Causality: A Rigorous Examination

One of the most important concepts, one underlying science and any model one would use to predict outcomes in daily life, is that of causality. But what is causality, and what can we say about it?

As in any good rigorous examination of ideas, we begin with definitions:

The start state of a possible world is simply the precise arrangement of elements at the time we begin examining it (however, an imprecise version of the start state is also used (called, imaginatively, the "known start state" as opposed to the "true" start state), in which the start state is simply the totality of information we have on the actual start state). The outcome set of this possible world is the set of possible arrangements at the time we finish examining it, weighted by probability. The outcome is the actual arrangement at the time we finish examining.

Example: Suppose we consider a very simple universe consisting only of a number of x-dice (x is allowed to vary over the set of dice but is constrained to always be a natural number); each die has x+1 possible states: it can either be still and showing any of the numbers from 1 to x, or can be in the process of being rolled. This universe moves forward by discrete ticks, with rule that still dice do not change ever, and rolled x-die returns a random number, evenly weighted, from 1 to x. Now suppose we have the following start state: there are 3 indistinguishable 2-dice that are being rolled, a 3-die that is being rolled, and a 12-die that is still and showing 8. The outcome set of this world after one 'tick' is as follows (fractions in parentheses denote weight - these must add up to 1; other numbers are formatted in the following way (3*2[1], 1*3[3]) = "three 2-dice showing 1 and one 3-die showing 3"):

{

(1/24) (3*2[1], 1*3[1], 1*12[8]);

(1/24) (3*2[1], 1*3[2], 1*12[8]);

(1/24) (3*2[1], 1*3[3], 1*12[8]);

(1/8) (2*2[1], 1*2[2], 1*3[1], 1*12[8]);

(1/8) (2*2[2], 1*2[1], 1*3[1], 1*12[8]);

(1/8) (2*2[1], 1*2[2], 1*3[2], 1*12[8]);

(1/8) (2*2[2], 1*2[1], 1*3[2], 1*12[8]);

(1/8) (2*2[1], 1*2[2], 1*3[3], 1*12[8]);

(1/8) (2*2[2], 1*2[1], 1*3[3], 1*12[8]);

(1/24) (3*2[2], 1*3[1], 1*12[8]);

(1/24) (3*2[2], 1*3[2], 1*12[8]);

(1/24) (3*2[2], 1*3[3], 1*12[8])

}

I apologize for the overly complicated example, I seem to remember thinking it wouldn't look this bad. But you can see that (a) the fractions in parentheses do indeed add up to 1 and (b) every outcome (part after the weighting) is unique and it comprises the possibilities and probabilities that result from the scenario I described.

Now, what is a cause? A cause is simply an alteration in the start state of a universe. Its corresponding effect is a change in the outcome set of said universe.

Now let us examine what Free Will means in this context: Free Will states that outcomes can change independent of determinist and random factors. However, this terminology of cause and effect essentially covers determinist and random factors, leading me to conclude that in the real universe in which I (and thus Free Will) exist, there exist effects without causes. This is a baffling result to me. It means that with identical start states, we can have two different outcome sets. But using the traditional definition of probability, we see that two identical start states should necessarily have the same probabilistic outcomes, since the traditional definition of probability is "we run the experiment an infinite number of times; what is the density of result x? That density is the probability of result x".

Well, shit, it seems I have trapped myself in a logical contradiction and that some part of my theory will have to break. Let us go over my axiomatic system and see where I can change my system - I will either have to change my axioms or change my definitions, which are all basically axioms anyways.

Axiom of Axioms and Axiom of Logic: ...no, can't change those, because without those there are no other axioms, there is no logic, and there is no system.

Axiom of Utility: ...no, can't change that either in my system, since I quit philosophy if I become convinced that the Axiom of Utility is false.

Assumption that there always exists a start state... hmmm... it's true that by Heisenberg (Heisenberg to the rescue, as always) matter intrinsically has no location if its velocity is precisely known and vice versa (Heisenberg is NOT just a limit on our measurement capability) - this means that we are limited to start states with missing information...

In cases where a precise start state is not known, a change in start state can fly 'under the radar' of our knowledge and still have an effect. What about in cases where some information is physically nonexistent? Can we have changes in information that isn't there in the first place? It doesn't make sense, really, since this probability cloud ought to be the start state itself, but I am forced to conclude that either (a) this doesn't make sense and thus the Axiom of Utility is false, so I should quit philosophy, or (b) the probability cloud of a particle can change in physically immeasurable ways based on an unknown process ('process' = wrong word; can't really think of the right one now), thus changing the outcome set without changing our start state. I conclude (b) until I am forced into (a).

I know this doesn't make sense. It's like a free gift to my perennial opponent on the free will debate, since I just uncovered a serious weakness in my system and published it (the other option is intellectual dishonesty, good if you want to get something and are arguing for it but bad in philosophy where I want to strengthen my system - it's not going to patch up its own holes if I just shut my ears off and pretend I don't notice the holes). However, my logic that I should still believe in my own free will is rock solid, so I keep up my belief (hell, it's like Pascal's Wager, only without the bullshit theology, false dichotomy, or implication that I should automatically trust someone who promises infinite happiness). I'll study quantum mechanics and come back with an answer that makes sense later.