The Rosen Modeling Relation
“I have been, and remain, entirely committed to the idea that modeling is the essence of science and the habitat of all epistemology.”
– Robert Rosen, Essays on Life Itself
- Natural and Formal Systems
- Entailment Structures
- Building the Modeling Relation
- References and Footnotes
This commitment to the central role of modeling, as embodied in the quote above, began for Rosen during his early days as a student, and eventually manifested in his development of the Modeling Relation.  To better understand Rosen’s statement above, we need to understand the Modeling Relation.
The Modeling Relation is described primarily in Rosen’s books, Anticipatory Systems and Life Itself. The former, in particular, devotes many chapters to defining, describing, and illuminating the consequences of the Modeling Relation. Therefore, what I describe herein only scratches the surface of the detail and depth to which Rosen examined the Modeling Relation.
In its most general terms, the Modeling Relation is a way to compare synonymies between a system in one form and another system in another form. In other words, to establish a congruence relation between two systems.
Natural and Formal Systems
What is a system? For Rosen, systems are of one of two types: natural, or formal. Roughly speaking, the distinction is that natural systems are selected portions of the external world, while formal systems are based in symbols, syntax and rules of symbol manipulation – a formalism, for short.
In discussing natural systems, we must first make the distinction between our self and the world outside our self. This is the dualism between the inner world and the ambience. [1a] This is not something we can know with certainty, but it is what we experience and take to be the case. What is called the ambience we further consider to be an actual world outside of us, the external world, where objective reality and objective phenomena exist – where material objects exist. [1b]
A natural system is some portion of the external world that is, at root, based upon a collection of our sensory impressions, or percepts . This portion of reality is something we actively choose: there are no a priori guidelines to tell us what aspects of the external world constitute the boundaries of a ‘system’.  Not surprisingly, the remaining portion of reality outside what has been chosen as the system is called the environment. We take these percepts to represent, at least to some extent, qualities of the external world. Indeed, unless we do so, science itself cannot exist. Not only do these percepts present various qualities of the external world to our mind, we also appear to comprehend various relations between percepts – relations that our organizing human mind actively builds, but which also appear to reflect in some way actual relations in the external world.  This brings us to the definition of a natural system:
“Specifically, we shall say that a natural system is a set of qualities, to which definite relations can be imputed. As such, then, a natural system from the outset embodies a mental construct (i.e., a relation established by the mind between percepts) which comprises a hypothesis or model pertaining to the organization of the external world.”
“In what follows, we shall refer to a perceptible quality of a natural system as an observable. We shall call a relation obtaining between two or more observables belonging to a natural system a linkage between them. We take the viewpoint that the study of natural systems is precisely the specification of the observables belonging to such a system, and a characterization of the manner in which they are linked. Indeed, for us observables are the fundamental units of natural systems, just as percepts are the fundamental units of experience.” 
An important point to emphasize is this: a natural system is itself a kind of “hypothesis or model” of the external world. At first glance, this might seem to invoke some kind of circularity in the Modeling Relation, in the sense of having the system under study be itself a model; however, the situation is more subtle than this. In particular, we must distinguish between a natural system and a material system.
The first subtlety is to recall Rosen’s philosophy (from the quote at top) that modeling is the “habitat of all epistemology”. That is, modeling is not merely the province of explicit Modeling Relations, but of all our ways of knowing, including innate or intuitive modes of knowing. As Rosen says: “…indeed the modelling relation is a ubiquitous characteristic of everyday life as well as science.” [5a] Therefore, when we subjectively define a natural system, we are engaging in a mental modeling act. As noted in the quote above, the relations between percepts are mental constructs. Further, it should be noted that even percepts are not to be regarded as some kind of direct knowing of the external world; but rather, percepts are themselves a kind of mental model (“sensory impressions”) of qualities we can only deem as plausible to actually have their source in the external world. [5b] Taken together, percepts plus relations comprise a mental model of the external world.
The second subtlety is that it would be impossible to engage in building a Modeling Relation without first carefully delineating the 1) observables and 2) linkage relations of the natural system. Otherwise we would have no well-defined basis from which to generate other models or to compare models. This act of delineation is itself an act of abstraction: a willful selection of certain finite number of both observables and relations among observables from out of the entirety of possible observables and relations. Rosen laments that contemporary science tends to be so prejudicial in its choices when defining natural systems that the result has been to isolate just those subsets of material systems in the external world that adhere to a mechanistic Newtonian notion of the world. [5c]
The third subtlety relates to realization. Realization is the process of working from a formal system to a natural system. [5d] It is essentially the notion of going from a blueprint to a working material version. When realizing a system, the process does not involve every intimate detail of its material structure. Instead, it is sufficient that the resultant material version embody the criteria from the formal model (the “blueprint”). So, the congruence relation established is only between the elements and relations specified in the formal model and a corresponding certain finite number of observables and relations in the material realization. As noted in the second subtlety, those identified observables and relations are abstractions. Whatever other additional material characteristics the realization might have, to the extent that they do not affect the congruence relation, such additional characteristics are not part of the natural system. Therefore, a natural system is not synonymous with a material system. Instead, a natural system is some subjectively defined subset, or abstraction, of an actual material system.
The definition Rosen gives of a formalism, or formal system, is as follows:
“We shall understand by a formalism any such “sublanguage” of a natural language, defined by syntactic qualities alone. That is, a formalism is a finite list of production rules, together with a generating family of propositions on which they can act, without any specification or consideration of extralinguistic referents. Thus, a formalism, as a fragment of natural language, could be “about” something (i.e., endowed with extralinguistic referents), but it need not be. A formalism, by its very nature, carries with it no “dictionary” associating its propositions with anything outside itself. It is propelled entirely by its own internal inferential structure, as embodied explicitly in its production rules. These and these alone determine the relations among the propositions of the formalism, which we have called inferential entailment.” 
The most typical type of formal systems encountered – particularly in modeling in science – are mathematical in nature. However, mathematical systems are only one class of formalisms. Other types of formalisms, meeting the conditions above, can and do exist.
The last sentence in the quote above refers to our next consideration: entailment structures. Entailment structures are certain types of relations between elements of a system. In the case of formal systems, entailment relations come in the form of implication or inference. That is, to say that “P entails Q” in a formal system, is to say that “P implies Q”, or that “Q is inferred from P”, via the production rules of the formalism. 
In a natural system, entailment takes the form of causality.  We understand intuitively, and experientially, that there does seem to exist entailment relations between phenomena in the external world. As such, we take causal entailment to be a real property of the external world, and hence, an entailment structure that is embodied in natural systems.
In both types of systems, entailment structures play a fundamental role in the organizational structure of the systems. Without inference rules, a formal system could generate no consistent set of propositions; without causal relations, the external world would have no discernable order at all. Therefore, any attempt at modeling must, of necessity, incorporate modeling of the entailment structure of the system under study.
Building the Modeling Relation
As stated at the outset, the idea of the Modeling Relation is to establish congruence between two systems; specifically, between the elements of each system and between the entailment structures of each system. By accomplishing both of these aspects, the orderly nature of one system can be be made to correspond to another system, to the extent that the two systems have a degree of correspondence.
In science, for example, the typical scenario involves creating formal models of aspects of the external world. Such formal models are often mathematical in nature, and the validity of these models is based upon the congruence of the system elements and the entailment structures, and also involves the processes of measurement and prediction. Rosen gives a generalized description of the way in which we relate a formal system (here called “F”) to a natural system (here called “N”):
“The essential step in establishing the relations we seek, and indeed the key to all that follows, lies in an exploitation of synonymy. We are going to force the name of a percept to be also the name of a formal entity; we are going to force the name of a linkage between percepts to also be the name of a relation between mathematical entities; and most particularly, we are going to force the various temporal relations characteristic of causality in the natural world to be synonymous with the inferential structure which allows us to draw conclusions from premises in the mathematical world. We are going to try to do this in a way which is consistent between the two worlds; i.e., in such a way that the synonymies we establish do not lead us into contradictions between the properties of the formal system and those of the natural system we have forced the formal system to name.”
“Another way to characterize what we are trying to do here is the following: we seek to encode natural systems into formal ones in a way which is consistent, in the above sense. Via such an encoding, if we are successful, the inferences or theorems we can elicit within the formal systems become predictions about the natural systems we have encoded into them; consistency then means that these predictions will be verified in the natural world when appropriately decoded into linkage relations in that world. And as we shall see, once such a relation between natural and formal systems has been established, a host of other important relations will follow of themselves; relations which will allow us to speak precisely about analogy, similarity, metaphor, complexity, and a spectrum of similar concepts.” 
What we see is that the two systems are related via the encoding and decoding arrows. Encoding is the process of measurement: it is the assignment of a formal label (such as a number) to a natural phenomenon . Decoding is prediction: it is the taking of what we generate via the inferential machinery of the formal system into representations of expected phenomena . Additionally, the arrows for inference and causality represent the entailment structures of their respective systems .
As Rosen described in the above quote, the modeling relation provides us with a way of ascertaining congruence between the natural system, N, and the formal system, or model, F. What determines successful congruence is that the diagram, as a whole, commutes. That is, such that the numbered arrows meet the condition:
(1) = ( 2) + ( 3) + (4).
This means that our measurements (2), when run through the inferential machinery (3) of our model, will generate predictions (4), which will agree (when verified) with the actual phenomena (1) occurring in N. 
It bears mentioning that any encoding (measurement) from N to F is an abstraction . As such, the features of N that are represented in F will be condensed as a result of the particular encoding schema (and measuring instrumentation). There is nothing new in this; however, it is often overlooked that the process of doing measurement and experimentation involves a willful act of abstraction. 
The Modeling Relation thus provides us with a methodology for studying one system in terms of another system. This framework is quite remarkable and flexible. In the example above, we saw a Modeling Relation between a formal system and a natural system, something quite typical of activities in science; although the modeling process has rarely been so explicitly examined as Rosen has done.
Using the same concepts as in the above example, a Modeling Relation can just as readily be established between two formal systems. Or it can be established between multiple natural systems that encode into one same formal system, where both natural system are then analogs of each other.  Or a natural system might have Modeling Relations with multiple formal models, and one could then consider the prospect of Modeling Relations between those formal models. And so on.
Finally, it might be useful to note that the modeling relation diagram is essentially a category-theoretic diagram.  The systems are essentially categories, the entailment structures are essentially morphisms, and the encoding and decoding arrows are essentially covariant and contravariant functors, respectively.
Now, certainly it will not be possible to force just any system to be congruent with another system in a Modeling Relation. It requires that we find systems that, together, will satisfy the conditions above, given the appropriate encoding and decoding “dictionaries” to translate back and forth between the two systems, consistently. Since these dictionaries do not exist prior to attempting to establish the Modeling Relation, nor is there any rote method for constructing such dictionaries, they must be created. In this way, modeling is as much an art, as it is a methodology. 
As a result, the Modeling Relation contains semantic elements  that cannot be replaced with syntactic elements alone. In other words, the Modeling Relation is itself complex.  This aspect of the Modeling Relation is discussed further in “The Modeling Relation as a Complex System“.
References & Footnotes
EL: Rosen, R. 1998. Essays on Life Itself. Columbia University Press
LI: Rosen, R. 1991. Life Itself. Columbia University Press
AS: Rosen, R. 1985. Anticipatory Systems. Pergamon Press
FM: Rosen, R. 1978. Fundamentals of Measurement. Elsevier Science
 EL, p. 197
[1a] LI sec. 3C
[1b] EL p. 160
 AS p. 45
 LI p. 41
 AS p. 46
 AS p. 47
[5a] AS p. 86
[5b] AS p. 45
[5c] LI p. 203-204
[5d] AS p. 84
 LI p. 44
 LI p. 46
 LI p. 53
 AS p. 73-74
 LI p. 60-61
 LI p. 60
 LI p. 63
 LI p. 54
 LI p. 54
 EL p. 159
 EL p. 138