Indeterministic Causal Relations

In her paper ‘Causality and Determination’, G.E.M. Anscombe gives an example of an indeterministic causal relation that was mentioned by the physicist Richard Feynman. It goes as follows:

“ . . . a bomb is connected with a Geiger counter, so that it will go off if the Geiger counter registers a certain reading; whether it will of not is not determined, for it is so placed near some radioactive material that it may or may not register that reading” (p. 145) (Link in references).

As Anscombe notes, if the bomb goes off, there won’t be any question that it was caused, most immediately by the geiger counter reading, and that in turn by the radioactive material.

Yet how are we to understand such a causal relation? Intuitively, this causal relation is one where the effect does not have to follow from the cause. And since the effect does not have follow from the cause, whether or not it occurs or not is not determined from the cause. It is caused, but nondeterministically so. Such is the indeterministic causal relation. Anscombe’s answer as to how we are to understand such relations goes along this route, where what is to be denied are necessary connections: “Indeed, I should explain indeterminism as the thesis that not all physical events are necessitated by their causes” (p. 145).

I am going to spend the rest of this article dissecting the intuitive notion that the indeterministic causal relation is one where the effect does not have to follow, and show that it is an absurd notion. What we end with is a necessary connection of a special kind.

That there is a necessary connection, even here, has to be correct if my previous post has anything going for it. There, I observe that given a cause, it’s not that anything just follows. Intuitively, causes place a constraint what occurs–given even radioactive material and geiger counters, there will be only a particular range of radioactive emission that can result (For this post, see The Hallmark of Necessary Connections in Causal Relations ). Now, of course, one particular and narrow reading of the geiger counter will not be necessary. So perhaps this is enough to show that the effect is not necessary after all. Yet I think this is fast and rather fallacious reasoning. Let me next show why this is, and then account for how we are to understand the effect in indeterministic causality.

First, let me present an argument against the intuitive notion that the effect does not have to follow from an indeterministic cause. First and foremost, note that, given some radioactive material, it is not that any particular reading from a geiger counter is the effect of the cause, since there will be different effects, given many occurrences of such a cause. That is, holding fixed the cause, there can be any particular of an extensive number (though confined to a range) of geiger counter readings, and these possibilities manifests into actualities given multiple occurrences of the cause. Once this is noted, it is easy to understand why one or another particular geiger counter reading does not have to follow from the radioactive material, since none of these are the effect of such radioactive material. There is no one effect given radioactive material.

On the other hand, of course there will be one such reading that will occur, and won’t this in turn be the effect, and won’t it be non-necessary holding fixed the cause? Still, obviously not–the claim goes too far– it won’t be the effect, given that many effects follow holding fixed the cause.

With this noted, there is of course something intuitively right about a similar, although substantially different, claim that goes: given that this radioactive material (let’s call it A) can produce the specific geiger counter readings of X, Y, or Z, then neither X nor Y nor Z is necessary given A. Thus, when X occurs, X was not a necessary result of A. This is correct. Yet it’s telling in what way it is correct. Note the language again: “X was not a necessary result”; that is, X was not a necessary effect. This would be in line with the previous observation that A has many effects, not just one. So of course no one effect is necessary. Yet I want to take this idea further. I want to say that, more than being unnecessary, X considered absent a larger causal context of which it is one possibility among others (here, Y and Z) is no effect of A at all. It’s not that X is not an effect–it is–but it is one because it is part of a greater effect, one that necessarily follows from the cause. It is this effect, if any is, that will end up being the effect of A.

To begin illustrating all of this, let’s suppose that X, Y, and Z is exhaustive of the range of radioactivity given A. This is as much to say that given A, what follows will be radioactivity, particularly of X, Y, or Z. Now this is a necessary connection on what follows. That is, it will be this range of X, Y, or Z, and not some other range, and it won’t be something else entirely different. This is my first result, that there is a way to understand what causally results from A in the form of necessary connections. I also think it’s obvious how this result generalizes to apply to any indeterministic causal relation whatsoever, so I will move on from this point.

Next it is important how to understand what I am proposing results from A, this X, Y, or Z. To start, I am using a logical connective, ‘or’. Yet I am not saying that what follows is a proposition. No, what follows is an event. So how can ‘or’ be the proper way to describe an event? This is actually the key to all that I’m saying. First, which version of ‘or’ am I using? Answer: ‘or’ is used in the exclusive sense. I might have used it inclusively, such as when X, Y, and Z overlap for being a general specified range of radioactivity. However, given the prior examples involving X as a non-necessitating effect of A, this is not the sense I meant. To have radioactivity of the X amount is thereby to not have it in the Y amount or Z amount. X, Y, and Z are exhaustive and non-overlapping amounts of radioactivity given A.

Next, I will try and further explain and justify my use of the logical connective. I use ‘or’ as a connective in the exclusive sense between possible events, given a cause. Given an indeterministic cause, the way to describe what results from such a cause is that there is a honing in on one and only one of the possible events that can happen as a result of it. The way that it is honed in is sometimes weighted toward the possibilities in differing ways, such that some have a better chance of occurring than others. This honing in, in either varied or equal weight, is causally irreducible–that is, there is no further causal relation to be found between things such that from such causal relations, there is but one outcome possible for the “indeterministic” cause. Epistemologically speaking, when it’s certain that the causes are accounted for as fixed, and yet the effects are sometimes different, then there is an element of indeterministic causality. Indeterministic cause and effect is a one to many relation. Particular instances of the causes fixed can yield differing results. So I connect such possible events via an exclusive ‘or’ since one and only one of the possible events will occur.

Now, propositions express a truth condition, and the truth condition is satisfied or not depending on whether or not things stand in the way specified by the truth condition. Causes do not, typically speaking, express truth conditions, since causes are on the side of how things stand. This just suffices to say that the disjunction of events connected by my exclusive ‘or’ is not to be understood as a proposition. Rather, it is itself an event. This is fitting, since the connective is used to connect events as atoms and to be itself a composite event, akin to disjunctions being propositions alongside their atomic disjuncts. That when a propositional disjunction is true, one and only one of its disjuncts is true is akin to the fact that when a disjunction that is an indeterministic event occurs, one and only one of its disjuncts will occur.

The largest divergence from a propositional exclusive ‘or’ comes from the fact that X, Y, and Z are, as I said, weighted, probabilistically speaking. If a propositional exclusive disjunction is true, then this is not because we are, speaking loosely, to roll a die to determine which disjunct is true. It is true in virtue of the fact that for one and only one of the disjuncts, how things stand is how things are expressed by the disjunct. If no disjuncts are true, or if more than one is true, then the disjunction is false. So this is a notable difference. Let me explain the difference on the side of the indeterministic cause.

There is a probabilistic weight given to X, Y, and Z in connection with A, even if the weight is evenly split between them. That one and only one of them will occur is certain, and is weighted at a probability of 1. That the event as a disjunction of events has a probability of 1 is just to say that there are a range of possible events given the cause, which can be separated into disjuncts, of which one and only one will occur when all’s said and done. What does not have a probability of 1 are the constituents of the disjunctive event. They are constituents since their combined probabilities add up to 1. Weighted evenly, the probability that either of X, Y, or Z occurs is 1/3 for each. There is a weight given to each one, evenly distributed. And when the weight is unevenly distributed, then they will each be unevenly disposed accordingly among them to be the one to occur. The way I think to describe such an event is that given A, X, Y, and Z are evenly (or varyingly) disposed to occur. The disposition, however, is once more understood irreducibly. If you ask what me is meant by ‘irreducible’ here, it’d have to be something that is not just itself some sort of unseen determinism. Reflecting what Anscombe said, “It is difficult to explain this idea any further” (p.145). However, perhaps I can shed more light on the thought of an irreducible indeterminism by turning to when a cause that is thought at first to be indeterministic may well turn out to be deterministic. If there are a set of causes in a vicinity, and they are thought to cause a range of events when fixed, and so thought to be indeterministic causes, but then there are found to be yet more causal workings within what we took to be the causes, then, while perhaps fixed, the causes were not fully accounted for. The original causes were made up additional causes as parts which, depending on how they factor in as causes, may account for the different effects. So when I say that the causes are fixed, I take that to include any causal constitutive components of them as well. If the causes are indeed fixed in this way, and yet many results are possible, then this is indeed an indeterministic causal relation, one that will not succumb to a deterministic reduction.

Given A, there is a honing in on one and only one of X, Y or Z, in the way explained above. I symbolize this disjunctively as, given A, (X or Y or Z). If X results given A, then X is an event whose appearance is explained causally. If we are to explain how X came to be, one way to explain it is that it came to be, causally, because of A. But if we further ask why it came to be causally because of A, the proper explanation is that, given A, there is a range of possible and probabilistically constrained events, of which one and only one must occur. That is, understanding how X came to be is not just to understand a causal relation involving A and X alone, but also of X as part of a larger effect of A of which it is a part, and which occurred because one (and only one) of them has to. It fell to X and not the others due to an irreducibly indeterministic move on X, given probabilistic propensity in its favor in relation to the rest (indeed this idea is difficult to explain “further”, but I think this does service enough!). X, Y and Z constitute (X or Y or Z), where they are disposed probabilistically and where one and only one must occur. If this is a complete description of what occurs given A, then I consider this a big win. This indicates that we have found the effect of A after all! And indeterministic causal relations can be understood in terms of necessary connections.

Given A, the effect is the disjunctive event of the sort described above, (X or Y or Z).

Resources

Anscombe, G.E.M. Causality and Determination. 1971. URL:               http://www.calstatela.edu/sites/default/files/dept/phil/pdf/res/anscombe.pdf

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