Recall the definitions from the 2011 synthesis paper:

My understanding of transitivity is rather simple. If A entails B and B entails C then A entails C transitively. That is the case for all the entailment transitions in either half (left or right) of the diagrams, including the identity diagram (b). A closed loop of causation cannot exist in either domain it has to go through the opposite domain in what I suppose we can call an inverse transitivity. For the relations in the holon, we have to write “If A is related to B and B is related to C than A is related to C”, and that looser form of transitivity is also true as long as we understand that relation is information which means that it is an abstraction for a previous set of conditions to establish new boundary conditions for what can happen on the dynamical side. It therefore does not specify the dynamics, it changes their constraints so the system behaves differently, perhaps with a different set of attractors.

While the realized entailment is a necessity according to the laws of a given domain, relations involve selection and interpretation by possibly different systems because contexts are non-local and non-discrete, and thus always shared to some degree, whereas within the realized domain entailments can be exact because states are discrete and the rules are defined for one domain of realization. More specifically, when we write that f entails f contextually, we have to say that f makes f possible, not that it necessitates f. Sustainability of the identity then becomes a genuine question that can be address, involving other relations.

Hence, f:(A,s) defines an efficient/material entailment in category theory [using the new conventions above]. We can also write f  |– s, meaning “f entails s“.  I’ll use words from now on because it is easier. Between holons we can use the traditional category theory to construct open-ended entailments. But, if we try to implement the Schroedinger/Rosen hypothesis (see Rosen’s book, Essays on Life Itself) that a material result (inertial system) can become an efficient cause (gravitational system) we should be able to write that s entails f. If we then construct these hierarchically, remaining in the realized world of efficient entailments, we have a description of f transitively entailing itself, which is indeed a logical contradiction in the mechanistic notion of the realized world, because that kind of organization cannot exist in the realized domain alone. This is a flaw in the mechanistic world view, or as Rosen calls it, “the Newtonian paradigm”.

But for that reason we must propose a way for a material result to become an efficient cause. In other words, how a ‘thing’ becomes an agent, not just saying that it does paradoxically. This is done in R-theory by considering inverse contextual relations. In plain language, a ‘thing’ placed in a different context, acquires new functions according to the nature of the context. This is directly analogous to saying a function causes a material result according to the nature of the material. Clearly the same force applied to wood produces a different result when applied to steel. In the same way a structure produces a different function in different contexts. For example: a screwdriver taken from the shop to the kitchen can acquire a new function for opening cans.

Note that in Rosen’s work, and Louie’s that followed, this contextual entailment was not proposed. It is an addition I made in R-theory, in fact what distinguishes R-theory from its predecessors. The Rosen complexity school simply worked with the contradiction of a closed loop of efficient entailment IN the realized domain, supposing that it is simply indicative of what can really happen in the natural world except that it was overlooked by mechanistic science. We know that a massive object has both inertial and gravitational properties, and so it is simply a fact that a material result in the efficient map (at the head of an open arrow) can also be the start of a closed headed arrow (efficient cause). But that merely throws a monkey wrench into quantitative science because it can’t be handled in any equation to create a loop from result to cause. Therefore, a context is required to save computation of observed temporal properties while still allowing for complex loops. That was the resolution R-theory came up with and it profoundly implies that no physical event takes place except within a context that determine its universal constraints; and that allows the possibility that those constraints can vary, or be altered by the system. That allows us to model obvious ways they are altered, such as by thought, but also un-obvious ways we have discovered empirically, as in quantum theory, relativity, ecology, sociology …. all fields that are characterized by complexity.

I label the inverse entailment by using dashed instead of solid lines, and because it is inverse the arrow conventions also switch positions. Thus you have s entailing f. Because the transitive relation between f and itself happens through context there is no logical contradiction — the problem of transitivity between a structure and a function has thus been removed because context inverts structure via context to entail function.

There is no difference between identity and interactive holons in this regard. The only difference is that identities define a special system and context that are associated with the origin of a specific holon, whereas the links shown externally (for convention) define a systems interactive properties as a whole. F in diagram (b) above has four facets that entail or relate each other, and thus define the identity in precisely the same way that they act between holons.

Relations are not a necessity but a possibility, as discussed above. For example, fish occurring out of water in the realized domain may place them in an unsuitable context (of existence factors) for their survival.  That contextual relation will then not result in any realizations. However, if there is some pre-adaptation that allows some fish to survive well enough to reproduce and evolve, then there may indeed be realizations of a new kind of amphibian. The relational arrows in this case do not say that a new kind of fish must necessarily emerge from that contextual matching, but that one can emerge. How we analyze that possibility can vary. We might give it probabilities or suitabilities, or we might try to analyze the internal physiology to see if survival is possible. But we do not know of a set of entailments that will necessarily predict the result – that is the whole point of relation vs. entailment. If we did know of such efficient maps that would explain the phenomenon adequately, that could be represented entirely in the realized domain because its very existence means that it shares a common context. The loop through context still theoretically exists, but it makes no difference in that special case. Again, this is not a flaw in the causal architecture, but a description of natural complexity and way of analyzing it.

Holon relations can then have multiple forms depending on how they are combined. There are many ways to combine them — I have not counted the ways. I showed these three in the paper [left], and additionally dealt with the case of shared contexts. I call identity a 1st-order closure and the hierarchical entailment you see at the bottom is a 2nd-order closure (meaning it closes two holons with each other), which is necessarily complex because it then has a dualistic context. The sequential composition in the middle is a 1.5 order holon. It is not fully complex because its context is not a dualism but a temporal sequence. Instead of being dual, the context is bifurcated according to our traditional concept of time. This is a mechanistic sequence where s1 is transitively related to s2 by a time domain. But notice there is no smooth transition between s1 and s2, the sequence of states is described in discrete steps as a result of the causal loop. Similarly, if we were describing the state-bifurcated context of a quantum particle — its various energy states — the sequence would depend on energy transitions between various quantized state potentials.

 I further argue that all natural systems have at least two fundamental contexts; that of their origin and that of their operation (which is the reason I also show the way holons interact with internal vs. external links). These are never commensurable contexts, so reality is complex. Naturally any syntactical interpretation of the holon loop that does not take into account contextual relations as an expression of semantics, must reveal a contradiction; but the contradiction is the result of the method of analysis. Specifically, any reductive method of analysis will reveal a contradiction in the logic, which is really a contradiction of, and in a natural sense disproof of, reductionism. The combination of like-headed arrows (see legend) is therefore quite unlike the forward and inverse entailments where unlike headed arrows are combined. The former is an information encoding and decoding between categories – the real meaning of ‘relation’ – whereas the later is an entailment within a category.