A Biology with Biologists In It

C. A. Hilgartner


In this paper, I intend to describe briefly a major flaw that has plagued biology for at least the past half century, and to show how progress in general semantics has begun the process of removing the fundamental logical-and-empirical error that underlies that flaw.


The flaw consists of the by now nearly universal consensus among biologists that they cannot define key terms such as life or living at all, with the unstated corollaries that no one else can now or ever will manage to do so. Back in the 1930’s and 1940’s, when the leading biologists of the day proposed rejecting these key terms, they appeared unaware of the way any theory-based discipline depends upon the ability to draw a boundary around the domain of discourse. Any scientific discipline includes a theory or theories which purport(s) to model the "doings" or "happenings" that take place within this domain, and a boundary which excludes those "doings" or "happenings" which occur within neighboring but incompatible domains. When the biologists rejected their key delimiting terms, they thereby rejected the possibility of drawing such boundaries. That not only eliminates the ability to exclude inappropriate topics from consideration – it also eliminates the possibility of testing, and perhaps disconfirming and rejecting, one’s own suppositions; for without a boundary, you have no delimited domain to reject them from. Thus by entering into this consensus, biologists destroyed –or rather, further delayed – biology as a science.


Starting about 165 years ago, biologists began the process of rejecting teleological explanations (in my opinion, appropriately). They replaced these with mechanistic ones. As a tactic for bringing experimental method to bear within biology, this substitution has turned out wildly successful. But as a method of study within the domain of biology, the program of the mechanistic/reductionist biologists unavoidably fragments biology. It focuses on bits and pieces of organisms (rather than on organisms-as-wholes-in-their-environments); and as with Humpty Dumpty, it cannot put these pieces back together again. After a century of research based on grinding up or otherwise fragmenting living organisms, biologists came to reject not only the teleological way of explaining biological observations, but also some of the key observations that teleology had purported to explain. The language and tactics they adopted from physics actively disallow the topic of apparently-purposive "doings" or "happenings," and so fail to account for important aspects of the domain it purports to account for. In the process, more or less unawarely, they destroyed biology as a theory-based discipline.


Ever adaptable, biologists became sub-specialist chemists and physicists. In their newly devised fields, they encountered so-called "borderline cases" – filterable viruses, blood platelets, and other systems, which function in ways that seemed odd, escaping or violating the older categories and generalizations. In the 1930’s and 1940’s, after a century of remarkable successes in their empirical investigations and perhaps half a century of study of various unsettling "borderline cases", biologists rejected the central defining terms of their discipline: life and living.


Exponents of general semantics can, as I will show below, offer new explanations, with new underpinnings, to replace both older views. Once solidly grounded, biology and biologists may do more than they can today to improve their discipline, and the prospects for the survival of our grandchildren.






(See Figure 0)


Alfred Korzybski (1879-1950), on the page facing the title page of the first edition of Science and Sanity, lists some "Volumes in Preparation," and also some other titles for which, he promises, "The Names of the Authors … To Be Announced Later." On that page alone, Korzybski outlines an ambitious program for his new discipline of general semantics.


I and several associates have, over the past thirty-plus years, generated an alternative frame of reference which synthesizes Korzybski’s non-aristotelian premises, other aspects of general semantics, and contributions from a number of other workers, including the anatomist and cyberneticist Gerd Sommerhoff. In a number of domains, including that of biology, we have considerably advanced Korzybski’s program.


In this frame of reference, I treat theories and theorizing as forms of human behavior –human behaving-and-experiencing – and I treat human behaving-and-experiencing as intrinsically transactional. Thus in this frame of reference, any theory, e.g. a theory of biology, must describe and deal with some kind of interchange which produces fundamental, irreversible alterations. To represent this peculiar requirement in a Western Indo-European (WIE) language such as English or the mathematical theory of sets, I invoke some pair of related terms, e.g. organism and environment, and treat them as a mutually-defining polar term-pair, where the transacting in question occurs as some kind of two-way interchange – "contacting" or "abstracting" – between the members of this term-pair. As Perls, Hefferline & Goodman put it,


We speak of the organism contacting the environment, but it is the contact[ing] that is the first and simplest reality. (Perls, Hefferline & Goodman, 1951, p. 227)


Thus I treat the construct of the environment as designating the other side of the organism’s skin, and treat the organism as designating the other side of the environment’s skin.


Transacting affects both participants, both poles. This kind of setting allows no "unmoved movers."




Within the available space of this paper, I cannot state my presuppositions in notation. But I can give relevant citations. Furthermore, for the frame of reference I use here, I can, and do, verbally summarize the setting, the undefined terms, a cluster of inter-defined terms, and the postulates that underlie it.





"blind creeds which cannot be elucidated further at a given moment…."







Environment (territory)

Abstracting (map-making)

Abstraction (map) and

An ordering on abstracting (or the notion of "logical levels").




Presume that no structuring, ordering or relationing stands as identical with any structuring, ordering or relationing (including itself).

"Our designated observer holds that the map IS NOT the territory it stands for."


Presume that no structuring, ordering or relationing can represent all the aspects of any structuring, ordering or relationing.

"Our designated observer holds that no map includes a representation of ALL aspects of the territory."


Presume that no structure, order or relation exists free of aspects which refer to itself and/or to the organism which elaborates it.

"Our designated observer holds that any map includes a representation of the map-maker."




(see Figure 1)


Let me start by characterizing the current condition of biological theory, as I see it.


My desk dictionary defines the term biology as follows:


biology n. 1. The science of life and life processes, including the study of structure, functioning, growth, origin, evolution and distribution of living organisms.


However, today’s biologists assert that terms such as life or living lack rigor, so that no one can offer satisfactory definitions for them. What difference in "the study of … living organisms" would an exponent of general semantics find, when the practitioners abandon the key terms used to define – set the boundaries of – their discipline?


This brings up a major issue of theory: I do believe that vitalism merited rejection and replacement. But so does mechanism/reductionism. Starting in about 1830, increasing numbers of biologists adopted mechanist/reductionist dogma to replace teleology. In so doing, they declared organisms "nothing but" mechanisms, each in effect "equal to the sum of its parts" and so fully describable from within the terminology and linguistic habits of nineteenth-century physicists and chemists. They thereby threw out the baby with the bath-water: They rejected not only the unsupportable dogma of teleology and vitalism, but also threw out the OBSERVATIONS which earlier biologists had EXPLAINED by means of that dogma. For logically speaking, teleological/vitalist explanations invoke a hierarchical structure, in which, for example, what happens on one "level of organization" makes possible what happens on another, higher "level". As an explanatory principle, of course, such views cannot further account for what they invoke. In contrast, logically speaking, a mechanism or collection of mechanisms remains "flat" – it has no way of generating a hierarchy of "logical levels", nor logical structures which can occupy these various "logical levels" (such as those invoked by teleological/vitalist dogma). In other words, the dogma of mechanist/reductionist biologists disallows multiordinal relations.


My approach to biological theory, which I frame in mathematical languages of known structure (such as the mathematical theory of sets), can and does define these terms, in a manner that makes them consistent with my chosen premises. Furthermore, the theory-based line which it draws between the constructs of living and non-living runs about where most people (including biologists) would want it to run.




(See Figure 2)


In Figures 2 and 3, I spell out my view of how we humans generate new knowledge, such as the findings about filterable viruses and other "borderline cases" that so excited and troubled biologists, and cast such doubt on the generalizations and terminology of biological theory.


In Figures 4, 5 and 6, I outline some of the presuppositions which I find encoded in the grammar of WIE languages, which have gotten in the way of generating logically-and-empirically satisfactory theory in biology.


Readers will, I expect, find that these figures express their points without need for further explanation.



(See Figure 3)



(Violates Non-identity)

(See Figure 4)



(See Figure 5)



(See Figure 6)



(See Figure 7)


By the 1940’s, biologists had concluded that the definitions for terms such as life or living had collapsed. For example, in 1937, the Cambridge University biochemist N. W. Pirie, published a paper entitled "The Meaninglessness of the Terms Life and Living." Eleven years later, the Nobel Prize laureate virologist Wendell Stanley (the first person to crystallize a virus) justified this stance as follows:


With the realization that there is no definite boundary between the living and the non-living, it becomes possible to blend the atomic theory, the germ theory, and the cell theory into a unified philosophy, the essence of which is structure or architecture. The chemical, biological and physical properties of matter, whether atoms, molecules, germs or cells are directly dependent upon the chemical structure of matter, and the results of the work with viruses have permitted the conclusion that this structure is fundamentally the same regardless of its occurrence. (Stanley, 1948; emphasis mine)


Thus, Stanley jubilantly proclaims the ultimate success of the mechanist /reductionist program.




(See Figure 8)


We humans have had half a century to get used to that conclusion, which has become widely accepted.


But please look again at what (as I see it) Stanley asserts:

"With the realization that there is [within the non-verbal territory] no definite [verbal-level construct of a] boundary between [the verbal-level constructs of] the living and the non-living,…


I regard the constructs of the living, the non-living, and the boundary between them as TERMS, belonging to the domain of map – and further, constructs central to how we define the term biology.


Stanley, however, expresses himself as if he expects to find these verbal-level TERMS within the non-verbal TERRITORY that biological science purports to describe. He declares the territory DEFICIENT because he can’t find the map already pre-formed within it.




(See Figure 9)


This stands as a fundamental error – a logical-and-empirical error, which contaminates (poisons) many of the broad conclusions drawn from otherwise useful findings.


Logically speaking, mathematicians (e.g. Frege) criticize this kind of logical maneuver as confusing name with thing named. Logicians (e.g. Turbayne) call this a sort-crossing error. General semanticists (e.g. Korzybski) call this confusing levels of abstraction. I call it map-territory identity. We concur in calling it A MISTAKE.


Scientifically speaking, Stanley holds his own map (mechanist/reductionist view of biology) as "True", and declares the territory DEFICIENT because it "fails" to match his map.


Stanley’s dictum offers an excuse: His conclusion PARDONS himself and his colleagues for failing to provide theory which accounts for what biologists set out to account for.


Further, it contains an implied dogma: Since (in his view) the territory ‘IS’ deficient, he implies that ‘Nobody else will EVER manage to offer logically and empirically satisfactory definitions for terms such as life or living.’




(See Figure 10)


I regard Stanley’s quoted conclusion as INCOMPETENT – logically and scientifically – and I REJECT IT.


I also regard it as HAZARDOUS TO LIFE ON EARTH.


In my opinion, a scientist may NOT allow her/himself to explain any mismatch between theory (map) and observations (territory, or evidence for a territory) by declaring the TERRITORY deficient.


Nevertheless, the conclusion which Stanley expresses got widely accepted. After about ten years, it passed out of the realm of discussion and into that of dogma, becoming part of the non-verbal practice of what now passes for "biology". Most of today’s biologists unquestioningly, even unawarely, regard the topic of definitions for terms like life or living as a dead issue. Today’s biologists cannot draw a theory-based line between the living and the non-living. Hence their "science" has nothing to say about living organisms, about biology.




(See Figure 11)


I observe a particular situation, which includes a North American lawn by daylight, with air containing about 20% oxygen at roughly one atmosphere of pressure, moist soil, grass, a population of earthworms (Lumbricus terrestris), and a male robin (Turdus migratorius). Robins belong to the order of passerine birds, whose primary gait for non-airborne locomotion consists of hopping. However, a hunting robin often moves by placing one foot before the other – he strides or stalks, without much head-bobbing. Here-now, our robin stalks about through the grass; he freezes, turns his head this way and that, and then very suddenly thrusts his beak into the ground (into a worm-hole). After a vigorous display of tugging behavior, the bird straightens up with an earthworm grasped in his beak; he shakes it vigorously, mashes it with his beak and otherwise abuses it, and then swallows it.


Here I regard the robin as fulfilling the role of the organism which I observe, and the other aspects of this situation as making up the robin’s environment. That uses two members of my ring of five terms.


Let me focus particularly on the moment, very shortly before he pecks, when our robin-organism cocks his head as if "listening". (I have heard that robins find their prey mainly by auditory cues.) I take this postural display as the outward and visible sign that the robin has engaged in abstracting so as to generate an abstraction – which I could express in English as the assertion, "I HAVE LOCATED A WORM". That uses two more of my ring of five terms.


By acting on his abstraction – thrusting his (partly-opened) beak into the ground – our robin-organism treats his abstraction as a guess or behavioral hypothesis, and puts it to test. Then when he captures a worm, he must render it unable to crawl back out of his crop and escape, and must swallow it, or else he will derive no biological advantage from his capture. If he had come up with no worm, he would have to discard that behavioral hypothesis, and set about to generate another – which means, resume stalking earthworms.


By using these terms to describe what I observe, I indicate that I attribute to the robin (or to any other living organism) the ability (non-verbally) to abstract (to generate a ‘map’ of "what goes on in and around this organism"), and the ability (non-verbally) to distinguish between (non-verbal) map and (non-verbal) territory. I can state this point in more abstract terms as (i) the ability to generate a map or abstraction, and (ii) the ability to generate an ordering on abstracting. That, then, completes the circle of five terms I started with.


In analyzing any other detail of the performance of our robin-organism transacting with his environment, I would again use all five members of this polar term-cluster. Then in order to present my proposals for biology in a comprehensive fashion, still within the set-theoretic approximation to the non-standard frame of reference, I would have to treat the other topics listed in Figure 11 in a similarly detailed and loving fashion.



STEP-DIAGRAM – Displaying the Logic of Opposites

The step-diagram in Figure 11 gives a graphical representation of these "doings" or "happenings". I discuss it in detail in the Appendix.



(See Figure 12)


In 1950, the British anatomist and cyberneticist Gerd Sommerhoff performed a logical analysis of the traditional biological construct of the apparently-purposive activities of living organisms. He generated a mathematically-defined model of it, which he calls directive correlation. He used it to analyze the major teleological terms, and to provide logically and empirically satisfactory definitions for them.


In 1962, W. Ross Ashby provided a summary of a Bourbaki algebraic set theory notation, and translated Sommerhoff’s construct into that notation. Further, he provided a theorem – an element-free expression – which generalizes Sommerhoff’s definition. In 1965, using Ashby’s notation, I incorporated Sommerhoff’s construct into my developing theoretical system. In order to do so, I had to reinterpret it in light of Korzybski’s premises in general, with special attention to the Postulate of Non-identity.


Let me return to the example of the robin and worm, discussed above – this predator/prey encounter which I observed. To describe this biological occurrence in terms of the modified construct of directively correlated, I must partition my picture of these "doings" or "happenings", and assign the various aspects of this setting to the relevant parts of the construct. (I began that process when I designated robin as organism and worm, etc., as environment.) Now, considering the various aspects separately and/or together in accord with this interpreted model, I must show what they do.


In order to do this, I'll resort to a mathematical vocabulary, but in a way that will allow persons with little mathematical background to follow the argument. In particular, I'll need two main mathematical constructs: variable and function.


a) The term variable refers to a kind of mathematical "blank check," to which you can assign various values.

b) The term function designates a kind of "logical machine" which, when you give it one value, returns you another value, according to some "rule."


Then to spell out the construct of directively correlated, I require at least two variables and three functions.



In this instance, these coenetic variables resemble a Gestalt in that they include background factors (the lawn-worm-robin complex), and foreground factors (for the robin: whatever interoceptive and/or sensory cues that led the robin to start hunting ["hungry", "stalking"], and whatever sensory cues that led him to peck into the ground at that spot ["I detect a worm"]; for the robin’s environment: harder to specify but still in principle specifiable).


In this instance, the specified observer would regard this condition as satisfied if and only if the robin obtains a worm to eat, and eats it.


The construct of focal condition implies-and-assumes a broader construct, Outcomes (Oc), of which FC forms a part (FC Ì Oc , the focal condition forms a subset of outcomes). Let oc signify a particular value of Oc , e.g. "ending up obtaining a worm", or "ending up not-obtaining a worm." Then oc might or might not satisfy this criterion, and so might or might not belong to FC .



As a set ("collection of the effects of various values of CV (e.g. effects of various sensory cues, etc.) on the organism"), f signifies "what the organism does" – which includes the making of maps (abstractions) and the process of guiding itself by these maps, or in other words, the activities which I call abstracting.


Then f(d) mathematically signifies "What the organism does with the disturbance d " – e.g. the organism generates some particular maps of or abstractions about what in particular goes on in and around itself, at the moment(s) in question, e.g. "hungry", "stalking", and "detecting a worm".


As a set ("collection of the effects of these factors on the environment"), g signifies "what the environment does."

Earthworms form a part of our robin’s environment, so "what the environment does" includes "what earthworms do."


Then g(d) signifies "What the environment does with the disturbance d " – here, it includes what this particular earthworm does. Among other things, when an earthworm in its burrow feels something grab it, it extrudes a set of thick bristles built into its hide into the surrounding burrow, so that it sticks there rather like Velcro.


As a set, y signifies "How what the organism does and what the environment does articulate."


Then y (g(d),f(d)) signifies "How what the robin does this time meshes with what the environment (including the earthworm) does this time" (so as to arrive at an outcome).



A directively correlated (apparently purposive) sequence, then, over the interval t0 to t2, involves the following "doings" or "happenings" (as our designated observer sees them):


At t0: d lawn-robin-worm-etc. system ...

At t1: f(d) and g(d) robin detects earthworm,..., pecks...

At t2: Y (g(d),f(d)) = oc Î FC struggle,..., robin eats earthworm


So far, so good – when presented in this verbal fashion and deployed in some detail, the construct of directively correlated gives a convincing enough accounting for this predator/prey encounter between robin and worm. From this point on, those with scant background in "Let’s Keep Track of What We Say" types of mathematics will have to take my word for it to an increasing degree, while I translate this verbal presentation into a set theoretic definition and three theorems.




(See Figure 13)


The construct of directively correlated (as our designated observer sees it), as defined in a Bourbaki algebraic set theory notation, goes as follows:


Given: Spaces O , E , CV Ì O × E , and Oc Ì O × E , together with an onto function g:CV® E and a function y :E × O® Oc ; then, with FC a subset of Oc, we have


DEFINITION: A function f:CV® O qualifies as directively correlated with respect to g , y and FC if and only if

" d Î CV : y (g(d),f(d)) Î FC. Sentence (1)


The graph drawn in Figure 13 may help in visualizing this construct.




(See Figure 14)


The main point of describing a situation of interest in a mathematical language lies in the support for further developments which the mathematical language provides.


As presented, Sentence (1) refers to specific situations (as our designated observer sees them), such as a robin-and-worm – predator-and-prey – encounter. As displayed, it shows both sets (e.g. CV , f , g , Y , etc.) and some of the elements of these sets (e.g. d , f(d) , g(d) , the ordered pair (g(d),f(d)) , etc.).


The tools of set theory make it possible to rearrange a definition like Sentence 1 to bring any one of its constituent terms under scrutiny (in the process, concealing the elements). Here, I scrutinize the term f , "what the organism does (with d )". This entails generalizing Sentence 1 so as to describe not just the structure of specific predator/prey encounters, but rather, to specify the part played by the chosen key component of the construct of directively correlated in relation with the other components.



THEOREM: In the notation of sentence (1), f qualifies as directively correlated with respect to g , y , and FC if and only if


f Ì y -1(FC) o g . Sentence (2)


PROOF. A proof appears in the Appendix, and also in Hilgartner & Randolph (1969a), p. 336.




Biologically Integrated Relations between

Directively Correlated Activities

(See Figure 15)


Sommerhoff defines the construct of integrated in the biological sense as follows:


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 (as viewed by our designated observer) qualifies as directively correlated in this notation. I define P Ì O × E so that (1) and (2) hold, while Q Ì O × E involves a space J Ì O × E , an onto function n:J® E , a function m:J® O , and a function g :E × O® Z , with H Ì Z . Then m qualifies as directively correlated with respect to n , H and g if and only if


" j Î J : g (n(j),m(j)) Î H . Sentence (3)


Then by reasoning like that displayed for sentence (2),


m Ì g -1(H) o n . Sentence (4)


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 , or in other words, FC = J .


Then, given a mapping, the projection onto the O axis PjO: E ´ O ® O , then the relation between P and Q qualifies as ‘integrated’ (a ‘directively correlated’ relation between two ‘directively correlated systems’) if and only if


f Ì y -1(m-1 o PjO o g -1(H)) o g . Sentence (5)


PROOF: A proof appears in the Appendix; and first got published in A Non-aristotelian "Rosetta Stone" (Hilgartner, 1971).




Living System

(See Figure 16)


Sommerhoff defines the term living organism (for which I substitute living system, (LS)) in terms of integrated relations between directively correlated systems.


A living organism may be described as a compact physical system of mechanically connected parts whose states and activities are related 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), pp. 195-6; cf. also 161 ff.)


Here, besides the multiordinal usages of directively correlated (as our designated observer sees it), Sommerhoff also specifies multiordinal usages of focal condition (and thus, by polar relations, multiordinal usages of coenetic variable as well). In my non-standard frame of reference, I take Sommerhoff’s definition as spelling out the conditions someone must satisfy in order to classify something as living.


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 (5a) to define the set of living systems, (LS). I make this explicit in Sentence 6 below.


GIVEN: a designated observer who regards the following situation as given:

A system which I shall call our (supposed) 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 (1) and (2) hold, and where FC1 È FC2 È FC3 ... È FCn Ì Z ; and given that our supposed organism contains another sub-system Q for which sentences (3) and (4) hold and for which J = FC1 × FC2 × FC3 × ... × FCn ; 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) throughout some finite time-interval (Hilgartner & Randolph, 1969a, p. 312):

Then, in notation.,


O Î LS Û [fi Ì y -1(m-1 o PjO o g -1(Pr)) o gi]i Sentence (6)


Proof. A logical proof of (6) would closely follow the lines of the proof of (5).


That completes the description of the mathematical construct of directively correlated.




(See Figure 17A)


As a test case, I propose to use my frame of reference to reexamine enzymic reactions.


A bit of background: Biologists who study enzymic reactions traditionally start by grinding up living things. In 1897, for the first such reaction ever performed in a laboratory, the Büchner brothers mixed a dollop of yeast with some sand, and ground up the mixture with a mortar and pestle. When, under the microscope, they no longer could see any intact yeast cells, they filtered out the sand, collecting the liquid. Then they put a sample of their liquid extract into some kind of flask or test-tube, poured in some sugar solution, and swirled the mixture. While they watched, bubbles (of carbon dioxide gas) formed. They inferred that in vitro (in their glassware), and in the absence of intact yeast cells, the extract had fermented the sugar – in violation of then-current dogma, to the effect that fermentation can take place only by the agency of the vitalists’ life-force, as embodied in intact cells.

Later workers both greatly improved the methods of obtaining and purifying enzymes (from formerly living cells); and increased their understanding of the chemistry of enzymic reactions (as these occur in vitro).

These workers also made key inferences, to the effect that (a) enzymic reactions also occur in vivo, inside intact cells (which remains an inference, no matter how much supporting evidence we may adduce). They further inferred that (b) these reactions in vivo differ in NO particular from the enzymic reactions as they occur in vitro – a dictum which expresses the mechanist/reductionist dogma that has supplanted vitalist dogma. And as a further inference, these workers (c) arranged the numerous enzymic reactions they discerned within cell extracts into (plausible) sequential metabolic pathways, which allegedly operate in vivo - without mentioning the teleological, or at least apparently-‘purposive’, construct of integrated (in the biological sense). Thus they disallow regarding enzymic reactions in an intact cell or intact organism as evidence for an apparently-‘purposive’ configuration – one which, in a hierarchically-ordered fashion, contributes to the survival of a cell and, perhaps, in turn, of an entire organism.


I reject mechanist/reductionist dogma. Taking the role of designated observer, I propose to treat the enzymic reactions which (we infer) occur in vivo, within intact organisms, as apparently ‘purposive’, and therefore as the kind of "doings" or "happenings" that I can represent by means of the construct of directively correlated. In order to do that, I must follow the procedure outlined in Figure 12: Using specifiable criteria in each step, partition my non-verbal picture of what happened, and assign the resulting "pieces of what happened" to the relevant parts of the construct, etc. In general, I must


(i) assign aspects of "what happened" to key terms of the model, e.g. the polar terms organism and environment,

(ii) specify the coenetic variables CV ("initial conditions which affect both organism and environment") and the focal conditions ("outcomes which appear favorable from the point of view of the organism"),

(iii) run this "interpreted model" and

(iv) compare what I obtain with my initial non-verbal picture of what happened.


As an example, consider one step in the Embden-Meyerhof pathway of anaerobic glycolysis – the one catalyzed by the enzyme phosphohexose isomerase, which reversibly inter-converts between glucose-6-phosphate and fructose-6-phosphate. This reaction changes only the three-dimensional configuration ("shape") of the substrate molecule.

(See Figure 17B)


In vivo, e.g. within a cell, any aqueous enzymic reaction presumably takes place within a small volume of water, which contains molecules of the relevant enzyme E , the substrate ("starting material") S , various inorganic ions I+/I-, various non-ionic substances N , etc.


(I) On this "logical level," I assign the enzyme to the role of organism; and relegate the other parts of the system to the role of environment.


(II) The coenetic variables here include suitable concentrations of the enzyme, [E] , of the substrate [S] , of small-molecular-weight co-factors [CoF] (if the reaction requires any), and appropriate concentrations of the various ionic and non-ionic substances [I+/I-] and [N] ; and suitable values for physical-chemical variables such as temperature (T), acidity (pH) , ionic strength (m ), etc.


(III) The focal conditions include at least two items:


As described in the literature of biochemistry, this reaction both can and does achieve these focal conditions. Therefore we may classify it as directively correlated.


At this level of analysis, if one enzymic reaction qualifies as directively correlated, so will any other which we may examine.


This drastic revision of enzyme chemistry brings that biological field of study within the domain of directively correlated in general, and of the three theorems presented above, in particular.


That leads, with full mathematical rigor, to various multiordinal instances of a key conclusion:

This multiordinal conclusion suffices to show that my alternative frame of reference provides acceptable definitions for terms such as living, on both logical and empirical levels, and so restores to biologists the ability to delimit their own field of inquiry.


Kuhn (1962) offers a vocabulary, a rhetoric and models for discussing scientific material which deviates from earlier formulations. Further study of my specific proposals and general approach may leave this work looking like an early step in a scientific revolution. This study may even become the basis for a general paradigm shift.







(See Figure 18)


Biologists, for the most part innocent of general semantics, have fallen prey to one or more of the fundamental errors outlined in Figure 18, each of which (as seen from within my frame of reference) stems from a reliance on the Postulate of Tacit Identity encoded in the grammar of WIE languages and Aristotelian frames of reference.


Competent exponents of general semantics should find it possible to rely on the

non-aristotelian premises of Korzybski and so avoid falling for these fundamental errors.




(Figure 19)


In this figure, I set forth the details of the claim I make for the theory of biology which I have proposed.


To go on from there and express what I see as the needed next step in the development of a general theory of biology, let me sketch out an analogy.

In the General Theory of Relativity, Einstein predicts that, in the vicinity of a massive body like the sun, space-time becomes more curved. Consequently, a ray of light from a star which passes near such a body will get "bent" enough to make the star appear displaced by some 1.75 seconds of arc. Newtonian theory predicts that under gravity, light will get "bent" only half that much.

In 1919, the British sent expeditions to Sobral and Principe, in Africa, to photograph a total eclipse of the sun so astronomers could see how much the stars nearest to the eclipsed disk of the sun seemed displaced. Their figures (1.98" + 0.12" and 1.61" + 0.30") matched the relativistic prediction and failed to match the Newtonian one. Subsequent investigators have made and published other, similar measurements, all of which have fallen within the range of Einstein’s prediction.


In biology, we need similar accomplishments: At least one testable prediction (publishable and published), and one or more experimental tests of that prediction (publishable and published). Perhaps the biological theory proposed here may yet fill the bill, delivering specific (rather than generalized) hypotheses, or predictions which someone can and will test (rather than "postdictions" tested against already-published findings). If that proves so, and if by applying scientific method we can shoot holes in my theory of biology, by all means let us proceed to DO so. And if it should turn out that (for a while) we cannot shoot down such a theory, we need to know that too.


Anyone who can avoid the fundamental errors outlined in the previous figure can play. Since the tools of general semantics facilitate disclosing precisely these errors, competent exponents of general semantics may find they have an advantage in this epistemological-biological domain. Moreover, no one person has a monopoly on inventiveness. To generate an insight, you don’t need first to earn an academic degree. In principle, every human has SOME special knowledge, some particular observations, which can serve as a springboard and loft us into action in this domain.


Throughout my entire lifetime – in my personal as well as my professional life – I have functioned as what I sometimes call "an obligatory collaborator." I don’t want to, and probably can’t, "go it alone." I request that anyone who has ideas, or even a glimmer, in this needed direction, please seek me out and discuss them with me. Who knows – we may even find ways to make something in this arena actually work.


I request help – I request the participation of my fellow humans.





In this section, I hope to clear up some of the problems I have heard expressed about my frame of reference, and also to consider some of the advantages of this method of doing biology.


I. Each of us learned practical grammar, sentence structure, etc. – and that means, a set of logical-and-linguistic presuppositions by which to segment our experiencing so we can fit it into the template of our sentences – in the very early phases of distinguishing self from other. The grammar which we created-and-learned, in turn, guided and continues to guide how we language our experiencing and pattern our "doings" or "choosings".


To the extent (a nearly global extent, at that) that we Westerners form as well as formulate our experiencing in WIE terms (subject/predicate, noun-phrase/verb-phrase, static-and-unchanging/ephemeral), we project the linguistic structure of our native tongues, imposing upon that territory composed of "what goes on in and around us" a map not similar in structure to that territory. And we do so without examining or even noticing the mismatch. Furthermore – and I consider this a crucial point – at the level of grammar, we systematically fail to distinguish between the verbal and the non-verbal, e.g. between "name" and "thing named", or between ‘map’ and ‘territory’, etc. How could we do otherwise? Our grammar has no built-in logico-linguistic tools that require us, every time we speak or write, to make such distinctions. Thus (as I point out in Figure 4) we tend unawarely to project our linguistic conventions onto the cosmos: we often regard what we say (or write) within such a framework as indicating "the way things REALLY ARE, independent of any observer."


II. A number of people have reported that they find it difficult to understand how any non-human organism or part of an organism can "abstract." This becomes less difficult when you remember that I regard abstracting or map-making as a process fundamental to living organisms, and that it does not require human languaging.


I build on Korzybski’s insights on the topics of assuming and abstracting. (Hilgartner, 1963) So far as I know, in his writings he did not state the whole pattern in any one place. But at least tacitly, he holds that any organism – from ameba to human – any organism operating as-a-whole-in-its-environment-at-a-date arranges to get what it needs, and to avoid getting clobbered, by the on-going process of making some kind of ‘maps’ of the changing ‘territory’ composed of "what goes on in and around this organism," and then using these maps to guide its subsequent "doings" or "choosings" (which ultimately means, its subsequent abstracting or map-making).


For example, when I reach for a glass of water on the table, I do not form an allegedly ‘mental’ "sentence" in the form of "I must use such-and-such a group of muscles to raise my arm twenty cm, those muscles plus some others to move my hand-and-arm forward precisely thirty cm, open my fingers using a different group of muscles, close them around the glass with a precise amount of tension, ... etc." As infants, we learned the whole complex of movements which we adults call "reaching" and "grasping" as non-verbal "doings" in response to non-verbal maps we made of ourselves-in-our-territory (which frequently includes things we want to pick up and hold). In similar fashion, other, less "complex" systems, such as cells, non-verbally map "themselves and their environments" in order to select and take in nutrients, excrete waste, etc. Or in other words, they too abstract.


In point of fact, when Randolph and I stated the non-aristotelian premises of Korzybski, and the contrasting Aristotelian premises, in a set theory notation (Hilgartner & Randolph, 1969a, pp. 295-7; -----, 1969b, pp. 353-6), in effect we performed a set theory analysis of the construct of abstracting, as it occurs in context. Later, I disclosed the necessity of specifying the setting for this frame of reference, and the necessity of postulating a designated observer who has an ongoing task: to describe what she observes within an ordering on abstracting, by writing down (perhaps in notation) markings which represent in detail what she observes. (Please remember to refer to the observer as "she".)


In the text and figures of this paper, I use the results of that extended set theory analysis; but nowhere else in the paper have I explicitly described these results, not even in diagrammatic form. I shall do that now.





To express the "internal structure" of any abstraction (or of any process of abstracting) in a sentence:


Our designated observer holds that the process of abstracting generates dynamic abstractions with an "internal structure" made up of least two poles, which she could suggest with paired terms such as "I and it" or "I and thou", etc. In particular (as I said above, ms p. [[21]]) in the domain of biology, abstracting entails "organism and environment, as viewed by a designated observer who operates in the fashion I have described above.


When using the construct of abstracting in an unfamiliar context:


Anyone who deals with a biological problem in such a way as NOT to invoke this whole-structure therefore demonstrably does not use THIS theory.


To say that something lives implies that this something abstracts.


III. In spite of the spectacular gains which experimental biologists have made, many biologists have decided that most problems of real interest in biology "are" too complex to handle. To compensate for this difficulty, they seek to start at the "bottom" and work "up" – to examine the smallest, "least complicated" systems (for example, to use the social behavior of ants or of rats as a basis for understanding that of humans). Or, if no such "simple" system exists, they break down a "more complicated" system into its "component parts" – "things" – without much thought for what they may have lost or destroyed in the process. Then, after having characterized the "mechanisms" of some component, they attempt to simulate, to reconstruct, the capabilities of the more complex intact systems by "adding" other fragmentary abilities – "mechanisms" – to those displayed by that first component. I assert that they get the impression of unmanageable "complexity" precisely from their procedure. The linguistic/mathematical relation of "plus" does not accurately or adequately model the interconnections within an intact organism-as-a-whole-dealing-with-its-environment-at-a-date, much less an intact ecosystem, or biosphere, etc. Hence their modeling does not work, they get no predictability – but they do not declare their hypotheses, and the theory from which they derived them, disconfirmed. Instead, they declare the success of their mechanist/reductionist viewpoint in dealing with a field of study which appears "too complex" to handle in any other way.


One rub arises in that, throughout three to four centuries of modern Western science, the findings of our scientists, on every level from the sub-atomic to the sidereal, have given evidence only for "processes", and none at all to indicate the empirical "existence" of anything like a static-and-unchanging "thing".


Throughout the history of biology up to the present, workers have sharply criticized the views of rival "schools" of biological opinion. But they have shown little inclination to spend their efforts critically scrutinizing the assumptions underlying their own theories, the terminology they themselves employ, etc. Instead, they show a reverential attitude toward their own traditional presuppositions, categories, and approaches. In this respect, their symbolic performance resembles that of the ancient Greek philosophers, who pose a new question, a new KIND of question, addressed to "dumb nature": "What ‘is’ the arche (a r c i )– the first principle or cause of all things?" Then in answering this question, they relied on the available categories and framed their reply in terms of one or another of the four elements: Earth, Water, Air, or Fire. (Korzybski characterizes this kind of answer as elementalistic.)


I suggest that biologists might make the theories they generate, and work from, more nearly similar in structure to the non-verbal "doings" or "happenings" they claim to want to study – in other words, they might increase their ability to generate predictions that would survive testing – by basing their theorizing on premises which start out with a whole-picture of the (suitably delimited) domain of discourse, and then yield a template in terms of which to understand subsidiary details. In this light, the theoretical framework I rely on in this paper functions more in a "top-down" than in a "bottom-up" fashion. Using it, we can account for the "doings" or "happenings" which make up the most complex organisms we know of, namely, humans. Then in order to simulate the "doings" or "happenings" of less complex, non-human organisms, we progressively constrain the theory so as to remove from consideration those "doings" or "happenings" which humans demonstrably can engage in but (so far as we now know) non-human organisms cannot.


IV. This paper comes from work in progress, and raises fundamental, self-reflexive questions: How does this way of doing biology alter biological science? In what ways does it fit into, enhance, and make more useful the current methods and developments in biology, and in what ways does it reorganize the study? Does it require throwing out contemporary work? What will happen to traditional biology? Etc.


At this juncture, I believe that the answers to such questions will become apparent only with further study. I invite those of my readers who have relevant knowledge, opinions, reactions, etc., to get in touch with me so we can discuss, and possibly implement, them.





Over sixty years ago, Korzybski (1933) called for a fundamental revision of the structure of human knowledge. He also proposed that to produce, and then live from, such a revised framework would require us to assimilate, and operate from, fundamentally revised premises, such as those which he specified (Korzybski, 1941). Although many scientific disciplines have revised their assumptions to include some of Korzybski's suggestions, they seem to have done so virtually by accident. In the field of biology, however, nothing of the sort has occurred. In fact, biology stands among the most regressive, fragmented, and theoretically unsupported of all the sciences.


In this paper, I have



In summary, I have proposed a principled way of revising biological theory, using general-semantics principles to further the work begun by Korzybski, Sommerhoff, and others. I suggest that the basic structure of biology must include the requirement that it define and delimit its own field – the study of living organisms, as manifested in their apparently purposive behavior – and rigorously alter its methods to make them as nearly similar in structure to the objects of study as possible. If further scrutiny of the theory proposed here does not disconfirm my claims, then this work may indeed come to look like a step towards a theory of biology capable of conferring enough predictability to enable us to live within the biosphere in a sustainable fashion.





American Heritage Dictionary, Second College Edition (1982). American Heritage Publishing Co. Inc., & Boston/New York/etc. : Houghton Mifflin Company.


Ashby, W. Ross (1962). "The Set Theory of Mechanism and Homeostasis." Technical Report No. 7, Electrical Engineering Research Laboratory, University of Illinois, Urbana, IL 61803 (cf. Hilgartner & Randolph (1969a), p. 336ff).


Bourland Jr., D. David (1965/1966). "A Linguistic Note: Writing in E-Prime." General Semantics Bulletin Nos. 32 & 33. Reprinted in D. David Bourland, Jr. & Paul Dennithorne Johnston, To Be or Not: An E-Prime Anthology. San Francisco: International Society for General Semantics (1991).


Dewey, John & Arthur F. Bentley (1949). Knowing and the Known. Boston: Beacon Press.


Hilbert, David (1898-9). Grundlage der Geometrie. Authorized English translation, E. J. Townsend, translator. LaSalle IL: Open Court, 1902. Reprinted in 1959. Second edition, 1971.


Hilgartner, C. Andrew (1963). "General Semantics, Psychotherapy, and the Logic of Science." Unpublished ms, 1963; revised 1967. Abbreviated version, ETC.: A Review of General Semantics 25:315-324 (1968).


Hilgartner, C. Andrew (1970). "Metabolic 'Control' in Mature Erythrocytes." (Submitted for publication.)


Hilgartner, C. Andrew (1971). "A Non-aristotelian ‘Rosetta Stone’." Privately printed for the Institute for Contemporary Education, Inc., 1510-16 East 87th Street, Chicago, Illinois 60619. pp. 49-52 & 52-3.


Hilgartner, C. Andrew (1972). "The Notions of 'Living System', 'Abstracting', and the 'Map'-'Territory' Analogy." (Submitted for publication.)


Hilgartner, C. Andrew (1978). "The Method in the Madness of Western Man." Communication 3:143-242.


Hilgartner, C. Andrew & Joseph Di Rienzi (1995). "A Non-aristotelian View of Quantum Theory." In press, Physics Essays 8(4):xx-yy (December 1995)).


Hilgartner, C. Andrew, Ronald Harrington & Martha Bartter (1989). "Anomalies Generated by Contemporary Physics." Bulletin of Science, Technology, and Society 9, 129-43.


Hilgartner, C. Andrew, Ronald Harrington & Martha Bartter (1991). "The Conventions for Symbolizing" ETC.: A Review of General Semantics 18(2), 2-19.


Hilgartner, C. Andrew & John F. Randolph (1969a). "Psycho-Logics: An Axiomatic System Describing Human Behavior. A. A Logical Calculus of Behavior." Journal of Theoretical Biology 23, 285-338.


Hilgartner, C. Andrew & John F. Randolph (1969b). "The Structure of ‘Unimpaired’ Human Behavior." Journal of Theoretical Biology 23, 347-374.


Korzybski, Alfred (1921). Manhood of Humanity: The Science and Art of Human Engineering. E. P. Dutton, 1921. 2nd Ed., (1950), (M. Kendig, ed.), Institute of General Semantics, Lakeville CT.


Korzybski, Alfred (1933). Science and Sanity: An Introduction to Non-aristotelian Systems and General Semantics. International Non-aristotelian Library Publishing Co., Chicago. 4th edition, Institute of General Semantics, Lakeville CT. 1958.


Korzybski, Alfred (1941). "General Semantics, Psychiatry, Psychotherapy, and Prevention." in Collected Writings, 1920 - 1950, pp. 297-308. (M. Kendig, ed.), Institute of General Semantics, Englewood, N. J. 1990.


Kuhn, Thomas S. (1962). The Structure of Scientific Revolutions. Chicago: University of Chicago Press. Second edition (enlarged), 1970.


Mayr, Ernst (1982). The Growth of Biological Thought Diversity, Evolution, and Inheritance. Cambridge MA: The Belknap Press of Harvard University Press.


Miller, S. (1953). Science N. Y. 117, 258.


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


Pirie, N. W. "The Meaninglessness of the Terms Life and Living." Perspectives in Biochemistry. Joseph Needham & D. E. Green, eds. London: Cambridge University Press, 1937, pp. 11-22.


Sillén, L. S. (1967). Science N. Y. 156, 1189.


Sommerhoff, Gerd (1950). Analytical Biology. London: Oxford University Press.


Stanley, Wendell M., quoted in White, Handler, Smith & Stetten, see below (1954), pp. 7-8. These authors attribute this quote to Stanley (1948), American Scientist 36: 59-68. However, I do not find this passage in that review article.


Turbayne, Colin (1962). The Myth of Metaphor. New Haven & London: Yale University Press.


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Whyte, Lancelot Law (1969). "On the Frontiers of Science: This Heirarchical Universe" Korzybski Memorial Lecture, 18 April, 1969. Printed in General Semantics Bulletin, No. 36,

7-14, p. 12b.







STEP-DIAGRAM – Displaying the Logic of Opposites (Figure 11)


I use this diagram to present that version of the Logic of Opposites which obtains within a non-aristotelian frame of reference, where by the construct of logic of opposites I express "the relations which hold between a term and its negation or contradictory or opposite or complement." (Elsewhere, I present an account of two other versions of the Logic of Opposites, which, below, I refer to as "the Aristotelian version" and "the refinement introduced by Zermelo." See Hilgartner & DiRienzi (1995), footnote 9.)


I invite you to regard this drawing as depicting the process of abstracting, in the setting of transacting and within the context of that ring of five polar terms.. Every visible and namable detail of this drawing represents some distinction (some non-identity) from within this Non-identity-based theoretical system (frame of reference), as viewed by a designated observer ("I", the implied "speaker" of this paper and the "drawer" of the drawing.


The "steps" signify a hierarchical ordering (a series of adjacent "logical levels", which make up one aspect of an ordering on abstracting). The arrows to the right of the two middle, complete "steps", labeled T1 and T2 , indicate that adjacent positionings in this hierarchical ordering (adjacent "steps") do not qualify as "synchronous" or "simultaneous", but rather "occur" or "exist" in such a way that they show a spatio-temporal ordering – it takes a while to get from one "step" "up", or "down", to the next. (That forms another aspect of an ordering on abstracting.)


Given any pair of adjacent "steps", I can designate the "lower" one as ‘territory’ and the "higher" one as ‘map’.


On the "lower" (‘territory’) "step", the irregular circle labeled H signifies that part of the ‘territory’ which I call our ‘organism’. The irregular circle labeled Y signifies that part of the ‘territory’ I call his (‘external’) ‘environment’. The smaller internal circles signify that part (of ‘organism’ or ‘environment’) in principle detectable by the ‘organism’. The exterior rings signify that portion (of ‘organism’ or ‘environment’) in principle not detectable by the ‘organism’.


The arrow labeled s signifies that relationing I call self-referential abstracting; the one labeled r signifies hetero-referential abstracting. (Taken together, these two constructs cover another member of my ring of five terms.)


The irregular circle drawn on the "upper" ("map") "step", not labeled, signifies the organism’s ‘map’ (abstraction – the fifth member of the ring)) of the adjacent ‘territory’. Here I show it as divided into Self (Sf) and Other (Ot) "components". The inner half-circles signify those portions of the ‘map’ directly obtained by abstracting from the ‘territory’, whereas the outer half-rings signify those portions of the ‘map’, unavoidably present, which do not in any sense derive from the ‘territory’, but rather, which have to do with the ‘organism’ who does the abstracting.


In order to transform this graphical way of representing the construct of abstracting into an explicit version of the Logic of Opposites, I need only re-interpret the irregular circle on the top "step". On the configuration which expresses the structural biological issue of "Self" vs. "Other" or "organism" vs. "environment", superimpose a representation of a Gestalt, with a figure which focally interests our organism, against a (back)ground which (at present) does not.


When I transform the argument into my non-standard notation, my premises (a variant of the non-aristotelian premises of Korzybski) require me to make explicit how (in the opinion of the designated observer) the abstraction arises, and where it occurs. When I approximate that notational argument in English, the fact that I use that ring of five terms this way appears merely arbitrary.


Then with reference to this modified step-diagram, I can approximate the non-identity-based version of the Logic of Opposites in English as:


In the opinion of our designated observer:


1. Our incompletely informed and inaccurately-informed and self-referentially-informed (symbolic) ‘organism’,


2. Consists of spatio-temporally-ordered "doings" or "happenings" which occur within a (delimited) overall setting known as transacting.


3. By his abstracting, our ‘organism’ elaborates a ‘map’ framed as a ‘gestalt’ (abstraction) composed of

a) a ‘figure’ which focally interests the ‘organism’

b) specified against a ‘background’ which does not (at present) interest him.


4. Any ‘gestalt’ further consists of two ‘components’

a) one of which tells about one aspect of the territory, namely, the ‘external’ ‘environment’ and

b) the other tells about another aspect of the territory, namely, the ‘internal’ environment’ – the ‘organism’ which elaborates the ‘gestalt’.


5. In negating a ‘gestalt’, our ‘organism’ interchanges the ‘figure’ and the ‘background’, and alters none of the other considerings listed here.


Allow me to remind you that this frame of reference shows an intrinsic spatio-temporal ordering. Consider the process of negating a Gestalt. First we have the Gestalt Gt at an initial moment t0 , which I could write as G0 . To negate it takes a while; I could write the result as . Re-negating it also takes a while, and yields G2 . And by the Postulate of Non-identity, G0 G2 . Thus, when you negate and then re-negate a Gestalt, that process does not bring you back to your starting-place. Although this way of writing the Gestalten appears to emphasize the changed values of the spatio-temporal index, the "content" of the Gestalten changes also, for, by the Postulate of Self-reflexiveness, each of these Gestalten (maps) includes some kind of representation of the map-maker or of the process by which it came into existence.


If nothing else has already brought the point home, this last finding should make it palpable that this specific delimited version of the logic of opposites differs radically from either the Aristotelian version or the refinement introduced by Zermelo.








DEFINITION: (cf. ms pp. 10-11 (Figure 13)) Given: Spaces O , E , CV Ì O × E , and Oc Ì O × E , together with an onto function g:CV® E and a function y :E × O® Oc ; then, with FC a subset of Oc, we have


DEFINITION: A function f:CV® O qualifies as directively correlated with respect to g , y and FC if and only if


" d Î CV : y (g(d),f(d)) Î FC. Sentence (1)





THEOREM: In the notation of sentence (1), f qualifies as directively correlated with respect to g , y , and FC if and only if


f Ì y -1(FC) o g . Sentence (2)


Proof: First let f qualify as directively correlated. I define a set A by


A = {(e,o)| $ d Î D: e = g(d), o = f(d)} .


That makes A a subset of the domain of y and, from (1), y (A) Ì FC . Also


A = {(e,o)| $ d Î D: (d,e) Î g, (d,o) Î f}

= {(e,o)| $ d Î D: (e,d) Î g-1, (d,o) Î f}

= f o g-1 Ì E × O by Definition 4.1 of Hilgartner & Randolph, 1969a, p. 329.


Thus y (f o g-1) = y (A) Ì FC , so that, from sentence (5.5) of Hilgartner & Randolph, 1969a, p. 334,


f o g-1 Ì y -1(FC) ,




f o g-1 o g Ì y -1(FC) o g .


Since g:D® E comprises an onto function, then g-1 o g É ID , so that


f o g-1 o g É f o ID = f .


From that, (2) follows.


Conversely, assume an f such that (2) holds. Then


f o g-1 Ì y -1(FC) o g o g-1 .


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


f o g-1 Ì y -1(FC) and then y (f o g-1) Ì FC .


Select any d Î D . Then (d,g(d)) Î g, (d,f(d)) Î f and thus


(g(d),d) Î g-1 , (d,f(d)) Î f . Consequently,

(g(d),f(d)) Î f o g-1 , so that

y (g(d),f(d)) Î y (f o g-1) Ì FC


for each d Î D ,which by (1) means that f qualifies as directively correlated with respect to g , y and FC .





THEOREM: I display two systems, P and Q , each of which qualifies as directively correlated in this notation. I define P Ì O × E so that (1) and (2) hold, while Q Ì O × E involves a space J Ì O × E , an onto function n:J® E , a function m:J® O , and a function g :E × O® Oc , with H Ì Oc . Then m qualifies as directively correlated with respect to n , H and g if and only if


" j Î J : g (n(j),m(j)) Î H . Sentence (3)


Then by reasoning like that displayed for theorem (2),


m Ì g -1(H) o n . Sentence (4)


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 , or in other words, FC = J.


Then, given a mapping PjO: 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 Ì y -1(m-1 o PjO o g -1(H)) o g . Sentence (5)


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


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


Hence R comprises a subset of the domain of g and, from (3), g Ì H or


R Ì g -1 (H) Ì E × O . Then

PjO Ì PjO o g -1(H) Ì O ,


and since m-1 Ì O × J ,


m-1 o PjO Ì m-1 o PjO o g -1(H) Ì J .


As I show for (2),


f o g-1 Ì y -1(FC) ;


but since J = FC , then


f o g-1 Ì y -1(m-1 o PjO o g -1(H)) ,


from which (5) follows.


Conversely, assume a P and a Q such that if J = FC , then (5) holds. Then


f o g-1 Ì y -1(m-1 o PjO o g -1(H)) o g o g-1 .


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


f o g-1 Ì y -1(m-1 o PjO o g -1(H))


I have already demonstrated that m-1 o PjO o g -1(H) Ì J.


If J = FC , then


y -1(m-1 o PjO o g -1(H)) = y -1(FC) ,


and so f o g-1 Ì y -1(FC) .


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




j Î m-1 o PjO o g -1(H) Û m(j) = o Î PjO o g -1(H) Ì O ,




o Î PjO o g -1(H) Û PjO-1(o) = (e,o) Î g -1(H) Ì E × O ,


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


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


we can see that


R = m o n-1 Ì E × O ,


and thus


m o n-1 Ì g -1(H) .


But as I demonstrate for theorem (2), given an expression of that form, then m qualifies as ‘directively correlated’, with respect to n , H and g .


Thus if (5) holds, then both P and Q quality as directively correlated systems.

But 5 holds if and only if FC = J , where FC 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 (5) holds and J = FC , then the relations between P and Q qualify as integrated. That completes the proof.




THEOREM: Given a system which I shall call our (supposed) 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 (1) and (2) hold, and where FC1 È FC2 È FC3 ... È FCn Ì Oc ; and given that our supposed organism contains another sub-system Q for which sentences (3) and (4) hold and for which J = FC × FC2 × FC3 × ... × FCn ; 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) throughout some finite time-interval (Hilgartner & Randolph, 1969a, p. 312):

Then, in notation.,


O Î LS Û [fi Ì y -1(m-1 o PjO o g -1(Pr)) o gi]i Sentence (6)


Proof. A logical proof of (6) would closely follow the lines of the proof of (5).