Non-Aristotelianism is the primary premise for a specialized field of study called General Semantics: a difficult subject to understand as it is totally different from almost everything else we ever learn. Aristotle wrote that a true definition gives the essence of the thing defined (in Greek, literally "the what it was to be"). This is such a core concept in almost all our understanding of the world, including science, mathematics and logic, that we have trouble recognizing how it hobbles our ability to perceive reality more accurately.

A Non-Aristotelian System from Alfred Korzybski, the Basis of General Semantics
A Non-Aristotelian System from Alfred Korzybski, the Basis of General Semantics

General semantics therefore denies the existence of such Aristotelian 'essence.' It also denies such 'essence' is necessarily recombinant in any algebra of standard mathematics, because there is no necessity that concepts, words, and reality should be directly correspondent to each other, but could in actuality, be only loosely inter-related.

Hence, numbers themselves, which are often regarded as some kind of pure conception, are no more than one model amongst many alternatives which could be equally true.

The Basis of Aristotelian Logical Relationships. Non-A Thought Suggests this to be just one Possible System of Many
The Basis of Aristotelian Logical Relationships. Non-A Thought Suggests this to be just one Possible System of Many

For example, the union of A and B does not necessarily consist of exactly what would exist if A by itself, and if B by itself, are combined. Non-Aristotelian mathematics may with perfect validity add or subtract interdimensional properties upon any unidimensional logical combination, for example.

Also, we may have a concept of 1, and a concept of 3, but adding 1 to 3 may produce something other than 4. This is a philosophically interesting approach, because it makes possible the existence of an infinite number of non-Aristotelian mathematical systems--a possibility mostly avoided in the West, because it challenges ideas of standard common sense too much.

In general semantics, it is always possible to give a description of empirical facts, but such descriptions remain just that--descriptions--which necessarily leave out many aspects of the objective, microscopic, and submicroscopic events they describe. Any attempt to isolate and purify a concept may produce interesting deductions, but they are necessarily reductionist, as the original descriptions had to be simplified in order to produce them. Hence also, all language, natural or otherwise (including the language called 'mathematics') can be used to describe the taste of an orange, but one cannot give the taste of the orange using language alone. The content of all knowledge is based upon structure, so that language (in general) and science and mathematics (in particular) can provide people with a structural 'map' of empirical facts, but there can be no 'identity', only structural similarity, between the language (map) and the empirical facts as lived through and observed by people as humans-in-environments (including doctrinal and linguistic environments).

This method of human understanding produced the famous statement 'the map is not the territory.' No matter how many markers one puts around a region, and no matter how finely grained the resolution of the numbers used for their placement, the territory described by a map is never exactly what the map states it is.

Non-Aristotelian Thought

The logic, ethics, and cognitive science of non-Aristotelian thought ~ Progressing from the very basics, then; in set theory: {A} refers to items included in a set, and null{A} refers to items not included in the set {A}. Consequently. {A}\{A} represents the 'relative complement' of A with itself--That is, the remainder when {A} is subtracted from {A}. Should the remainder be 0, or null{A} ?

An Aristotelian answers 0, because null{A} exists whether you subtract {A} from {A} or not, since it is the set of all things not in (A} and so does not care.

A non-Aristotelian. on the other hand. answers the result could be null{A}, because if {A} is removed (which is the same as subtracting {A}) then null{A} could appear (or be produced) as the new condition, as the absence of {A}.

For example, if one removes all the molecules of air from a jar, a vacuum remains. In this case, the vacuum refers to the absence of atoms. The vacuum is defined by absence of something which could be there, in other words, it is defined by a null set. This has been very important in physics, because a vacuum is an abstract concept which never really exists. However by assuming a null set, one can simplify calculations of electromagnetic and subatomic particle movement in the void, by removing various unnecessary calculations~~as they are nulled by the absence of air.

Thus, there is a difference between nullification and zero in physics. And likewise, there is a difference mathematically. Mathematicians often factor out zeroes from equations to simplify them, glibly considering a 0 and a null quantity to be the same. Semantically however, the mathematician is transforming the zeroes into nulls, in order to avoid calculating zero quantities unnecessarily. Pure mathematicians would like to define a difference between null and a zero quantity in conventional mathematics, but it transpires to be very complex, requiring the definition of some new kind of definition through which zeroes pass when transformed semantically and nullified. To make the distinction, it is necessary to consider all simple numbers not to be linear values, but rather multidimensional, and in effect, completely defining a new mathematical system.

Now to consider the implications of this example at a more abstract level, for Aristotelian logic. Aristotelian logic states that which is {A} is necessarily separate from that which is not {A}. That is the basis for all pure mathematics. A and not-A do not overlap, and any value can be combined quantitatively in various progressions to make 2A, 3A....A*A, A*A*A...A/A....etc.

By contrast, Non-Aristotelian thought states such an artificial separation of A and not-A is a simplification which only leads to error. Similarly, progressions need not be linearly progressive, such that adding an infinite number of values 'A' does not necessarily calculate in the same way as adding an infinite number of values 'B.'

In reality, general semantics states, it is not necessarily possible to perform such total separations or apply such uniform rules. Therefore, the logical deductions based upon such separations and combinations must contain semantic simplifications, because the definition of a separation of A from B, or applying consistent rules of combination across multiple cases, itself is an act of simplification, because in reality such separations and continuations are not necessarily pure in nature.

The simplifications therefore can cause results which are necessarily incomplete, possibly paradoxical, or possibly anomalous. The results can contain inexplicable terms, such the imaginary numbers (the "square root of minus one", which cannot exist as a real number, but which can be created on both sides of an intermediate equation so that it can cancel itself out, thus removing the imaginary number and reducing the final equation's complexity). There can be multiple but incompatible results to the same equation. And there can be inconsistent conclusions about various kinds of infinities, nulls, and zeroes. For example, in some cases, a set of infinite values contains the same number of values as...an infinite number of sets each containing an infinite number of values; and in other cases, these are different infinities. To add to the confusion, there is also the set of all infinites, which cannot include itself, or does include itself, depending which set of all infinities one is considering at any time, and which may contain the same number of values or not, depending how one chooses to consider it.

Advocators of pure Aristotelian systems can only excuse such anomalies as unfortunate limitations in our ability to understand 'pure' mathematics.

General Semantics: Applying Non-Aristotelian Thought to Language

Non-Aristotelian thinkers instead indicate that the paradoxes in mathematics (which is meant to be a pure language) are miniscule in scope compared to the paradoxes arising in ethics and cognitive science, because in such fields, the intermediate descriptions between our deeper cognition and the words we use to describe them are far more complex.

Saying "abortion is wrong" isn't the same as stating "abortion is not right," because ethical judgments are non-Aristotelian.

In natural language, words have far too many shades of meaning and interacting factors to permit any necessary exclusive concluding condition that, even if proposition (A) is always true, that one can necessarily conclude proposition (B) is always not true. As one very simple example, to state "a law against abortion is wrong" is NOT the same as stating "a law against abortion is not right," despite the fact that, in Aristotelian terms, the two statements could hardly be more equivalent. Moreover, instead of abortion, one could substitute many other ethical debates in these two propositions, and the choice as to which of the propositions would be better will vary.

Even so, from such a basis, non-Aristotelian thinkers do draw substantive conclusions by adding more scope, qualification, and specificity to ethical and cognitive propositions, improving conciseness and accuracy in such fields. To do so, non-Aristotelians start with guidelines refined from their first conception in the 1920s. These make a layered distinction between physical reality, our deeper unexpressed perception, the concepts to which they relate, the descriptions of the concepts, and the words connected to those concepts. By distinguishing between the layers of mental activity, and the resulting words used to express the inner thought, non-Aristotelians hope to improve comprehension and expression.

A common difficulty with understanding the early writings in general semantics is that the neurosciences have advanced significantly since Korzybski first advocated the ideas in the 1930s and 1940s. Somewhat strangely, one discovers though, Korzybski's own ideas of cognition's layers permit a relative correlation of his terms with current scientific knowledge. Here he is describing his layered semantic theory with a 3D physical model.

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Reactions to Non-Aristotelian Thought

While this methodology may appear, intuitively, to offer many possibilities,it has not been successful for two main reasons.

First, the broad group of people who find the concept appealing tend to be those with fundamental problems with Western civilization as a whole. Many of them are altogether incoherent. At first I was puzzled why the main advocator of this field, Alfred Korzybski , ended up working in psychotherapy. After attempting to discuss the topic in groups interested in semantics, I was not surprised. Empirical observations were frequently intermingled with raging rants against the social order, advocations of anarchism, etc.

Second, the narrow group of professionals who one would think most interested in expanding this field, theoretical mathematicians, were all, without exception, more interested in disproving the concept that non-Aristotelian logic is not already part of existing formal mathematical systems, than in expanding the implications of a layered non-A system. After extended discussion, frequently such individuals cite set theorems which they studied at college demonstrating that any non-A system must collapse into Aristotelian logic,and are so adamant that the premise is therefore flawed, further discussion of its implication to semantics is impossible.

I tried to indicate the latter group that , even if their conviction could be shown as absolutely true, that the benefits of expanding a formalized logical system based on non-A thought could provide better models of observed phenomena in certain situations. To no avail.

Similar objections appeared in mathematical circles as an initial reaction to neural networks, which are now an accepted concept, and certainly a simplified derivative of the above proposed rational system. So I remain hopeful maybe one day this field of study will result in serious progress for fields where absolute statements are difficult to make, such as social psychology, politics, and ethics.