomenon at least in relation to those systems of logic in which it is most evident.
6. THE EQUIVALENCE IN ALL SYSTEMS OF LOGIC
Quite apart from possible conclusions from deep future philosophical and/or neurological studies, the equivalence is a mysterious invention of our mind that gloriously gives us a triplecheck on our conclusions. Thus our early ancestors and classical logicians reasoned as follows: How do I know that the moon (or the truth) is the moon? The first evidence I have is that I am consistently observing something that I have chosen to call the moon. The second piece of evidence is that I do not see anything which contradicts the evidence I have assembled. Therefore, until contrary evidence is brought forward, I can rest assured that I am actually observing some individual event that I have chosen to call the moon—and I am not observing a salmon. That gave certainty to Greek thinkers—no less than to our earlier ancestors.
Primordial Logic, as we have seen, is based on a very basic equivalence relation that automatically includes the principle of identity and respects the principle of noncontradiction. The syllogistic structure of Classical Logic forms sets of equivalence relations. There is more. Greek thinkers began to use the equivalence relation in many fields, and its applications have unmistakably multiplied ever since: A syllogism is an equivalence; a system of equations is an equivalence; trigonometry is based on an equivalence; regression analysis is based on an equivalence; many religions are based on an equivalence (The Father ≡ The Son ≡ The Holy Spirit); all systems of logic are based on an equivalence: Rational Logic (see Fig. 4) is based on the equivalence of A to Z to Infinity; Dialectic Logic is based on the equivalence of A to Z to P; Eastern Logic is based on the equivalence of Yin to Yang to ... the night, the day, the universe, etc.
Perhaps the point that regression analysis is based on equivalence relations deserves to be elucidated somewhat through the use of geometry, in the following fashion:
Given point A and point B, regression analysis is able to estimate point C, by projecting segment AB and making (alternative) estimated guesses as to where point C might lie on the horizon. These three points themselves form an equivalence; but the equivalence, as suggested by the circles drawn around each point, is much more complex than that. The equivalence is actually assumed to exist among the "situations" around those points—and indeed determining each point. The greater the knowledge of conditions determining points A and B, the better the chances of estimating C correctly.
7. THE TRANSITION TOWARD RELATIONAL LOGIC
The transition toward Relational Logic that has taken place within the work of this writer perhaps will remain forever enshrouded in the mysteries concerning the inner movement of thought. The spark that illuminated and indeed established the circuit between "moral" and "physical" sciences so barely outlined above occurred while reading that physicists—with Newton at the head of the parade—proceed without a definition of gravity.
The spark can be simply put in this fashion. There are three “terms” in what physicists observe: Action and Reaction are the first two; the third term is the Relation between the two. The apple falls to the ground because the earth has a tremendously more powerful force of attraction within itself than the apple; but, though miniscule, the apple also has a force of attraction within itself. It is the play of these two forces that determines the direction of the action. Let us put it a little more technically: If the apple had no power of attraction within itself, it would fall to the ground at a minimally faster rate of speed—at the limit, instantaneously.
The spark was ignited by the knowledge of Dialectic Logic. That knowledge created the following definition of gravity: Gravity is the process of action and reaction; namely
Action ↔ Reaction ≡ Gravity
Action and Reaction are the first two terms; the notation ↔ is the invisible link that fuses the first two terms together, in a fashion that we synthetically call “gravity.”
At the distance of a few months from this discovery, the breakthrough occurred. The foundation of Relational Logic had been laid out.
8. THE FOUNDATION OF RELATIONAL LOGIC
The foundation of Relational Logic is composed of the following axioms, which are presented in a literary and geometric fashion. These axioms read as follows:
I Axiom: A point is equivalent to a line, and both are equivalent to infinity.
II Axiom: A point is equivalent to a circle, and both are equivalent to a sphere.
III Axiom: A point is equivalent to a sphere, and both are equivalent to infinity.
The whole of Relational Logic can be enclosed in the following geometric representation:
Only three observations concerning Fig. 9 are permitted here. First, this figure encloses within itself all the essential elements of each system of logic observed above. Second, these elements are indeed enclosed here in an "organic" fashion. Third, Fig. 9 can be progressively interpreted as representing a single cell, many things in between, or the entire universe.
Mathematically, in fact, Relational Logic can be synthetically expressed with the following equivalence (Gorga, 2010):
0 ≡ 1 ≡ ∞
or, alternatively,
0 ↔ 1 ↔ ∞
Zero is equivalent to one, and both are equivalent to infinity. All three elements are complete systems in themselves; and yet, they can be fully understood only when observed in relation to each other—as in this expression: the infinitely small is equivalent to a blade of grass and both are equivalent to Infinity.
8.1 The Principles of Relational Logic
As it can be seen from the above, there has never been any disagreement about the validity of the Principle of Identity. The Principle of NonContradiction has also been implicitly or explicitly accepted as valid by all systems of logic, except (dynamic, longterm) Dialectic Logic. Indeed, both principles have been subjected to much refinement. They are accepted as given by Relational Logic.
The struggle has always occurred on the validity of the third principle which must link the former two together. Explicitly, the Principle of Excluded Middle was replaced by the Principle of Indifference and both by (for short) the Principle of Process. Relational Logic suggests that any such principle be replaced with the Principle of Equivalence. Relational Logic provides the intellectual means for a cumulative set of proofs.
But an identity relation is itself an equivalence relation (cf. Suppes 1957, p. 218). Is Relational Logic simply repetitive? Not at all. The identity relation directs the observation inward, always deeper and deeper into the "event" itself. The equivalence relation, instead, directs the observation outward, linking the event always closer and closer up with the outside reality.
8.1.1. Some Application of Relational Logic
Mathematics respects religiously the dictates of the principle of identity and the principle of noncontradiction; mathematics is especially all built on equivalence relations (Gorga, 2010). Physics still benefits enormously from the application of this methodology (Gorga, 2007). Through the “remorseless” application of this methodology, economics morphs from mainstream economics into Concordian economics (Gorga, 2009b); political science escapes the doldrums of the individualism vs. collectivism paradigm and falls into the embrace of Somism, the theory and practice of the social man, the civilized person (Gorga, 2014); theology is affirmed (Gorga, 2009d); and philosophy is rejuvenated (Gorga, 2009c).
9. A READER'S GUIDE
The issues treated in this paper are so numerous and cover so wide a field that it might be useful to reduce the paper to a synopsis of two or three topics that might be of more direct and immediate interest to the reader who specializes in economics.
9.1 On Friedman's Instrumentalism
Friedman's Instrumentalism is a valid system of logic that might be reduced to the following equivalence:
Economic Reasoning ↔ Useful Conclusions ≡ Economic Theory
Friedman's Economic Reasoning is not abstract. It is reasoning spurred by the continuous study of observable facts; but the facts are not observed through the lenses of a rigid system of economic theory. Theory for Friedman is a conclusion; it is a derivative. Rather, he uses all available theories to help him study the facts in order to reach Useful Conclusions. And he wants to be judged not on the Economic Reasoning that precedes his conclusions, nor on the "Theory" that results from his efforts, but simply on the validity of his Conclusions.
This methodology might not—an indeed it does not—satisfy everybody. But, as Boland makes clear, Friedman does not want to be judged on this issue. He wants to be judged on the validity of his conclusions. Besides, since he does not suggest that his methodology should be imposed upon others, others should not want to impose their methodology on him.
This paper reinforces the appropriateness in choosing one's own methodology. With so many systems of logic available, one indeed has the duty to choose the methodology that best fists one’s own temperament and purposes in life, as well as one’s own intellectual, spiritual, and aesthetic capacities. The issue of the choice among systems of logic is more complex than can be indicated in this context. It of course starts with the knowledge of the existence of that many systems of logic. Indeed, this paper tries to establish an even more important point. It tries to foster tolerance of all systems of logic by pointing out that, provided they are consistently and thoroughly developed and applied, they are all equivalent to each other.
9.2 On the Equivalence of Systems of Logic
From the most simple to the most complex, all systems of logic perform the same function. They all provide rules of correct, disciplined reasoning.
But rules pertaining to one system can be entirely contradictory when compared to rules of other systems. That is predictable and immaterial. What is important is only that rules should not be contradictory when compared to other rules belonging to the same system of thought.
Hence methodological discussions across systems of logic are totally pointless. Consequently, this issue leads to the thorniest questions of all: What is the relationship between systems of logic and economic theory? Finally, how to judge the value of each economic theory?
9.3 On Systems of Logic and Economic Theory
Each system of logic leads to a peculiar vision of the world. This vision affects all of man's activities, from practices to theories—and hence, unavoidably, economic theory.
Even though the issues require much elaboration, we have briefly seen how Classical Logic led to a deemphasis of economic theory, or how Rational Logic led to what might be called Rational Economics, or how Dialectic Logic led to Marxist Economics. These correspondences might of course be a result of happenstance. But they are too strong to lead to any other conclusion. Systems of logic provide basic structure to thought, and economic theory is automatically influenced by that structure.
Relational Logic has led to Concordian economics. This is an entirely new paradigm, which, though presented in a number of readily available books and papers published in peerreviewed journals, is still in its infancy. The field is wide open; everyone is welcome.
So far, so good. The question, however, remains: How can we compare economic theories? This is the thorniest question of all.
9.4 On the Comparison of Economic Theories
The question of how to compare theories is thorny. Indeed, it is insurmountable. Peace of mind is acquired only when one accepts the reality: Namely, recognize the existence of many theories; accept their validity, if they are internally consistent; and then refuse to compare them.
9.5 The Real Issue
The real issue is a different one. The real issue concerns the fruits of each theory—hence the intrinsic validity, again, of Instrumentalism. While it is invalid to compare a pear tree to an apple tree (a tree is a tree), it is entirely appropriate, valid, and indeed necessary to compare a pear to an apple. Then the discourse flows. Then the discourse is profitable. This writer's standard is to compare economic theories in relation to the solutions they propose for the problem of poverty. Other people might prefer a variety of other standards: efficiency, creation of wealth, preservation of the ecosystem are possibilities that easily come to mind.
CONCLUSION
The above is a schematic presentation of various systems of logic, from Primordial Logic to Relational Logic. The implications—for economics, no less than for many other mental disciplines—are vast, and can only be explored through subsequent efforts. Two observations should suffice to bring this paper to a close.
First, this work proves that men and women of—at least—40,000 years ago were reasoning the same way as we have been reasoning ever since. Clearly, there has been an astonishingly constant use of identical principles of logic all throughout “recorded” history. The questions this observation raises are innumerable. They have to be left out of this presentation.
Second, all complete systems of logic are equivalent to each other. And what does this rather cryptic expression mean? It means that each system of logic, if it is consistently developed and applied, enables the practitioner to reach an understanding of the world—or portions thereof—which is valid and true. That understanding can withstand all tests of validity. That understanding is true because the truth simply is: it was, it is, and it will be. What changes is the "amount" of truth that is grasped at each time. Certainly, the more "complex" the system of logic used to reach that understanding, the "greater" the amount of truth can be expected to be grasped. But each amount is equivalent to all others. The truth is the truth. A "larger" truth is not truer, even though it may be more useful than a "smaller" truth. Knowledge necessary for celestial navigation is not truer than knowledge for terrestrial navigation. Therefore, mastery of the knowledge of systems of logic leads not only to understanding. It also leads to tolerance.
Acknowledgments
This paper received great initial encouragement in 1979 from such notable figures as Buckminster Fuller, “I find your paper… to be extremely interesting"; Moses Abramovitz, "I find your article…of considerable personal interest.... I am... very grateful to you for letting me see it.... I would want to show it to my students"; Charles T. Wood, a Professor of Medieval History at Dartmouth College, “Thank you (David Wise) for your letter of November 25 and the fascinating piece it enclosed.... Since I'm really supposed to be grading final exams this morning, I haven't had a chance to devote as much time to Gorga's paper as it clearly deserves. I do know, though, that I am in sympathy with many of the points he makes: e.g., on the inability to communicate across logical systems, on the equivalent merits of the various systems, etc. I'm not sure that I have as yet mastered the intricacies of organic logic [later classified as Relational Logic], which obviously needs a lot more thought than I've had a chance to give it, but the basic proposition makes sense." Equally encouraging have been a few other readers to whom this paper has been shown over time. Joan M. Gorga saved me from some serious grammatical oversights. Special thanks are owed to invaluable editorial assistance by Dr. Peter J. Bearse.
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ABOUT THE AUTHOR
Carmine Gorga, president of The Somist Institute, is a Former Fulbright Scholar and recipient of a Council of Europe Scholarship for his dissertation on the Political Thought of Louis D. Brandeis. Using ageold principles of logic, he has founded Concordian economics, Somism, and Relationalism. Dr. Gorga has fundamentally transformed the linear world of economic theory into a relational discipline in which everything is related to everything else—internally as well as externally. He was assisted in this endeavor by many people, notably for 27 years by Professor Franco Modigliani, a Nobel laureate in economics at MIT, and 23 years by Professor M. L. Burstein, a professor of economics at York University. Mr. Gorga is the author of numerous publications, including The Economic Process: An Instantaneous NonNewtonian Picture, 2002, a book that was reissued by The University Press of America in an expanded paperback edition in 2009. He can be reached at cgorga@jhu.edu. At times, he blogs at New Economic Atlas, Modern Moral Meditations, and A Party of Concord.
