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This is my explanation of Ca, shown here in Slot X:
Okay, those should be in the order "Configuration, Affiliation, Perspective, Extension, Essence". That's how they're typically listed, including on the page that explains them, which can be found here. If you find that page confusing, though, here's my attempt at an explanation.
You'll notice that Ca isn't grayed out, which means that it is required in all formatives. It marks five things, which the language evidently all considers very important. But what are any of them?
I have boiled down these five categories into series of questions. My intent is that answering these questions can naturally take you to any of the possible values for that category. Here they are:
Configuration: Does it consist of one, two, or more parts? If there are more than two, are they similar or different? Are they connected or not? If so, do they form something new when put together? Are none of these applicable, as the parts comprise a fuzzy set instead?
Affiliation: Is the concept of a "greater purpose" applicable or relevant? If so, do these parts have a shared purpose or varying purposes? In the former case, are their roles complementary?
Perspective: Is it viewed as a whole; or is it uncountable, immeasurable, or inaccessible in full? Is it something "in general" as opposed to something specific?
Extension: Is all of it relevant, or are we really referring to part of it? Is it specifically the beginning or end? Does it begin or end all at once or bit by bit?
Essence: Is it really there, or just hypothetical?
And now for answering all of those!
(Note: I don't have as many examples as I would like. I recommend looking at the page on the website, at least for the examples.)
This category is the largest, with nine possible values. Configuration is the difference between a single tree, a grove of trees of one type, a more diverse forest, an entangled mass of trees, etc.
Answering the questions on my list gives you these possibilities:
(Note: Quijada lists these in the order UNI, DPX, DCT, AGG, SEG, CPN, COH, CST, MLT.)
This one is perhaps best understood as not being any of the others. It's used for a single thing that isn't made up of constituent parts. Of course, "made up of constituent parts" is relative; note that Quijada calls this a "single, contextual embodiment" (emphasis mine). His examples, later contrasted with other Configurations, include "(to be) a tree", "(to become) a person", etc. (as opposed to a forest, an orchard, a crowd, a couple...)
This one draws comparisons to the dual number found in some languages (as in, a special form for exactly two of something, as distinct from singular and plural), but is in fact different. Quijada defines it as denoting a "related binary set" whose members are "identical or complimentary", such as one's hands, two people in a relationship (Ithkuil has roots for many kinds of relationships), a set of salt and pepper shakers, etc. For verbs, Duplex denotes two-sided actions such as a tail wagging back and forth, a light flashing on and off, etc.
The Discrete is distinguished by describing similar or identical things that are not physically linked. As with all Configurations, the things can be anything: objects, actions, experiences. Quijada notes that when using the Discrete Configuration for verbs, the individual actions being repeated can themselves be Uniplex (as in walking on stairs, i.e. taking one step over and over) or Duplex (as in swinging a hammer, i.e. swinging it up and then down over and over).
The Segmentative is like the Discrete, only the parts are linked. They may be physically touching, or they may just act as if they are (Quijada offers the image of dolls on an assembly line). The difference between SEG and DCT is a bit trickier with verbs; the best example Quijada can give is that the former describes the way a machine gun shoots, and the latter is how a "single-fire" gun shoots repeatedly.
The Coherent configuration, like the Composite below, describes something whose parts are "connected, fused or mixed" to form something new and distinct. In the Coherent's case, those parts are the same or similar. Examples might be a sculpture made of cans, or, as per Quijada's examples, a bookcase (similar shelves that form something new) or a phalanx (the same but with soldiers). In the realm of verbs, he suggests that buzzing or glowing might be thought of as Coherent, if one agrees that they consist of individual noises or flashes respectively.
The Aggregative is like the Discrete (i.e. disconnected things) but with dissimilar members. (The line between "similar" and "dissimilar" is, of course, somewhat arbitrary.) It might be used for a pile of mixed candy or a group of dogs of different breeds.
The Componential is used for dissimilar things that are linked to each other. Just like with the Segmentative above, they can be linked physically, or they can just act as if they are. If AGG describes a group of assorted dogs, CPN would describe those dogs on parade.
The Composite describes dissimilar parts that combine to form a new and distinct entity. Quijada points out that if one wanted to talk about a recipe, a skeleton, or a melody in Ithkuil, the words for those would be the CST forms of ingredient, bone, and musical note respectively.
This one describes a fuzzy set, which is a set with degrees of membership. A word in the Multiform Configuration means something like "a group in which some members are more like a given prototype, and some are less like that prototype".
As described up here, Affiliation is about how the parts, described above with Configuration, relate to a shared purpose, or as Quijada puts it, "subjective function, state, purpose or benefit".
This is the default affiliation, used when the concept of a "shared purpose" isn't important or relevant. Things in the Uniplex configuration don't have any interacting parts, and are thus typically (but not necessarily) in the Consolidative affiliation.
The Associative affiliation describes things whose "subjective function, state, purpose or benefit" is shared by design. (Take care to distinguish it from the Coalescent below.) Quijada notes that the difference between CSL and ASO is the difference between a grove and an orchard (which would each be comprised of trees in the Discrete configuration). Keep in mind that any affiliation can apply to any noun or verb; using ASO on something in the Uniplex calls attention to its specific purpose, the integrity of its construction, its single-mindedness, et cetera.
This is VAR.
The Coalescent is easily confusable with the Associative, in that the purpose of the members is shared; the difference is that with the Coalescent, the members have complementary roles. Or in simpler terms, they do different things for the same purpose. An example might be an RPG party or sports team in which different players have different roles (such as focusing on attack, defense, support...), but crucially work "in furtherance of some greater unified role".
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