Contingency means that "either P or not-P might not be actual." — aletheist
So to complete the pattern, 2ns would need to be characterised by a failure of distribution of the principle of identity. And you are saying 2ns is really to be labelled contingency rather than actuality?
I can't really agree with your framing here as my point was that P is only truly actualised to the degree that not-P (as its generic 3ns context) is also actualised.
Returning to my original remark about a "tension", vagueness is pure contingency while 2ns is constrained or contextualised contingency. That is, 2ns is about actualised degrees of freedom - a degree of freedom being a determinate direction of action, or an existent with a predicate.
So it is confusing to call 2ns contingency when 1ns is usually regarded as the maximally contingent. Really, 2ns is contingency limited, regulated, contextualised.
My argument has been that the principle of identity makes a claim that a thing is the same as itself by definition - it appears no context or larger relation is needed and no contingency or uncertainty could be involved.
The laws of thought make identity the concrete and completely uncontingent starting point for then reasoning about the particular. Particularity is claimed as an atomistic fact and so off logic can merrily trot to derive its further two laws.
Peirceanism then stands against that with its holism. Now the concrete particularity of identity - the category of the actual - is instead the emergent intersection of the possible and necessary, the local potential and a wider context of constraint.
So actuality or 2ns becomes the transition zone. It reflects the mixing of the polar extremes of being - vague possibility and crisp generality. Or total freedom vs total constraint. Actuality is actualisation - the process of coming to be framed by the limits on being. A developmental arc is being described that (for me, if not for you) goes from vague 1ns to generic 3ns via the concrete foothold or symmetry breaking which is bald 2ns - a difference that can make a difference in that it does serve to construct, or at least continue to reinforce, some large state of 3ns habit.
The emergent nature of 2ns or the concretely particular is what makes for ambivalence. We are talking about actuality - but the concreteness is secured by the 3ns it anticipates. Context is what gives the particular its definite character, what allows it to be seen and remembered as an occasion that is the same as itself/different from aught else. And this in turn means the particular has been sharply formed by the pruning away of all unnecessary possibilities. Identity is arrived at apophatically.
So there is a pattern to be completed. The law of identity ought to have an exact apophatic definition in the "true meaning" of 2ns, or actuality.
And what does identity presume most? It presumes brute existence instead of emergent development. It presumes a pure state rather than a mixed state. It presumes it stands at the beginning rather than arriving at the end.
Yet then "actuality" in semiosis requires the wholeness of 3ns (the 3ns that incorporates the 2ns and 1ns). So the identification of 2ns as actuality - or better yet, actualisation - has to be understood in that light.
Thus in terms of your logical formalisation - "contingency means that "either P or not-P might not be actual" - it seems to me rather that we are talking not about contingency but about actualisation. So the category of contingency reduced to its deterministic minimum by the constraint of a generality - ie: a freedom that has a direction.
You simply seem to be re-stating the fact that the PNC does not apply to the vague (the vague being the radically contingent).
So Vagueness means that "both P and not-P are possible." = Vagueness means that "either P or not-P might not be actual."
Although, as I say, vagueness defined directly is the degree to which P and not-P are co-jointly not actual. This captures the anticipated 3ns which is the further rule that actuality is irreducibly contextual. So it takes a matching degree of P and not-P for actualisation. And a matching absence of P and not-P for there to the maximum indistinctness or lack of identity.
In vagueness, P is indistinguishable from not-P. In actualiity, they are as distinct from each other as possible. And in generality, that actualised counterfactuality is not merely a one-off event but a habit, a law, a routine state of affairs, an irreversible fact of history.