More logic: Indeterminate probability and the conjunction paradox

Source: PixaBay / PIRO4D

Aka, God and the conjunction fallacy. I recently participated in a survey aimed to gauge rational thinking against supernatural beliefs. One of the questions raised an unusual situation that I will discuss in this post.

Firstly, the question went something like this:

Q: Which has the greater probability of being true?
A) At least one god exists and the earth is approximately spherical
B) At least one god exists

I answered A, and the quiz marked my answer incorrect because of something called the Conjunction fallacy. In this article, I will discuss the conjunction fallacy and then the effect of including an indeterminate case like the probability that god exists.

A conjunction is a statement that requires two (or more) things to be true in order for the whole statement to be true. Think of it as X AND Y must be true for a statement to be true. A conjunction fallacy takes the form that:

Q: Which statement is more likely to be true?
A) X
B) X and Y

Well, obviously B right? Well no, it is actually A. We usually think that multiple terms have a greater chance of being true, but that is misinterpreting the logical AND for an inclusive OR. In mathematical terms, a logical AND of probabilities has the effect of multiplying probabilities. However, think about it for a second, B requires two things to be true at the same time.

So far, so good. It looks like I answered incorrectly, except it matters that God was the B term. Umm what?

My personal beliefs are deeply agnostic, which means I believe that the existence of God is unknowable. I am so deeply agnostic that I refuse to attach any probability to the existence of God. Probability is a real number between 0 and 1 inclusive and my conception of what it would take to make a God means he would have to transcend numbers. I think it is a category error to assign a probability to the existence question. What’s a category error? Well, in this particular case it means to try and measure something using a scale that doesn’t work. Imagine trying to measure your weight in kilometres; it just doesn’t work.
You may have heard of the Dawkins Scale which measures theistic probability. Here it is, from the God Delusion.

1. Strong theist. 100% probability of God. In the words of C.G. Jung: "I do not believe, I know."
2. De facto theist. Very high probability but short of 100%. "I don't know for certain, but I strongly believe in God and live my life on the assumption that he is there."
3. Leaning towards theism. Higher than 50% but not very high. "I am very uncertain, but I am inclined to believe in God."
4. Completely impartial. Exactly 50%. "God's existence and non-existence are exactly equiprobable."
5. Leaning towards atheism. Lower than 50% but not very low. "I do not know whether God exists but I'm inclined to be skeptical."
6. De facto atheist. Very low probability, but short of zero. "I don't know for certain but I think God is very improbable, and I live my life on the assumption that he is not there."
7. Strong atheist. "I know there is no God, with the same conviction as Jung knows there is one."

Dawkins places himself as a 6.9. I cannot place myself on this scale because I cannot assign a probability to the existence of God. Now, how does this affect the conjunction fallacy?

Assigning a probability to the existence of God is a category error, and this undid the logical conjunction. Undoing the conjunction (ignoring the existence of god clause) collapsed the question to a comparison between nothing and the earth being roughly spherical. The question of whether nothing is more probable than something that exists is a philosophical one.

A responder did say that we should ignore the indeterminate probabilities:

I'm not sure I agree, I would have thought that if you have some gods for which you can assign a probability, and some that it is indeterminate then for those indeterminate ones you would leave those out.

Leaving out existence probabilities for members of the set of gods means we are not considering the existence of those gods whose probabilities are excluded. That will not give a complete answer to the clause.
However, as soon as the big G kind of god is includable in a set of some (at least one) gods, then the probability of the existence of any random member of the set becomes indeterminate due to the category error that goes with the big G god.

The responder disagreed:

I'll give a reduction to the absurd.
There exists at least 1 proposition in the category of natural objects for which the probability is indeterminate; therefore the probability that at least one natural object exists is indeterminate.

I abstract the situation to highlight how my interpretation differs from the responder:

P1 The set S has two members: S = { Y, Z }
P2 Each member of S may have an attribute "x" with a probability P().
P3 The probability of Y having the attribute "x" is a specific real number between zero and one inclusive: 0 =< P(Y.x) =< 1
P4 The probability of Z having the attribute "x" is indeterminable: P(Z.x) = indeterminable

So, what is the probability that at least one member of S has the attribute "x"?
A) Ignore P(Z.x), answer = P(Y.x)
B) A real number cannot be compared with an indeterminable value, answer = MAX( P(Y.x), P(Z.x) ) is indeterminate

I think the responder intends A, while I hold that B is correct. However, if I was doing the same thing and all set members had determinable existence probabilities, then a comparison is possible, and that comparison is to take the maximum.

Set R has two members{ U, W } with the probability P() that the member has attribute "x" (mirrors P1 and P2)
P(U.x) = 0.7, P(W.x) = 0.5
The probability that at least one member of R has attribute "x" is the maximum of 0.7 and 0.5, which is 0.7.

Applying that back to set S, a real number cannot be compared to an indeterminable value so determining a maximum is not possible and the "at least one" is then indeterminable. This matches with my interpretation, which is B.

I abstracted the responder’s case for an important reason; that we might be distracted because we know that some natural objects exist with 100% certainty. If we can know that some member of the set has a determinable probability of 100%, then we can ignore the indeterminate probabilities of other set members. Think of it a bit like an indeterminate probability could be represented by a range between 0-100% inclusive and this indeterminable probability cannot be greater than the other member that has a 100% probability, so yeah, in that case, we could determine the probability.

The reverse case is also true: if asked for the probability that at least one member of a set does not exist and if we knew for certain that one member of the set did not exist then we can answer affirmatively even if the set contains indeterminate existence probabilities.

I think the distraction is why the responder labelled their case as absurd because we know with 100% certainty that some natural objects exist. However, absolute certainty is a special case when dealing with indeterminate existence probabilities. So, where does this lead us?

The probability that at least one god/God exists is going to be either indeterminate or 100%. “At least” demands the maximum. If we know with absolute certainty that at least one god/God exists then we can ignore the gods/Gods that have indeterminate existence probabilities. But, since we do not know this with absolute certainty, then the existence of God is indeterminate.

So, somewhat complicated, but I hope you were able to follow.