C. A. Hilgartner

The 'general semantics' of Korzybski suffers from at least one serious drawback: It remains non-notational stated in a discursive (or "natural") language. And to the best of my knowledge, no linguist or logician has yet characterized the structure of even one discursive language. Therefore, I must conclude, the 'general semantics' of Korzybski shares the fate of any doctrine which long remains stated in a discursive language: We must regard it as non-rigorous (by the standards of 1971), or in other words, we must regard it as containing at least one hidden assumption which serves to keep it incompatible with formal mathematical treatment (by the mathematics available at a given date).

But now, I claim there exists well worked out formal (notational) theory in general semantics a formal 'psycho-logics', in Korzybski's (1933) terminology, sufficient to fulfill the first prerequisite.

Quite apart from any other publication, this booklet suffices to put to test this claim (and, as its readers will see, does not disconfirm this claim).

*Please note that this criticism may well cut the mathematics available at a given date as well as, or even instead of, the doctrine in question: Repeatedly in the history of science, formal theory in a given field has hinged on the invention (availability) of new and perhaps entirely unprecedented mathematical languages. (I think of Newton and the calculus or Einstein and the non

Euclidean geometries and the tensors calculus.) In 1933

the date of publication of Korzybski's Science and Sanity

algebra set theory such as I have used to elaborate formal theory in general semantics did not yet exist The publication dates for the famous texts on algebraic set theory listed by Lipschutz (1964; cf. below) range from 1950 to 1960, viz. , after Korzybski's death.

Moreover, "traditional" algebraic set theory by itself does not suffice for framing a formal 'psycho

logics' In my hands set theory has undergone

SECOND PRE

REQUISITE:: Anyone who would try to make use of formal theory in general semantics as way to rebuild his own life more to his own liking must first 'comprehend' the theoretical constructs in both their 'logical' and their 'empirical' aspects, where

a) the term 'logical' refers to the ordered relations which make up the structure of the constructs themselves (viz., their definitions and their inter

relations); and

b) the term 'empirical' refers to the relations existing between the constructs ('Names') and the 'Things Named' the constructs refer to (viz., the ways one can use the constructs to describe "happenings", including the "happenings" which make up one's own life).

In one sense at least, one can 'comprehend' a theoretical system if and only if he approaches it "on its own terms". If a prospective student encounters some obstacle to 'comprehending' no matter what its source, then he must sooner or later overcome the obstacle , or else quit.

Formal theory in general semantics on first encounter say well soon Inaccessible, even overwhelming.

This booklet has as its main purpose to make it easier for students to approach formal theory in general semantics.

I deem it best to begin by acknowledging the magnitude and the nature of the difficulties any student

even one who already has some familiarity with "traditional'' general semantics

will face in trying to 'comprehend' a formal psycho

logics.

Polanyi (1964), with reference to the example of "a medical student attending a course in the X

ray diagnosis of pulmonary diseases," aptly describes the dilemma of a student who undertakes to study a (logical

and-empirical) theoretical previously unknown to him:

.

demonstrable modification (reinterpretation), in ways which make my notation demonstrably consistent with the non

Aristotelian premises of Korzybski

a condition which (as I prove below) does not hold for "traditional" set theory.

He watches in a darken room shadowy traces on a fluorescent screen placed against a patient's chest and hears the radiologist commenting to his assistants, in technical language, on the significant features of these shadows. At first the student is completely puzzled. For He can see in the X

ray picture of a chest only the shadows of the heart and ribs, with a few spidery blotches between them the experts seem to be romancing about figments of their imagination; be can see nothing that they are talking about. them as he goes on listening for a few weeks, looking carefully at ever new pictures of different cases a tentative understanding will dawn on him; He will gradually forget about the ribs and begin to see the lungs. And eventually, if He perseveres intelligently a rich panorama of significant details will be revealed to him: of physiological variations and pathological changes of scars, of chronic infections and signs of acute disease. He has entered a new world. He still sees only a fraction of what the experts can see but the pictures are definitely making sense now and as do most of the comments made on than He is about to grasp what he is being taught; it has clicked. thus, at the very moment when be has learned the language of pulmonary radiology, the student will also have learned to understand pulmonary radiograms The two can only happen together. Both halves of the problem set to us by an unintelligible text, referring to an unintelligible subject, jointly guide our efforts to solve them, and they are solved eventually by discovering a conception which comprises a joint understanding of both the words and the things. (Polanyi,1964, p. 101; italics mine)

Against a context of "comprehending" or "understanding the logical

empirical relations among a set of 'Names' and of 'Things Named', this passage documents the main analogy which underlies my formal psycho

logics: can explain the behavior of 'organisms' by regarding them as "mechanisms for generating and testing behavioral hypotheses", where I use the term 'behavioral hypotheses' to signify "guesses concerning how to obtain what the 'organism' "needs" and/or how to avoid catastrophe".

I acknowledge that at times at least the "happenings" which make up my own life have seemed to me "an unintelligible subject" or perhaps on which I didn't dare to make sense of); and I infer that every other human also has had experiences which fit this description.. Furthermore, I recall that at one time I had no familiarity at all with the constructs which make up 'naive set theory', and no fluency with an algebraic set theory notation; and before that, I recall having no command of the construction which make up "traditional" general semantics. Had I at those times, encountered a well worked out formal psycho-logics, I predictably would have found it "an unintelligible text".

But it so chances 'comprehending' some of 'comprehending' some that over and over again I have had the experience aspect of the constructs of "traditional' general semantics, and in the same breath making sense of some more or less troublesome and/or satisfying aspect of my own life. Eventually I became curious about how this mysterious process of "making my own life more to my own liking" works. Then, having elaborated (in discursive language) an answer of sorts to this question (an answer which compared human behavior to the logic of science), I eventually became curious about the logical structure of my own answer. Now I find myself the author of a formal psycho-logics, which I claim to comprehend (at least in part, but increasingly). And so, understandably enough, on the basis of my own experience, I believe that anyone else (who wants to) can come to comprehend these (developing) constructs, at least in part but increasingly.

This booklet cannot do "the whole job" for anyone. But its intelligent use can make certain things especially the logical aspects if the formal system accessible which otherwise might not seem so. (Seminars or "self-study groups", by contrast, seem more likely to make the empirical aspects of the formal system accessible, along with the "my own life" referents.)

SEQUENCE OF TOPICS

The text of this booklet will approach the formalized versions of the constructs recursively and gradually, each succeeding section requiring more command of set theory than the previous one. I shall

1. List each of the key constructs I later define in notation;

2. Provide a "strictly verbal" definition for each;

3. Provide a simple but formal definition for each of four of the constructs;

4. Define three of the constructs by means of formal theorems, with proofs;

5. Provide formal theorems, with proofs, which show the inter

relations between the different constructs, and which define two more of the constructs

6. Provide an account of the Premises and the Counter

Premises of a non

Aristotelian system; and

7. Provide a succinct account of my set theory notation, defining the relevant mathematical constructs in words in terms of Venn diagrams, and in algebraic form.

ALGEBRAIC SET THEORY

For those already familiar with the constructs which make up 'naive set theory', a brief check on my notational conventions should suffice to permit access to my formulations, at least in their 'logical' aspect.

For those with no previous exposure to set theory, I point out that the constructs which

make up 'naive set theory seem closely akin to our intuitive notions concerning the process of "classifying things", and once you become familiar with set theory, you may well discover that only the notation itself proved unfamiliar.

Perhaps the best way to become familiar with any body of mathematical constructs, including 'naive set theory', involves doing the problems in an elementary text. I recommend Seymour Lipschutz Set Theory And Related Topics, Including 530 Solved Problems (New York McGraw Hill, Schaum's Outline Series, 1964, $2.95). For the sake of initial orientation, here I verbally summarize most of the mathematical notions I utilize. I point out that the distinction between 'set' and 'element' resembles the distinction between 'Name' and 'Thing Named'. Furthermore,

.. each; in a foursome of golfers has a "set" of golf clubs and the "union" of these four sets consists of all clubs being carried down that fairway The suggestive symbol ( is used for "union" and this turned over ( () means "intersection". For instance if T is the set of dishes used on the table and C the set of cooking utensils in a home than T ( C is the set of dishes used for both purposes, and it might be an "empty" set, unless there are Pyrex dishes or the family has very casual customs concerning serving dishes. In general, the various kinds of unions and intersections of sets formalize the basic experience of discovering that two superficially dissimilar objects or notions prove on closer examination to have certain similarities (elements in common). We use various punctuation marks which are no different in principle from the orthographic conventions used in written English. Ordinary graphs make use of Cartesian coordinates; a Cartesian product space stands as a generalization of ordinary Cartesian coordinates, in which every element of the space is composed of an ordered pair of elements selected from two other designated sets. the notion of a mapping generalizes ordinary algebraic equations, or functions, e.g., y = x2 the inverse of a mapping is obtained by turning the mapping around the other way, so as to obtain all the elements of x which correspond to a particular value of the mapping of x , e.g., x = (y .

Therefore, if there be anyone unfamiliar with set theory but who takes heart from these encouraging remarks our advice would be to go over the mathematical statements of these fundamental notions in the Appendix until they no longer seem alarmingly strange and alien, and then to put them to use by reading the set

sentences of the theory, OUT LOUD. More detailed understanding of set theory will cone hand

hand with familiarity with how the set theory is used in this analyses. Understanding of the fine points of set theory, is necessary to write set

theory analyses or to prove theorems or perhaps even to follow proofs of theorems, but these represent later accomplishments, not prerequisites. (Hilgartner & Randolph, 1969a, pp. 291

2)

1. KEY CONSTRUCTS "TRADITIONAL" GENERAL SEMANTICS

Structure, order, relation(s)

'Map'

'Territory' analogy

Non

Identity

Non

Allness

Self

Reflexiveness

Abstracting, abstraction, abstracted

from

Order of abstraction

Level of abstraction

Multiordinal

Similar in structure

Directively correlated

Identifying

Consciousness of abstracting

Korzybski (1933) used, and more or less (informally) defined, each of these terms, with one exception: the term directively correlated', originally defined and us used by Sommerhoff (1950, 1969), translated into an algebraic set theory notation by Ashby (1962), and incorporated into the logical superstructure of our formal psycho

logics (Hilgartner, Randolph, 1969a,b,c,d).

This listing does not qualify as 'exhaustive', but it does seem 'sufficient'. It does not include a number of useful terms, e.g., semantic reaction', 'objectifying', 'elementalizing' nor several somewhat more intractable terms e.g., 'internal silence' ("silence on the objective levels"), nor 'sanity' ("natural order of evaluating"), etc. Nor have I given explicit discussion of the topic of the 'extensional devices' (altho those already familiar with them will see that I use them).But the terms listed above have received explicit definition, in notation; and furthermore they have provided a sufficient basis for interpreting and/or criticizing various published papers or books (Hilgartner, 1970c, d, e; 1971, a, b, c).

2. "STRICTLY VERBAL" DEFINITIONS

(In this section I use only three terms from set theory, namely, 'set', 'element' and 'belongs'. These terms comprise the undefined terms of the mathematical theory of sets

and from within the viewpoint of ("traditional") set theory we cannot further specify these term in words. But I point out that as my criterion of correct usage, I require that for any 'element, we can without ambiguity determine whether or not it belongs to a given 'set'.)

OVERALL CONTEXT

Every statement in a non

Aristotelian system refers immediately and/or eventually to a 'context' which I can describe by means of the expression, "the dealings of a human organism

his environment

a-date", where I use the term 'dealings. in the sense of "interactions", or better (after Dewey & Bentley, 1949), "transactions".

STRUCTURE, ORDER AND RELATION(S)

These terms comprise the 'undefined terms' of a non

Aristotelian system which I can perhaps define in terms of each other but cannot further specify in words. (Thus I can define 'structure as 'ordered relations or 'related orders'; but I decline to try to say in words what I mean by that.)

As well as considering them in their 'logical' roles, as 'undefined terms' not further specifiable in words, I must also consider these terms in 'empirical' roles' as designations for some Thing(s) Named' or 'territory. And (colloquially) I could describe the 'territory' designated by these terms by the expression, "'reaction(s)' or 'response(s)' of an 'organism' to his 'external' and/or 'internal' 'environment ".

'MAP'

' TERRITORY' ANALOGY

When I hypostatize an 'organism' capable of making a 'response' to his ('external' and/or 'internal') 'environment, I have already made a fundamental distinction: I have distinguished between the 'response and the 'responded

to ' (or 'environment ). As a shorthand way of indicating this distinction, I could refer to any 'response' as analogous to a 'map' (or 'Name') and the 'responded

to' as analogous to the 'territory' (or 'Thing Named') which the 'map' (allegedly) "represents" or "refers to".

NON

IDENTITY NON

ALLNESS, SELF

REFLEXIVENESS

These comprise the names of the non

Aristotelian postulates of

Korzybski (1933, 1943). I can express these in words as follows:

NON

IDENTITY: No structure shows the relation of 'identical with ("the same in all respects", "qualifies as a 'point

point perfect replica of'") any other structure.

NON

IDENTITY: No structure qualifies as 'identical with' that structure I could call 'itself

a-different

instant'.

In the terms of the 'map'

'territory' analogy, I can state Non

Identity as,

"No 'map' qualifies as 'identical with the 'territory it represents."

NON

ALLNSSS: No map' can show all the elements which make up the 'territory' it represents; no 'map' remains free of elements which do not in any sense refer to the 'territory' it represents.

SELF

REFLEXIVENESS: No action or utterance of any organism exists free of self

reference.

In the terms of the 'map'

'territory' analogy, I can state Self-Reflexiveness as' "No 'map' remains free of elements which refer to the 'map

maker' rather than to the 'territory' which the 'map' represents."

VALIDITY OF THESE PREMISES

Under no circumstances may we regard a self

consistent axiomatic system as showing any greater degree of "validity" than that we attribute to the premises from which it starts.

(This truism from traditional logic' leaves unspecified the criterion or criteria by which we judge the 'degree of "validity" we attribute to an axiomatic system or a set of premises. By reminding readers that a non

Aristotelian system must meet both 'logical' and 'empirical' criteria, and by the remarks which follow, I hope to foreshadow my own criteria of "validity".)

The non

Aristotelian postulates of Korzybski stand as denials (negations) of the famous "Laws of Thought" attributed to Aristotle (384

322 BC).

Since each of these postulates stands as a negative proposition (which I could phrase in terms of an "IS NOT"), a single valid counter

example would suffice to disconfirm any claim to its 'universal validity'.

But in the 38 years which have passed since Korzybski proposed the postulate of Non

Identity, or the 28 years Since he collected all three postulates into one place (Korzybski, 1943) no critic has produced even one counter

example.

As far as I know, no one has even dared to try.

Indeed, as Korzybski (1933, p. 10) points out, a burden of impossible proof rests on the man who would deny these denials of the "Laws of Thought" of Aristotle, for he would have to adduce at least two things which qualify as "absolutely the same in ALL respects"; or a 'map' which includes ALL the elements of the 'territory for which it stands, and no other elements

or an action or utterance of some organism which exists FREE of self

reference.

Perhaps these remarks give some sense of the utterly unprecedented security of conclusion obtainable by basing an axiomatic system on un

deniable negative premises. Using the older, "positive" mode of statement, I conclude that until someone somewhere somewhen somehow produces a convincing counter

example we may regard the non

Aristotelian premises as 'universally valid' for dealing with a 'universe' which includes human organisms.

ABSTRACTING, ABSTRACTION ABSTRACTED

FROM

Definition 1. Abstracting. The verb

form 'abstracting' designates that set of "fundamental life

processes" by which an 'organism selects ("responds to") certain elements of his 'external' and/or 'internal' 'environment', and ignores other elements The term ''abstraction'' emphasizes the "ignoring"; at least as much as the "selecting".

Definition 2. Abstraction. The noun

form 'abstraction' designates any structure or set (map, 'picture', 'construct', 'symbol', 'Name', "Gestalt', etc.) elaborated by means of the processes of 'abstracting'.

Definition 3. Abstracted

from. The notions of 'abstracting' and 'abstraction' singly and together, imply also a third term signified by the noun

form 'abstracted

from' (or 'territory', 'environment ', 'Thing Named', etc.).

ORDER OF ABSTRACTION

The notion of 'order of abstraction' provides a means of relating one or more given sets to the triadic relation expressible in the terms specified by Definitions 1, 2, and 3; or in other words, provides a means of relating one or more given sets to the postulates of Non

Identity, Non

Allness and Self

Reflexiveness.

I shall express the notion of 'order of abstraction' initially in terms of the 'map'

'territory' analogy:

Definition 4. Order of abstraction. Given two sets, one of which we may regard as an abstraction from the other (such that one comprises a 'map' and the other comprises the 'territory' it refers to): then these two sets belong to different 'orders of abstraction'.

LEVEL OF ABSTRACTION

The notion of 'level of abstraction' provides a means of making self

consistent distinctions between different but related aspects of 'the dealings or a human organism

whole

environment

date'', namely, those aspects I might colloquially describe as "non

verbal" (or 'silent

level') activities versus those I might call "verbal" ('symbolic

level' or 'verbal

level ') activities.

In Section 3, with reference to a demonstration borrowed from Korzybski (1951), I develop sub

distinctions within the 'silent levels' (viz., 'event

level', object

level') and within the 'verbal levels' (viz., 'label', description, 'lower

order inferences', 'higher

order inferences', etc.).

MULTIORDINAL

The notion of 'multiordinal' refers to a problem of interpretation which requires that we regard each member of one class of our terms as

ambiguous, or infinite

valued, in general, and each has a definite meaning, or one value, only and exclusively in a given context, when the order of abstraction can be definitely indicated. (Korsybski, 1933, p. 433)

If ignored, this ambiguity of a 'multiordinal' term inevitably leads to fundamental theoretical error and thus to in tractable confusion; but when explicitly taken into account, the problem of interpretation which requires that we posit a class of 'multiordinal silent

level terms also underlies our ability to elaborate increasingly general formulations, etc.

Given: A specified 'context' which includes a series of (at least two) related sets of successively higher orders of abstraction (viz., a series of structures (sets, prop, oppositions, etc.) each later member of which stands as an abstraction from the previous member); and

Given: Some term, taken in one and only one 'dictionary sense'; then

Definition 5. Multiordinal. The given term belongs to the set of multiordinal terms' it and only if, as judged by the usual canons of word

usage, one can apply this term (in this sense) to the set of lowest order of abstraction, and can also apply this term to the set of the next

lowest order of abstraction (etc.).

Importance: Within the specified constraints (a single 'dictionary sense' a specified context), a multiordinal term when applied on different orders of abstraction, will undergo definite and distinguishable changes in significance. Thus, as Korzybski (1933) notes, any multiordinal term shows a fundamental ambiguity, which makes it analogous to a 'variable, rather than to a 'constant' in logic or mathematics.

Theorem. Given a multiordinal term, taken in one 'dictionary sense' but, within a specified context, applied on two different orders of abstraction: then the act of interchanging these usages (viz., failing to distinguish between them, or otherwise treating them as 'identical') qualities as a fundamental error.

SIMILAR IN STRUCTURE

The notion of 'similar in structure' implies that our 'organism' engages in the process of somehow "comparing'' two "things" which belong to different orders of abstraction (viz., "comparing" a 'map' with the 'territory' it (allegedly) represents). The notion further implies that our 'organism' does this "comparing" on the basis of some (more or less) explicit 'criterion' and for some (specified or specifiable) 'purpose(s)'.

As I show in detail (in notation) in Section 3, the notion of 'similar in structure' function as a judgment made on a given 'map', based on "evidence" ("past experience", viz., the outcome of at least one test of a prediction, based on this 'map'); end further the notion of 'similar in structure' serves as a generalization from this "evidence or in other words functions as a prediction2 concerning the probable outcome of further tests of this prediction1, and/or concerning the probable outcomes of tests of other predictionsla, based on this 'map'.

Without the ability to make predictionsi which subsequent events fail to disconfirm, no organism' (human or sub

human) could survive (persist) for even a few moments; and further, by engaging itself in activities which at least might result in its own survival for at least a few moments more, an 'organism' displays its own (non

verbal) predictionj that the processes of abstracting by which it generates and tests predictionsi at least might result in its own survival for at least a few moments more; etc. Therefore the issues summarized under the rubric of 'similar in structure' quality as fundamental to the phenomena we colloquially call "life".

Given. An 'organism' engaging in the processes of "evaluating" (abstracting in a survival

related context); and

Given: Two related sets, one of which constitutes an abstraction from the other, such that we might designate one as a 'territory' and the other as our 'organism 'map' or that 'territory'; and

Given: An independent criterion of 'sufficient (or "good enough fit"), by which our 'organism' can 'judge the 'outcome' of his actions; then

Definition 6. Similar in structure. Our 'organism' may regard his 'map' as 'similar in structure' to the 'territory' it allegedly represents if and only if, tacitly and/or explicitly our 'organism' satisfies the following conditions:

That he uses the 'map' to make one or more definite (disconfirmable) predictions concerning the territory;

That he performs one or more tests which, in principle, so judged by the criterion, could disconfirm the prediction(s); and

That upon examining the results of his test(s), our 'organism' finds that, as judged by the criterion, the results do not disconfirm the prediction(s).

I shall defer until Section 3 the discussion of the ways the notion of 'similar in structure' functions as a generalization (second

order prediction ).

DIRECTIVELY CORRELATED

Any discussion of the relations between an 'organism' and its 'environment' must deal somehow with the pervasive phenomena we colloquially refer to as "apparently 'purposive' activities". This topic permeates the 'life sciences' and the 'human psycho

cultural sciences' on all levels from those of molecular structure, e.g., that of heme molecules (Caspari, 1962), to those of 'psycho

cultural evolution' (Huxley, 1953) or 'time

binding' (Korzbyski, 1921). Gerd Sommerhoff (1950, 1699) first provided a rigorously

defined construct which (he claims) gives a logically adequate and empirically correct model for "apparently 'purposive' activities". This construct, the relation known as 'directively correlated stands as a rigorously

defined form of "joint

causation". It involves a temporally

ordered sequence coenetic variables ("initial conditions" which affect both 'organism' and 'environment'), followed by responses of 'environment' and 'organism' to these coenetic variables; responses and these responses in turn interact so as to achieve (or in the negative case, not achieve) the focal conditions ('goals') of the 'organism', outcomes which appear somehow "favorable" from the point of view of the organism'.

Any "definition" of this construct which becomes stricter than the above immediately becomes formal, utilizing mathematical constructs more intricate than 'set', 'element' and 'belongs', and so falls outside the scope of this section. But generally, in "strictly verbal" terms, in specifying a 'directively correlated' sequence we must distinguish between 'coenetic' variables, 'directively correlated variables, and 'focal' variables.

*Having placed Sommerhoff's 1969 paper in silent

level section devoted to the topic of "the properties of open systems" F. E. Emery (1969) offers a comment which cite" one of Sommerhoff' as predecessors:

If it seems a little strange to the reader that an adequate solution to the theoretical problem of the 'properties of open systems' existed almost before the problem was posed (and thence generally ignored as if it did not exist), then he might also consider the fact that the logical structure for Sommerhoff's (1950) solution had been publicly formulated several years before by the Americal philosopher, E. A. Singer (1946) (for an equally appreciative audience). (Emery, 1969, p. 58)

The technical terms Korzybski develops (e.g., 'abstracting' and silent

level 'similar in structure' among others) refer to the topic of "the dealing of an organism

environment

date", and so they imply "apparently 'purposive' activities. But altho in his writings Korzybski devotes many passages to issues from biological theory, during his lifetime ho did not develop an explicit model which specifies the structure of "apparently 'purposive'". Hence when in the fall of 1965 I succeeded in relating my own 'psychological' constructs based on the non

Aristotelian postulates of Korzybski to a formal definition of 'directively corre1ated' in effect I disclosed and eliminated a ''hidden assumption" in "traditional" general semantics. Below, in Section 5, I show in notation the relations between the formal we version of the notion of 'abstracting' and the formal version of the notion of 'directively correlated'.

IDENTIFYING

3. SIMPLEST FORMAL DEFINITIONS

(Definitions 1

4 refer to Premise 2 (cf.. 6. PREMISES AND COUNTER

PREMISES), and so require only two notions from set theory beyond those of 'set', 'element' and 'belongs' namely the notion of 'subset' and the notion of 'mapping or 'functions'. In 7. NOTATION, sections 2

5 and 2

6 define the notion of 'subset' and sections 5

1 and 5

2 define the notion of 'mapping' or 'function'.)

ABSTRACTING, ABSTRACTION, ABSTRACTED

FROM

I shall define these three terms (on verb

form and two noun

forms) in words and also in our notation, always with reference to our non

Aristotelian 'context ('the dealings of a human organism

his-environment

date') and to our 'principal set', O ( E.

Definition 1. Abstracting. The verb

form 'abstracting designates that set of "fundamental life

processes' by which an organism selects (''responds to") certain elements of his 'external and/or 'internal' 'environment' and ignores other elements. The term 'abstracting' emphasizes the "ignoring" at least as much as the "selecting".

In the terms of Premise 2, the operator ( stands equivalent to the notion of 'abstracting'. In this sense, I treat this operator as a 'mapping or 'function', which fact I can signify in notation by the symbolism (: Y ( Z , which treats ( as an 'onto' function, or by (: E ( O, which treats it as a 'into' function.

Definition 2. Abstraction. The noun

form 'abstraction' designates any act ('map', 'picture', 'construct', 'symbol', 'Name', 'Gestalt', etc.) elaborated by the processes of abstracting'.

In the terms of Premise 2, the set Z stands equivalent to the notion of 'abstraction', as defined with respect to the mapping ( and to the set Y .

Definition 3. Abstracted-from. The notions of 'abstracting' and 'abstraction, singly and together, imply also a third term, signified by the noun

form 'abstracted

from'.

In the terms of Premise 2, the set Y stands equivalent to the notion of 'abstracted

from, as defined with respect to the mapping ( and to the set Z .

ORDER OF ABSTRACTION

The notion of 'order of abstraction' provides a means of relating one or more given sets to the triadic relation expressible in the terms specified by Definitions 1, 2, and 3; or in other words, provides a means of relating one or more given sets to the notational forms of the postulates of Non

Identity, Non

Allness and Self

Reflexiveness.

I shall express the notion of 'order of abstraction' initially as a binary relation.

Definition 4. Order of abstraction. Given two sets one of which we may regard as an abstraction from the other: then these two sets belong to different 'orders of abstraction'.

In the terms of Premise 2, the sets Y and Z , as related by the mapping ( , belong to different 'orders of abstraction' (specifically, Z belongs to a 'higher' order of abstraction then does Y ).

She postulate of Non

Identity, as stated in Premise la, expresses the fundamental relation existing between two related sets which differ in 'order of abstraction':

Y + Z ( (

Formally speaking, the notion of 'order of abstraction' seems analogous to a measure, or in other words, provided a systematic and consistent rule for "ignoring differences". Thus given two sets, P and Q , such that we may regard Q as an abstraction from P (regardless of the "content" of each), then I can indicate the abstract relation between them by writing

P ( Y , Q ( Z ; (3:1)

and, given that relation between them I may substitute into Premise la:

P + Q ( ( . (3:2)

LEVEL OF ABSTRACTION

The notion of 'level of abstraction' provides a means of making self

consistent distinctions between different but related aspects of 'the dealings of a human organism

environment

date', namely, those aspects I might colloquially describe as "non

verbal' (or 'silent-level') activities versus those I might call "verbal" ('symbolic 'level or 'verbal

level') activities. Moreover, since (as far as I know) only humans engage in 'verbal level' activities, I can also use the notion of 'level of abstraction' to make distinctions between the structure of the activities of human versus sub

human organism.

Any discussion of the topic of the dealings of 'a human organism

environment

date' (or of any other topic) can take place only and exclusively by means of: that subset of 'abstractions' which I might designate by the term 'symbols'. Furthermore, the 'symbols' which make up any 'discussion of this topic must perforce somehow represent the triadic elation between 'abstracting', 'abstraction' and 'abstracted

from'. And in discussing this topic, we can avoid immediately falling into fundamental error if and only if we manage to avoid treating any two of those terms (and/or the "Things Named'' they refer to) as interchangeable ('identical', "the same In all respects"). This we can do by means of the distinctions between 'silent levels and verbal levels'. (These distinctions, formally speaking, define another measure; but one which, ordinarily, I shall not render in notation. Where it becomes unavoidable to represent these distinctions, I shall do so orthographically, by changes of type

face, e.g., the use of italic or bold

face.)

In keeping with the fact that I do not indicated the distinctions between 'silent levels' and 'verbal levels' in notation, I shall not offer 'formal definitions' of these notions, but rather colloquial descriptions.

Description A. Silent levels (event-level, object level). I could describe my non

verbal waking experience as a set of "moments of 'awareness'" (a set of abstractions), elements of which (from moment to moment) shift in emphasis from one 'sensory modality to another (e.g., visual, proprioceptive, etc.) and shift in focus from aspects of my 'environment' to aspects of 'my own functioning' and back again.

After Korzybski (1933), I could design ate this act of "moments of 'awareness" as the 'object

level' (or in ether words the set of 'macroscopic transactions' or first

order abstractions'), and could regard this as the set of my abstractions from an (inferred) event-levels (or the set of 'sub

microscopic events', or the 'abstracted

from).

Please note that, taken in one sense, the notion of 'object

level contrasts with 'event

level'; but taken in another sense, 'object

level' comprises a subset of 'event

level' for, according to the non

Aristotelian postulates of Korzybski those 'Things Named. which I design ate "moments of 'awareness'", and which (I say) 'exist' on the 'object

level', also show an (inferable) 'sub

microscopic structure. and so 'exist' also on the 'event

level. For example the process of talking takes place on the 'silent levels'.

In other words, the terms 'object

level' and 'event

level' qualify as multiordinal (q.v. ).

Finally, please note that both 'event

level' and 'object

level qualify as members of the 'silent levels'.

Description B. Verbal levels (label, description, first

order inference, higher-order inferences, etc.) In order to make the following distinctions intelligible, I must involve each member of my audience in a little demonstration (stolen from Korzybski 1951):

Instructions: Between thumbs and fore finger of one hand, firmly pinch one finger of the other hand.

Than, out loud, make some remark about the experience of pinching your own hand.

To insist upon the obvious (and in the process to exemplify the distinctions spelled out above), the comment about the experience "IS NOT" the non

verbal experience itself. Instead, I would say that the comment constitutes an abstraction from the non

verbal ('object

level') experience. Thus the 'experience and the 'comment' differ in 'order of abstraction', the former belonging to the silent levels and the latter to the 'verbal levels.

In order to illustrate the notion of different 'verbal levels', say that the demonstration of pinching his own hand elicits the following soliloquy from one person:

(a) "Ouch!

(b) "That kind of hurts.

(c) "That seems an odd thing to do, to pinch yourself.

(d) "You seem a little odd yourself to make us do a screwy thing like that.

(e) "What are you, some kind of sadist or something?"

In the terminology of general semantics, (a) represents an immediate response or 'label (L) for the non

verbal experience (O) (which in turn represents an abstraction from the (inferred) 'event

level' (E) ); (b) represents a 'description' (D) of the experience; (c) represents an 'inference (I1) from the 'description'; (d) represents an 'inference (I2) from the first 'inference'; and (e) represents a still higher

order 'inference' (I3) from the second 'inference.

Each of these "remarks" (propositions, sets) differs from the preceding occurrences (and also differs from the subsequent occurrences) in 'order of abstraction;. And as you can see, when I indicate this sequence of operations by means of Venn diagrams, the resulting figure turns out as on form of the 'Structural Differential' (Korzybski, 1924, 1933) (Figure 1. about here )

Please note that this sequence of operations shows a space

time order.

Furthermore, as I indicate in Figure 1., I can apply the mapping ( successively as many times as I may wish

which means that the terms 'territory' and 'map', or 'abstracting', 'abstraction', and 'abstracted

from (or their notational equivalents) all qualify as multi

ordinal (q.v.), a conclusion I prove in Section 5 (the 'Map'

'Territory' Analogy and Multi

Ordinal).

SELF

REFERNTIAL

In a "dictionary" sense a 'self

referential' construct somehow "refers to itself". For example, in traditional set theory a relation R defined on a set A qualifies as 'self

referential if and only if it qualities as reflexive: for each a ( A , [a R a] .

In a non

Aristotelian system we regard any 'construct' as a member of the set of abstractions; and we regard any particular abstraction Z j as a "part" of our 'organism': Z j ( O j ; and further, we regard any abstraction Z j as derived from some 'territory' Y i , by a (temporally ordered) process of abstracting: Z j ( Y i . Hence, in general, I could define the notion of 'self

referential' by saying, "An abstraction qualifies as 'self

referential' if and only if it contains at least one element (or component) which 'represents or 'refers to' that 'territory' which I could describe by the phrase, "'reactions' or 'responses' of an 'organism' which elaborates an 'abstraction'"."

Furthermore, by the postulate of Self

Reflexiveness, we must regard any abstraction Z j as a subset of an intra

organismic Cartesian product space, (self

other), e.g. , Se ( Oe , where in the terms of Premise 2 and sentence (6:5),

Z j ( Cs, Cs = Se (Oe ,

Cq1 (Z j) = Se (Z j) = [Z j - ( (Q i)] ,

Cq2 (Z j) = Se (Z j) = [( (Q i)] ,

U (Se(Z j)) = Oe (Z j) . (6:5)

Finally, in the process of defining the

intra

organismic Cartesian product space, which we designate as the (self ( other) space (Hilgartner & Randolph, 1969a, p 296)

I explicitly distinguished between two components of any abstraction, namely, between the "focal 'awareness'" (or the Gestalt) and the "subsidiary 'awareness' of the processes of the Self, by which the focal 'awareness is organized":

We use the symbolism "x i / (E

x) i " , including the quotation marks, to mean a Gestalt, the "focal 'awareness of (an environmental object or situation) x ( E via sensory modality i ; and we use "Sf i", including the reversed quotation marks, to mean the subsidiary 'awareness of the processes of the Self, by which the focal 'awareness' is organized (Hilgartner & Randolph, 1969a, p 297).

So altho previously I have not explicitly defined an operator which shows where the component [Z j

( (Q i)] = Se(Z j) = "Sf ii" "comes from", from the start I have left no doubt that it "refers to or represents" those "'reactions or 'responses by which the 'organisms' elaborated the abstraction Z j " .

In order to make this origin explicit I could define a mapping ( , analogous to ( , which performs the operation in question:

"Sf i j" = ( i(O i) ;

or, stated as an relation,

"Sf i j" (i O i .

In light of these constraints, by envisioning the notion of 'self

referential' as a matter of degree I can bring out another fundamental point of a non

Aristotelian system. This I can do by defining three cases, "not self

referential" (NS) , "partially self-referential" (PS) , and "entirely as self

referential" (ES) .

Definition 5a. Not self-referential. A structure S2j qualifies as "not self

referential" (NS) if and only if

It qualifies as an abstraction S 2j ( Z j (viz., it derives from another structure

S1i ( Y i thru a process of abstracting: S 2j ( S1i ); and

b) It qualifies as "hetero

referential", viz. , the structure S1i from which it derives does not consist of "'reactions' or ''responses' of an 'organisms'": S1i ( O i ; and

c) It contains not even one element (or component) which I can describe by the phrase "a representation of a reaction' or 'response of the 'organism' which doss the abstracting":

( (Oi) = Cq1 (S2j) = Se (S2j) = ( .

S 2j ( NS ( [[S2j ( S1j] ( [S1j ( O i] ( [S2j + Oe(S2j) = (]] . (3:3)

But the phrase of sentences (3:3) which contains the condition ' = ( ' expresses a relation forbidden by the postulate of Non

Allness (Premise 4a). In other words, the set of "not self

referential constructs" qualifies as void:

NS = ( . (3:4)

Definition 5b. Partially self-referential. A structure S2j qualifies as "partially of self

referential" (PS) if and only if

a) It qualifies as an abstraction: S2j ( S1i ; and

b) It qualifies as "hetero

referential": S2j ( O i

c) It contains at least one element (or component) which I can describe by the phrase "a representation of a 'reaction' or 'response' of the 'organism which does the abstracting':

( (O i) = Cq1 (S2j) = Se (S2j) '( ( .

S2j ( PS ( [[S2j ( S1i] ( [S1i ( O I] ( [S2j + Oe (S2j) ( (]] . (3:5)

According to the non

Aristotelian postulates of Korzybski, every member of the set of abstractions qualifies as at least partially self

referential.

Definition 5c. Entirely self

referential. A structure S2j qualifies as "entirely self

referential" (ES) if and only if

a) It qualifies as an abstraction'' S2j ( S1i ; and

b) It qualifies as "not hetero

referential", viz. , ('the structure S1i from which it derives consists of "'reactions or 'responses' of an 'organisms'": S1i ( O i ; and

c) It contains at least one element (or component) which I can describe by the phrase, "a representation of a ''reaction' or 'response' of the 'organism' which does the abstracting" ,

( (O i) = Cq1 (S2j) = Se (S2j) ( ( .

S2j ( ES ( [[S2j ( S1i] ( [S1i ( O] ( [S2j + Oe (S2j) ( (]] . (3:6)

Like any other abstraction, an "entirely self

referential construct" comprises a "part" of the 'organism: [S2j (ES] ( O j . Also it comprises a subset of a Cartesian product space made up of ordered pairs of elements taken from two different sets which comprise two different kinds of representation of the 'organism' at a previous moment:

[S2j ( ES] ( ( (O i) ( (O i). (3:7)

4. DEFINITIONS IN THE FORM OF THEORETIC

(From this point on, I must freely make use of the mathematical notions defined in 7. NOTATION.)

MULTIORDINAL

As I point out below, I may treat the operator ( (signifying 'abstracting') as a 'set', e.g.

( ( Y ( Z or ( ( E ( O . And this fact allows me to make a fundamental point.

In an expression like Premise 2 (which has the form of a definition) viz.,

Q ( Y , ( (Q) = {z ( Z |(y ( Q, z = (Y)} (Premise 2), mathematicians (e.g. Ashby, 1962, p. 84) point out that the set ( appears in two different senses, one of which 'exists' at the level of elements and the other at the level of 'sets' ; or otherwise stated, one of which we define on the set Y ( E so that it generates a unique subset Z ( O (in notation, ( ( Y ( Z or ( ( E ( O ), and the other of which we define on the set of all subsets of E into the set of all subsets of O . If I use the notion of the 'power set on A ', indicated by ((A) , to designate the set of all subset of a set A ' (Lipschutz, 1964, p. 5), then using this notation I can specify the set ( at the level of 'sets' as ( ( ((E) ( ((O) .

But by Definition 4, the notion of the 'power set on A ' qualifies as a construct of higher 'order of abstraction' than that of the set A itself. So here in a specified context (namely, in the definition labeled 'Premise 2'), we find ourselves able to apply the operator ( on two distinguishably different 'orders of abstraction'. Therefore, in line with distinctions specified by Tarski (1965, pp. 4

5), we must regard the notion of 'the set ( ' as analogous to a logic 'variable' rather than to a logical 'constant':

In view of the fact that variables do not have a meaning by themselves, such phrases as:

x is an integer

are not sentences, altho they have the grammatical form of sentences; they do not express a definite assertion and can be neither confirmed nor refuted. From the expression:

x is an integer

we only obtain a sentence when we replace "x" in it by a constant denoting a definite number; thus, for instance, if "x" is replaced by the symbol "l" , the result is a true sentence, whereas a false sentence arises on replacing "x" by "1/2" . (Tarski, 1965, pp. 4

5)

In principle, even in the specified context of a definition, (e.g. , Premise 2, only from clues in the context in which it appears can we tell whether to interpret a given usage (example) of 'the set ( ' as signifying ( ( E ( O or ( ( ((E) ( ((O) .

Here the notational context makes the ambiguity relatively easy to see; and as Ashby (1962) comments in this specific notational context,

experience has shown that the use of the same symbol (() to represent [two different set] is convenient and rarely a source of confusion. (Ashby, 1962, p. 84)

But hero we have our finger on a fundamental issue, the significance of which, as a infer from his comment, escapes Ashby (1962) (among other mathematicians), but which Korzybski (1933, pp. 433

442) correctly analyzes, and which applies both in mathematical and in linguistic contexts: Korzybski points out the existence of a class of constructs each member of which, taken within a specified context and within one 'dictionary meaning', we can apply on two or more 'orders of abstraction'. He coined the term 'multiordinal' to designate this class of constructs:

The words 'yes', 'no', 'true', 'false', 'function', 'property', 'relation', 'number', 'difference', 'name', 'definition', 'abstraction', 'proposition', 'feet', 'reality', 'structure', 'characteristic', 'problem', 'to know', 'to think', 'to speak', 'to hate', 'to love', 'to doubt', 'cause', 'effect', 'meaning', 'evaluation', and an endless array of the most important terms we have, must be considered as multiordinal terms. (Korzybski 1933, p. 433)

Korzybski succinctly states the problem of interpretation which requires that we take into account the notion of 'multiordinal':

There is a most important semantic characteristic of these multiordinal terms; namely, that they are ambiguous or infinite

valued, in general, and that each has a definite meaning or one value, only and exclusively in a given context when the order of abstraction can be definitely indicated. (Korzybski, 1933, p. 433)

Furthermore, Korzybski provides an explicit criterion by which we may distinguish which terms qualify as 'multiordinal'

The test for the multiordinal of a term is simple. Let us make any statement and see if a given term applies to it ('true', 'false', 'yes', 'no', 'fact', 'reality', 'to think', 'to love', etc.) If it does, let us deliberately make another statement about the former statement, and test if the given term may be used again. If so, then it is a safe assertion that this term should us considered as multiordinal (Korzybski, 1933, p. 433)

Finally, as I mention in the text, (Korzybski, (1933, p. 440

1) points out that to neglect the intrinsic 'ambiguity' of 'multiordinal; terms necessarily leads to serious confusion, which we can see when we take into account the fact that we can apply any multiordinal terms to itself. When we do this, he writes, we find three main classes of 'second

order effects': With one subset of 'multiordinal' terms (ones which we might conventionally regard as referring to 'positive' or 'desirable activities), the 'second

order usages' reinforce the 'first

order usages' into designations for highly valuable activities, as in 'curiosity about curiosity', analysis of analysis', 'reasoning about reasoning', etc.; with a second subset (terms referring to conventionally 'negative' or 'undesirable' activities), the 'second

order usages' reinforce the ''first

order usages' and in the process convert then into designations for morbid activities, as in 'worry about worry 'fear of fear', 'ignorance of ignorance', etc. and with a third class of terms, the activities designated by the 'second

order usages' reverse and annul he activities designated by the 'first

order usages', as inhibition of inhibition', 'doubt of doubt', etc..

Thus as Korzybski put it,

The main point about all such multiordinal terms is that in general, they are ambiguous, and that all arguments about then, 'in general lead only to identification of orders of abstraction and semantic disturbances, and nowhere else. (Korzybski 1933 p. 434)

In order rigorously to define the notion of 'multiordinal' find I must find some way to translate Korzybski's "test for the multiordinality of a term" into our notation. In order to do so, I shall provide specific preposition1

and a specific proposition2 about my preposition1, and shall test each against one term which I suspect as 'not

multiordinal' ("uni

ordinal"?) and against another which I suspect as 'multi

ordinal; and then I shall translate this linguistic procedure into notational form.

PROPOSITION1: "The creature before me shood barring nor way and barking at me in an exceedingly hoarse voice."

PROPOSITION2: "My friend informed me that the creature before me, barring my way and barking, belongs to the breed of Basenje hounds, and that Basenje hounds uniformly have exceedingly hoarse voices."

As my "uni-ordinal" term, I take the term 'dog', in the 'dictionary meaning' of ''a carnivorous domesticated mammal"; and as my 'multiordinal' term, I take the term 'true', in the 'dictionary meaning' of "conformable to fact; correct; not erroneous, inaccurate or the like".

Clearly proposition1 designates (describes) a "thing" which I could label as a 'dog'; but proposition2 designates (describes) the act of 'classifying' the "thing" designated by proposition1 and I cannot legitimately label an 'act of classifying' as a 'dog'. Hence the term 'dog, by this test, does not qualify as 'multiordinal'. However, I could consider proposition1 as 'true' and could also consider preposition2 as also 'true'. Hence the term 'true', by this test, qualifies as 'multiordinal'.

In order to translate these operations into notation, I shall make use of the following conventions:

欀ˈ

A NON-ARISTOTELIAN "ROSETTA STONE"

TABLE OF CONTENTS

INTRODUCTION

SEQUENCE OF TOPICS

ALGERBRAIC SET THEORY

KEY CONSTRUCTS FROM " "TRADITIONAL" GENERAL SEMANTICS (from

Elwood Murray)

2. "STRICTLY VERBAL" DEFINITIONS

OVERA

a) Let MO signify the set of 'multiordinal' terms. Then

EMBED Equation.2

signifies the "not

multiordinal" ('uniordinal' remainder of our constructs.

b) In line with the requirements for temporal ordering, let Re signify one 'dictionary meaning' of a 'uniordinal' term (e.g. 'dog`), and let Qe signify one 'dictionary meaning of a 'multiordinal term (e.g., 'true'). In notation, Re (

EMBED Equation.2

and Qe ( MO .

c) Let Te signify some particular 'territory' ("abstracted

from"), and let ( f = ((Te) signify a particular proposition (e.g., preposition1) and finally, let (i = (((f) = ((((Te) = (2(Te) signify a particular proposition about (f (e.g. , proposition2').

d) In order to indicate the activities of our 'organism', in line with conventions specified in Hilgartner & Randolph (1969a), let M (at some specified time t ) signify some particular 'map' (or 'abstraction' for 'awareness' or 'consciousness'), and let At (at some specified time t ) signify the act of 'attending' ("in which the organism uses its motor apparatus in such a way as to prevent habitation. For example, visual scanning, tactile exploring (caressing), "cocking an ear", sniffing, etc." (Hilgartner & Randolph, 1969a, p. 306)) Furthermore, after sentence (20) of Hilgartner & Randolph (1969a), I shall regard a 'map' as a subset of Cs ('consciousness'), and shall further regard it as having the logical structure of a Cartesian product space, Se ( Oe (self-cross-other):

M ( Cs, Cs = Se ( Oe

Cq1(M) = Se(M)

Cq2(M) = Oe(M)

U(Se(M)) = Oe(M)

1. "A given term either applies or else does not apply to a given statement" becomes

[(f ( Re] ( [(f ( Re] (4:1)

and the notion of ''our 'organism' seeing if a given term applies to a given statement" becomes

Of: (f = ( (Te)

Og: At((f )g = Mh ,

Cq2(Mh) = Oeh : "[(f ( Re] ( [(f ( Re]"h . (4:2)

2. ''If it does (apply), let us deliberately make another statement about the former statement and test if the given term may be used again."

Oi : (i = ( ((f) ,

Oj: : At((i)j = Mk ,

Cq2(Mk) = Oek : "[(i ( Re] ( [(i ( Re]" k . (4:3)

Then according to Korzybskis definition of multiordinal, given that Re (

EMBED Equation.2

and Qe ( MO , it follows that

[(f ( Re] ( [(i ( Re]

[(f ( Qe] ( [(i ( Qe] .

Then a formal definition of the notion of multiordinal becomes

Definition 6: Multiordinal

MO = {Qe ( MO ( ( (Te, (f, (i ): (f = ((Te), (i = (((f),

[[(f ( Qe] ( [(i ( Qe]]}. (4:5)

THEOREM. In the notation of Definition 6,

Qe ( MO ( [Qe:(f ( Qe] + [Qe:(i ( Qe] ( (. (4:6)

Proof: Presume first a Qe such that Qe ( MO .

Ex hypothesei

(f = ((Te),

(i = (((f) = (2(Te) .

By Definition 4 (Order of abstraction, Rosetta p. 13, p. 20), (i occupies a higher order of abstraction than does (f . In other words,

(f ( Y , (i ( Z ,

and therefore, substituting into Premise 1a,

(f + (i ( ( .

But the ex hypothesei conditions exactly fit Definition 6; so given the presupposition that

Qe ( MO , then

[(f ( Qe] ( [(i ( Qe] .

This conditional proposition implies two distinguishably different sets designated by the symbol "Qe", namely,

[Qe:(f ( Qe] and [Qe:(i ( Qe] .

And since (i occupies a higher order of abstraction than does (f , then the set [Qe:(i ( Qe] must also occupy a higher order of abstraction than does Qe:(f ( Qe] , which means that

[Qe:(f ( Qe] ( Y , [Qe:(i ( Qe] ( Z ;

and this leads to the desired result.

Conversely, presume a Te , an (f , a (i , and a Qe such that

[Qe : (f ( Qe] + [Qe : (i ( Qe] ( (.

From Premise 1a and Definition 4 (Rosetta p. 13, p. 20, Order of Abstraction), we know that

[Qe : (f ( Qe] and [Qe : (i ( Qe] differ in order of abstraction; and since these two distinguishably different sets Qe differ in order of abstraction, so must (f differ in order of abstraction from (i .

But Definition 6 specifies the set of multiordinal constructs as composed of a set of elements Ae for which there exists at least one related series of sets Be , (f , (i , such that (f and (i belong to different orders of abstraction, viz.,

(i = (((f) , (f = ( (Be)

and such that each element Ae functions as a superset to which the set of higher order of abstraction (i belongs if the set of lower order of abstraction (f belongs to it, viz., such that

[(f ( Ae] ( [(i ( Ae]

Since the presumption with which we start satisfies these conditions, therefore the superset Ae meets the criteria for classification as an element of the set of multiordinal constructs, or

Ae ( MO .

This completes the proof.

Corollary 1: In Premise 2

[( ( E ( O] + [( ( P (E) ( P (O)] ( (

This relation holds for any mathematical construct which we could define by means of an expression of the form ((A) = {(, = ((a) } .

SIMILAR IN STRUCTURE

The notion of similar in structure implies that our organism engages in the process of somehow "comparing" two "things" which belong to different orders of abstraction (viz., comparing" a 'map' with the territory it (allegedly) represents). The notion further implies that our organism does this "comparing" on the basis of some (more or less) explicit 'criterion' and for some (specified or specifiable) 'purpose(s)'.

The notion of similar in structure functions as a judgment made on a given map, based on "evidence" ("past experience", viz., the outcome of at least one test of a prediction1 based on this map); and further, the notion of similar in structure also serves as a generalization from this "evidence", or in other words functions as a prediction2 concerning the probable outcomes of further tests of this prediction1 and/or concerning the probable outcomes of tests of other predictionsla based on this map.

Given: An organism engaging in the processes of abstracting; and

Given: Two related sets, one of which constitutes an abstraction from the other, such that we might designate one as a 'territory' and the other as our organism's map of that territory; and

Given: An independent criterion of sufficient (or "good enough fit"), by which our organism can judge the outcome of his actions; then

Definition 7: Similar in structure. Our organism may regard his map as similar in structure to the territory it allegedly represents if and only if, tacitly and/or explicitly, our organism satisfies the following conditions:

(a) That he uses his map to make one or more definite (disconfirmable) prediction(s) concerning the territory;

(b) That he perform one or more tests which in principle, as judged by the criterion, could disconfirm the prediction(s); and

(c) That upon examining the results of his test(s), our organism finds that, as judged by the criterion, the results do not disconfirm the prediction(s).

I can express the temporally-ordered relation of similar in structure in the set theory calculus of the (organism ( environment) field by means of the following conventions: Let S designate the relation of similar in structure. Then I read the expression

Tf S Mg (4:8)

as, "The set M (at time tg ) qualifies as similar in structure' to the set T (at time tf )."

The following derivation refers to a simple example:

Suppose I note certain "funny feelings" in my middle somewhere, makethe guess that "I hunger," locate and eat some food, and then feel replete. (Hilgartner, 1968, 1971b)

I shall distinguish between silent levels' and 'verbal levels' by underlining those expressions which refer to silent-level operations, and by enclosing in double quotes those "expressions" which refer to 'verbal-level' operations. Please note that within the silent levels I can still have sets existing on different orders of abstraction, e.g., Mg = ((Tf) . Furthermore, within the 'verbal levels' I shall (redundantly) distinguish between statements of observation, e.g.,, "Oci " j , which combine underlining with a time-index for the whole expression equal to or later than the time-index for the individual term, and statements of expectation, e.g., "Oci " g , using Roman characters with a time-index for the whole expression earlier than that of the individual term.

a) Let the unmodified symbol T stand for "the territory". Here Tf designates that set of happenings I referred to as an organism with funny feelings in his middle somewhere. (Thus Tf signifies the silent-level territory, and "Tf"e or "Tf"g signifies the notion of "silent-level territory", expected or observed.) But in another sense, an organism with funny feelings in his middle somewhere already comprises a subset of the O ( E field; and to bring in the terminology of 'directive correlation' (cf. below), Tf stands equivalent to d ( D .

b) Let the unmodified symbol M stand for "a map of this territory." Here Mg = ((Tf) designates the hypothesis (perception) which I verbalized as, "I hunger." (Thus Mg signifies the silent-level map of this silent-level territory, while "Mg"e or "Mg"g signifies the notion of "a map of the territory", expected or observed.)

c) Let the unmodified symbol A designate the notion of all possible interactings of our organism with his environment. Then ah ( A designates some particular interacting (at some specific times th ). Here, for convenience, I shall utilize the notation Ah = {ah | ah ( A} to designate the procedure of locating, approaching, seizing and ingesting some particular item of food (at times th). (Thus in the terminology of formal deductive systems, Ah signifies the silent-level act of performing a test of the hypothesis Mg ; and "Ah"g signifies a verbal-level expectation, the symbolic operations of "planning a test of the hypothesis Mg", while "Ah"j signifies verbal-level operations with the observed "test of the hypothesis Mg ".

d) Let the unmodified symbol Oc designate the notion of all possible outcomes of the interactings A of our organism with his environment.Then oci ( Oc designates some particular outcome (at some specific times ti ) of some particular interacting ah . Here, for convenience, I shall utilize the notation Oci = {oci | oci ( Oc} to designate the particular outcome of a particular (concerted) interacting Ah . (Thus Oci signifies the silent-level results of the test Ah , and "Oci"g or "Oci"j signifies the notion of "silent-level results", expected or observed.)

e) Let the unmodified symbol G ( Oc stand for the notion of "the (sub)set of outcomes which prove somehow favorable from the point of view of the organism." Then oci ( G ( Oc designates some particular favorable outcome (at some specific ti). Here, for convenience, I shall utilize the notation Gg = {oci | oci ( G} to designate the notion of "an organism FREE of funny feelings, free of restlessness, free of detectable muscle tensions, free of heightened or restricted breathing, etc." Thus Gg signifies a criterion of 'sufficient', viz., "replete", and so exists on verbal levels. Hence I may not present Gg as italicized (though I may enclose it in quotes).

f) Finally, let the unmodified symbol At (attending) designate that set of procedures by which our organism examines (classifies) something. (Thus Atj signifies the silent-level act of examining (at specific times tj), while "Atj"g or "Atj"k signifies the verbal-level notion of "examining", expected or observed.)

In the terms of these conventions, and also those of sentence (20) of Hilgartner & Randolph (1969a, p. 307, quoted above in Appendix II: Premises), I can express the various constructs which make up the definition of similar in structure as follows:

1. "The results, as judged by the criterion" becomes

[Oci ( Gg] ( [Oci ( Gg], (4:9)

and the notion of "our 'organism' examining the results (as judged by the criterion)" becomes

Oj : At(Oci)j = Mk ,

Cq2(Mk) = Ock : "[Oci ( G] ( [Oci ( G]"k . (4:10)

2. "A prediction concerning the territory" becomes

Cq2 (Mg) = Oeg : [A ( Oci ( Gg]g , (4:11)

and the notion of "our organism using the map to make a prediction concerning the territory" becomes

Og : ((Tf) = Mg ( Og ,

Cq2 (Mg) = Oeg : ["Ah ( Oci ( Gg "]g . (4:12)

3. "The prediction gets disconfirmed" stands as

Cq2 (Mg) = Oeg : [Ah ( Oci ( Gg ]g ,

Cq2 (Mk) = Oek : [Oci ( Gg ]g , (4:13)

Then the notion of "a definite disconfirmable prediction" (rendered as "our organism (at times tg) imagining finding the prediction disconfirmed (at later times tk)") comes out as

Og : ((Tf) = Mg ( Og ,

Cq2 (Mg) = Oeg : ["Ah ( Oci ( Gg "]g

( ["Ok : At(Oci)j = Mk ,

Cq2 (Mk) = Oek : ["Oci ( Gg "]k"]g . (4:14)

4. Then condition (c), "Upon examining the results of his test (at time tk), our organism finds that, as judged by the criterion, the results do not disconfirm the prediction", comes out as

Og : ((Tf) = Mg ( Og ,

Cq2 (Mg) = Oeg : ["Ah ( Oci ( Gg "]g ,

Oj : At(Oci)j = Mk ,

Cq2(Mk) = Ock : "[Oci ( G]"k . (4:15)

The form of the verbal definition of similar in structure requires that I explicitly state S in terms of the structure of the temporally-ordered behavioral sequence by which our organism comes to the judgment that his map qualifies as similar in structure to the territory, viz.,

Og : ((Tf) = Mg ( Og ,

Cq2 (Mg) = Oeg : ["Ah ( Oci ( Gg "]g ,

Oh : Ah ,

Ah ( Oci ,

Oj : At(Oci)j = Mk ,

Cq2(Mk) = Oek : "[Oci ( G]"k ,

Ol : At(Oek) = Ml ,

Cq2(Ml) = Oel : ["["Oci ( Gg "]k , (Tf S Mg"]l , (4:16)

whereas the complementary judgment becomes

Cq2(Mk) = Ock : "[Oci ( G]"k ,

Ol : At(Oek) = Ml ,

Cq2(Ml) = Oel : ["["Oci ( Gg "]k , (Tf

EMBED Equation.2

Mg"]l , (4:16a)

So this notational formulation does appear to me to define similar in structure as a judgment (our organisms ''conclusion", at time tl ) concerning the observed relation between his prediction and the observed outcome of a particular test of his prediction.

But the importance of the notion of similar in structure lies not so much in its role as a judgment concerning past occurrences, but rather in its role as a PREDICTION2 concerning future occurrences: If the map qualified as similar in structure to the territory in at least one past instance, we say, then future replications of the test actually performed will again fail to disconfirm the original prediction1; and furthermore, given other, future predictions1a (members of the ''host of yet hidden implications which his discovery [the 'map' Mg) will reveal in later days to other eyes" (Polanyi, 1964, p, 64), then as yet unimagined tests of those "yet hidden implications (predictions1a)" will themselves also fail to disconfirm those predictions1a.

Translated into a more diagrammatic form, then, the notion of similar in structure consists of three pairs of double-conditional propositions: First, our organisms "conclusions" at time t1 :

[["Ah ( [Oci ( Gg]"]k ( [Tf S Mg]l]l ,

[["Ah ( [Oci ( Gg]"]k ( [Tf

EMBED Equation.2

Mg]l]l . (4:17a)

Second, our organisms prediction (at time tl) concerning future replications of the test already performed once (replications performed at times tm tp):

[[Tf S Mg]m ( ["An ( [Oci ( Gg]"]p]l

[[Tf

EMBED Equation.2

Mg]m ( ["An ( [Oci ( Gg]"]p]l . (4:17b)

And third, our organisms prediction2 (at time tl ) concerning future tests (performed at times t q tt) of "yet hidden implications" (which I might symbolize as [Cq2 (Mg)q = Oeq : ["Ar ( [Ocs ( Gq]"]q]l ):

[[Tf S Mg]q ( ["An ( [Oci ( Gg]"]q]l

[[Tf

EMBED Equation.2

Mg]q ( ["An ( [Oci ( Gg]"]q]l . (4:17c)

Since in writing sentences 4:17b and c I have reversed the positions of antecedent and consequent as originally stated in sentence 4:17a , perhaps I can generalize in such a way as to subsume all three sentences by writing a single biconditional proposition. In doing so, I shall indicate generality by dropping the time-indices for the whole expression, and for each of its component clauses (though not for the individual terms), e.g.,

[[Tf S Mg] ( ["Ah ( [Oci ( Gg]"]] (4:18)

In so doing, I show that this generalized expression comprises not a specific proposition, but rather a propositional function' (Korzybski, 1933, p. 136; Tarski, 1965, pp. 4-5).

DIRECTIVELY CORRELATED

In the set theory calculus of behavior of Hilgartner & Randolph (1969a,b,c,d) we treat human behavior

experience ('behavior' a, viewed by others, 'experience' as viewed by oneself) as a special case of the "apparently 'purposive' activities" of 'living systems'. In order to do this, we take as basic the relation of 'directively correlated' (Sommerhoff, 1950, 1969), a rigorously

defined form of "joint

causation". This construct consists of a temporally

ordered sequence: coenetic variables (initial conditions which affect both 'organism and 'environment'), followed by responses of 'environment' and 'organism' to the coenetic variables; and these responses in turn interact so as to achieve (or in the negative case, not achieve) the focal conditions ('goals') of the 'organism', outcomes which appear somehow 'favorable' from the point of view of the 'organism'. A mathematical definition of this construct goes as follows:

Given: Spaces O , E , D ( O ( E , and Oc ( O ( E , together with an onto function g: D ( E and a function (: E ( O ( Oc; then, with G a subset of Oc , we have

Definition 8. Directively correlated. A function f: D ( O qualifies as 'directively correlated with respect to g , ( and G if and only if

((g(d) , f(d)) ( G for each d ( D . (4:19)

Speaking intuitively, f interacts with g via ( in such a way that whenever a disturbance d which meets certain constraints and thus belongs to D influences both the 'environment' E and the 'organism' O , then the outcome of this interaction proves somehow 'favorable from the point

of view of the 'organism' and thus qualifies as a member of G ( Oc . The accompanying diagram (Figure 2) may help in visualizing this mathematical model.

f and g function mathematically as first and second projections, specifying temporally

ordered functions from D onto O and onto E respectively. Thus as a set f represents "what the 'organism' does" while g represents "what the 'external' and/or 'internal' 'environment' does" in the class of "apparently 'purposive' activities" which they refer to.

A. Element

free expression: In the notation of Definition 8, f

qualifies as 'directively correlated' with respect to g , ( and G if and only if

f ( ( -1 (G) ( g . (4:20)

Proofs of this theorem appear elsewhere (Ashby, 1962, pp. 91

92; Hilgartner & Randolph, 1969a, p. 336; Hilgan1970a). Since a proof of this theorem involves most of the considerations used below to drove theorem (5:15), which relates 'directively correlated' (f) to 'similar in structure (S) (or rather, to its inverse, S -1), and involves no other considerations, I shall not repeat the proof of the element

free expression here.

'INTEGRATED' RELATIONS BETWEEN DIRECTIVILY CORRELATED ACTIVITIES

As I mentioned in Section 2, the topic of "apparently 'purposive' activities" permeates the 'life sciences' and the 'human psycho

cultural sciences' on every level. But to say that suggests the possibility that someone somewhere somewhen somehow might manage to frame a logically

empirically satisfactory 'invariant relation." which not only qualifies as similar in structure to those 'Things Named' we colloquially call "apparently 'purposive' activities", but which also we can apply on any level whatsoever

or in other words which also qualifies as multiordinal.

According to Sommerhoff (1950), those activities which we refer to as "biologically 'integrated"' offer a case in point, involving what I might call a directively correlated relation between a set of directively correlated relations1:

A set of organic activities is 'integrated' in the biological sense if the activities are directively correlated and if these correlations are again directively correlated inter se (e.g., if their respective focal conditions may in turn be regarded as a set of directively correlated variables. (Sommerhoff 1950, p. 195)

I display two systems, P and Q , each of which qualifies as 'directively correlated' in this notation. I define P ( O ( E so that (4:8) and (4:9) hold, while Q ( O ( E involves a space J ( O ( E , an onto function n: J ( E, a function m: J ( O , and a function (: E ( O ( Z , with H ( Z . Then m qualifies as 'directively correlated with respect to n , H , and ( if and only if

((n(j), m(j) ) ( H for each j ( J . (4:20)

Then by reasoning like that displayed for theorem (4:20),

m ( ( -1 (H) ( n . (4:22)

Simplest case: Let us suppose that the set of 'favorable outcomes. (focal conditions) for P consists of the same elements as does the set of 'disturbances' (coenetic variables) for Q . viz., G = J .

Then, given a mapping Pj0: E ( O ( O , the relation between P and Q qualifies as 'integrated (a 'directively correlated' relation between two 'directively correlated systems') if and only if

f ( ( -1 (m -i ( Pj0 ( ( -1 (H))

g . (4:23)

Proof. First let P and Q qualify as 'directively correlated' systems, with G = J , and let the relation between P and Q qualify as 'integrated'. I define a set R by

R = {(e, o) (j ( J: e = n(j), o = m(j)} .

Hence R comprises a subset of the domain of ( and, from (4:21), ((R) ( H or

R ( ( -1 (H) ( E ( O . Then

Pj0 (R) ( Pj0 ( ( -1 (H) ( O .

and since m -1 ( O ( J ,

m -1 ( Pj0 (R) ( m -1 ( Pj0 ( ( -1 (H) ( J .

As I show for (5:15), f ( g -1 (((-1 (G) ; but since J = G , then

f ( g -1 ( ( -1 (G) = ( -1 (m -1 ( Pj0 ( ( -1 (H)) ,

from which (4:23) follows.

Conversely, assume a P and a Q such that if J = G , then (4:23) holds. Then

f ( g -1 ( ( -1(m -1 ( Pj0 ( ( -1(H)) ( g ( g -1 ,

Again I use g ( g -1 = IE to obtain

f ( g -1 ( ( -1 (m -1 ( Pj0 ( ( -1 (H)) .

I have already demonstrated that m -1 ( Pj0 ( ( -1 (H) ( J . If J = G , then

( -1 (m -1 ( Pj0 ( ( -1(H)) = ( -1 (G) ,

and so f ( g -1 ( ( -1 (G) .

But I have already demonstrated that given an expression of that form, then f qualifies as 'directively correlated' with respect to g , G and ( .

Moreover ,

j ( m -1 ( Pj0 ( ( -1 (H) ( m(j) = o ( Pj0 ( ( -1 (H) ( O ,

and

o ( Pj0 ( ( -1 (H) ( Pj

0-1 (o) = (e, o) ( ( -1 (H) ( E ( O ,

where w = n(j) . Then from the set R defined by

R = {(e, o)| (j ( J : (e, j) ( n-1 , (j, o) ( m}

we can see that R = m ( n -1 ( E ( O , and thus

m ( n -1 ( ( -1 (H) .

But as I demonstrate for theorem (5:15), given an expression of that form, then m qualifies as 'directively correlated' with respect to n , H , and ( .

Thus if (4:23) holds, then both P and Q qualify as 'directively correlated' systems.

But (4:23) holds if and only if G = J , where O designates the set of focal conditions for P , and J designates the set of coenetic variables for Q .

And by Sommerhoff's definition of "'integrated' in the biological sense", if (4:23) holds and J = G , then the relations between P and Q qualify as 'integrated'. That completes the proof.

A more general case: Suppose we have a finite set of 'directively correlated' systems,

P1, P2, P4, , Pn , for each member of which its fi qualifies as 'directively correlated' with respect to its gi , Gi , and (i . Further, suppose that the focal conditions of this array of 'directively correlated' systems stand as 'directively correlated' inter se, via Q . This could come about provided that a) each Gi comprises a disjoint subset of Z (or Oc) , so that

G1 ( G2 ( G3 ( ( Gn ( Z ;

and still further, provided that b) the set J of 'coenetic variables' which figures in Q comprises a subset of a Cartesian product space G1 ( G2 ( ... ( Gn , every element of which thus consists of an ordered n

tuple composed of one element from the Gi of each directively correlated system Pi Then the 'integrated relation between one particular Pi and Q becomes

fi ( (I-1 (m -1 ( Pj0 ( ( -1 (H)) ( gi . (4:23a)

Proof. The proof follows exactly the lines or the proof of the simplest ease of this theorem.

'LIVING SYSTEM'

Sommerhoff (1950) defines the term 'living organism' (for which I substitute 'living system') in terms of 'integrated' relations between 'directively correlated' system:

A living organism may be described as a compact physical system of mechanically connected parts whose states and activities are relate by an integrated set of directive correlations which, over and above any proximate focal condition, have the continued existence of the system as an ultimate focal condition. Death may be described as the breakdown of these directive correlations. (Sommerhoff, 1950, p. 195

6; cf. also pp. 161ff)

In the notation of 'directively correlated systems' and of 'integrated' relations between directively correlated systems, I can express the sense of Sommerhoff's formulation by using sentence (4:23a) to define the set of 'living systems' (LS) .

Altho I do not explore those ramifications in detail here, this endeavor will bring us to the interface between logical' and 'empirical. relations in human and sub

human biology.

Given a 'system' which I shall call our (putative) 'organism' O , which contains a finite set of 'parts' (sub

systems which involve the 'states or activities of our 'organism'), namely, P1, P2, P3, ... , Pn , for each of which sentences (4:19) and (4:20) hold, and where

G1 ( G2 ( G3 ... ( Gn ; and given that our putative 'organism' contains another sub

system Q for which sentences (4:21) and (4:22) hold, and for which J = G1 ( G2 ( ... ( Gn ; and finally, given that the 'focal condition' H which figures in Q refers to what Sommerhoff calls "the continued existence of the system (as an ultimate focal condition)", which I have rendered as

the ultimate focal condition of all organisms the preservation-and-growth of the organism (Pr) thruout some finite time

interval (Hilgartner & Randolph,1969a, p. 312):

Then, in notation,,

O -1 ( LS ( [fi ( (I-1 (m -1 ( Pj0 ( ( -1 (Pr)) ( gi] i .

Proof. A 'logical' proof of (4:24) would closely follow the lines of the proof of (4:23).

From a non

Aristotelian viewpoint, sentence (4:24) qualifies not only as a logical' structure (an empty form composed of (empty) mathematical symbols), but also and perhaps more importantly qualifies as an 'empirical' assertion, a 'map' from which we can perhaps derive testable predictions concerning the (putative) 'territory it allegedly 'represents'. Altho we can examine the 'logic' of an 'empirical assertion', and can perhaps 'prove' that it does (or that it does not) imply

assume our stated premises, we cannot (by means of any manipulation of mathematical symbol) 'prove that an 'empirical assertion' qualifies as 'similar in structure' to its 'territory'. Instead, I can use the assertion as the basis for one or more definite disconfirmable predictions, and can try to find some way(s) to put the prediction(s) to test.

Since I have addressed this booklet to the task of making it easier for prospective students to approach formal theory in general semantics, I shall not here engage in detailed examination of "evidence", but instead shall outline only one possible strategy: I could treat both the 'if' and the only if' senses of the biconditional proposition (4:24) as 'predictions', and could put. them to teat by considering the null

hypothesis, "Each of the 'predictions' derived from theorem (4:24) will turn out devoid of members (equivalent to the empty set)."

The prediction in the 'if' direction states in effect that it we select a putative 'organism' and examine its 'states or activities', then we will find at least two sub

systems which will qualify for classification as 'directively correlated systems', and further, we will find that the relation between those two sub

systems will qualify for classification as 'integrated'.

In a study already performed (Hilgartner, 1970a), I examine the putative class of 'organisms. known as "mature mammalian red blood cells" (as represents in the biochemical literature), and find there at least six different biochemical 'systems', each of which functions in the Embden

Meyerhof pathway of anaerobic glycolysis, and each of which qualifies for classification as a 'directively correlated system'; and furthermore, the relations between certain pairs of these 'directively correlated systems' qualify for classification as 'integrated'. These findings establish the 'if' sense of sentence (4:24) as non

void.

The prediction in the 'only if' direction states that if upon examining some 'system' we discover in it at least two sub

systems which qualify for classification as 'directively correlated systems', and further, if we find that the relations between (certain pairs of) these sub

systems qualify for classification as 'integrated', then this entire 'system' qualifies for classification as an element of the set of 'living systems'.

The only evidence I have so far considered in order to test the prediction in the 'only if' direction (Hilgartner & Randolph, 1969a, footnote p. 309) still remains in part speculative so I shall not discuss it here.

5. INTERRELATIONS BETWEEN CONSTRUCTS

(Depending on our purposes, we can consider a 'mapping' alternatively as a 'function', as a 'binary relation', as a 'set', etc. Above, under the rubric of MULTIORDINAL, I treated the mapping ( (signifying 'abstracting') as a 'set'. Now, by considering this mapping as a 'binary relation and as a 'function' I can bring out other fundamental points concerning the structure of a formal non

Aristotelian system.)

THE MAPPING ( AS A 'BINARY RELATIONS AND AS A 'FUNCTION'

Paraphrasing Premise 2, I can express the set theory operator ( (signifying 'abstracting') as a 'binary relation', viz., Z ( Y , which I read as, "The set Y stands in the relation ( to the set Z ." In words, I can express this relation as, "(Y) comprises the 'territory' represented by the 'map' (Z) ." Then the inverse relation ( -1 , as in Y (-1 Z , designates the relation, "(Z) comprises a 'map' which represents the 'territory' (Y) ."

Elsewhere (Hilgartner, 1970d) I showed that considered as a binary relation., ( does not comprise any kind of 'equivalence relation'; nor, considered as a 'mapping' or 'function' signify, does it qualify as any kind of a 'l

to-1 function'. In notational terms if I signify the notion of 'equivalence relation by [=] and the relation of 'abstracting' by ['('] , then

[[=] = [(] ( [Z ( Y] = [Z=Y]] . (5:1)

But that eventuality lead' to a contradiction:

[Z =Y] ( [Z + Y = (] , (5:2)

as forbidden by the postulate of Non

Identity.

In general, a binary relation R defined on a set A belongs to the class of 'equivalence relations' if and only if it qualifies as reflexive, symmetric, and transitive, which I can show in notation as:

Reflexive: For each a ( A , [a R a] .

Symmetric: For each a, b ( A , [b R a] ( [a R b] .

Transitive: For each a, b, c ( A , [b R a] ( [c R b] ( [c R a] .

But the relation ( does not make muster as either reflexive or symmetric (and probably not as transitive either) :

Not reflexive: [Z ( Z] (5:3)

Not symmetric: [Z ( Y] ( [Y ( Z] . (5:4)

Therefore, considered as a binary relation, ( demonstrably does not qualify as any kind of 'equivalence relation'.

Similar reasoning applies to ( considered as a mapping or function and 'l

1 functions'. By definition, there exists a class of binary relations f ( X ( Y such that, given any element x , there corresponds one and only one element Y for which the couple (x, y) belongs to f :

[(x, y1) ( f and (x, y2) ( f] ( y1 = y2 .

I refer to f as a 'function`, and designate this class of binary relations as 'single

valued' functions. (In elementary mathematics, we customarily indicate a single

valued function by the notation y = f(x) .)

In general, the inverse of a function comprises a binary relation but does not necessarily comprise a single

valued function: the function y = f(x) = x2 , the inverse of this becomes

f-1(y) = +

EMBED Equation.2

.

However, given a function f ( X ( Y such that its inverse f -1 (Y ( X also qualifies as a function, then

[(y, x1) ( f -1 and (y, x2) ( f -1] ( x1 = x2 .

I designate binary relations of this type as 'one-to-one (briefly, 'l

l') functions. Then a function f ( X ( Y which maps from A ( X ( Y onto B ( X ( Y qualifies as 1

1 on A onto B if

[a1 ( A , a2 ( A , a1 ( a2] ( f(a1) ( f(a=2) .

Moreover, I make use of the notion of functions of functions: If y = f(x) (where f ( X (Y) and z = g(y) (where g ( Y ( Z ), then the substitution of the first relation into the second gives

z = g(f(x)). This operation specifies a composition function, which I designate, by g ( f ( X ( Z . Thus g (f = g(f) .

From this it follows that the composition functions of a 1

1function with its inverse become

f -1(f(a)) = a for each a ( A ,

f(f -1(b)) = b for each b ( B .

Or, regarding f and f -1 as sets, these composition

functions of a 1

1 function and its inverse specify identity-mappings:

f -1 ( f = {(a, a)(a ( A} = IX ( X ( X ,

f ( f -1 = {(b, b)(b ( B} = IY ( Y ( Y .

Given these mathematical relations,, the mapping or function ( ( E ( O fails to qualify as a 'l

1 function' (or even as a 'single

valued function'):

( qualifies as '1

l' ( ( -1( ( = {(e, e)| e ( E } ( IE ( E ( E

( ( ( ( -1 = {(o, o)(o ( O] ( IO ( O ( O . (5:5)

Given that condition, then

[(-1 ( ( ( IE] ( [IE(Y) = Y] . (5:6)

But that, condition too leads to a contradiction:

[( -1 ( ( ( IE] ( [Y + ( -1 ( ((Y) = (] ,

[( ( ( -1 ( IO] ( (Z + ( (( -1(Z) = (] , (5:7)

as forbidden by the postulate of Non

Identity.

Therefore, considered as a 'function', ( demonstrably does not qualify as any kind of '1

to-1 function'.

THE RELATION BETWEEN 'ABSTRACTING' (() AND 'SIMILAR IN STRUCTURE' (S)

I have treated the mapping ( ( E ( O as an 'into' relation, at times writing it as

Z = ((Y) . I have consistently regarded this mapping as temporally

ordered, and could make this temporal ordering explicit by writing the sentence as Zg = ((Yf) . I propose now to specify the relation between ( ('abstracting') and S ('similar in structure'), or rather S -1 , the inverse of 'similar in structure'.

Presume a Zg such that Yf S Zg . By 7

2 of 7. NOTATION,

Yf = S(Zg) , (5:8)

and by 6

2,

Zg = S -1(Yf) . (5:9)

Both S and S -1 refer to the relations between a 'map' and the 'territory' it allegedly represents. Therefore, by Premise 3a,

Yf + S(Zg) ( ( , (5:10)

and by Premise 4a,

Zg + S -1(Yf) ( ( . (5:11)

From these relations I conclude that, stated as a mapping, S -1 has as its domain Y and as its range ((Q) ( Z :

S -1 : Y ( ((Q) ( Z , (5:12)

or, treated as a set,

S -1 ( ( ( E ( O . (5:13)

Here I understand S -1 as signifying the relation which holds between 'territory' and 'map' in "that subset of 'abstracting' which "results" in 'maps' which qualify as 'similar in structure' to the 'territory' they represent (as judged by some specified criterion)." {Here I use the colloquial verb

form "results" to subsume in one term the various possible temporal orderings, as outlined in sentences (4:17)a, b, c.} By contrast,

( - S -1 = (

EMBED Equation.2

)-1 (5:14)

signifies the relation which holds between 'territory' and 'map' in "that subset of 'abstracting' which "results" in 'maps' which do not qualify as 'similar in structure' to the 'territory they allegedly represent (as judged by this specified criterion)."

I shall return to the relations specified in sentences (5:13) and (5:14) in order to show the significance of S -1 and (

EMBED Equation.2

)-1 when I take as the 'specified criterion' the condition of "complies with the postulate of Non

Identity".

'SIMILAR IN STRUCTURE' and 'DIRECTIVELY CORRELATED'

As I intimated above, by drawing the relations between the terminology of "traditional" general semantics and Sommerhoff's notion of 'directively correlated in effect I disclosed a "hidden assumption" in "traditional" general semantics, and showed how to eliminate it.

Before I can relate the terms 'similar in structure' and 'directively correlated', I must reconciled the notational conventions they utilize. Above I stated that in our O ( E notation we treat human behavior

experience as a special case of the "apparently 'purposive' activities" of 'living systems'. Hence it seems plausible to assume that we can interpret an elaborate "behavioral sequence "such as that represented in (4:16) as a 'directively correlated' configuration.

Sommerhoff (l950, p. 194) points out that in any 'directively correlated' sequence, we must

distinguish between 'directively correlated' variables, 'coenetic' variables' and 'focal' variables. (Sommerhoff, 1950, p. 194) In terms of the definition of 'directively correlated' (4:19), the expressions ((g(d), f(d)) , and d ( D , and G ( Oc respectively fill these roles.

As I mentioned above, in the definition of 'similar in structure' (e.g., sentences (4:17) a, b, c or (4:18), Tf stands equivalent to the 'coenetic variable(s)' d ( D in (4:19). Moreover, both definitions make use of the notion of 'outcomes' (Oc) , and so implicate 'focal' variables' viz.,

[Gg ( Oc] = [G ( Oc] .

And by a similar kind of reasoning, Mg = Og stands equivalent to o ( O . If now I interpret the set Ah: [Ah ( Oci] in (4:16) as equivalent to the 'directively correlated' variables ((g(d), f(d)) in (4:19), then I can proceed to state and prove the following theorem:

THEOREM. Tf S Mg ( S-1 ( (-1(Gg) ( g . (5:15)

Proof. First, presume an S , a Tf and an Tg such that Tf S Mg .

According to the above reconciling of notations, Tf = df ( D and Mg = og ( O , which means that

[Tf S Mg] = [df S Mg] .

I may regard the relation S ( O ( D also as a mapping, S : O (D , and by 6

2 of 7. NOTATION,

S : O ( D ( S-1 : D( O .

I display a set B defined by

Bh = [(e, o) g | ( df ( D, eg = g(df), og = S-1 (df)} .

Hence B comprises a subset of the domain of ( : E ( O ( O ( E , and from (4:19)

((e, o) h = oci ( Gg ( Oc , or

((Bh) ( Gg ( Oc .

Also

B-h = {(e,o) g(( df ( D , (df, eg) ( g , (df, og) ( S-1}

= {(e, o) g| ( df ( D , (eg, df) ( g-1, (df, og) ( S-1}

= S-1 ( g-1 ( E ( O by 7

1 of 7. NOTATION.

Thus ((S-1 ( g-1) = ((Bh) ( Gg so that, by 7-3 of 7. NOTATION

S-1 ( g-1 ( ( -1(Gg), and

S-1 (g-1( g ( ( -1(Gg) ( g .

Since g : D ( E qualifies as an onto function, then g-1, g ( ID , so that

S-1 (g-1 ( g ( S-1 ( ID = S-1 ,

Hence

S-1 ( ( -1(Gg) ( g ,

which leads to the desired result.

Conversely, presume an S such that S-1 ( ( -1(Gg) ( g . Then

S-1 ( g-1 ( ( -1(Gg) ( g ( g-1 .

This time I use the fact that g, g-1 = IE to obtain first

S-1 ( g-1 ( (-1(Gg)

and then.

((S-1 ( g-1) ( Gg .

Select any d ( D, for instance df . Then (df, g(df)) ( g , (df, S-1(df)) ( S-1 and thus

(g(df), df) ( g-1 , (df, S-1(df)) ( S-1 .

Consequently,

(g(df), S-1(df)) g = oci ( ((S-1( g-1) ( Gg for each df ( D .

That depicts a temporally

ordered process, which I could also represent as

((g(df), S-1(df)) g ( (((S-1 ( g-1) g) h ,

(((S-1 ( g-1) g) h ( Oci ( Gg for each df ( D .

I display a set Ah , where

Ah = (((S-1( g-1 ) g) h ,

which means I could substitute for the latter half of the temporally

ordered

sentence the expression [Ah ( [Oci ( Gg]] .

Sentence (4:l7a), translated into the present notation, asserts that

[Ah ( [Oci ( Gg] ( df S og ,

and that leads to the desired result and so completes the proof.

Slightly paraphrased; theorem (5:15) tells us that

S-1 = f ( (-1(G) ( g , (5:16)

which means that the relation of 'directively correlated, stands equivalent to the inverse of the relation of 'similar in structure'.

'ABSTRACTING' and ''DIRECTIVELY CORRELATED'

By paraphrasing (5:13) and combining it with (5:16), I immediately obtain an expression which relates 'directively correlated' (f) and ''abstracting'' (() , viz.,

f = S-1 ( ( ( E ( O (5:17)

or, in words, the relation of 'directively correlated' comprises a subset of 'abstracting', comprises and moreover, comprises specifically that subset of 'abstracting' which "results" in 'maps' which ((as judged by some specified criterion) qualify (as 'similar in structure' to the 'territory' they refer to.

'MAP'

'TERRITORY' ANALOGY AND MULTI

ORDINAL

In the specified context of Definition 5, the notion of 'organism' occurs in at least three different senses, where these different senses occupy different orders of abstraction. For by Definition 4, since

oj ( di ,

therefore di and oj differ in order of abstraction. In fact, if

di ( o , o = Pj1(o ( e) ,

then

oj ( ((o) , ((o) = Pj1(((o) ( ((e)) . (5:18)

Furthermore, if I treat ((o) as equivalent to ( , so that

((o) = ( , (

Pj1(( ( E) ,

then

( ( ((() , P(() = Pj1(((() ( ((E)) . (5:19)

And thus, demonstrably, in Definition 5 the notion of 'organism' occurs on at least three different orders of abstraction, namely, o , ((o) = ( and ((() .

Substituting into Theorem (4:6), I can express these findings in notation as

O ( MO ( [O : o ( O] + [O : [((o) = (] ( O] ( (

( [O :

( ( O] + [O : [((o)] ( O] ( ( . (5:20)

These considerations suffice to establish that the notion of 'organism', whether represented in words or in notation, qualifies as multi

ordinal.

Furthermore, in the notation of Definition 5, if di (oi (and given that th signifies some 'territory'), then di = ((th) , and so di itself qualifies as a 'map'. Therefore reasoning similar to the above suffices to establish that the notion of 'map' qualifies as multi

ordinal.

Furthermore, if in the expression di

( (th) I designate di as a 'map', then th occupies the role of 'territory'; whereas in the expression oj = ((di) , if I designate oj as a 'map', then di occupies the role of 'territory', Therefore, reasoning similar to the above suffices to establish that the notion of 'territory', whether represented in words or in notation, qualifies as multi

ordinal.

Still further, sentence (4:7) {Corollary 1 to Theorem (4:6)} suffices to demonstrate that the notion of 'abstracting', whether represented in words or in notation, qualifies as multi

ordinal.

Finally, for the terms o. the 'map'

'territory' analogy, I can substitute various roughly synonymous constructs, including the ones I have defined in notation, viz., for the terms 'map', 'territory', 'map

maker' ('organism'), and 'map

making' ('abstracting'), I can substitute respectively the terms 'abstraction', 'abstracted

from, 'abstractor' (organism'), and 'abstracting'; or the terms 'Name', 'Thing 'Named', 'Namer' and 'Naming'; etc.

Therefore, each member of each of these sets of terms qualifies as multi

ordinal.

6. PREMISES AND COUNTER-PREMISES

In a non

Aristotelian system, the end presupposes the beginning and the beginning presupposes the end.

Less tersely stated, a formal (notational) non

Aristotelian system codifies a "World

View" consistent with, but more general than, that of the non

Euclidean geometries and the non

Newtonian physics. The non

Aristotelian premises of Korzybski (1879

1950), which he codified and then found that they turned out as contradictories to the famous "Laws of Thought" attributed to Aristotle (384

322 BC), differ from any previously codified premises. Therefore the structure of a formal non

Aristotelian system will differ drastically from that of any formal system previously written out. On an a priori basis, therefore, we may not uncritically accept the undefined terms the postulates, or the rules of inference of any other formal system. (However, as a first approximation to a formal non

Aristotelian system, I have found it possible to make use of the formulations of 'naive set theory'. But I have proved that my assumptions, as I do so, differ from these of Zermelo & Frankel (1908, 1925) and/or any of their later colleagues or critics.)

Every statement in a non

Aristotelian system refers immediately or eventually to a 'context' which I can describe by means of the terms 'the dealings of a human organism

environment

date, where I use the term 'dealings' in the sense of "interactions" or better (after Dewey & Bentley, 1949), "transactions".

The 'undefined terms' of a non

Aristotelian system., which we can define in terms of each other but cannot specify further in words, comprise structure, order and relation (taken as "noun

forms" and/or "verb

forms"). (Thus I can refer to 'structure' as composed of 'ordered relations' or as 'related orders'; but I decline to try to say in words what I mean by this.) But I can say one more thing: In a non

Aristotelian system, we must consider any statement of promises from at least two points of view, which I might differentiate by calling one 'logical' (in the sense of "concerned with the relations existing between 'terms' or 'Names"') and the other 'empirical' (in the sense of "concerned with the relations existing between 'Name', and 'Thing Named"'). So I point out that in as well as considering the terms 'structure', 'order', and 'relations' in a 'logical' sense, as 'undefined terms' not further specifiable in words, we must also consider each of these terms 'empirically', as designating some 'Thing(s) Named'. And since every statement in a non

Aristotelian system refers immediately or eventually to the 'context' specifies above, I must (after Bourland (1952)) regard the dealings 'Things Named' designated by my 'undefined terms' as examples of what I might colloquially call reaction(s)or response(s)' of an 'organism' to his 'external' and/or 'internal' 'environment. " (Below, under the rubric of 'multiordinal', I shall make explicit the structure of the "shift in point of view" implied by the distinction between 'logical' and 'empirical.)

In my particular (formal) non

Aristotelian system, I shall make use of a term to designate a subset of those 'Things Named' which I might refer to by means of my 'undefined terms': Some specific examples of the 'Things Named' which I might call "'reactions' of an 'organism' to his 'environment"' I shall designate by the term abstraction(s). Furthermore, I shall regard any 'Thing Named' which (from at least one point of view) I might call an 'abstraction' as a manifestation of (product of) the (inferred) ''fundamental life

processes" of the 'organism'. And for the colloquial term "fundamental life

processes" I shall substitute the technical term 'processes of a abstracting'

or simply abstracting'. Then I can define the term 'abstracting' in its moat general sense by using my undefined terms: 'abstracting' signifies "the means by which, given one structure, an 'organism' generates another structure."

Below, in Premise 2, I shall define an operator, signified by ( dealings to stand for the notion of 'abstracting'. Then a fair part of the labor of writing out a formal non

Aristotelian system will go toward specifying the structure of my operator (

or in other words, specifying the structure I attribute to those 'Things Named' ("dynamic processes") I designate by means of the term 'abstracting'.

The non

Aristotelian postulates of Korzybski go by the names of Non

Identity, Non

Allness and .Self

Reflexiveness (Korzybski, 1943; Hilgartner & Randolph, 1969a, pp. 295

7).

In order to express these postulates in notation, I hypostatize a Cartesian product space, which I call (organism

environment) and signify by O ( E . In other words, I presume a structure ('principal set') every element of which comprises an ordered pair (o, e) composed of elements taken from two other structures, one of which I call O (our 'organism') and the other I call E (his 'environment'). (In interpreting the resulting formulations 'empirically', I consider each (o, e) couple as representing an interaction ('transaction') between 'organism' and 'environment'.) Furthermore, I represent this 'principal set' as temporally ordered, using right superscripts to indicate the index set ("time

index"), e.g., Yt, t ( T . (Where I write a term without a time

index, I imply a generalization:

Y = [ ( Yh ( Yi ( Yj ( Yk ( ] .)

Then, allowing yi ( O ( E ( O ( E to signify 'any structure whatsoever', I can state the postulate of Non

Identity as

Non

Identity: No structure exists 'identical with' ("the same in all respects as") any other structure (including 'itself

a different

date').

[y1, y2 ( O ( E ( O ( E, t1 ( t2] ( y1 + y2 ( ( . (Premise 1)

Or. granted that Y comprises some kind of subset of E and Z comprises some kind of subset. of O , and granted a mapping defined by

Q ( Y , ((Q) = {z ( Z( (y ( Q , z = ((y)}, (Premise2)

I can state Non

Identity as

Non

Identity: Y + Z ( ( . (Premise la)

(Mathematicians who start from premises different from mine might want to translate the postulate of Non

Identity into a more "conventional" form, viz., given a set A of 'experiences' and given that x1 ( A and that y2 ( A , then

[x1 = y2] ( [x = y] ( [t1 = t2]

Put that expression implies that

y1 ( yi

which asserts that "y1 qualifies as 'identical' with itself." I grant that any construct, including the notion of 'Non

Identity', taken on 'logical' levels, operates by the "logic of opposites". So the notion of 'Non

Identity', as defined on some specified or clearly understood 'universal set', implies the 'logical existence' of its complement, which I might call the notion of 'Identity'. But in a non

Aristotelian system, the notion of 'Identity' occupies one and only one role: as a designation for fundamental error. In traditional mathematics, a great many proofs end up in a way I could caricature as, "And therefore, as any fool can plainly see, A ( B ." A non

Aristotelian system allows different but equally convincing punch

lines for proofs: "And therefore, as any fool can plainly see, A ( B ''; or else, "Put that implies that

A ( B, as forbidden by the postulate of Non

Identity." And if, in the latter expression, we should substitute "A" for "B , the conclusion still remains forbidden by the postulate of Non

Identity.)

In terms of the mapping defined in Premise 2, I can specify two difference relations, [Y - ( -1 (Z)] and [Z

((Q)] . Then the postulate of Non

Allness expresses the 'extent' of these two difference relations:

Non

Allness: No 'map' can show all the elements of the 'territory' it represents; no 'map' remains free of elements extrinsic to (elements which in no sense "represent") the 'territory' which it represents.

Y - ( -1(Z) ( (, (Premise 3)

Z - ((Q) ((( . (Premise 4)

In Premises 3 and 4, these (sub)sets subtracted from the (super)sets Y or Z (namely, [[( -1(Z)] = Q] or [ ((Q)]) at the very least comprise subsets of Y or Z respectively:

( -1 (Z) ( Y , (2:1)

((Q) ( Z ; (2:2)

or at most they prove equivalent to Y or Z respectively:

[[( -1(Z) ( Y] ( [Y ( ( -1(Z)]] ( Y [(-1(Z)] , (2:3)

[[((Q) ( Z] ( [Z ( ((Q)]] ( Z = [((Q)] . (2:4)

Then the postulate of Non

Allness asserts that these subsets do not turn out equivalent to their respective supersets (do not turn out as improper subsets of their respective supersets). This fact I can express also in the notation of symmetric differences:

Non

Allness: Y + ( -1(Z) ( ( , (Premise 3a)

Z + ((Q) ( ( . (Premise 4a)

This version of the postulate of Non

Allness appears of the same form as the postulate of Non

Identity (viz., derived from it by substitution).

On first encounter, the postulate of Self

Reflexiveness seems the "strangest" of the three. I can state it by reference to the mapping ( defined by Premise 2; but the significance of the formulation will require some explication.

Self

Reflexiveness: No action or utterance of any organism exists free of self

reference.

[((Q)] ( [Z

((Q)] . (Premise 5)

For the sake of initial comprehension, I interpret these postulates as follows: Korzybski explicitly distinguished between 'Name' and 'Thing Named' (or between 'map' and 'territory'), whereas others, e.g., Aristotle, did not so distinguish. Furthermore Korzybski showed that failure to distinguish between 'Name', and 'Thing Named' (in some sense or other) underlies 'fundamental theoretical error'. So (as I intimated above) the postulate of Non

Identity serves as a "guideline for the avoidance of fundamental error".

In the terms of Premise 2, Y stands for some structure which I can regard as a part of the 'environment' of our 'organism', while Z stands for some other structure which I can regard as our 'organism's' 'map. of this 'territory' Y . And then ( stands for that set of "fundamental life

processes "('processes of abstracting') by means of which our 'organism' forms his 'map', Q , then, designates that subset of Y which our 'organism' call in principle detect by means of his (unaided) sensory receptors. Furthermore, I interpret [((Q) ( Z] as our organism's 'Gestalt' of Y (where the term 'Gestalt' refers to a structure composed of a figure of focal interest to our 'organism', bounded by a ground or context more or less empty of interest). Then [Z

((Q)] stands for that subset of Z which bears no direct relation to the 'territory' Y (but which forms an intrinsic part of Z ), namely, for that set of activities by which our 'organism' puts together his 'Gestalt'. In other words, an organism can become focally 'aware' of Y if and only if he maintains a subsidiary 'awareness' of those processes by which he puts together his Gestalt (Polanyi, 1964, pp. 55

65).

Then the postulate of Self

Reflexiveness asserts, "Gestalt if and only if a subsidiary 'awareness' of the "fundamental life

processes" ('abstracting') by which cur organism puts together his Gestalt."

This polarity of focal and subsidiary 'awareness' implies an intra

organismic Cartesian product space, which I designate as the (self

other) space, and symbolize by Sf ( Ot or Se ( Oe , depending on the context. (In other words' any 'abstraction' in part "refers to" or "represents" some 'territory', and in part "refers to" or "represents" the 'organism', the 'abstractor'.) And presuming for the moment that I can represent the intra

organismic relations of "our organism's 'considered picture' of the territory Y " as Z ( Cs , a subset of our organism's 'consciousness' Cs (Hilgartner & Randolph, 1969a, sentence 20, p 307), then the relations which specify this intra

organismic Cartesian product space become

Z ( Cs , Cs = Se ( Oe ,

Cq1(Z) = Se(Z) = [Z

((Q)] ,

Cq2(Z) = 0e(Z) = [((Q)] ,

U(Se(Z)) = Oe(Z) . (2:5)

As originally stated, " the postulate of Self

Reflexiveness refers to the structure of intra

organismic relations" (Hilgartner & Randolph, 1969a, p. 297). But to mention 'intra

organismic relations' presumes first that we distinguish between 'Name' and 'Thing Named' (or between 'map' and 'territory'). Thus considered in an even more general sense, this postulate states the dual requirement that we distinguish between 'Name' and ''Thing Named', and that we distinguish between our organism's 'Gestalt' and the intra

organismic 'processes of abstracting' ("fundamental life

processes") by which he puts together his Gestalt. Putting these requirements into notational form, I paraphrase this postulate as

Self

Reflexiveness: Y + ((Y) ( 0 , (Premise 5a)

[((Q)] + [Z - ((Q)] ( 0 . (Premise 5b)

Or, to make the second sentence slightly more succinct,

Se + U(Se) ( 0 . (Premise 5c)

This form of the postulate of Self

Reflexiveness also appears of the same form as the postulate of Non

Identity (viz., derived from it by substitution).

COUNTER

PREMISES

As I commented above, in a non

Aristotelian system the notion of 'Identity' occupies one and only one role, as a designation for fundamental error: Korzybski showed that failure to distinguish (in some sense or other) between 'Name' and 'Thing Named' underlies 'fundamental theoretical error(s)'; and furthermore, that whereas Korzybski explicitly made this distinction, others, e.g., Aristotle, did not. The "Laws of Thought" attributed to Aristotle rest on this 'failure to distinguish' (but of course, in their traditional forms they do not express it directly). By reference to a 'universal set' ( which contains one proper subset Z , I can express a 'traditional' form of the "Laws of Thought" in LL CONTEXT

STRUCTURE, ORDER AND, RELATION(S)

'MAP'

TERRITORY ANAL0GY

NON

IDEN'I'I'I'Y, NON

ALLNESS, SELF

REFLEXIVENESS

VALIDITY OF THESE PREMISES

ABSTRACTING, ABSTRACTION, ABSTRACTED

FROM

ORDER ON ABSTRACTING

LEVEL OF ABSTRACTION

MULTIORDINAL.

SIMILAR IN STRUCTURE

DIRECTIVELY CORRELATED

INDENTIFYING

SIMPLEST FORMAL DEFINITIONS

ABSTRACTING, ABS'I'RAC'I'ION, ABSTRACTED

FROM

ORDER ON ABSTRACTING

LEVEL OF ABSTRACTION

Figure: SEQUENTIAL ABSTRACTING

SELF-REFERNTIAL.

4. DEFINITIONS OF TE`

Times New Roman

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Microsoft Equation 2.0

DS Equation

Equation.2

set theory notation as:

Identity: Z = Z, or (2:6)

Z + Z ( 0 . (2:6a)

Contradiction: Z (

EMBED Equation.2

= ( . (2:7)

Excluded Middle (x : x ( Z (

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(2:8)

(where ( signifies "for each" or "all")

By analogy with our statement of the non

Aristotelian postulates of Korzybski w ten translated into the notation of the OX E field the "Laws of Thought" of Aristotle become the postulates of Identity, Allness and Linearity:

Identity: (Aristotle did not explicitly distinguish between 'Name' and 'Thing Named'.)

Y + Z (

EMBED Equation.2

M (2:9)

(In words, the symmetric difference between the sets Y ('territory') and Z ('map' ) remains negligible (subliminal, (

EMBED Equation.2

M), so small as to escape detection) the direct contrary of the postulate of Non

Identity. Here the condition '(

EMBED Equation.2

M ('remains negligible') comprises et slightly less stringent condition than ' = ( ' , which would 'remains negligible (non

existent, equivalent to the empty set).)

Allness: (Since Aristotle did not explicitly distinguish between 'Name' and Thing Named', he also failed to take into account the concurrent and related problems of the 'incompleteness' of maps' and of the inclusion in 'maps' of elements extrinsic to the 'territory'.)

Y + ( -1(Z) (

EMBED Equation.2

M , (2:10)

Z + ((Q) (

EMBED Equation.2

M . (2:11)

(In words, the symmetric difference between Y and inverse

( of Z proves negligible (subliminal); the symmetric difference between Z and ((Q) proves negligible (subliminal): the direct contrary of the postulate of Non

Allness.)

Some implications of these formulations illuminate the contrast between Aristotelian and non

Aristotelian premises. For example in terms of our mapping ( ,

( -1 (Z) = Q ( Y (2:12)

According to the postulate of Allness, Y + ( -1(Z) (

EMBED Equation.2

M , from which it follows that

Y + Q (

EMBED Equation.2

M (2:13)

or, stated in words, the symmetric difference between the (sub)set of the 'territory' which our organism can in principle detect. by means of his (unaided) sensory receptors (Q) and the (super)set of the 'territory' itself (Y) remains negligible (subliminal): the assumption of a 'gross empiricist'.

Furthermore, Cq (Z) = 0e (Z) = [((Q)] . But according to the postulate of Allness,

Z + [((Q)] (

EMBED Equation.2

M , from which it follows that

Z + Oe(Z) (

EMBED Equation.2

M . (2:14)

Then, stated in the notation of symmetric differences, the postulate of Allness becomes:

Allness: Y + Q (

EMBED Equation.2

M , (2:15)

Z + Oe(Z ) (

EMBED Equation.2

M . (2:16)

So if the postulate of Allness holds, viz., if the 'territory remains practically devoid of elements undetectable by the (unaided) sensory receptors of our organism and if our organism's 'map' of the 'territory remains practically devoid of elements extrinsic to the 'territory', then in effect our organism's 'map' of the 'territory approximates a "point

point perfect replica of" the 'territory' In set theory, we can render the notion of a "point

point perfect replica of" (or tautology, or y = y relation) by means of the notion of 'identity

mapping(which implies a l

1 function: see 7. NOTATION).

IE = {(y, y) y ( Y ( E}: IE(Yi) = Yi (6:17)

Then, given that ( -1(Z) = Q and that ((Q) = Oe ( Z and that ( -1 = {(y, y) y ( Y ( E} = IE , the relation Z + Oe (

EMBED EquationCHINCAL TERMS IN THE FORM OF THEOREMS

MULTIORDINAL.

SIMILAR IN STRUCTURE.

DIRECTIVELY CORRELATED

INTERGATED RELATION BETWEEN DIRECTIVEY CORRELATED ACTIVITIES

Graph: DIRECTIVELY CORRELATED

LIVING SYSTEM'

5. INTERRELATIONS BETWEEN CONSTRUCTS

THE MAPPING ( AS A BINARY RELATION' AND AS A FUNCTION

THE REALTION BETWEEN 'ABSTRACTING' ((()) AND SIMILAR IN STRUCTURE

SIMILAR IN STRUCTURE AND 'DIRECTIVELY CORRELATED'

'ABSTRACTING AND 'DIRECTIVELY CORRELATED

'MAP'

TERRITORY' ANALOGY AND MULTIORDINAL.

6.

PREMISES AND COUNTER

PREMISES

COUNTER

PREMISES

ARISTOTELIAN "ROSETTA STONE"

C. A. Hilgartner

INTRODUCTION

This booklet addresses those who wish to build their own lives more to their own liking, and who see themselves as willing to try to find ways to make use of formal (notational) theory in general semantics as one means of doing so.

In particular, this booklet addresses those who have participated in, and/or those who intend to take part in, one or more seminars or "self

study groups" oriented around the particular theoretical system outlined here. Since different people "ill differ also in terms of their interest in, or tolerance of, formal theory, I expect that some people will find more pages of this booklet of interest to them than others will.

FIRST PRE

REQUISITE: There must exist a 'formal theory' to make use of, before anyone can make use of it for guidance in the process of living his own life (or for any other purposes).

Everything which follows here depends crucially on the life and work of Alfred Korzybski (1879

1950). Korzybski made his first contribution, at the age of about 41 by elaborating a new vision of 'humanity'; or, more precisely stated, by offering an explicit definition for the species

term 'Man'. Korzybski (1921) points to the observable fact that we humans accumulate 'human knowledge' at exponential (viz., "compound

interest") rates: "We start from where our fathers end...". And in defining 'Man' as "a time

binding class of life", he takes t

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Aristotelian system (Korzybski, 1933), in which he codifies a 'World

View' consistent with, but more general than, the non

Euclidean geometries and the non

Newtonian physics. He calls his system 'general semantics'; and by the end of his life, he had put it in the form of a (non

notational) axiomatic system.

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So the postulate of Linearity comprises the direct contrary of the postulate of Self

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By the use of this set of Counter

Premises, I claim we can account for the structure of any observable human error.

Hilgartner

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C. A. Hilgartner

A NON-ARISTOTELIAN "ROSETTA STONE"

TABLE OF CONTENTS

INTRODUCTION

SEQUENCE OF TOPICS

ALGERBRAIC SET THEORY

KEY CONSTRUCTS FROM " "TRADITIONAL" GENERAL SEMANTICS (from

Elwood Murray)

2. "STRICTLY VERBAL" DEFINITIONS

OVERALL CONTEXT

STRUCTURE, ORDER AND, RELATION(S)

'MAP'

TERRITORY ANAL0GY

NON

IDEN'I'I'I'Y, NON

ALLNESS, SELF

REFLEXIVENESS

VALIDITY OF THESE PREMISES

ABSTRACTING, ABSTRACTION, ABSTRACTED

FROM

ORDER ON ABSTRACTING

LEVEL OF ABSTRACTION

MULTIORDINAL.

SIMILAR IN STRUCTURE

DIRECTIVELY CORRELATED

INDENTIFYING

SIMPLEST FORMAL DEFINITIONS

ABSTRACTING, ABS'I'RAC'I'ION, ABSTRACTED

FROM

ORDER ON ABSTRACTING

LEVEL OF ABSTRACTION

Figure: SEQUENTIAL ABSTRACTING

SELF-REFERNTIAL.

4. DEFINITIONS OF TECHINCAL TERMS IN THE FORM OF THEOREMS

MULTIORDINAL.

SIMILAR IN STRUCTURE.

DIRECTIVELY CORRELATED

INTERGATED RELATION BETWEEN DIRECTIVEY CORRELATED ACTIVITIES

Graph: DIRECTIVELY CORRELATED

LIVING SYSTEM'

5. INTERRELATIONS BETWEEN CONSTRUCTS

THE MAPPING ( AS A BINARY RELATION' AND AS A FUNCTION

THE REALTION BETWEEN 'ABSTRACTING' ((()) AND SIMILAR IN STRUCTURE

SIMILAR IN STRUCTURE AND 'DIRECTIVELY CORRELATED'

'ABSTRACTING AND 'DIRECTIVELY CORRELATED

'MAP'

TERRITORY' ANALOGY AND MULTIORDINAL.

6.

PREMISES AND COUNTER

PREMISES

COUNTER

PREMISES

ARISTOTELIAN "ROSETTA STONE"

C. A. Hilgartner

INTRODUCTION

This booklet addresses those who wish to build their own lives more to their own liking, and who see themselves as willing to try to find ways to make use of formal (notational) theory in general semantics as one means of doing so.

In particular, this booklet addresses those who have participated in, and/or those who intend to take part in, one or more seminars or "self

study groups" oriented around the particular theoretical system outlined here. Since different people "ill differ also in terms of their interest in, or tolerance of, formal theory, I expect that some people will find more pages of this booklet of interest to them than others will.

FIRST PRE

REQUISITE: There must exist a 'formal theory' to make use of, before anyone can make use of it for guidance in the process of living his own life (or for any other purposes).

Everything which follows here depends crucially on the life and work of Alfred Korzybski (1879

1950). Korzybski made his first contribution, at the age of about 41 by elaborating a new vision of 'humanity'; or, more precisely stated, by offering an explicit definition for the species

term 'Man'. Korzybski (1921) points to the observable fact that we humans accumulate 'human knowledge' at exponential (viz., "compound

interest") rates: "We start from where our fathers end...". And in defining 'Man' as "a time

binding class of life", he takes this observable tact of our behavior as the 'defining mark' of the species. Thereafter, In the process or working out the implications of his definition of 'Man', he elaborated the first non

Aristotelian system (Korzybski, 1933), in which he codifies a 'World

View' consistent with, but more general than, the non

Euclidean geometries and the non

Newtonian physics. He calls his system 'general semantics'; and by the end of his life, he had put it in the form of a (non

notational) axiomatic system.

But the .2

M becomes

Z + IE(Y) (

EMBED Equation.2

M , (5:18)

(As noted, this presumes that the mapping ( qualifies as 'l

l'.) In other words, the postulate of Allness implies

assumes the postulate of Identity. Otherwise stated according to the postulate of Allness our organism's 'map' of the 'territory' approximates a "point

point perfect. replica of'' the 'territory'

a conclusion (premise) which no 'evidence' could possibly fail to disconfirm, as judged by any 'criteria' whatsoev

er (except the "criterion" of the presupposition of 'Identity').

I showed above that the postulate of Self

Reflexiveness states the dual requirement that we distinguish 'Name' from 'Thing Named and that we distinguish between our organism's Gestalt and the intra

organistmic processes by which he puts together his Gestalt.

If, however, we admit the postulate of Identity than we transform any account we may generate concerning the structure of the transactions between an 'organism' and his; 'environment', or the transactions of an 'organism' with 'himself' in such a way that a priori we regard

a) our organism's 'Name' or 'map or 'gross perception as a "point

point perfect replica of" the 'Thing Named' or 'territory' or 'thing perceived', and

b) the activities by which our organism puts together his Gestalt as a "point

point perfect replica of" his Gestalt.

Stating these 'requirements' in notation, the postulate of Linearity becomes

Linearity: Y + Z (

EMBED Equation.2

M ( ((Y) + IE(Y) (

EMBED Equation.2

M (6:19)

( U(Se) + IE(Y) (

EMBED Equation.2

M . (6:20)

So the postulate of Linearity comprises the direct contrary of the postulate of Self

Reflexiveness.

By the use of this set of Counter

Premises, I claim we can account for the structure of any observable human error.

Hilgartner

PAGE

PAGE \# "'Page: '#'

PAGE \# "'Page: '#'

~ñâ,$ALGEBRAICANALOGYIDENTITYIDENTIFYINGABSTRACTIONREFERENTIALTECHNICALINTEGRATEDDIRECTIVELYRELATIONnon-euclideannon-euclideanPolanyi, 1964althoughpositKorzybskiAmericanalthoughcorrelatedREFERENTIALalthoughthroughalthoughproposition1proposition1stoodBassettBassettproposition2uni-ordinaluni-ordinalproposition1Hilgartner 1970aDIRECTIVELYalthoughthroughoutRandolph, 1969aAlthough 1 function non-euclideanposith i Premise 22organismic

52

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