Probability of a deterministic world

Every now and then the discussion at PH has turned to the question: Is our world deterministic or not? I say it is.... and at the same time it isn’t. I mean I find most compelling the kind of a model that determinism exists, but it is not necessary. Once in PH I boasted that I can ”mathematically” prove using David Lewis’s many worlds semantics that the probability for a ”fully determinismic” world is practically zero. Fully deterministic world means here a kind of world where every event is necessarily deterministically linked to each other... So in order not to loose face, I will put my proof here.

But before doing that, I suppose I should first define what I mean by determinism. I would use the definition from Wikipedia: Determinism is the philosophical proposition that every event, including human cognition and behavior, decision and action, is causally determined by an unbroken chain of prior occurrences [1] [2]. I think it would be also important to note that with causation I mean here the philosophical causation, not just a logical causation. In my understanding the logical causation is limited just to formal logical language, which is made of statements like A entails B (A => B), which in words means that if A is true then B is true, when A and B are logical sentences, within the formal language of the logic. But philosophical causation is a more complex thing since A and B are not just sentences of a formal language, but events or states in an actual world (or universe). I have heard at least two kind of definitions for causal relation: B is causally dependent from A (A -> B) means either that
a) A and B are events in an actual world and If A occurs, then B occurs too
b) A and B are events in an actual world and If A doesn’t occur, then B doesn’t either

So now that we are proving something about our world, we would be obviously be talking about causation in a philosophical sense…but which variety a) or b) should we use? Both definitions have their proponents among philosophers... suspicion rises that something fishy is going on with causality….but I think I don’t necessarily need to go into more details about the differences of a) and b) type of causality for my proof (I might be wrong though =).

So here are my assumptions:
1) Only logically possible worlds can be actual worlds.
- It means that I assume that logical reasoning holds in any possible world that we can actually exist. Like 2+2=4 holds for all possible worlds
2) Indeterministic events are logically possible
- which I think entails that coincedental events should be assumed logically possible and we should use modal logic as our formal language

To help thinking how the probabilities could be calculated I hypothesize a kind of a Laplace’s demon (L-demon) that knows every event that has happened and is happening at the present moment in the universe and it has an unlimited, even unworldy, efficient calculation power to solve equations based on the initial conditions.

First let’s assume a “full indeterminism”. It would mean that none of the events A and B in the world are causally related. The fact that things are as they are at the present is just a huge coincidence. I would then say that it’s possible but the likelyhood is 1 / [the sum of all the individual events that have occurred and are occurring at the present], which is a very small number indeed. But if we assume that our actual universe has only a finite age (like for example big bang theory suggests) then there has been only a finite number of events so far and the likelyhood for full indeterminism with our present knowledge is > 0. Let’s denote that number as P[full indeterminism]. Now our L-demon would know what this number is….but since full indeterminism holds also to the future, L-demon cannot say anything about the future! Future is completely incalculably open and all futures are equally possible. Thus the number P[full indeterminism] at the present time is still a finite number. Future does not add anything to that

Secondly we could assume that there are both causally dependent events á la A -> B, which can form long chains A->… -> Z ->…. that coexist at the same time with smaller chains or even individual indeterministic events O that are not causally linked to any other events at all. I would imagine that these events O go actually completely unnoticed within the world, (except that our L-demon would know them). I would say that the likelyhood for the existence of all such worlds among all possible worlds is huge. It could contain an infinite amount of individual causally unrelated events O arranged different ways. the Causal chains could have been combined or separated in a myriad way during the history of universe, yet resulting to the exact state of affairs things are at the present time. The mechanism how the causal chains separate and combine is different depending which type of causality a) or b) you prefer to use, but regardless it is easy to think examples where different series of events lead to same states of affairs as end result (redundancy) or where one event kicks of several causal chains. (For a learned contemporary discussion about this I recommend [3]). So the probability for our actual world being one of such worlds where both determinism and indeterminism coexists is P[partial indeterminism] = (integral over causally linked chains + sum of all causally unrelated finite event chains and single events) – P[full indeterminism] – P[full determinism]. Seems like a huge number….our L-demon again would know what this number and it would be even able to make very good predictions to the future. Even deterministic chaos theory would not bother it’s calculations since it knows exactly all the initial conditions and has an infinite calculation power…but the indeterministic events future that will occur in the future would eventually destroy the accuracy of L-demon's predictions. Little by little the indeterministic events and even chains that pop up into the world will make the demon’s predictions only probabilistic and over time the prediction power will go to zero. Thus the causality holds only to one direction. Past is defined, but future is uncertain and an infinite number of possible futures exist also for all partially indeterministic worlds.

Thirdly how big is P[full determinism] then? As far as the past concerns the likelyhood should be the same as for full indeterminism, in other words 1 / [the sum of all the events that have occurred and are occurring at the present]. A small number, but finite….so far! But if full determinism holds it means that also all the future events are defined. Our L-demon can calculate the state of affairs in any given instance in the future, which means that only one future is possible. But then how to separate the arrow of causation? I doubt that it’s not possible, which would then imply that the probability P[full determinism] at the present moment should be multiplied also with a figure 1 / [integral over logically possible futures] which approaches infinity when time approaches infinity. Thus the probability for P[full determinism] at the present time and according to L-demons calculations should be 1 / ([the sum of all the events that have occurred and are occurring at the present] * [integral over logically possible futures] ….which approaches 0 if the future is infinitely long. Thus it’s even more unlikely than full indeterminism!!!

And I haven’t even treated possibilities that world is deterministic or indeterministic up to a certain time t and then changes….all combinations just push the likelyhood for full determinism further away….QED

[1] internet page, http://en.wikipedia.org/wiki/Determinism , feb 6th 2009,
[2] Van Inwagen, Peter, 1983, An Essay on Free Will, Oxford: Clarendon Press.
[3] Causation and Counterfactuals, edited by John Collins, Ned Hall, L. A. Paul, MIT Press, 2004

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In fact I didn’t come up with the idea for a proof on my own, but it is probably been presented somewhere, since I have heard the elements of this general idea in a discussion….but I haven’t found it written down anywhere before….

Anyways, some say that fully deterministic world is intuitively right for them. But I don’t feel that way at all and I could make a long list of arguments based on our current knowledge of science and mathematics why I think assuming the existence of indeterministic events is far more intuitively appealing to me. Full determinism would imply to me that the whole existence of our actual world is actually only one event with fully determined past and predestined future. If even just one quantum at anytime over the whole lifetime of our world would appear spontaneously – or perhaps leak from another world, it would mean the destruction of the fully determined future for our world. Seems very unlikely to me….infinetely unlikely even!

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Sorry for typos and all….I will publish this already as a draft . So if you see changes later, it is not that I would like to change the content, but just make it more readable.

Quint