In quantum mechanics, there are at least two main  categories of measurement outcome/truth value contextuality.

Kinds of contextuality

1. environmental

2. algebraic

However I have also come across ontic contextuality (Clifton 1993, Redhead 1 and Heywood 1983), independent state contextuality, and strong contextuality.

It is said that independent state contextuality is independent of the (A)state and (B) tensor product expansion of basis. Thus I would presume (A) denotes environmental contextuality, ie a different state under fine graining or due to different events that have interfered with the same state in the past, although the state remains teh same),

and (B) by tensor product I presume it means the overall expanded measurement basis state,ie algebraic contextuality. For example, all of the other commuting measurements that are made alongside the measurement in question which is what was originally meant by contextuality, I thought, in KS

So independent state contextuality is thus distinct from both of 1 and 2, at least in their original guise as disguised by Shimony (1984). Is that correct?

I presume the original KS theorems depended on algebraic contextuality meaning that the same observable or particle that is measured for (in the same component), and which is prepared in the same way may give different experimental results depending on how its tensor product state can be expanded (or rather how, or which?, other observables are measured along with it).

I presume that independent state contextuality has something more to with just the singular observable itself and how the way in which the detection devices are used to measure it, where different experiemental results may occur, even if the environment is the same, and observable, is measured alongside the same observables (whose operators commute with that of the measurement in question) which are themselves measured for in the same way and in the same component.

If we call the system or observable or 'measurement of the observable' of interest x, then

are these versions of contexuality to be further distinguished, in virtue of (1)  which observables with commuting measurement operators (if they were measured  although they may not be)alongside observable x, exist at the same 'time' (whatever that means exactly- perhaps that is a form of environmental contextuality), or

(2)which observables, particles, are entangled with the observable in question (perhaps a form of environmental contextuality or algebraic), or

(3)which other observables (alongside the measurement in question) are measured alongside xfor (algebraic),

\(4), the other observables alongside are x, are such that they are prepared, and measured, but measured along a different axis component (the apparatus is set to a distinct angle or component for these other observables- some form of algebraic or logical contextuality, or perhaps independent state contextuality) or with regards to whether  spin up or down is measured for passively or actively, depending on how the registration devices are set (logical contextuality or close to strong contextuality) WIth

(5) being the individual system described above, which is either strong contextuality or something stronger or different again (strong perspectivalism)

 Is this correct? any help here would be of use: or perhaps:

I presume that logical contextuality is just a form of contextuality which entails that a certain measurement must take a different value in distinct measuring or that some one of the measurement must, although it does not specifiy an individual one. By measuring context I mean (when measured alongside two distinct sets of commutting operators denoting distinct observables that are measured, or the same observables but measured alongside a different component)

I presume this is distinguished from contextuality simpliciter or weak contextuality where its merely possible or there is a certain probability that some one of the measurement may take a different value. And perhaps by strong contextuality it is just meant that one can say definitely of a singular system, when measured in a different context, that it must take a different value. This being opposed to logical contextuality where one can one say that some one of the measurements must take a different value, but not which particular one. Perhaps instead this is what is meant by weak, strong, and logical contextuality which are really again a kind of modal grade and not a different category unlike perhaps independent state contextuality

What I called Categories, I distinguish from modal grade, mechanism and kind; or rather (Counter-factual modal grade now, if the above contention is correct, and thus there is a fifth distinction). Where these three others groupings as mentioned are (and I have one more question here in the second grouping)

(1)modal grades (single case, single case counter-legal, and the more trivial backtracking subjunctive counter-legal form and two distinct but operationally equivalent events tested together but each alongside other measuremet), or

(2) kind (preparation, apparatus, measurement, ie at what point in the process is the context distinct-presume these are distinct from the above although spekkens speaks of transformation contextuality, so perhaps his three categories embody envion-mental of a form, algebraic, and independent state)? Ie is spekkens actually referring to the three categories as I mentioned before algebraic, environment and independent state contextuality by these terms or is just some division of how or at what point an environmental or algebraice-y contextual system is 'contextual'.

(3) the mechanism of contextuality and or/or the kind of contextual hidden variables or how said contextual hidden variables are contextual; is it again just backtracking and thus not really a case of a distinct context creating a distinct measurement, because only one (or a a certain set of) hidden variables or hidden causes/bohmian parameters, is compatible with each measurement/environment context,

(A)The hidden variables in-deterministically produced, or (2) is it causal in that the hidden variable or hidden cause, of the measurement outcome is caused to be different by the different context ( i presume this is what is meant by causally contextual hidden variables or hidden causes as opposed to standard backtrackingly-contextual hidden causal variables), or

(3) does the hidden variable stay the same and its function is contextual, the same hidden cause, cause, or become or translate into, different things in different scenarios, ie conditional upon and after measurement and context, I presume these are partial or latent hidden 'entities' and do not really constitute any standard form of realism at all, nor are they what are normally meant by contextual hidden variables or causes or causally contextually hidden variables or causes but something else (aspectual, contextual function, (  there are lot of categories here, causal, ontic, existential, standard, retro-causal, perspectival, interactive,aspect; where by causal I mean that a distinct hidden variable or hidden causal variable, such as a bohmian parameter is caused by a distinct context as opposed to the fact that ) Perhaps this is the ontological contextual hidden variable type or the perspectival, creative contexts, or aspectual view? (I dont really know)