Conventions for Symbolizing

Contact Us
Whenever the most beautiful is perceived, ugliness arises, the least beautiful. Whenever good is perceived, evil exists, its natural opposite.
Thus, perception involves opposites: reality and fantasy are opposing thoughts; difficult and simple oppose in degree; long and short oppose in distance; high and low oppose in height; shrill and deep oppose in tone; before and after oppose in sequence.
The truly wise accept this and they work diligently without allegiance to words. They teach by doing, not by saying; are genuinely helpful, not discriminating; are positive, not possessive; do not proclaim their accomplishments, and because they do not proclaim them, credit for them can never be taken away.
For members of Western cultures—native speakers of one or more of the Western Indo-European (WIE) languages—the above passage from the Tao Te Ching seems somewhat alien. We Westerners regard "natural opposites"—pairs of polar terms—not as closely related and mutually-determining, but rather, as disparate, separable, as if their oppositeness provides the ONLY relation between them. As Alan Watts puts it,
By and large Western culture is a celebration of the illusion that good may exist without evil, light without darkness, and pleasure without pain, and this is true of both its Christian and secular, technological phases. Here, or hereafter, our ideal is a world in which "there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain; for the former things are passed away." (Revelations 21:4) 2
In the present paper, we deal extensively with the polar terms theory and practice. Initially we focus on theory, disclosing and developing inter-connections between the "parts" of an axiomatic system. Then in the Discussion section, we consider some of the implications for practice, the "practical" repercussions, of these developments of theory.
The moment we mention the topic of theory, we risk losing most of our audience. In accord with the so-called Western cultural "illusion," good, practical people scorn theory—"mere" theory. It has no bearing (we say) on the pragmatic concerns of realistic, practical folks. It just doesn't matter.
We sympathize with this attitude. We have developed more than a passing acquaintance with what passes for theory among Westerners, and a great deal of it seems not to make a difference in practice. Furthermore, we suggest why not. Characteristically Western theories systematically eliminate the observer from consideration. And by so doing, such theories deny that any transacting occurs between an observer and her/his theories. But if the theorist has no effect on his theories, and/or the theories have no effect on the theorist, it should come as no surprise that such theories "just don't matter."
(The fact that the revolutionary physicists of the beginning of the Twentieth Century first introduced the notion of "taking the observer into account," and found ways of building this notion into their theories, means only that their theories remain inconsistent. What relativity and quantum theory give with one hand, they take away with the other.)
In the present paper, we do not use Western theories (nor Eastern ones either). Instead, we utilize an entirely non-traditional theoretical system, based on assumptions created, or generated (or whatever-one-does-to-produce-assumptions) by a particular person—Alfred Korzybski (1879-1950)—in a particular time-and-place (mostly Chicago, during the period 1921-1943) rather than by anonymous "tradition." Our theoretical system takes the observer into account in a systematic fashion. Hence we DO show the transacting that occurs between observer and theory. We deal with a new class of theories: lived theories, which do make a difference to the people that use them.
We invite our readers to join us in exploring territory which seems unfamiliar to everyone, ourselves included—and find out just what kind of a difference it makes to rely on lived theory.
For a very long time, humans have "known" in principle that the Earth we live on orbits the Sun, has the shape of a flattened spheroid, rotates on its own axis, etc. But not until someone got off the surface of the Earth and went far enough out to see it as having an uninterrupted contour did anyone know these things experientially. Those first photographs of the Earth made from off-planet provided a polar-opposite, a contrast to our earthbound experience which irreversibly altered our perception of, and our relation with, terra firma. Experientially, it became a planet—our home planet—and we became inhabitants-of-a-planet. The human race has yet to finish assimilating this major alteration of our perceptions, and working out its consequences.
Similarly, we humans live as social organisms embedded in a matrix called both culture and language. This matrix consists ultimately of human "doings" in the human environment, the Earth's biosphere. The terms culture and language serve to name aspects of these "doings". Language comprises the self-reflexive sub-category of these "doings," within which we humans can get explicit about the "doings" (including "language") which make up the matrix. This matrix appears highly diverse, with many different cultures (which students of culture classify by geographical region, by the level of technology employed, etc.) and many different languages (which students of language classify into linguistic families, super-families, etc.).
Here too, in principle we "know" a lot about this matrix—e.g., about language and about the reciprocal relations between humans and language. But a part of what we "know" appears somehow askew. For example, in English, language functions as a noun, "the name of a person, place, thing, etc." In our unguarded moments at least, we treat the term language as if its referent, "that which the term language designates," comprised a static-and-unchanging THING. When we look candidly at what we DO, however, it seems more apt to represent the referent as a process or processes, using a verb, to language, created for the purpose. Then we can say, succinctly, that we humans language, that we engage in languaging.
The term languaging includes our speaking-and-listening, writing-and-reading, signing-and-receiving-signing, etc. It also includes the bodily movements which accompany our speaking-and-listening.
As long as a human continues living, her/his body-parts—e.g. chest, left thumb, head, right knee, etc.—make small movements. Any given movement begins, continues for a while, and then stops or else changes direction. Also, the process of speaking itself consists of making a series of small movements. When a human speaks, any locution s/he emits consists of a string of "syllables." When observed in sound movies made at 24 frames per second and studied with a time-motion analyzer, the small bodily movements which accompany speaking maintain a precise relationship to the syllables which make up the locution which the speaker emits. Each of these observable movements starts in precisely the same frame of the film as does the beginning of a syllable, and ends in precisely the same frame as does the end of a syllable. Some movements last for only one syllable; others span several syllables. But each movement starts in the same frame as does a syllable, and each ends in the same frame as does a syllable. As Condon & Ogston put it,
... The body dances in time with speech. 3
As observed in sound movies using a time-motion analyzer, the bodily movements of a person listening to a speaker occur in precise synchrony with the syllabic structure of what the speaker says. Each movement of the listener starts in the same frame as does the beginning of a syllable of what the speaker says, and ends in the same frame as does the end of a syllable.
... Metaphorically, the three interactants [in the filmed behavioral sequence under discussion] looked like puppets being moved by the same set of strings. 4
We interpret the small motions of a listener's interactional synchrony with a speaker as an outward and visible sign of the process of listening, of languaging. 5
Even the inclusive-sounding hyphenated terms we use to describe human languaging turn out misleadingly discrete and isolating—whereas the activities they refer to permeate, influence, and form a part of just about everything we humans do. We find it almost impossible to separate our activities—ranging from our largest, most central ventures to the smallest and most trivial—from the ways we language. We humans get born into a social (and that means cultural and linguistic) context, grow up there, produce and raise offspring, engage in social relations with our fellow-humans, and, after we die, the corpses get buried, cremated, etc., in accord with the patterns given in the languaging of our speech communities. Our languaging makes up the core of human living, providing a central, supporting structure for our relations with ourselves, with our fellows, with humans from other communities, with members of other species, with the entire human species, and with the biosphere as a whole.
However, our pictures, our perceptions of these processes and these activities, have lacked opposition. No matter how many contrasting languages we may have, we have had no contrast to those ways of languaging rooted in the distant past. Our species has developed an almost incomprehensible diversity of traditional patterns, but no alternative to the traditional. And as a Taoist might tell us, this lack of opposition has limited what of our matrix we could discern.
In the Western family of cultures, we have created a social institution with the job of developing new oppositions, new perceptions. The exponents of the social institution of science have not only gradually and progressively generated new insights, new knowledge—and have done so at exponentially-increasing rates—but also have led the rest of the human race to adopt these innovations, and so to relinquish more traditional ways of doing things. As Polanyi puts it,
... The discoveries of science have been achieved by the passionately sustained efforts of succeeding generations of great men, who overwhelmed the whole of modern humanity by the power of their convictions. 6
But the exponential increase of Western scientific knowledge has occurred only within a specific cultural and linguistic setting. The anthropological linguist Benjamin Lee Whorf asserts that
What we call "scientific thought" is a specialization of the Western Indo-European type of language .... 7
And since we humans have had no contrasting alternative to science as a specialization of the WIE type of language, we have remained unable clearly to perceive the structure of presuppositions which our dependence on the WIE type of language, and the grammar common to the WIE languages, has entailed. For example, the WIE languages, and therefore the Western cultures, grant a central place to the logical construct of identity (defined as "entire and absolute agreement or negation of difference"8): These languages utilize this construct to 6generate the distinction between the two main "parts of speech"—nouns and verbs—which form the bulk of their vocabularies. 9 They treat any noun or noun-phrase as subject to Aristotle's Law of Identity, and therefore, as identical with itself or self-identical. Conversely, they treat any verb or verb-phrase as not-subject to the Law of Identity—as not-self-identical. 10
Then our classical scientists posit a "world" made up of distinct, discrete "objects" or "things" with inherent "properties" or "attributes," which enter into more or less transient "relations". Further, they treat the "objects" and "attributes" as precisely suitable to represent by means of self-identical noun-phrases, and treat the "relations" as precisely suitable to represent by means of not-self-identical verb-phrases, combined in agreed-upon patterns. In so doing, they grant a privileged position to the grammar common to the WIE discursive and formalized languages—and thus to the at least partially unknown presuppositions encoded in that traditional grammar. The privilege which they grant amounts to an unwillingness to recognize or acknowledge the presuppositions encoded in that grammar, nor to allow themselves to question them, nor to permit anyone else to do so.
Particularly in the last couple of centuries, exponents of geometry, logic, physics, etc., have framed "heretical" viewpoints which in certain ways break with the WIE tradition. But even the non-euclidean geometers, the modern set theorists and the exponents of relativity and quantum theory still maintain tradition in a wider sense: by continuing to utilize identity-based mathematical and logical languages in which to frame their "heresies," they still grant a privileged position to the WIE grammar.
One recent series of studies has systematically broken with tradition. To date, it has yielded an alternative frame of reference which delivers a revised world-view, an alternative grammar derived from known, non-traditional premises, a non-standard notation of the "Let's keep track of what we say" type built up on that derived grammar, and an axiomatized theoretical system. These studies start with the work of Korzybski. In effect, Korzybski makes a quick tour of the universe, asking, "In a cosmos which has human observers in it, when and where may we LEGITIMATELY use the logical construct of identity? Where and when does the notion of absolute sameness in all respects, or negation of difference prove valid? Under what circumstances does it apply?" His considered opinion: Never. Under no circumstances does identity survive scrutiny. 11 And then Korzybski makes an outrageous suggestion: Since identity never holds, he says, let's not RELY on it.
Korzybski proposes that we reject identity—disallow it as a valid "relation."
To state the matter in more general terms, Korzybski discloses that there exist not one but two ways to handle the paired terms identity and non-identity: a) One can "like" identity as a foundation, and "dislike" non-identity (as our linguistic forbears did); or else b) one can "like" non-identity and "dislike" identity. Once a choice becomes possible, any human who uses this term-pair at all must choose which way to use it. Korzybski himself takes a stand, declaring himself one of the company—the first of the company—of those who explicitly prefer non-identity.
Korzybski thus creates a choice with consequences—a fork in the road, or a turning-point, for the human race—where no choice had previously existed.
Furthermore, by way of taking his own suggestion, Korzybski develops a coherent system or world-view, known as general-semantics, and reaches two end-points: a) He brings out into view, and states in English, the fundamental (I might even say metaphysical) premises on which his system rests, including three undefined terms and three non-aristotelian postulates (discussed in more detail below); and b) he develops methods by which to enable others to USE the new system, in their own living, rather than just to TALK ABOUT it.
In our own research project, we take non-identity seriously, relying on Korzybski's non-aristotelian Postulate of Non-identity as our most fundamental postulate. We deliberately generate an axiomatic system—an alternative theoretical system radically different from the WIE frame of reference; from any other traditional language or family of languages; and from the "common-sense" of any traditional culture or family of cultures.
Using this alternative frame of reference, we have developed unexpected contrasts, new questions, new methods of analysis, unexpected insights into the familiar. Of central importance here, we have disclosed two opposing structures of assumptions: one which forms the basis for taking the observer into account, as opposed to another which forms the basis for eliminating the observer from consideration. Then, knowing how it works, we systematically take the observer into account. In the present study, we extend the construct of "taking the observer into account" so as to make a new contribution to the knowledge of axiomatic systems, human languaging, and also of the Gestalt viewpoint.
In the remainder of this paper, we back up these claims by disclosing a general and fundamental pattern which I call the conventions for symbolizing. We shall
  1. State the conventions for symbolizing in general terms, expressing them as fundamental parts of any axiomatic system.
  2. Particularize this general pattern for WIE languages, e.g. for written or spoken English and for written notational ("formalized") languages such as the mathematical theory of sets.
  3. Particularize this general pattern for the non-standard notation we have framed on the derived grammar we have generated.
To facilitate making the comparison between the WIE frame of reference and notation and our own, let us review the general outlines of what we call an axiomatic or postulational system. One difference shows up immediately: In the WIE view, an axiomatic system "exists" as a thing, independent of any observer, logician, or other human. In contrast, what we call "an axiomatic system" "exists" or "occurs" solely and exclusively as HUMAN ACTIVITIES—"something someone does." Thus, someone (a logician or "observer" or "organism") chooses a) some setting or other and b) a small number of undefined terms; s/he selects c) some postulates, which s/he expresses by means of the undefined terms; s/he arranges for d) rules of inference, and e) standards of proof; etc. With these "pieces" in place, s/he then derives and proves one or more theorems, or formal conclusions. In so doing, s/he satisfies our minimal criterion for the "completeness" of an axiomatic system.
Except for the difference of opinion concerning the "locale" of an axiomatic system, we expect that exponents of standard WIE axiomatic systems will agree with this way of naming the "parts" that make up such a system.
However, we discern some key connections between these "parts" not previously described. For example, in keeping with our views on the "locale" of an axiomatic system,
  1. We regard the setting, and also the undefined terms, as human "doings"—something someone DOES. Specifically, we hold that each of these constructs operates as a special kind of postulate: namely, as a silent postulate, the tenets of which the person who relies on it cannot state in words. 12 S/he must, however, know how to USE these constructs correctly.
  2. We posit a further connection between setting and undefined terms, such that for one to occur requires the occurrence of the other. In other words, we posit a polar relation between them.
    1. In choosing a setting, our logician (to use a characteristically WIE image) chooses the way in which her/his frame of reference "slices up the world." 13
    2. In selecting undefined terms, our logician puts limitations on the "slicing-up" process, so as to get "elements of the lexicon"—terms or "words" or "sentence-parts" which will fit into the "slots" of the template or grammar of the notation in question.
    3. In articulating the setting and the undefined terms, our logician produces a version of the logic of opposites, which amounts to a pattern for handling defined terms—in particular, one which specifies the connections between a term or construct and its opposite or contradictory or negation or complement. 14
  3. We maintain that our logician obtains the grammar by somehow inter-defining the undefined terms.
Thus, taken together, setting and undefined terms specify the conventions for symbolizing.
From our point of view, anyone writing any text whatsoever (in any language)—even a laundry list—takes a rhetorical stance: presumes a speaker, addresses an audience, and posits a dramatic situation. When we use our alternative theoretical system, we take a rhetorical stance different from that which we attribute to users of the WIE languages.
In general, texts written in a WIE language "posit" some dramatic situation which centers about one or more "thing(s)" interacting with one or more "thing(s)." The term interacting (modified from Dewey & Bentley15), suggests a kind of "one-way causality":
A causes B, and B causes C, and C causes ...
As an example of the language of interacting, consider Newton's second law of motion:
f = ma (1)
"Force," "mass" and "acceleration" exemplify the kind of "things" known as quantities, while "equals" and "times" or "multiplied by" exemplify the kind of "interactions between things" known as operations or relations. This famous equation describes the "motion" of physical "bodies," in a world thought to consist of more or less discrete objects ("bodies") of different sizes, which follow trajectories and sometimes collide. Newtonian physicists regard the "happenings," the interacting which this equation describes as taking place independent of any observer. (The descriptions generated by Newtonian mechanics turn out quite accurate, as long as the dimensions of the "bodies" or "things" or "objects" examined remain within a few orders of magnitude of the dimensions of a human body.)
One interpretation of (1) into English goes, "To exert a force f on a mass m causes an acceleration a ."
In our non-standard frame of reference, in general we posit "doings" or "happenings" which consist of the two-way transacting between a human organism-as-a-whole and her/his environment, over a specific period—as viewed by a particular observer, who writes out her/his observations in our alternative notation. The term transacting (modified from Dewey & Bentley) suggests a kind of two-way interchange between organism and environment which profoundly alters the living system(s) involved, in some way that affects its/their further living. For example, humans A and B meet as strangers, and have a conversation which matters to each of them: Then they can never again meet as strangers. Or consider the transacting between a weekend sailor and her/his sailboat, before and after s/he takes a course on the physics of sailing—one which includes applications of Newton's second law of motion. When we humans learn how to use a tool skillfully, as Polanyi points out, we come to handle it as a part of ourself. 16 Practicing what s/he has learned in the course, our sailor improves her/his handling of her/his tool, her/his sailboat. This obviously alters the sailor. That this alters the environment as well—alters the way that wind and water and channels, etc., occur for her/him—becomes apparent when a sudden squall hits, and s/he brings the boat safely into a harbor in whose dangerously narrow entrance a skipper with her/his previous degree of skill would have wrecked the boat.
In our frame of reference, we regard the obvious "structural" features of a symbolic system such as a grammar as evidence which manifests what the speakers/writers of this symbol system assume.
As noted above, a WIE discursive language such as English has a vocabulary composed mostly of two main categories of terms, known respectively as nouns and verbs. Between them, these two categories make up some 70 to 80 percent of the entries in a big dictionary. Furthermore, we distinguish between the two categories relationally, by regarding and treating the nouns as "identical with themselves" and the verbs as "not-identical with themselves." In so doing, we define the binary relation of identical with as signifying "entire and absolute agreement or negation of difference." Identical with itself, then, comes out meaning "permanent" or "persisting" or "really existing" or (as little as we may like to admit it) "unchanging."17 To form a complete sentence, we place at least one noun—or, better, noun-phrase—next to at least one verb-phrase:
The cat grinned.
A WIE formalized language such as a symbolic logic or one of the mathematical theories of sets also has a vocabulary composed of two main classes, which we variously call things and relations (between things), or quantities and operations, etc. Again, we tell the classes apart by regarding and treating the things or quantities, etc.—the terms which substitute for nouns—as "identical with themselves," and the relations or operations—substitutes for verbs—as "not-identical with themselves." To generate a well-formed formulation, then, we place at least one noun-substitute next to at least one verb-substitute:
x [element of] A.
The WIE grammar makes no provision for distinguishing between "map" and "territory." For example, it includes no mandatory prefix, suffix, infix, no special grammatical forms, etc., by which a speaker/writer can keep track of this distinction. Of course, a speaker/writer CAN arrange to differentiate "map" from "territory," by secondary means, whenever s/he wants to (or remembers to)—but underneath such locutions, the primary structure of the WIE grammar still does not make this distinction. By analogy, a video tape record of an "event" SUGGESTS the past "existence" or "occurrence" of that event. But the structure of English, or set theory, provides no built-in way to distinguish between the taped "event" and the actual event (if any).
In our frame of reference, the construct of "does not distinguish between" appears equivalent to "holds or treats as identical with." In other words, where by presupposition the WIE grammar provides a speaker/writer with no primary means to distinguish between "map" and "territory," by that presupposition it constrains this speaker/writer to hold that "the territory"—ultimately, "the world" or "the Universe" or "the Cosmos"—has a structure which precisely matches that of the WIE grammar. "The world," it says, REALLY DOES consist of (a) static-and-unchanging "objects," precisely suitable for anyone to represent in "language" by means of self-identical noun-phrases; and of (b) more or less transient "relations between objects," precisely suitable for anyone to represent by means of not-self-identical verb-phrases.
In WIE mathematical terms, we can express this tenet by means of the construct of an exhaustively complete, entirely accurate one-to-one relation, such that every point of "the territory" or "the world" gets represented by one and only one point of our speaker/writer's "map" of "the world," and that our speaker/writer's "map" contains no extraneous points (no points which do NOT represent or refer to points of "the world").
We designate this presupposition as map-territory identity or tacit identity. Elsewhere, we have demonstrated that this (silent) postulate of tacit identity underlies the standard logical sense(s) of the term identity (or the relation of identical with). We designate this latter construct as explicit identity.
Our frame of reference and notation contrast sharply with the WIE discursive and formalized languages at precisely this point. The WIE frame of reference and the various WIE notations include among their premises some kind of postulate of tacit identity, the construct of explicit identity, and either Aristotle's Law of Identity or the modern Logical Axiom of Identity. Hence we refer to these languages as identity-based or as generically Aristotelian. Our frame of reference and notation stem from a central premise known as Korzybski's Postulate of Non-identity, which rejects (disallows as valid) the construct of identity in any guise or form, explicit or tacit. Hence we refer to these languages as non-identity based or generically non-aristotelian.
We regard English as a traditional discursive language of the Western Indo-European family. It did not arise as a deliberately generated axiomatic system. For example, no one consciously chose a setting or undefined terms or postulates for it, or otherwise specified its "antecedents." If someone holds (as we do) that English does have a postulational structure, with "antecedents" or "premises" and "consequents," s/he has made an assumption not shared by the speech community as a whole. In contrast, each of the WIE mathematical languages, e.g. each of the various mathematical theories of sets, does stand as a deliberately generated axiomatic system, whose innovators do specify undefined terms and—after 1902, the date of Russell's paradox concerning sets which do not belong to themselves, 18 a setting, etc., for it. However, lacking a contrasting system, each of those innovators relies in an unquestioning fashion on the grammar common to the WIE languages (both discursive and formalized). None examines that grammar for "conventions for symbolizing" or other assumptions, nor characterize such conventions in general terms (as we do here). Thus the antecedents we outline here occupy a "position" logically prior to any of the antecedents discussed by WIE logicians and linguists.
By analogy with the revised pattern for axiomatic systems laid out above, let us now frame English or set theory as that kind of axiomatic system. To do that, we infer the missing "parts," and then fit them into place. In the process, we account for (or at least incorporate) the features of WIE languages mentioned above, in our overview of the identity-based WIE pattern.
  1. Setting: As noted, we regard setting and undefined terms as human "doings", silently postulated by the person who relies on them. The setting for a WIE language, in our view, consists of a distinction—a pair of opposites, an "empty" dichotomy which we may indicate generically as x and not-x.
  2. Undefined terms: As undefined terms, we propose noun, verb, and the copula (e.g. 'is').
  3. Interconnections between setting and undefined terms: The interconnections in question generate the conventions for symbolizing of WIE languages such as English or set theory.
    1. (a) Slicing-up: Someone who chooses the above setting thereby "slices up the world" into two "KINDS"—into the "empty" dichotomy mentioned above, x and not-x . Or, to use an image that begs fewer questions, s/he CREATES two disparate "kinds." In any case, s/he treats them as polar; or in other words, treats her/his "process of slicing" or "process of creating" as if it necessarily generates both "kinds." Further, s/he utilizes this process of "slicing in two" over and over again within the grammar of a WIE language, on various logical levels, producing a dichotomously branching logical (or grammatical) "tree."
    2. Putting on constraints: Someone who chooses the above undefined terms (noun, verb, and 'is') thereby puts constraints on the "process of slicing" just described. S/he treats one of the two "kinds" (say, x ) as in some sense "the domain of words," and subdivides it into noun and verb (where noun 'is' noun and it is not the case that verb ("not-noun") 'is' verb). In the same breath, s/he treats the other "kind" (say, not-x) as in some sense "the domain of not-words," and subdivides it into two sub-classes, one of which s/he treats as precisely suitable to represent by means of self-identical noun-forms or noun-phrases, and the other of which s/he treats as precisely suitable to represent by means of not-self-identical verb-forms or verb-phrases.
      These arrangements not only yield terms or "words" or "sentence-parts" which will fit into the "slots" in the template or "grammar" of the notation or discursive language in question—they also support the conviction that there "really exist" two disparate domains: one which has something to do with "words" or "language," and another which seems somehow "non-verbal," e.g. a "reality" or "world" that "exists" "out there," independent of any "speaker" or "observer" or "organism," and also independent of any "words" the "speaker" may utter. Finally, these arrangements support the conviction that these "sentence-parts" uniquely "represent" this non-verbal "reality".
    3. Logic of opposites: As noted, the logic of opposites amounts to a pattern for handling defined terms (in a WIE language, nouns or noun-substitutes), one which specifies the connections between the noun-phrase in question and its negation (or other polar-opposite). The WIE tradition has included at least two distinctly different versions of the logic of opposites, commonly known as undelimited and delimited.
      1. Undelimited The undelimited version of the WIE logic of opposites utilizes two key terms, namely, some (defined) noun-substitute x and its negation, not-x . When expressed in a discursive language, this version of the logic of opposites contrasts "a thing" (e.g. day) against its negation (not-day), 19 which eventually consists of "everything else." Combined with the discursive language example, a simple Venn diagram makes this relationship clear:
        Critics point out that the construct of "everything else" proves unsatisfactory, undefinable—for the term day, it not only includes "night," but also "pollywogs," "foreign policy," "Cantor's transfinite cardinal numerals," etc. But that makes "everything else" unsuitable as a basis for a mathematics capable of generating "secure" inferences.
      2. Blank delimited: This version of the WIE logic of opposites utilizes three key terms, namely, a delimited domain D , within which "exists" the noun-substitute x and its negation not-x . Here not-x comprises "everything else within the domain D ". No one has yet modified a WIE discursive language so as to make use of the blank delimited version of the logic of opposites, so we have examples only from WIE notational languages. 20 A Venn diagram displays these relationships clearly:
        In this version of the WIE logic of opposites, not-x proves readily definable, avoiding most of the logical difficulties of the earlier version.
  4. Grammar: The pattern for a "complete sentence" or "well-formed formula," requires someone to place at least one noun-phrase or noun-substitute next to at least one verb-phrase or verb-substitute. But since any noun 'is' 'identical with' itself, this grammar consists precisely of the undefined terms inter-defined.
  5. Postulates: The postulates of the WIE frame of reference include (for the undelimited version of the logic of opposites) the "Laws of Thought" of Aristotle of Samos (384-322 BC). Aristotle called them laws of thought; today, it becomes clear that they spell out the rules for naming or "nouning":
    • The law of identity: Whatever is, is.
    • The law of contradiction: Nothing can both be, and not be.
    • The law of excluded middle: Everything must either be, or not be.
    For the delimited version of the logic of opposites, replace the "Law of Identity" with the modern Logical Axiom of Identity:
    For all x which belong to the delimited domain D ,
    x º x .
  6. Rules of inference, and
  7. Standards of proof: We have little to add to the traditional characterization of these topics.
"Things" interacting with "things".
In a larger sense, the frame of reference here consists of a dualism, e.g. framed as 'mind' vs. 'matter', where the "words" (which belong to 'mind') stand for or represent "things" (which belong to 'matter'), with perfect accuracy.
As noted above, we treat the constructs of setting and undefined terms as designating human activities: "something someone does."
  1. Specific delimited setting: We can express the setting for our non-standard notation by means of a run-on phrase, such as "One particular human-organism-as-a-whole-dealing-with-her/his-environment-at-a-date." More succinctly, we can express it by means of single terms such as "transacting" or "contacting" or "living." With the Gestalt therapists Perls, Hefferline & Goodman, we say,
    We speak of the organism contacting the environment, but it is the contact which is the first and simplest reality. 21
    This makes "I" and "it," or "I" and "thou," into INFERENTIAL "ENTITIES" on our specific delimited setting, rather than into "the Really Real."
  2. Undefined terms: As undefined terms, we follow Korzybski's example and choose structure, order, and relation. 22 In our non-standard notation, we signify these by means of S , O , and R . When we write these in English, we treat them as verb-forms: (to) structure, (to) order, (to) relation.
  3. Interconnections between setting and undefined terms: When dealing with the present non-standard notation, the image of "slicing up the world" has seriously misleading implications. It suggests an already-existing dualism, e.g. a dichotomy between "the world" and "the one who slices." Such a construct works well as an image for an identity-based WIE frame of reference; it does not work for the present one. Instead of regarding "organism" and "environment" as separate, disparate "things" which occasionally collide with each other, the present frame of reference treats "the environment" as the other side of the organism's skin, and treats "the organism" as the other side of the environment's skin.
    1. Creating a pattern: When we choose the above specific delimited setting, we thereby create the main pattern which the non-standard notation expresses. By adopting this non-standard notation, a user thereby restricts her/himself to a single point of view. S/he then can discuss "doings" or "happenings" solely and exclusively from a standpoint of this created pattern, which we can indicate by some run-on phrase such as the one stated above concerning organism-and-environment, or by a single term such as transacting.
    2. Putting on constraints: When we choose the above undefined terms, we thereby put constraints on the pattern generated by the setting. In particular, when we use an undefined term, we designate, or postulate, some example of "a transacting" or of "the dealings of an organism-as-a-whole with its environment at-a-date."
    3. Logic of opposites: As noted above, the logic of opposites amounts to a pattern for handling defined terms. In our non-standard frame of reference, this pattern consists of five points:
      1. Our incompletely-informed and inaccurately-informed (symbolic) 'organism'
      2. Consists of spatio-temporally ordered "doings" or "happenings" which occur within a (specific, delimited) setting known as transacting.
      3. By her/his abstracting, our 'organism' elaborates a 'gestalt' composed of
        1. a 'figure' which focally interests the 'organism'
        2. specified against a 'background' which does not (at present) interest her/him.
      4. Any 'gestalt' further consists of two 'components'
        1. one of which tells about the 'environment', and
        2. the other tells about the 'organism' who elaborates the 'gestalt'.
      5. In negating a 'gestalt', our 'organism' interchanges the 'figure' and the 'background', and alters none of the other considerations listed here.
      Please note that when we negate a 'gestalt', and then re-negate it, that does not bring us back exactly to our starting-place.
      EXAMPLE: Consider the positive degree of an ordinary English adjective vs. its negated negation: not unhappy does not qualify as an exact synonym for happy.
  4. Grammar: We obtain the grammar of our notational system by inter-defining the undefined terms. For example, where we designate or posit a structuring S , that already gives us a "bridge" from the non-verbal to the verbal. Whenever we want to give more details about that S , we use the other two undefined terms. For instance, we can specify the structuring S in terms of two relationings R ordered O somehow, namely,
    SRRO .
    Or, equally comfortably, in terms of two orderings O relationed R somehow, namely,
    SOOR .
    By similar 'reasoning,' when we use R or O as the bridge from the non-verbal to the verbal, that gives two more pairs of possible expressings composed of undefined terms, or a total of six such expressings:
    SRRO or
    SOOR or
    ORRS or
    OSSR or
    ROOS or
    Where we use an undefined term twice in an expressing, we intend that no one take the two usages as 'identical'.
  5. Postulates: In a formal presentation of an axiomatic system, one uses the undefined terms to state the postulates. Here, for the sake of intelligibility, we state Korzybski's non-aristotelian postulates in terms of the map-territory analogy, which holds that to say that an organism lives means that it makes some kind of maps of (or guesses about) that territory composed of "what goes on in and around our organism"—and then it guides its "doings" or "choosings" by these maps. Stated colloquially, then, these postulates become:
    • Non-identity: The map is not the territory it stands for.
    • Non-allness: The map represents not all of the aspects of the territory.
    • Self-reflexiveness: Any map contains some kind of representation of the map-maker (organism).
  6. Rules of inference: For the present purposes, it suffices merely to name the rules of inference of the non-standard notation: Generalizing, Particularizing, Componenting, and Specifying.
  7. Standards of proof: Again, for the present purposes, it suffices to say that to "prove" an expressing, one demonstrates that s/he can both derive (particularize) it from an expressing composed solely of undefined terms, and can "decompose" (generalize) it back into an expressing (or a number of expressings) composed solely of undefined terms.
"An observer observing the observed," as viewed by another observer ("logician"), who writes down her/his observations. 23
The present findings, or creations, extend and in a sense complete the revised world-view delivered by our alternative frame of reference. They summarize the accomplishments to date of what I might call non-identity 'reasoning', and (as mentioned above) offer a new domain for the construct of taking the observer into account.
As noted above, Korzybski opens the way to the revised world-view by making a quick tour of the universe, asking self-reflexive questions concerning humans and how they—we—fit into the observable scheme of things. His procedure contrasts with that of Western philosophers and other workers, who for thousands of years have asked certain self-reflexive questions, e.g. "What IS man?" In conformity with the presuppositions encoded in the WIE grammar, they have framed their queries so as not to distinguish between "map" and "territory"—so these queries lead not to testable hypotheses, but rather to answers which presuppose and so conform to the fixed, culturally-determined views of the asker. In the Western cultures in particular, two answers to such questions have predominated. As Korzybski puts it,
One of the answers is biological—man is an animal, a certain kind of animal; the other answer is a mixture partly biological and partly philosophical—man is a combination or union of animal with something supernatural. An important part of my task will be to show that both of these answers are radically wrong and that, beyond all things else, they are primarily responsible for what is dismal in the life and history of humankind. 24
In the course of his own inquiry, Korzybski re-frames the self-reflexive questions so that they yield disconfirmable answers, in the process questioning the shared "map" and so distinguishing between "map" and "territory". He designs his answers for us to examine in the light of evidence and observation. He asks, in effect, "What do we humans DO that distinguishes us from other living organisms?" His query centers on a kind of dimensional analysis:
To know anything that is to-day of fundamental interest about man, we have to analyze man in three coordinates—in three capacities; namely, his chemistry, his activities in space, and especially his activities in time; whereas in the study of animals we have to consider only two factors: their chemistry and their activities in space. 25
Korzybski finds that we humans function "in time" as a time-binding class of life—we accumulate human knowledge (in the form of disconfirmable guesses that we test), at rates that depend on how much knowledge we already have. We gain our living in the biosphere by cooperating to apply what we know, in the process coming to know more.
Increases in knowledge, in this alternative view, occur in the setting of a shared linguistic and cultural viewpoint; and they occur by a process of making new distinctions—ultimately, of distinguishing between "map" and "territory," at some point where the shared viewpoint previously available does not distinguish between the two. In other words, increases in knowledge come about by a process of eliminating usages of tacit identity. For example, consider how Newton, in devising his construct of universal gravitation, changes what he himself assumes.
His first Gestalt in the relevant sequence of "doings" has as its background the traditional view, which tacitly assumes map/territory identity; and has as its figure, a questioning of the traditional view, which in effect means distinguishing between map and territory: What if the commonly held view amounts to only a surmise, instead of expressing 'the way things really are'? This Gestalt then becomes the background of the next Gestalt encoded in Newton's insight. In expressing its figure, Newton devises an alternative map: What if the ball (or apple) and the Moon BOTH fall—and the apple reaches the surface of the earth whereas the Moon doesn't. How would I account for that?26
Newton completed this sequence by using his new theory of fluxions (or as we now call it, the calculus) to compute the centripetal and centrifugal forces on the moon, and compute the value for the period of the moon which would make these forces equal. His first rough calculation gave a period close to the true value, about 27.25 days. 27 Thus a procedure which hinged on eliminating a usage of tacit identity led to a major gain in the ability of physical theory to deliver accurate predictions.
Our view holds that any usage of an undefined, or defined, term designates or postulates some example of "a transacting between knower (organism) and known (environment)." Consequently, it treats any "increase of knowledge" as having two components: It not only models and so makes explicit some aspect of the environment of the human organism who tests (and fails to disconfirm) her/his hypothesis, but also makes explicit some aspect of her/his own "doings." With Newton's second law of motion, the construct focuses on and models the "behavior," the trajectories, of physical "bodies"; but it also models the behavior of organisms capable of using eyes, muscular system and nervous system to track and perhaps move so as to intercept physical bodies.
At the turn of the twentieth century, the revolutionary physicists, notably Einstein and the quantum theorists, introduced to the human race the construct of "Taking the observer into account." As noted above, the alternative frame of reference discloses two opposing structures of assumptions: the one which underlies taking the observer into account, in contrast to the one which underlies eliminating the observer from consideration.
  1. We know only one route to taking the observer into account: to distinguish between "map" and "what the map refers to," to postulate map/territory non-identity. This manifests itself in remembering to treat the map as more or less tentative and approximate, as incomplete, and as created from one's own point of view for one's own purposes; in remaining willing to test it for accuracy; and in holding oneself in readiness to revise it at need. These locutions stand as shorthand for "relying on the non-aristotelian postulates as one's most fundamental presuppositions." But to treat one's "maps" as created from one's own point of view for one's own purposes amounts to "taking the observer into account."
  2. In order to eliminate the observer from consideration, one need only assume map/territory identity. Let us express this in terms of an exhaustively complete, entirely accurate one-to-one relation between "map" and "territory." If every point of my "map" represents one and only one point of the "territory," no point of the "territory" gets left out, and and my "map" contains no extraneous points, then there exists no "room" in my "map" for any kind of representation of the map-maker. In other words, such a "perfect" map eliminates the observer from consideration.

The process by which any human transacts with her/his environment shows a special twist, for any human guides her/his transacting by means of lived theory. 28 The alterations which result from the transacting make both the human and the environment into a closer approximation of the picture of "what goes on" embedded in whatever theory s/he lives.
For example, a lived theory which tacitly posits that its tenets represent the territory "accurately," "completely," and "objectively" already eliminates the observer or logician or biologist from consideration. Consequently, since it systematically leaves out of account the observer, it also systematically leaves out any transacting between organism (observer) and environment; and so in turn it leaves out the environment too. Such a theory, then, provides the symbolic means to represent only inanimate mechanisms, e.g. the kind of theory that makes up an "isolated system" which maximizes a single variable.
Someone who lives such a theory may have the tools to account very well indeed for the non-living, the physical—or for such "variables" as power, or status, or profits, etc. Unavoidably, in the process of living that theory s/he makes her/himself resemble the kind of mechanism that seeks to maximize such variables—single variables. In general, when we humans live a theory which systematically leaves out of account the observer—ourselves-in-our-environments—we transform the environment so that it becomes less hospitable to transacting organisms, transacting persons, living systems, and more like a "mechanism" and so more hospitable to non-living mechanisms; while we transform ourselves so that we become less adept in transacting with a varied and healthy environment and more "mechanical" and "businesslike," more resembling the types of "mechanisms" depicted in the theory.
Contrariwise, if the lived theory stems from premises which systematically takes the observer into account, the alterations induced in the environment in principle make it less like a "mechanism" and more hospitable to transacting organisms; and the alterations induced in the human in question in principle make her/him less "mechanical" and more adept in transacting with the world of living systems.29
Since we regard the conventions for symbolizing outlined above as a specific example of human activities, to take them into account amounts an extension of the construct of taking the observer into account—namely, taking a previously neglected aspect of the observer into account.
Conversely, to leave the conventions for symbolizing unspecified amounts to eliminating the observer from consideration. Doing so requires relying on map/territory identity, so that there exists "no room" in the map for representing these aspects of the map-maker (observer). It requires assuming a dualism—assuming that some "core" of language—or languaging -- "exists" independent of any observer, and also that some kind of "external reality" "exists," independent of any observer.
To assume that some "core" of languaging "exists" independent of any human, and that "reality" also exists independent of any human, puts the origin of languaging beyond reach of human inquiry, into some kind of "transcendental" domain rather than into the domain of the evolution of living systems.
In contrast, to postulate map-territory non-identity leads to an evolutionary perspective. Once upon a time, we hold, no humans, and no human world-views, existed. In the process of evolving from non-human great apes to the first humans, our predecessors generated the first human world-views. Today, in choosing a setting we awarely and knowingly choose a way of slicing up human experiencing. In choosing our undefined terms , we choose a way of putting constraints on this "cutting up" process so that the "pieces" we obtain will fit into the "slots" in the template known as "our grammar." Further, we obtain a DERIVED grammar (rather than an inherited one) by inter-defining our undefined terms. We also use the undefined terms in stating our chosen postulates. Finally, in the notation and world-view which we elaborate upon these beginnings, we resolutely and consistently maintain that the 'maps` we generate on these beginnings do not "describe things the way they REALLY ARE," but rather, remain in principle inaccurate, incomplete and self-referential. Thus, on every level, we systematically take 'the observer'—ourselves—into consideration.

1 #The Book of Tao#, Chapter 2. MacHovec, translator. (Mt. Vernon NY: Peter Pauper Press, 1962), p. 33.
2 Watts, Alan W.: The Two Hands of God: Myths of Polarity. New York, Brazillier. Paperback edition, 1969, Collier Books, p. 48.
3 Condon, W. S. & W. D. Ogston (1967). "A Segmentation of Behavior". Journal of Psychiatric Research 5(3):221-235, p. 225.
4 Condon & Ogston, 1967, p. 229.
5 Hilgartner, C. A. (1965). "Feelings, Orientation, and Survival: The Psychological Dimension of the Current Human Crisis." Presented at the Ninth International Conference on General Semantics, San Francisco State College, August 1965.
6 Polanyi, Michael (1964). Personal Knowledge: Toward a Post-Critical Philosophy. University of Chicago Press, Chicago, 1958. Harper Torchbook, New York, 1964, p. 171.
7 Whorf, B. L. (1956). Language, Thought, and Reality: Selected Writings of Benjamin Lee Whorf. John B. Carroll (Ed.). MIT/Wiley, New York, p. 246.
8 Webster's (Second) New International Dictionary of the English Language Unabridged, second edition. Springfield MA:G & C. Marriam, 1961, p. 136.
9 Some 70 to 80% of the entries in a big dictionary get classified as either nouns or verbs.
10 Hilgartner, C. A. (1977/78) "Some Traditional Assumings Underlying Indo-European Languagings: Unstated, Unexamined, and Untenable." General Semantics Bulletin Nos. 44/45, pp. 132-154.
11 Korzybski, Alfred (1933) Science and Sanity: An Introduction to Non-aristotelian Systems and General Semantics. Non-Aristotelian Library Publishing Co., Chicago, pp. 187, 194-5.
12 Korzybski, 1933, p. 153.
13 Whorf, 1956, pp. 213-4, 240.
14 Hilgartner, C. A. (1978a). "The Method in the Madness of Western Man." Communication 3:143-242, pp. 159, 196-9, 202-3, 204-221, 235-6.
15 Dewey, John & Arthur F. Bentley (1949). Knowing and the Known. Beacon Press, Beacon. Paperback edition, 1960.
16 Polanyi (1964), pp. 55-60, especially 58-9.
17 Hilgartner, 1977/78, p. 140-1.
18 If you want to know more about Russell's paradox and how it affected twentieth century WIE mathematics in general and set theory in particular, see Guillen, Michael (1983). "Logic and Proof—A Certain Treasure." From Bridges to Infinity. Los Angeles: Jeremy P. Tarcher, Inc. For a succinct statement of Russell's paradox itself, see Cohen, Paul J. & Reuben Hersh (1967). "Non-Cantorian Set Theory." Scientific American 217:104-116, December 1967, p. 105.
19 Hilgartner, 1977/78.
20 Speakers of certain discursive languages outside the WIE family appear routinely to utilize a delimited version of the logic of opposites. In Sino-Tibetan languages, for example Mandarin, speakers do not treat opposing term-pairs, e.g. buy and sell, as "unrelated"; instead, they treat them as if defined on a common domain (e.g., the domain of trade). As Chu puts it,

...Ideas are often denoted [in Mandarin] by compound expressions composed of antonyms: for examples, "buy-sell" for "trade," "advance-retreat" for "movement," "rule-chaos" for "political condition," etc. The antonyms are not thought of as irreconcilable opposites but as being united to form a complete idea. One of the most important concepts in Chinese philosophy is expressed in a compound of antonyms, yin1yang2. These two terms denote two opposing but complementary forces in the universe, the interaction of which produces all things and the unity of which reposes in the Ultimate. (Chu, Yu-Kuang: Interplay between language and thought in Chinese. ETC.: A Review of General Semantics 22:307-329 (1965), p. 316.)

In this respect, then, their version of the logic of opposites closely parallels that of our non-standard notation. See Hilgartner, 1978a, pp. 199-203.
21 Perls, Frederick M., Ralph Hefferline & Paul Goodman (1951). Gestalt Therapy: Excitement and Growth in the Human Personality. Julian Press, New York.
22 Korzybski, Alfred (1943). "General semantics, psychiatry, psychotherapy, and prevention." In M. Kendig (Ed.), Papers from the Second American Congress on General Semantics. Institute of General Semantics, Chicago, pp. 93-108.
23 Examples of this kind of 'reasoning', done discursively, appear in Hilgartner, C. A. (1978b). "A Human Studying the Sensing of Chemicals by Bacteria." Acta Biotheoretica 27:19-43. Examples done in notation, in Hilgartner, C. A. (1978a). "The Method in the Madness of Western Man." Communication 3:143-242.
24 Korzybski, Alfred (1921) Manhood of Humanity: The Science and Art of Human Engineering. E. P. Dutton, 1921. 2nd Ed., (1950), M. Kendig, ed., Institute of General Semantics, Lakeville CT 06030, p. 3.
25 Korzybski, 1921, p. 67
26 Hilgartner, C. A., Ronald V. Harrington, & Martha A. Bartter (1986b). "Wanted: Scientific Science." (Submitted for publication.); also, note (8) of "Reply to Kenyon",General Semantics Bulletin No. 55, pp. 53-76 (1990), p. 71.
27 Bronowski, Jacob (1973). The Ascent of Man . Little, Brown & Co., Boston. Paperback edition, Futura, London, 1981.
28 Hilgartner, C. A., R. V. Harrington & M. A. Bartter (1984). "A Stably Unstable System with 4.5 Billion Participants." H. Chestnut, et al., editors, Supplemental Ways of Increasing International Stability. Oxford: Pergamon Press, pp. 223-9.
29 Hilgartner, C. A. & David L. Johnson (1988). "A Unifying Principle for Biology." Presented at the Annual Meetings of the Ohio Academy of Science, 29 April 1988. (Copies available from first author, at cost of copying.)