First Retroaction

Everything lately has been rush and weft, writing an impossibility. Nevertheless, I thought I’d drop by what I can muster; a few quick notes to report on the current [and still nascent] research finally underway here at the Institute for Contaminative Method [CM]. There has been much to occupy me following my recent forced relocation, a violent expansion of possible investigations, only exacerbated by the interventions of satellite investigators. Welcome though they are. In general, the tendency has been towards what one such agent has recently named mode délire: the conviction that the farther one extends beyond the bounds of ‘sense’, the more pressing the need for rigour in extending these vectors. The aim cannot only be the production of nonsense. Beyond the embrace of delirium as method, I am interested in something altogether steelier: method as delirium. Perhaps this is something around which all of us here at 0AZ can constellate.

However, method requires material. In this respect, a vital inducement has been the recent discovery of some scattered work signed ‘Yeter Çaba’ in the University of Canberra archives, an event already hinted at by Fi. Although some others at the Institute have remained sceptical of these fragments, me and |end| have been pursuing their consequences with mounting excitement. The Çaba papers, from what we have been able to discern thus far, are concerned with the development of some mongrel logic for retroaction [henceforth abbreviated to LoR, although see [note 1]. Its utility is certainly suspect, its coherence even more so. Despite [or perhaps because of] this, the notation has a certain appeal to it. At the very least, its apparent purpose - a minimal formalization of potential organisations of non-linear causality - is an urgent one. Excavation seems merited. Whilst questions remain as to the providence of the documents, we have largely bracketed these in favour of extending Çaba’s initial systematisations to their terminus.

Such an extension is however complicated by ambiguities of intent. At least in the documents we have examined, Çaba’s elaboration operates on the level of pure syntax. It develops a closed systems of arrows, each supposedly relating to a different temporal determination. However, semantic interpretations of the arrows remain contentious. We know a little of how they relate, but are less certain of what they ‘mean’ [or if such a question is valid]. This has led, for the sake of comprehension, to apply the basic heuristic of reading the notation through the lens of first-order logic. Vigilance is key here, for certain results already suggest that this might be an overcoding of LoR’s syntactic autonomy. Despite this risk, such a frame has the benefit of giving an initial sense to the formalism. We can take the standard material conditional [if ---> then] and invert it, tracing a retroactive conditional of the form:

then ---< if

This notational innovation is the core of Çaba’s logic, from which all other proliferations issue. We can then read the plethora of additional determinations in Çaba’s scribblings as following this pattern of implication. For example, the first-order biconditional, A <---> B, ‘if and only if’ [iff], the familiar mark of necessity, has an immediate retroactive correlate:

A >---< B , ‘then and only then’ [thenn]

Other logical combinations of arrowheads can then be introduced at will, such as >--->, a relation we have dubbed the ‘Robin Hood’ or ‘Looper’. Further mutations then occur with Çaba’s introduction of a function of ambiguity, ---), and its correlate, ---(. It is as yet unclear whether these new signs [ ) , ( ] notate concrete undecidability/indeterminacy or some decomposable register of potentiality [an interpretational dilemma that has significant implications when introducing negation into the system]. In either case, what is clear is that these parenthetical marks are an index of possible becomings, such that:
) = > , ) , .

( = < , ( , .

This can be read as an [exclusive] list of ‘options’ open to a given parenthetical arrow. The introduction of the dot [ . ] here highlights one of the unique features of LoR: the lack of a mark constitutes a determination. Put otherwise, the empty space ‘around’ the arrows marks a failure of [conditional] relation. In its most extreme form, this function of the system introduces the potential to mark the relation of ‘there is no possible relation’, syntactically notated and semantically interpreted at follows:

A --- B , ‘nilponens’

Perversely then, one can say that the condition of nilponency is simultaneously the most and the least determinate of all temporal relations. Çaba’s final act of proliferation is then to introduce the notion that the arrowheads ‘stack’ at the point, a situation that we are currently taking to mark causal overdetermination or contradiction [whether these two interpretations in fact coincide is a stake that has yet to be adequately explicated]. In any case, the simplest form of this stacking yields the basic relation of contradictory cause, ---><, or the 'dialethal’.

To summarize then, we have a list of 9 ‘basic’ arrowheads, from which Çaba’s menagerie of relations is constructed.

. , ‘no relation’ or ‘nilponens’
> , ‘ifthen’
< , ‘thenif’
) , ‘ifthenish’
( , ‘thenifish’
>< , ‘ifthenthenif’ or ‘dialethal’
>( , ‘ifthenthenifish’
)< , ‘ifthenishthenif’
)( , ‘ifthenishthenifish’ or ‘ambigoolian’

From this point, it is easy to concatenate the arrowheads into the 45 operators [see note 2] of Çaba’s logic, simply by working through the different combinations, such as >--->< , (----( , etc. This should hopefully suffice as an introduction to LoR’s formal construction.

Beyond these syntactical combinatorics however, interpretation becomes notably murkier. This is, after all, a map of time. Here we touch upon a tender point of articulation. The traditional logical conditional is a general schematic of dependence, cutting across various realms of application. In contrast, Çaba’s logic renders the conditional irrevocably temporal. Internal to its model then, at least, all arrows must be considered with a ‘hidden’ subscript acting as a temporal index:

t1--->t2 , t2---<t1 , etc.

This is to say that LoR injects time into the very syntax of logic. This is the intention of Çaba’s [seemingly aborted] project. Yet, as desirous as such a project is, it trails behind it havoc for all other principles of logical construction; concatenation, negation, disjunction and so forth. Over the following days and weeks I will hopefully have time to spiral down into some of the labyrinthine consequences of this, at least insofar as me and |end| have been able to trace them. And, beyond that, comment must be made on some [un]surprising isomorphisms between Çaba’s ‘logic’ and hyperstition... Consider this a first dispatch.

∇

[Note 1] Although the degree to which LoR is accurately termed ‘a logic’ remains a central object of investigation. The terminology is merely a placeholder for the purpose of present discussions.

[Note 2] The terminus of possible arrows at 45 may initially seem confusing, as a linear combinatory elaboration of arrows from the initial 9 signs could also be read as yielding a sum of 90. The key to the smaller sum of 45 lies in a enigmatic prohibition scrawled at the edge of one of Çaba’s papers: ‘No Mirrors’. Initially opaque, the sense of this declaration was admirably decoded by |end|. If we consider the arrows as relations severed from any determination by an object [the As and Bs of first-order logic], then the directionality of an arrow as whole does not differentiate it. So, for example, the arrows ---< and >--- have no functional difference until one introduces an object relation [whereas, say, A ---< B and A >--- B clearly map different conditions of relation]. Taking the logic in it’s ‘purest’ form, we can then eliminate half of the full elaboration of 90 arrows as functionally moot - specifically, those that simply mirror each other. Whilst my own proliferationist tendencies [surely an underlying factor in my approbation for Çaba’s project] make me reticent to assent to such a culling, I have yet to find any coherent reason for resisting it in this case. As such, we have decided to abide by Çaba’s prohibition, at least until some [perhaps ,kataphysical?] model is discovered which problematises it. That we have made this practical reduction is however a fact that should be borne in mind as research proceeds.