The set analysis you proposed doesn't work in my opinion, because it implies a law-governed universe whereas a universe that's not law-governed could still have causality.

Causality obtains simply when:
(a) one event, x, precedes another, y,
(b) x and y are contiguous (in terms of tactility),
(c) with respect to (a) and (b), x exerts forces that produce y, and
(d) y would not have occurred without (a), (b) and (c) being the case.

we call every situation where a dynamics inside the set A is always followed (wrt time) by a dynamics in the set B as "A causing B" — Babbeus

In your model universe and by your definition, everything causes everything that follows. That is, any given set of events cause any given events that follow (assuming that your universe is also non-relativistic, so that "follow" is objectively given). A butterfly flapping its wings in China causes a hurricane in the Caribbean, as long as one precedes the other.

x exerts forces that produce y — Terrapin Station

"x produces y" is a paraphrase of "x causes y". Your definition is even worse than circular: it appeals to a narrower notion of causation (one that specifically relies on forces) to explain causation simipliciter.

"x produces y" is a paraphrase of "x causes y". Your definition is even worse than circular: — SophistiCat

My entire definition is a synonym, only much wordier, of course, of the word "causality."

If you propose a definition that is not a synonym of causality, then it's no definition of causality. Your definition in that case would either (a) not capture some necessary feature of causality, or (b) propose additional features that are beyond what causality is.

So your criticism actually would underscore that you don't understand how definitions work in the first place. Definitions are necessarily circular in the sense that you're using that term, or they're no definitions.

Re "a narrower notion", that would be (a) in your opinion. However, in my opinion, anything broader than that is (b)--it's proposing additional features that are beyond what causality is.

"Produces" is indeed a synonym for "causes" in this context. Thus we can plug in "causes" in place of "produces" and the problem should then become obvious:

x causes y just in case A, B, C, ..., x causes y, D, E, ...

Again, definitions are synonyms, or they're not definitions. The synonym can be one word or many. It's not as if something isn't a definition just because it's a one-word synonym rather than being many words.

In your model universe and by your definition, everything causes everything that follows. — SophistiCat

In my model causality is defined between two sets of states, not between two states. It wouldn't make sense to say that the state A causes the future state B. When we say (according to my definition) that the impact causes the crash we mean that "impacts" are always followed by "crashes" (which is in fact not completely true).

Causality obtains simply when:
(a) one event, x, precedes another, y,
(b) x and y are contiguous (in terms of tactility),
(c) with respect to (a) and (b), x exerts forces that produce y, and
(d) y would not have occurred without (a), (b) and (c) being the case. — Terrapin Station

What does it mean that an event "exert forces"? We know that electromagnetic forces always exist between charges: they are not "triggered", does it mean that there is no causation when electromagnetic forces are the only one involved?

What does it mean that an event "exert forces"? — Babbeus

Basically, matter comes into contact with other matter and influences it.

We know that electromagnetic forces always exist between charges: they are not "triggered", — Babbeus

I wouldn't say that forces exist between charges if matter is not coming into contact with other matter and influencing it.

I wouldn't say that forces exist between charges if matter is not coming into contact with other matter and influencing it — Terrapin Station

Electrostatic forces for examples are described by Coulomb's law which doesn't require any contact and predict a non-zero force for any finite distance.

We'd be getting into whether I agree with taking particular interpretations of instrumental utilities to be correct ontologically (and the answer is that I don't with respect to quite a few normal interpretations with respect to the sciences) . Do you want to move the conversation in that direction? I wouldn't say that moving the conversation in that direction will have anything to do with causality in a general sense if that's what you want to discuss.

Donald Hoffman is a Game theorist who spent ten years running one computer simulation after another and studying the latest revelations on the neurology of the brain before finally concluding that, according to all the evidence, if the human mind and brain had ever resembled anything remotely like reality we would have already become extinct as a species. Which, of course, means that his discovery can be considered just more self-contradictory nonsense along the same lines as quantum mechanics that, nevertheless, appears to be much more useful than anyone's silly ideas about reality and causality. In recent decades these zingers have kept coming nonstop with the scientific evidence rapidly accumulating that life, the universe, and everything revolves around bullshit, or what's missing from this picture, and what we consider real and unreal, true and false, serious and funny, existent and nonexistent, causal and acausal, simply depends upon the context.

Without realizing that they had corroborated this, a group of mathematicians examining all of classical mathematics and physics concluded that any number of simple metaphors can describe physical causality equally well. If you want, you can take your pick from among countless explanations to describe everything as merely consisting of black holes, balls of string, bouncing springs, clockwork, or lotions in motion on vibrating rubber sheets for all I know! Along these same lines, a similar mathematical study established that only two dimensions are required as if, in reality, life were a cartoon. Like Donald Hoffman before them, the mathematicians have rigorously confirmed is that academia has traditionally relied upon what I like to call "Cartoon Logic" which is the logic of small children who will adopt whatever rationalization happens to appeal to them more at the time or conflicts less with reality as they know it.

We are using nature to study nature, while the void laughs back in our faces without the slightest compassion for all of our hard work, sacrifices, good intentions, deeply held beliefs, and thousands of years of progress and tradition.

Again, definitions are synonyms, or they're not definitions. — Terrapin Station

I am not sure why you are dwelling on this. If I accept a definition, then I henceforth take the definiendum to be synonymous with that definition. But I cannot accept your definition because it is defective, for the obvious reason that I have pointed out. I don't know what else to say.

In my model causality is defined between two sets of states, not between two states. It wouldn't make sense to say that the state A causes the future state B. — Babbeus

On second thought, I am less clear about what you are trying to say. I guess you are trying for what's called type-causation (typical, lawful, repeatable effects result in like causes), but I am not sure. If you are interested, I suggest you read about some already existing attempts to make sense of (or even to dismiss) causality, e.g. here. Perhaps you'll find something similar to your idea.

I am not sure why you are dwelling on this. — SophistiCat

Because your criticism, which persisted after I explained the problem with it, evidenced that you don't understand how definitions work.