The LW survey says about 1 in 4 of us is a communist

No, it doesn't. In the last LW survey, 0.7% of respondents identified as communists. Perhaps you're talking about the 30% that identified as "socialist"? But "socialism" in the survey was defined as support for a highly redistributive, socially permissive political regime, like they have in Scandinavian countries. That doesn't imply allegiance to Marxist doctrine, or knowledge of it.

As for the idea of "dialectic", Marx got it from Hegel, and a full understanding of Hegel -- if it is possible at all -- is not something that can be effectively communicated in a comment or short internet article, I think (this might help, but probably not). As a rough approximation, though, the dialectical method is basically just systems thinking.

It's presented as an alternative to the analytical method, which involves breaking a system down into parts and attempting to understand each part individually. The idea governing analytical thinking (says the proponent of dialectic) is that the intrinsic nature of the parts can be understood prior to figuring out how they fit together to form the whole.

Dialectical thinking, on the other hand, is based on the idea that the concrete nature of the parts cannot be understood without understanding the role they play in the whole, the relationships between them. So the analytic project, which focuses first on understanding parts considered individually, is doomed to failure, because it ignores the extent to which the overall context is essential for our understanding of the nature of the parts.

"Dialectical materialism" in Marxist thought is basically just an application of this dialectical thinking to economics. One could approach economics analytically by first, say, constructing a model of individual economic agents, and then trying to figure out what happens when these agents interact under certain conditions. The proponent of the dialectical method (like Marx) would, however, insist that this is mistaken. Human nature and human needs cannot be understood in isolation. They are a product of the socio-economic context, just as the socio-economic context is itself a product of human nature and needs, and the individual elements and overall context are constantly changing in response to one another. So to truly understand the dynamics of the economy, you need to approach it from a systems perspective. You need to start by understanding the historical dynamics of the interactions and relationships between elements of the system and how that effects the evolution of the natures of those elements, rather than starting with a static model of the individual elements and only then moving to an analysis of their interactions. The natures of individual elements are constituted by their participation in the system, and they change as the system evolves, so you shouldn't treat those individual natures as logically prior to the system.

So that's a quick and clumsy attempt at explicating what the dialectic method is all about. As for whether the method is useful: There is something right about the idea that focusing purely on the analytical method can lead to mistakes, but a complete repudiation of this very useful pattern of reasoning also seems to be a mistake. It seems to me that the dialectical method (and systems thinking in general) should be regarded as a useful complement (and often corrective) to analytical thinking, but not as a wholesale replacement.

I'm making my way through George Lakoff's works on metaphor and embodied thought; are familiar with the theory at all?

Unfortunately no, but from your description it seems quite like the theory of the mind of General Semantics.

Whereas what you're saying is starting with symbols, which I think would be the reverse of what he's saying?

Not exactly, because in the end symbols are just unit of perceptions, all distinct from one another. But while Lakoff's theory probably aims at psychology, logic is a denotational and computational tool, so it doesn't really matter if they aren't perfect inverse.

How does this connect to the map-territory distinction? Generally as I've understood it, logic is a form of map, but so too would be a model. Would a model be a map and logic be a map of a map? Am I getting that right?

Yes. Since a group of maps can be seen just as a set of things in itself, it can be treated as a valid territory. In logic there are also map/territory loops, where the formulas itself becomes the territory mapped by the same formulas (akin to talking in English about the English language). This trick is used for example in Goedel's and Tarski's theorems.

This is something that has always confused me, the probability definition wars. Is there really something to argue about here?

Yes. Basically the Bayesian definition is more inclusive: e.g. there is no definition of a probability of a single coin toss in the frequency interpretation, but there is in the Bayesian. Also in Bayes take on probability the frequentist definition emerges just as a natural by-product. Plus, the Bayesian framework produced a lot of detangling in frequentist statistics and introduced more powerful methods.

thank you to everyone for pointing me to Cox's Theorems again. I know I've seen them before, but I think they're starting to click a little bit more on this pass-over

The first two chapters of Jaynes' book, a pre-print version of which is available online for free, do a great job in explaining and using Cox to derive Bayesian probability. I urge you to read them to fully grasp this point of view.

And Richard Carrier's new book said that they're actually the same thing, which is just confusing

And easily falsifiable.

Also, am I doing it right for the one ontology and one interpretation that I've stumbled across, regardless of the others?

Yes, but remember that this measure interpretation of probability requires the set of possible world to be measurable, which is a very special condition to impose on a set. It is certainly very intuitive, but technically burdensome. If you plan to work with probability, it's better to start from a cleaner model.

Right, because in fuzzy logics the spectrum is the truth value (because being hot/cold, near/far, gay/straight, sexual/asexual, etc. is not an either/or), whereas with PTEL the spectrum is the level of certainty in a more staunch true/false dichotomy, right?

Yes. Fuzzy logic has an infinity of truth values for its propositions, while in PTEL every proposition is 'in reality' just true or false, you just don't know which is which, and so you track your certainty with a real number.

The other question I forgot to ask in the first post was how Bayes' Theorem interacts with group identity not being a matter of necessary and sufficient conditions, or for other fuzzy concepts like I mentioned earlier (near/far, &c.). For this would you just pick a mostly-arbitrary concept boundary so that you have a binary truth value to work with?

Yes, in PTEL you already have real numbers, so it's not difficult to just say "The tea is 0.7 cold", and provided you have a clean (that is, classical) interpretation for this, the sentence is just true or false. Then you can quantify you uncertainty: "I give 0.2 credence to the belief that the tea is 0.7 cold". More generally, "I give y credence to the belief that the tea is x cold".
What comes out is a probability distribution, that is the assignment of a probability value to every value of a parameter (in this case, the coldness of tea). Notice that this would be impossible in the frequentist interpretation.

*I keep seeing probability referred to as an estimation of how certain you are in a belief. And while I guess it could be argued that you should be certain of a belief relative to the number of possible worlds left or whatever, that doesn't necessarily follow. Does the above explanation differ from how other people use probability?

I believe you've defined an equivalent if unusual form (or rather, your definition can be extended to an equivalent form). You need the notion of the measure of a set (because naively it can be confusing whether one infinite set is bigger than another), and measure is basically equivalent to probability; "how certain you are of a belief" is equivalent to "the measure of the worlds in which this belief is true, relative to that of the worlds that you might be in now".

*Also, if probability is defined as an arbitrary estimation of how sure you are, why should those estimations follow the laws of probability? I've heard the Dutch book argument, so I get why there might be practical reasons for obeying them, but unless you accept a pragmatist epistemology, that doesn't provide reasons why your beliefs are more likely to be true if you follow them. (I've also heard of Cox's rules, but I haven't been able to find a copy. And if I understand right, they says that Bayes' theorem follows from Boolean logic, which is similar to what I've said above, yes?)

The only laws of probability measure I know are that the measure of the whole set is 1, and the measure of a union of disjoint subsets is the sum of their measures. I'm finding it hard to imagine how I could hold beliefs that wouldn't conform to them. I mean, I guess it's conceivable that I could believe that A has probability 0.1, and B has probability 0.1, and A OR B has probability 0.3, but that just seems crazy. I guess what convinces me is dutch-booking myself; isn't a dutch books argument precisely an argument that another set of probabilities would be more likely to be true?

*Another question: above I used propositional logic, which is okay, but it's not exactly the creme de la creme of logics. I understand that fuzzy logics work better for a lot of things, and I'm familiar with predicate logics as well, but I'm not sure what the interaction of any of them is with probability or the use of it, although I know that technically probability doesn't have to be binary (sets just need to be exhaustive and mutually exclusive for the Kolmogorov axioms to work, right?). I don't know, maybe it's just something that I haven't learned yet, but the answer really is out there?

I'm not aware of any flaws with propositional logic. If you reach a problem you can't solve with it then by all means extend to something fancier.

Those are the only questions that are coming to mind right now (if I think of any more, I can probably ask them in comments). So anyone? Am I doing something wrong? Or do I feel more confused than I really am?

I think you're trying to be too formal too fast (or else your title isn't what you're really interested in). Try getting a solid practical handle on Bayes in finite contexts before worrying about extending it to infinite possibilities and the real world.

I'll try to give you the formalist perspective, which is a sort of 'minimal' take on the whole matter.

Everything starts with a set of symbols, usually finite, that can be combined to form strings called formulas.
Logic is then defined as two set of rules: one that tells you which strings of symbols are considered valid (morphology), and one that tells you which valid formulas follows from which valid formulas (syntax).

Then there's the concept of truth: when you have a logic, you notice that sometimes formulas refer to entities or states of some environment, and that syntactic rules somehow reflect processes happening between those entities.
Specifying which environment, which processes and which entities you are considering is the purpose of ontology, while the task of relating ontology and morphology/syntax is the purpose of semantics.

As you can probably imagine, there are a myriad of logics and myriads of ontologies (often called models).
There's Hilbertian ontology, where the environment is a set or a class, the entities are its members and relations and functions between them, and semantics relates Tarskian truths to other Tarskian truths.
There's categorial ontology, where formulas are interpreted as object of a category and syntactic rules as arrow between them.
There's dialogical ontology, where formulas are states of a game and rules are ways for attacking or defending the current state.
There is possible world ontology, which you have described.
And so on and so on.

Hystorically, a specific set of rules and symbols emerged exceedingly often, and seemed to be particularly apt to capture the reasoning that mathematicians intuitively adopted. That is now known as classical logic (CL), and for a very very long time it was believed to be the only "true" logic, that is the logic of the atemporal and universal description of all things.
CL is very useful and can be adapted to a wide array of ontologies: for example, Boolean algebras (which are instances of the Hilbertian ontology) and possible worlds semantics. The latter in particular was ideated for an extension of CL, both in symbols and rules, known as modal logic.

Enter probability.
The determination of which concepts the word refers to has a long and heated history, but now the modern understandig is four-fold. On one side, you have those who refer to a stated property of the world, a not very well specified "long-run randomized frequency" (frequentist). On the other side, you have those who believe that probability does not refer to an objective property of a system, but to a degree of beliefs of an observer (Bayesian). Bayesians themselves are divided among those who thinks that probabilities are entirely subjective (subjectivist), relying on the De Finetti coherence theorem and pragmatic interpretation, and those who thinks that probabilites are the degrees of belief that an idealized rational agent has about a system. These folks we can call objectivist, but since they are the majority here on LessWrong, it's simpler to just call them Bayesian.
Bayesians rely on Cox theorem, that justifies the structure of probability from a small set of minimally rational requirements.
All three of them, frequentists, subjectivists and Bayesians, believe that the structure of probability is correctly described by the mathematical concept of a measure, as formalized by the Kolmogorov axioms.

Enter Jaynes.
He was a physicist, and wrote a book from a Bayesian perspective, showing that probabilities thus intended are an extension of CL. Where CL can be seen as assigning to proposition only two values, 0 and 1, PTEL (probability theory as extended logic) actually relaxes that restriction, assigning values in the [0,1] interval, following two simple rules derived from Cox theorem.
In his work, Jaynes greatly systematized and simplified the field, solved many of its paradox and brought probability theory to previously unexplored areas of application. He also talked about the ideal rational agent as a 'robot', it is thus no surprise that his book is regarded as a cornerstone for LW's understanding of Artificial Intelligence.

Now, on to your questions:

Does the above explanation differ from how other people use probability?

As you can see, you are just using one ontology (possible worlds) to justify one interpretation (Kolmogorov measure), but there are many more. The interpretation of choice here is PTEL, usually coupled with an Hilbertian ontology for the underlying CL or just left unspecified.

why should those estimations follow the laws of probability?

Because of Cox theorem, you can show that with a specified amount of initial information, you can do no better than following the laws of probability.

And if I understand right, they says that Bayes' theorem follows from Boolean logic, which is similar to what I've said above, yes?

Actually no. First, there is no Boolean logic: there is classical logic interpreted in the ontology of Boolean algebra, but that doesn't really matter. Cox theorem is based on CL, but makes additional assumptions (that of a minimal, ideally rational observer). It's not just pure logic.
Also there has been some exploration in the direction of extending Cox with other basal logics.

I'm familiar with predicate logics as well, but I'm not sure what the interaction of any of them is with probability or the use of it

As per above, PTEL is based only on classical logic, which is a particular first order, two-valued predicate logic. AFAIK, no successful extension of Cox has been made to other kind of logic.
Fuzzy logic resembles PTEL in the expansions of the set of truth values, but uses different rules than CL, so the resemblance is only superficial: PTEL and fuzzy logics are two very different beasts.

sets just need to be exhaustive and mutually exclusive for the Kolmogorov axioms to work, right?

No, not really. Kolmogorov axioms are defined on any (sigma)-algebra, wether it is the algebra of subsets of a measurable sets or some other things.

the answer really is out there?

Of course it is ;)

It's a quirk of the community, not an actual mistake on your part. LessWrong defines probability as Y, the statistics community defines probability as X. I would recommend lobbying the larger community to a use of the words consistent with the statistical definitions but shrug...

In The Princess Bride, Vizzini loses his contest of wits with Wesley because he doesn't question his assumptions, which leads to him dying (and falling over comically). It's not super high-level, but it could be a useful talking point, given that it's a popular movie and relatively kid-friendly.

As a variation on the explicit "I'm ending my talking turn", maybe use Aesop-style summaries as the marker, using some kind of agreed-upon lead-in, like "and that's why..." That could be useful for a number of reasons:

~It would definitively mark that the speaker was done talking

~It would encourage one's thinking to be more focused. If you can't briefly sum up what you've been saying, then maybe you've been trying to say too much at a time. This isn't always a problem, but it could lead to your conversation partner missing something.

~It serves to signal what the speaker themself thinks is important or more tangential, which may differ from the ranking the listener inferred. This could become a talking point in itself, either for clarification or for a more meta analysis.

Thoughts?