Articulation Is Not Realization

A formal chain may look complete before anything has happened. An action can be named, a continuation selected, and an expected result described. Yet the present form need not change because of this.

A relation remains between an articulated continuation and an achieved result. Notation can display this relation, but cannot guarantee it by notation alone. This is where a formalism most easily substitutes a description of realization for realization itself.

From Possibility to Result#

The general form of continuation can be written as follows:

Here, $F$ is the present form, $\Delta$ the operative distinction, $G_{\Delta}(F)$ the field of possible continuations, $g$ a distinguished continuation, and $F'$ the obtained form.

The arrows express possible relations of continuation. They do not assert that every transition has already occurred.

The field $G_{\Delta}(F)$ is not yet a concrete $g$. An articulated $g$ is not yet the obtained form $F'$. In the first case, continuation is merely possible. In the second, one way of continuing has acquired determinacy. In the third, the present organization has been changed or a new one has been produced.

The transition from the field of possibilities to a concrete $g$ may be determined while the relation from $g$ to $F'$ remains open.

Unclosedness as a Retained Status#

If a relation between two positions is not yet determined enough to be confirmed, let its unclosed status be denoted by

$X$ denotes a determinacy already available. The role of $Y$ is known from the general form of continuation, but the concrete relation to it has not yet been articulated sufficiently.

$\mathsf U(X,Y)$ is a status assigned to the proposed relation, not the negation of that relation. It does not assert that no continuation exists or that continuation is impossible. Its claim is narrower and more precise. From the present $X$, continuation to $Y$ is not yet determined enough for the formalism to regard the relation as established.

Unclosedness does not add a new point to the chain or introduce a special entity. It names the status of a relation between positions already distinguished. The formalism can thereby retain where continuation was expected, which next role remains undetermined, and exactly where a proposed connection still requires verification.

Articulation Is Not Realization#

For the relation between a continuation $g$ and an obtained form $F'$, this gives

$g$ may be a detailed plan, a program, a command, or a precise description of the required action. None of these forms by itself means that the action has occurred.

A blueprint is not yet a tool. A description of manufacture is not yet a manufactured object. A distinguished way of reorganizing is not yet a reorganized form.

Obtaining $g$ is nonetheless an actual result of some movement. A new determinacy has appeared. The proposed continuation is now known. If the task was to construct a plan or a program, $g$ may be its result. But if another form had to be changed, creating the means should not be mistaken for the accomplished change.

A Description of the Result Is Not the Result#

One common substitution is to treat a description of $F'$ as $F'$ itself.

An external system may report that an action has been performed, an object changed, or the required state reached. Such a report is indeed a new form. But it is first of all a form of report or description.

If a description was required, it may be a complete result. If the task was to change an object, a description of the change is not yet the change itself. Otherwise, the chain could be closed simply by generating its final sign.

It is therefore necessary to distinguish a description of the expected result from an articulation of the organization actually achieved. This does not require the world to be represented with final completeness. It requires only that a statement about a transition not be passed off as the transition itself.

Opaque External Execution#

Another substitution occurs when $g$ is passed to an opaque external executor and its response is accepted as the obtained form $F'$.

The external source may actually perform the required movement. Externality is not itself the problem. Every local formalization has a boundary and depends on conditions that it does not retain completely.

The problem begins when this boundary is concealed and the relation is presented as closed within the formalism. The formalism then retains only the beginning and the end. It does not distinguish whether the actions performed corresponded to $g$, what exactly changed, whether the movement can be reproduced, or whether the returned representation articulates a form actually obtained rather than merely describing the result.

If a formalism retains only the two ends of the relation, the relation remains unclosed even when something has actually happened in the external environment. To treat it as closed, the movement’s organization and its connection to the obtained form must be articulated with sufficient determinacy.

This does not require every condition of realization to be described without remainder. It requires that the very relation claimed to have occurred not be concealed.

A Candidate Does Not Yet Close the Relation#

A rule, an interpretation, or a newly obtained definition may propose a connection between $X$ and $Y$. But a proposed connection is a candidate for closure, not an already closed relation.

If the candidate is confirmed, the unclosed relation is closed by that transition. If the candidate is rejected, the candidate is excluded, but $\mathsf U(X,Y)$ may remain. If the available conditions are insufficient for confirmation, the relation remains unclosed.

Generating a new continuation expands what can be verified, but the continuation is not confirmed merely by appearing. The ability to propose a transition and the ability to establish that it occurred are different relations.

Unclosedness is not a criterion of truth or logical correctness. It indicates only that a particular relation has not yet been established. Whether it can be closed by proof, observation, or an accomplished change depends on the kind of relation involved.

Holding the Gap Open#

An unclosed relation is not a defect that a formalism must immediately conceal. The ability to retain such a relation is a positive capacity of the formalism. It shows the exact point at which the formalism’s available determinacy ends.

A stronger notation is not one that always depicts a complete chain. It is one that distinguishes established relations from continuations that have only been proposed.

$\mathsf U(g,F')$ does not solve how an articulated continuation becomes an achieved form. Nor does it assert that such a transition is impossible. It holds the relation open until the name of the action, the means of its realization, and the obtained result have been connected with sufficient determinacy.

The relation may later be articulated and confirmed. The unclosed relation is then closed by a concrete movement. Until that happens, the formal notation must retain the relation as unclosed.

Articulation gives form to a possible continuation. Realization changes or produces a present organization. By retaining the distinction between them, a formalism can relate a plan, a program, an action, and a result within one ontology without taking one for another.