Honestly, I do not think I can point to examples of universes like the ones described by 3 and 4, not least because it is not entirely clear which of the four our own universe is, let alone that there are any others. I suppose I could have tried to recall specific universes in fiction, though the interpretation freedom would still leave it difficult to definitively identify them with the attribute combinations from earlier.
I’m confident I cannot express the ideas in an even less technical way than until now, too. So the followig illustration will, alas, rather be more, technical. I try my best to keep things simple, in the hopes that it will be in total easier to understand and appreciate the difference I am getting at. If by now you do not have any more patience for this sort of thing, I understand. If you do, I ask to bear with me.
The universe’s state is some element of a set of all possible states. For purposes of simplicity we can assign an index to all the states, uniquely mapping them to, say, real numbers.
Setting relativity aside, and what implications it may or may not have on the ontology of time, let’s for the purposes of simplicity consider it just a numerical parameter. Whether time gets to branch could be considered important for questions about the existence of alternative futures and thereby libertarian free will, but for the context of this message of mine, we are discussing fatalism and determinism. I’m sure an extension for more complicated topologies of time could be rendered by even maths-savvier users than I, but I’ll treat it as a simple line here. In fact, I’ll treat time as a discrete parameter, whose values I’ll pick from natural integers, and ignore the consequences of how well that applies to physical spacetime in reality.
So, with time and states characterized as elements of number sets, we can formalize a universe’s history as a function f:\mathbb N\to\mathbb R,t\mapsto f(t) which assigns every moment t\in\mathbb N a corresponding state f(t)\in\mathbb R.
Now we are ready to talk about what fatalism and determinism are in this toy model, going through the same variants, albeit in a narratively hopefully more cohesive order. I note the number of the case from before just after the title:
Fatalism And No Determinism (scenario 2):
Suppose there is an infinite list, like f(1)=s_1,f(2)=s_2,f(3)=s_3,\ldots mapping every time to a specific state. Such a list would completely fix all of our toy universe’s history f. This is fatalism. Pick any natural integer, and there exists exactly one state the universe has at that time. If you ask God to name the state at that time, they can. It is entirely irrelevant if you live in that universe, or what the current time is for you, there is an unambiguous one-to-one mapping between points in time and points in state-space.
However, outside of what sets f maps out of and into, there are no properties demanded of f in this scenario. You ask God what the state at t=7 is, they tell you it’s, say, 68.176. You have learned nothing about f(3) or f(8) this way. The universe still has a very specific state at t=3 and at t=8, to be sure, but it cannot be deduced from a knowledge of f(3), because there are no deterministic rules to link the states to one another through time.
Determinism And No Fatalism (scenario 3):
Suppose there is a condition history needs to meet. Let’s say f is a polynomial function and satisfies \frac{\mathrm df}{\mathrm dt}(t)=m for all t with some specified number m\in\mathbb R. In that case, f(t+n)=f(t)+n\cdot m for all n\in\left\{1-t,2-t,3-t,...\right\}. Knowing what the universe’s state is at some reference time t, one can directly deduce its state at all other times.
However, we did not suppose that the state of the universe itself is specified at any time at all. In fact, assuming fatalism is false amounts to supposing the contrary. There is an equation that tells us how states evolve, but what state it is actually in is not specified for any time. If there is a God in a universe like this, they could not tell you what state the universe was in at any time at all, because there is no definitive answer to that question. If we’re lucky, maybe they could name some range of states, or a superposition (for lack of a better expression) of states, and sure enough one could specify the ranges of possible past and future states from that, but never one definitive state.
Determinism And Fatalism (scenario 1):
Once again let’s suppose the history function f is a polynomial that satisfies \frac{\mathrm df}{\mathrm dt}(t)=m, but in addition it also satisfies f(14)=3.3. Because the states evolve in accord with a deterministic rule, and we have what’s called an initial condition, the state at every time t is completely specified, as if it were non-deterministic fatalism, except now we can also deduce all of that mapping as soon as we know what state the universe has at any one point in time.
No Determinism And No Fatalism (scenario 4):
Now we suppose there are no conditions on f, but also no list of which states it maps to which times. f is completely undefined. Some would say there is no history of that universe, even. This is practically equivalent to deterministic non-fatalism, in that it is just as impossible to find out what state the universe is in at any time. The only difference is that there is no rules the evolution of states follows. If God were to try and specify a range or superposition of states at some referene time t, this would tell us nothing about the range or superposition of states at any other time.