8: Abstractors

The basic idea behind abstractors is to take a selbri/bridi and produce a new selbri. Note carefully that the abstractors come in two families, depending on whether they take selbri or bridi. What's the difference? A selbri has {ce'u} slots open, and thus has an arity; it's a relation, after all. A bridi, however, is a concrete state of values being related by a relation (terbri related by selbri). They aren't the same type of thing.

"Hold on," you might object, "but what about nullary relations?" On one hand, yes, okay, the two selbri {cei'i} and {gai'o} are also bridi, because of this. But they're the only ones! All other nullary relations are isomorphic to one of those two.

"Hold on again," you might continue, "but don't all NU syntactically permit {ce'u}?" Yes, that's a fair objection. However, it is a well-established community standard that {du'u} and {nu} do not capture {ce'u}, and further that they capture fully-applied predicates with truth values.

First, {ka}. We understand {ka} pretty well by now; when we informally talk of e.g. {mlatu}, we are talking of {pa ka ce'u mlatu ce'u}, or mlatu/2, the binary relation which is inalienable from the text {zo mlatu}.

Next, {du'u}. We haven't yet dug into these, but they are fully constructed bridi; {lo mlatu ku cadzu} is represented as {pa du'u lo mlatu be zo'e ku cadzu zo'e zo'e}. Folks like to imagine these as having definite truth values, and it's tempting to join in; after all, surely every relation either does or does not contain a given row, right? Well, if we do that, then we actually hobble our logical power! It seems strange, but classical logic loses the ability to consider certain kinds of infinite objects if it would take infinite time to determine their truth value in a true-or-false context. What we can give is something nearly as good: Every subset of some set S has a complement relative to S. When S is a Von Neumann universe (a powerset of an ordinal, basically), then the elements of S have a bounded set theory that does allow for Boolean logic.

"Hold on," you might object, "but this sounds like romantic hogwash." There are two ways to look at it. One way is computationally: We can't know whether some element is definitely in a set or not without running a program, and that program can take forever. We can't tell if a real number is equal to zero or any other real number without running a program forever, either. Another way is logically: Gödel et al. have rigorously showed us that there's no way to just assign a truth value to an utterance like {nei na fatci}. Either way, it's very solid, and Boolean logic can either be classical, and have a gaping hole where we can be convinced of things that we can't build, or constructive, and have a lot of times where things are true, false, or not false; zero, non-zero, or not non-zero; infinte, finite, or not finite. For extra fun, we can say "not" as "not yet known to be, but maybe if we run the program just a little longer..." This is Turing's undecidability.

Okay, that's where we've been. Now let's go to new places.

Amounts: {jei},

Despite incompleteness, truth values are still limited to "true" and "false"; a relation either has or doesn't have (or is incomplete on) a row. We can enrich any relation by adjoining some type and using its elements to decorate rows, and we can use that enriched data in our characteristic function. Now, a row isn't simply present or not present, but present and annotated or not present.

First, let's generalize presence. Some selbri will have a hidden column for the likelihood that the relation holds, expressed as a classical probability in the unit interval. These are fuzzy sets. Fuzzy sets happen to need a universe relative to which to establish their probability distributions, which happens to just be the way in which subsets were given the ability to have complements in my earlier rant.

For now, I can't find any gismu which would directly allow accessing fuzzy membership probabilities, so I'll hold off on axioms, but the idea is roughly that a {du'u}, which is a proof of membership, could be traded freely for a {jei}, which indicates the degree of membership.

For {ni}, the story could be much brighter. We could imagine bending around a wire from a {ka}, so that {pa ka ce'u broda ce'u} could become {pa ni ce'u broda}, with broda2 becoming the measurement somehow. Questions abound, including why broda2 and not broda1, whether the arity change is correct, and how arithmetic evaluation interacts with the whole shebang. Nonetheless, the core theory is the same: Treat a numeric column as a numeric per-row annotation, and create a new abstracted relation which bends the wires. This makes {ni} like a subtype of {ka}, although the as-parent-type and the as-true-as relation send {ni} to different {ka}.

Events: {nu}, {mu'e}, {pu'u}, {za'i},

Now for tougher food. What is an event? For starters, an event happens. We'll think of this again in terms of annotations. Let's imagine some sort of spacetime coordinates around which the event is happening. Formally, the event has a spatiotemporal neighborhood. Within that neighborhood, the event is happening; outside that neighborhood, the event is not happening. Immediately, there is the problem where we cannot tell if an event is one instance, paused and resumed, or two instances; it's not obvious just by looking at the neighborhood. So we will have to be better about quantification. The good news is that universal and existential quantification should still work; the bad news is that we will need to allow more than just {pa nu} formation, because an event might have more than one neighborhood within which it is occurring.

Also, {nu} doesn't capture {ce'u}, so immediately we know that it abstracts over bridi, not selbri.

CLL says that each of the different event abstractors could be replaced with {nu}, being just as true, just less structured. We could take that to imply that there's structure within each specialized event abstractor.

First, {mu'e}, which treats events as points in spacetime. On one shoulder, quantum mechanics tells us that point idealizations are bogus; on the other shoulder, from the definition of "neighborhood", if we zoom out far enough from an event then we should eventually find a reference frame from which the entire event is visible. It follows that {mu'e} asserts that the neighborhood of an event is connected; if it wasn't actually connected, then we're imagining the smallest ball/neighborhood which contains all of the pieces and is connected, but technically we should only be allowed to go one-way on this. When we go from {mu'e} to {nu}, we forget connectedness.

Next, {pu'u}, which breaks events into stages. Obviously, I am biased about how rich a composition of stages should be, but for starters, let's imagine serial algorithms. pu'u2 would merely be a sequence of component events which, when composed in order, yield the entire event. So, with processes, we're talking about events which are decomposable, regardless of whether or not they're connected. But, also, they're composed in an order. Spacetime's events are only partially ordered, so we're asserting that these particular components are actually causally related. Putting this all together, pu'u2 is a totally ordered sequence of events, and pu'u1 is the smallest neighborhood which contains them.

Guh, that one was tough. But {za'i} is much easier. CLL says that continuous states have sharp boundaries. So their neighborhoods also have boundaries.

Finally, we have {zu'o}, for activities. An activity recurs many times within its neighborhood, and although each zu'o2 action is clear, the entire zu'o1 might not be. To give an example, imagine stirring a mixture. A zu'o2 would be a full rotation of the stirrer, while a zu'o1 might contain multiple incomplete rotations. Just like with {pu'u}, zu'o2 is an ordered sequence; the activity of walking consists of two symmetric, but chirally distinct, actions, for example.

CLL 11.11 explains that event contours aren't available for all subtypes of event. The only one always available might be {ca'o}. We also should be able to zoom out and get {co'i}. Imagining a sheaf, we might expect that the subtypes have more contours. {pu'u} has all contours, while {mu'e} adds only {pu'o} and {ba'o}; CLL forbids {ca'o}, but if we can zoom out, then we can zoom in.

Finishing up, both {za'i} and {zu'o} allow {pu'o}, {ba'o}, and {ca'o}, and {za'i} also has {co'a} and {co'u}.

Minor abstractors: {li'i}, {si'o},

CLL lumps these in a "minor" page, and I guess I will too. {li'i} is a subjective {nu}; li'i2 is the experiencer of the event. Not much more to say for now; we'll have to get to subjective experiences for this to be useful.

{si'o} is similar, but vague. si'o2 is an agent who is imagining a bridi. Can {ce'u} appear underneath? Nobody knows.

{su'u} is theoretically essential for constructing abstractors, but CLL doesn't give us enough information for how to wield it. It's not even clear whether {ce'u} are bound, although many folks believe so.

Summary of Abstractors