The Argument against Determinism – A Probabilistic Calculation (Column 176)

Article Contents

With God’s help

A supplement to the previous column:

The Argument against Determinism – A Probabilistic Calculation

Introduction

In the last column I pointed out that conclusions forced upon a person have no claim to truth, that is, it is unreasonable to trust them. The reason was that there is no logic in assuming that a system built in an arbitrary way will yield a correct result. I gave an example of a computer whose contents are unknown to me (I do not know the program and do not know who the programmer was), and I enter two numbers into it and ask for their sum. The probability that what comes out is indeed the correct sum is, of course, negligible. There is no reason to assume that some arbitrary system will produce for me exactly what I asked of it. One may say that the probability that the result is correct is roughly on the order of the number of programs that calculate a sum divided by the total number of programs. Something truly negligible, of course. And therefore I wrote there that only conclusions that are the product of judgment deserve trust.

In a footnote toward the end of the column, I examined the application of this principle to the question of determinism itself. On the face of it, what follows from this is that a person who arrives at a deterministic conclusion is sawing off the branch on which he himself sits, because that very conclusion is forced upon him, and therefore, in light of the principle above, it should not be trusted.

But I explained there in the note that matters are not so simple. Even if this conclusion is forced upon him, that still means that he really is compelled (for if not, what is forcing this very conclusion on him?!). Therefore, either way, the conclusion is that he really is compelled. But I rejected that argument because it ignores the possibility that the person erred, namely, that the truth is libertarian and the deterministic conclusion was reached through faulty judgment. My bottom line was that arriving at a deterministic conclusion does not improve the prior probability that the world is indeed deterministic, and therefore there is no significance to the fact that the determinist has reached that conclusion. The question remains, at most, as open as it was before. See there for the full discussion.

The Goal

Here I want to examine this question and explain and sharpen my position using probabilistic tools. I will do this again through Bayes’ formula (see columns 144–5). The reason is that I am looking for a way to calculate the probability that I am right in my conclusion from the probability that I arrived at that conclusion. As we saw in those columns, this is exactly what Bayes’ formula does. It reverses the direction of the discussion. If we have probabilities in one direction, namely, if I am compelled or not, what is the probability that I will arrive at conclusion Y, I can now ask: if I arrived at conclusion Y, what is the probability that the truth is one way or the other (and in particular, what is the probability that I am right). Here, of course, I want to place the deterministic conclusion itself in the position of X.

Definitions

First, we need to define the events under discussion:

X – I arrived at the conclusion that the world is deterministic (that human beings have no free will or judgment).

Y – Determinism. As a matter of fact, human beings have no judgment, and we are compelled in all our conclusions.

I remind you that the notation ¬Y means negation, that is, that the world is not deterministic: libertarianism.

In these terms, my goal is to calculate the conditional probability P(Y/X), and of course also to compare it to the prior probability P(Y). In other words, I want to examine whether the fact that a person has arrived at the conclusion that the world is deterministic changes that prior probability in any way. Or, put differently, whether there is any significance to the fact that I arrived at some conclusion in this issue. I remind you that in the previous column I argued that there is not. I explained there that arriving at this conclusion changes nothing at all, and therefore a person who reaches a deterministic conclusion should ignore it. This does not mean that it is not correct, only that the fact that he reached this conclusion has no significance in the discussion.

The Probabilities in the Forward Direction

First, let us assume determinism for the sake of discussion, that is, that the world really is deterministic. In such a case, the conclusion I reach depends on what is yielded by the deterministic system imposed on me (the one that performs the calculations for me). The probability that an unknown deterministic system will yield the correct conclusion is:

P(X/Y) = ε

In fact, this is a tiny probability (the number of systems that yield a correct conclusion out of all possible systems, or the number of programs that add numbers out of all programs), and for practical purposes one may assume that ε is truly negligible.

Now let us assume libertarianism for the sake of discussion, that is, that the real world is not deterministic. In such a case, my decisions are made through judgment (and not by compelled calculation; see the previous column). The probability of arriving at a deterministic conclusion (which is a mistaken conclusion) depends on the quality of our judgment:

P(X/¬Y) = 1-q

If our judgment has quality (or reliability) q, then the probability that we err is 1-q. If this quality is high, then of course the probability that our judgment will lead us to a mistaken conclusion is low.

Beyond all this, we must use the prior probability that the world is deterministic. This is, of course, a philosophical question that is hard to answer, and it is quite clear that the answer may affect the outcome of the calculation. So let us assume here that, a priori, the probability of a deterministic world is:

P(Y) = p

for some p.

And from this, the probability of a non-deterministic world is:

P(¬Y) = 1-p

We now have all the probabilities in the forward direction, and we are ready to calculate the conditional probability.

The Calculation

Bayes’ formula for two possible states is:

P(Bi/A) = P(Bi)P(A/Bi) / (Σj P(Bj)P(A/Bj))

where Bi represent Y and ¬Y, and A stands in place of X.

Let us substitute what we obtained above, and we get:

P(Y/X) = pε / (pε + (1-p)(1-q))

If we do not want to assume the conclusion, then we must assume that the prior probability of a deterministic world and a non-deterministic world is equal, that is, p=1/2. In that case the expression simplifies, and we are left with:

P(Y/X) = ε / (ε + 1-q)

The Results

If the quality of our judgment in the libertarian case, q, is perfect, that is, q=1, then this probability is 1.^[1] If the quality of our judgment is extremely poor, the probability is approximately ε, that is, a negligible probability.^[2] The most plausible state, as is usual in the world, is that the quality of our judgment lies somewhere in the middle. We of course err sometimes, but we are also right quite often. That is the meaning of our trust in our judgment: it brings us to correct conclusions with a not insignificant probability. In any event, this is much higher than a shot in the dark. As noted, the probability that a compelled system is reliable, ε, is a shot in the dark, and therefore it is very small. As stated, our judgment sometimes errs, and perhaps even not infrequently, but still a reasonable estimate of our chances of being right is something like 50% (q=1/2). In such a case, the conditional probability above is still utterly negligible (almost to the same degree as with completely distorted judgment).

Notice that even if the quality of our judgment is extraordinarily high, equal to about 1-ε, the resulting conditional probability is 1/2. That is, even if we wildly exaggerate the quality of our judgment (something even the most optimistic libertarian would not imagine), we find that the fact that we arrived at the conclusion that the world is deterministic does not increase the probability that it is indeed so. And if we estimate it more realistically, as something around 1/2 (and the same is true for any value that is less than 1 by an amount significantly greater than ε, for example, 999999999/1000000000), the probability that the world is deterministic is immeasurably lower than its prior probability of being so.

Conclusions

The conclusion from this calculation is that our arriving at the conclusion that the world is deterministic significantly lowers its prior probability of being so. Not only does arriving at a deterministic conclusion have no significance, and not only should one not take it into account; arriving at it actually lowers the prior probability that this is the case. Exactly as I said, the determinist is sawing off the branch on which he himself sits.

The determinist can of course claim that the quality of our judgment (even on the assumption that we have judgment) is also negligible. But that is a purely skeptical claim. After all, he too does not cast doubt on what he sees or thinks; he merely claims that this is the result of calculation and not of judgment. That is, the skeptical option will not help him, since it would also negate his deterministic conclusion. Discussions of this kind are not conducted with skeptics.

Therefore, one who opposes libertarianism can at most cling to a skeptical position (as an unsupported hypothesis), but there is no significance whatsoever to the fact that a person has arrived at a deterministic conclusion. Determinism as such is not an option as the conclusion of philosophical thought and argument.

The only way to be a determinist is to argue that, a priori, the probability of determinism is almost 1, that is, that libertarianism is plainly unreasonable. Not as a result of calculation, argument, and conclusions, but as a prior assumption. On that prior assumption, the bottom line from the Bayes calculation is indeed that if you arrived at a deterministic conclusion, you may trust it. But, as stated, this is simply assuming the conclusion, and once again the argument and the arrival at the conclusion have accomplished nothing. That person assumed determinism a priori, and lo and behold, that is exactly what he got. There is no significance to the fact that he arrived at a deterministic conclusion, since he merely confirmed what he had assumed from the outset. This is not the result of the calculation but his assumption. Not a very impressive philosophical argument…

If one does not assume the conclusion, and at least a priori places the two options on equal footing (as I have done here), the bottom line is that a person’s deterministic conclusion has no significance whatsoever. Whoever arrives at it may ignore it. If he assumes determinism, then he will remain with his assumption, and if not, then not. His position is really dependent on his assumptions and not on the arguments in its favor.

A Potentially Confusing Side Note

How should the determinist relate to the calculation and the interpretation I have offered here? Are they too the result of thinking that is forced upon me? This problem accompanies the discussion of determinism, and indeed every discussion of our thinking in general. On the assumption that one is engaging in discussion, these are the conclusions of the discussion. If you are a skeptic and unwilling to conduct discussions, perhaps you are right. But then there is no point in conducting a discussion. If you open a discussion, you must be prepared to accept its conclusions.

1.

Footnotes

1. This is clear, because there is no chance that we exercised free judgment and reached a mistaken conclusion. If our conclusion is deterministic, then that must certainly be because determinism is true.
2. Because then there is a very high probability that our deterministic conclusion resulted from an error in free judgment.