A Previously Unnoticed Connection


C. A. Hilgartner



In what follows, I present an insight concerning human languaging, arrived at via the non-standard notation developed by Hilgartner & Associates. This insight, which suggests that we should regard the constructs of setting and undefined terms as polar (mutually necessary), discloses presuppositions which underlie logic and mathematics in particular and human languaging in general. These presuppositions operate at a 'deeper' level than any I have seen explicitly discussed elsewhere, e.g. in general-semantics or in traditional philosophies.


I report that I reached this insight during my first exposure to the constructed language known as Lojban (earlier name: Loglan). Something "outre" about the structure of Lojban showed up as a contrast to something I already knew, and sparked the insight mentioned above. As for the construct which I call setting, it seems a bit more general that its closest cognate within standard Western Indo-European (WIE) philosophy, namely, the construct of domain of discourse. The modifying terms delimited and undelimited sometimes get applied to both of the cognate terms, setting and domain.


The construct of setting had long troubled me. Our research group had established that our non-standard notation, and the non-aristotelian frame of reference of which it forms a part, presupposes a setting different from that of contemporary WIE symbolic logic or mathematical theories of sets. We label the standard "domain of discourse" as a blank delimited setting, and label the setting for our notation as a specific delimited setting. Furthermore, we characterize ours by a run-on phrase such as "ONE PARTICULAR organism-as-a-whole-dealing-with-its-environment-at-a-date". Or we abbreviate the run-on phrase by terms such as transacting (modified after Dewey & Bentley, 1949); or


(Dewey, John & Arthur H. Bentley (1949). KNOWING and the Known. Boston: Beacon Press. Paperback edition, 1960.)



contacting, (modified after Perls, Hefferline & Goodman, 1951):


We speak of the organism contacting its environment, but it is the contact which is the first and simplest [reality.



(Perls, Frederick S., Ralph Hefferline & Paul Goodman (1951). Gestalt Therapy: Excitement & Growth in the Human Personality. New York: Julian Press, p. 229.)



I kept trying to EXPRESS this setting in terms of bunches of "things" and some kind of "boundaries" or "domains." However, that didn't work -- it led to no further insights, and seemed instead like a "blind alley" -- and further, whereas every other detail of the non-standard notation which I and my collaborators had become familiar with differed drastically from the WIE formalized languages, this way of expressing our specific delimited setting seemed somehow jarring, incongruous -- it "smelled" far too close to the WIE way of handling the topic of "domain of discourse," which we had supposedly abandoned.


In order to state my insight, I relate the general semanticist Alfred Korzybski's view of undefined terms to the anthropological linguist Benjamin Lee Whorf's ideas about linguistically cutting up the world. In the process, I show how the logic of opposites applies to languaging, and how taking the observer into account functions in this context.


1. Korzybski and Whorf


Many therapists say that people often use jokes to express a "truth" too uncomfortable to state directly.


Well, some wag once said, "God created the numbers -- all else in mathematics is the work of Man."


Stating it directly, I take that aphorism as meaning that we scientists and mathematicians, we adherents to the WIE world-view, HOLD some core of our world-view as unanalyzable, or more accurately, unexaminable.


This supposition has broad implications. Remember that Whorf holds that


What we call "scientific thought" is a specialization of the Western Indo-European type of language ....


Whorf, B. L. (1956): Language, Thought & Reality: Selected Writings of Benjamin Lee Whorf. John B. Carroll, ed. New York/London: John Wiley/MIT Press, 246.)


In other words, the generalized grammar common to Dutch, French, English, German, Greek, Latin, Russian, Spanish, etc., also underlies our symbolic logics, mathematical theories of sets, analysis, topology, etc.


Assuming that Whorf's generalization about grammar actually holds, then that gives me grounds to juggle that jest a bit, extending it (heavy-handedly) to say something like "God created the WIE grammar -- all else in WIE logic, mathematics, science, philosophy, religion, etc., comes from the work of humans."


Like the original, the paraphrased jest suggests that we Westerners hold some part, some core, of our common viewpoint as "unexaminable." Let me say this in several ways. a) To use the philosophical terminology of an earlier century, both versions of the jest amount to calling that core transcendent. In other words, it doesn't require a person, or persons, to generate that core -- it "just exists," prior to and independent of any speaker, listener, observer, etc. b) As another way to say it, this tenet claims some kind of cosmic validity for WIE mathematics (or in my paraphrase, for the WIE world-view). c) In a more modern lexicon, for those fond of computer analogies, or electronic analogies, this tenet claims that God "hard-wired" this core of the WIE world-view in place in humans ( -- ALL humans!). (Actually, they would probably use the passive voice and so avoid naming the 'agent' ("God") -- "The WIE pattern 'IS' hard-wired into humans").


In order to show precisely WHAT we Westerners hold as unexaminable -- and what it means to hold that as unexaminable -- I need to bring together the promised insights from Whorf and Korzybski.


Let me start with Korzybski, who (unlike most of his predecessors) appears to posit a universe with human observers in it. He compares human behavior to a formal deductive (or axiomatic ) system: Humans assume -- we cannot not-assume. What we DO, then, follows from what we assume in roughly the way a theorem of an axiomatic system follows from the premises of the system.


Please remember the "elements" which make up the structure of an axiomatic system, such as Euclidean geometry: Euclid starts with a small number of undefined terms; with axioms and/or postulates (stated by means of these undefined terms); with rules of inference; standards of proof; etc. For example, in his plane geometry, Euclid utilizes terms such as point, line, plane, and parallel, which he makes no effort to define by means of other words.


Modern logicians, reacting to the "troubles" posed by Russell's Paradox (1902), discovered another "element" of axiomatic systems, called the domain or setting, -- a topic which Aristotle and Euclid (and their successors until 1902) do not discuss.


Korzybski's view of undefined terms comes up in the course of discussing the structure of interchanges between humans. He treats undefined terms as a part of any such interchange; and he describes such undefined terms as


... taken on faith. They represent some kind of implicit creed, or metaphysics, or structural assumptions.


(Korzybski, Alfred (1933). Science and Sanity: An Introduction to Non-aristotelian Systems and General Semantics. Non-Aristotelian Library Publishing Co., Chicago, p. 153.)


In other words, according to Korzybski, an undefined term requires a PERSON to hold it or rely on it; and where that occurs, the undefined term functions in the person's behaving-and-experiencing as a special kind of postulate, namely, a silent postulate, the tenets of which the person who relies on it cannot state in words. S/He must, however, know how correctly to USE her/his undefined terms.


2) Whorf indicates, e.g. by using the term speech community, that he too posits a universe with humans in it, and assumes that humans take part in interchanges carried on in part by means of what I call languaging. In discussing the fundamental structure of a large number of languages of widely different patterns, Whorf uses the image of "cutting nature up." (I might paraphrase this as "cutting up our own experiencing or transacting.") Whorf writes,


... We dissect nature along lines laid down by our native languages. The categories and types that we isolate from the world of phenomena we do not find there because they stare every observer in the face; on the contrary, the world is presented in a kaleidoscopic flux of impressions which has to be organized by our minds -- and this means largely by the linguistic systems in our minds. We cut nature up, organize it into concepts, and ascribe significances as we do, largely because we are parties to an agreement to organize it in this way -- an agreement that holds throughout our speech community. The agreement is, of course, an implicit and unstated one, BUT ITS TERMS ARE ABSOLUTELY OBLIGATORY; we cannot talk at all except by subscribing to the organization and classification of data which the agreement decrees.


(Whorf, 1956, pp. 213-4.)


You can check that out against personal experience. Ask yourself: When you engage in languaging -- speaking-and-listening, or writing-and-reading, both "on the job" and "at home" -- how much of a sense do YOU have of actively "dissecting nature," of generating the "categories and types," of ACTIVELY CREATING the "concepts" and "significances" with which you do your languaging?


I really don't know what you will answer to that question. YOU may have a very strong sense of actively dissecting nature, in order to get the "pieces" with which to speak-and-listen, write-and-read, and so transact with yourself-and-your-fellow-humans. But I bet you'll agree with me that MOST PEOPLE have very little sense of active creating. Many people appear to have no awareness that they can manipulate already constructed "ideas," much less that they can manipulate the basic units of the lexicon upon which "ideas" rest. Instead, they appear to hold that 'language' "just IS," already "ready to be used" -- they ACT as though they manipulate "prefabricated pieces" when they engage in languaging, rather than having a hand in "fabricating" these pieces. Similarly, it appears not to have come to the attention of psycho-linguists and others studying how children learn a first language that each child must learn how to "slice up the world" in the process of learning to language.


So much for the tools I'll need to use. The matter of immediate concern to me in 1989, when I first encountered Lojban, centered on the topic of setting, and in particular, how to visualize, and how to represent, the setting of my alternative frame of reference.


I had already established that I can display the setting for an axiomatic system in terms of the version of the so-called logic of opposites which that system posits. For Aristotle in particular, and for his successors, the exponents of WIE logic, mathematics, etc., (until 1902), in general, the logic of opposites contains only two terms, which I can symbolize as C and not-C . C in general stands for a noun-form (or noun-phrase), and not-C stands for the opposite or contradictory of C , namely, "Everything Else."


Let's represent this setting in terms of the construct of binary in/out choices: Draw a circle, label it B (for "black"), and consider a bunch of "colored" points or "marbles". Place the "black" ones INSIDE the circle, and the "not-black" ones OUTSIDE the circle. For any point, the classifying ends once you have made this one in/out choice.


When Russell proposed his famous Paradox, logicians and mathematicians concluded that the usual presuppositions they had depended on do not provide a satisfactory basis for the logic and mathematics of their day. In effect, the category of "everything else" appears inadequately specific: it contains the "not-black" "colored points," but it also includes "yellow marbles with pink and blue polka-dots," and "male sexual functioning," and "potatoes," and "Cantor's trans-finite cardinal numerals," and "foreign policy," etc.


Ernst Zermelo (1908) found a way to get around this difficulty, by positing a delimited domain for any term. (I call his construct a blank delimited domain, in opposition to the specific delimited domain of my own system.) In effect, he said to draw TWO circles, C and D , with C "inside" of D . Here, C represents the chosen term ("black") and D represents the "domain of discourse." Again, consider a bunch of "colored points." Then the process of classifying a given "point" requires two binary in/out choices. If the given "point" appears "black" or "white" ("not-black"), use the first choice to place it within the "outer" circle, the "domain" D ; otherwise (if it appears "red," "yellow," "orange," "green," "blue," "purple," etc.), place it "outside" of D . If the given "point" appears "black," use the second choice to place it "inside" C ; otherwise (if it appears "white"), place it "outside" of C (but "inside" of D ).


Zermelo's trick works -- avoids the invalidating consequences of Russell's paradox. Mathematicians proceeded to revise set theory, and then over the next three or four decades, they framed every branch of WIE mathematics as a specialized branch of set theory.


In about 1971-72, I rejected set theory (and all other WIE languages, notational as well as discursive), and addressed myself to developing a notation and a general frame of reference based "from the very beginning" on Korzybski's non-aristotelian premises (rather than on any traditional and partially unknown "philosophical grammar"). By 1989, I knew that structurally, the notation which Hilgartner & Associates had developed differed from the structure of WIE logic and mathematics on every level and in every detail that my collaborators and I had any awareness of. I also knew some things about its setting: As I said, in some way, the setting consists of positing "ONE PARTICULAR organism-as-a-whole-dealing-with-its-environment-at-a-date." Or I can abbreviate the run-on phrase by terms like transacting, contacting, etc.


That way of framing it makes the constructs of "I" and "it," or "I" and "thou," or "organism" and "environment," into INFERENTIAL entities   -- instead of taking these constructs as representations of "the really real" (the (allegedly) pre-existing "outside world," or whatever -- the way a WIE notation or language would do).


But somehow, until I encountered Lojban, I kept thinking of our specific delimited setting in terms of "bunches of points" and "nested circles" -- and then feeling uncomfortable, because that sounded too much like the WIE pattern, which I had supposedly abandoned.


In the Lojban materials to which I got exposed, I particularly remember a discussion of the term tavla, which means something approximating the English word "talk." In Lojban, terms of that type have a "predicate" structure -- each one has associated with it a certain number of "places." into which fit other words, where the "meaning" of the "predicate"-term consists of the relation between the "place"-terms. A Lojban speaker inserts a word into each of those "places," thereby making explicit the relation expressed by the "predicate"-term. Representing each "place" by a subscripted x , Lojbanists use tavla within the following "place-structure":


x1 tavla x2 x3 x4




x1 signifies "the speaker"

x2 signifies "the addressee"

x3 signifies "the topic"

x4 signifies "the language used"




coi. maixl i. mi tavla do loi bangu la lojbo


(shoy, Maich'l -- ee me tavla doh loi bangu la lojbo)


"Greetings, Michael. I talk with you concerning language in Lojban."


When I took in that linguistic pattern, all of a sudden I could see that a setting AMOUNTS TO a PATTERN. The WIE vocabulary of "noun" and "verb," or "things" and "relations," etc., specifies one such pattern and so expresses one particular setting -- but nothing compels me to follow that particular pattern. Nothing requires me to deal with 'things' such as "colored marbles", "nested circles or buckets", etc., and with 'relations' such as "in or out," as prescribed by the WIE pattern. Instead, in Whorf's metaphor, when you choose a setting you choose a WAY OF SLICING UP "THE WORLD" (or slicing up "your own experiencing") into manageable units.


So far, so good. In fact, another passage from Whorf supports this interpretation. He writes,


... We cut up and organize the spread and flow of events as we do, largely because, through our mother tongue, we are parties to an agreement to do so, not because nature itself is segmented in exactly that way for all to see. Languages differ not only in how they build their sentences but also in how they break down nature to secure the elements to put in those sentences. This breakdown gives units of the lexicon. "Word" is not a very good "word" for them; "lexeme" has been suggested, and "term" will do for the present. By these more or less distinct terms we ascribe a semifictitious isolation to parts of experience. English terms, like 'sky, hill, swamp,' persuade us to regard some elusive aspect of nature's endless variety as a distinct THING, almost like a table or chair. Thus English and similar tongues lead us to think of the universe as a collection of rather distinct objects and events corresponding to words. Indeed this is the implicit picture of classical physics and astronomy -- that the universe is essentially a collection of detached objects of different sizes.

(Whorf, 1956, p. 240)


By implication, other tongues, other languages (such as the alternative formalized notation of Hilgartner & Associates) might "lead us to think of the universe" in quite different fashions.


However, such familiarity as I already had with the new notation suggested that I hadn't finished with the new insight. Within a delimited setting (such as our specific delimited setting), terms usually come in pairs -- any "term" has a "complement." What term would serve as the complement for setting?


In context, there seemed little choice. According to the rhetoric of "axiomatic systems," when someone says, in effect, "I think I'll build up a new formal deductive system," s/he starts out with NOTHING (except that intention). So first, s/he chooses a setting and a few undefined terms, and --


So there we have it. For the role of the complement of setting, it looks like we have only one candidate: undefined terms. Extrapolating from Korzybski's and Whorf's comments, then, in choosing a setting, we choose a way of slicing up "the world," or "human experiencing," or some such; and in choosing undefined terms, we choose a way of putting constraints on this "cutting-up" process, so that the "pieces" we obtain will prove useful, will serve as "units of the lexicon," or "the elements to put in ... sentences."


This insight completes our alternative notation in a peculiarly satisfying fashion, as an alternative to the inherited WIE frame of reference.


a) The exponents of the inherited WIE frame of reference start from a non-evolutionary or 'prefabricated reality' perspective: they hold that some 'core' of 'our common viewpoint' JUST EXISTS, prior to and independent of any human. They further hold that 'the universe' also JUST EXISTS 'out there' -- as Whorf puts it, as "a collection of detached objects of different sizes," each with its inherent 'properties' or 'attributes' -- prior to and independent of any human. Thus they systematically eliminate 'the observer' -- themselves -- from consideration.

b) In contrast, we, the exponents of an alternative notation and frame of reference, start from 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 -- CREATED -- the first human world-views. Today, in choosing a setting (such as "one particular organism-as-a-whole-dealing-with-its-environment-at-a-date") we awarely and knowingly choose a way of slicing up human experiencing. In choosing our undefined terms (the undefined terms from Korzybski's non-aristotelian premises -- (to) structure, (to) order, (to) relation), 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. (Thus we may specify structure in terms of ordered relations or related orderings, or specify order in terms of structured relations or related structures, etc.) We also use the undefined terms in stating our chosen postulates (the postulates from Korzybski's non-aristotelian premises -- Non-identity, Non-allness, and Self-reflexiveness). 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.


Finally, taking the observer into account confers an increase in the ability to predict accurately the outcomes of our actions: We can then allow for the probable effects of "what we just did" on "what we intend to do next," increasing the likelihood that we can achieve our goals. Thus an organism which can take the observer -- him/herself -- into account has an advantage over an organism which cannot.