Btw, if one assumes the existence of the infinite in mathematics, an event CAN be classified as having both a probability of zero of occurring, and still be possible.

Let us, in the spirit of the attempt, use the greek letter Epsilon - usually left for the extremely small, as the variable assigned to said probability.

There would then be 6 possibilities to consider:

1) Epsilon=0, and no deities exist.

2) Epsilon=0, and at least one deity exists.

3) Epsilon is between zero and one, exclusively, and no deities exist.

4) Epsilon is between zero and one, exclusively, and at least one deity exists.

5) Epsilon=1, and no deities exist (Talk about false hope)

6) Epsilon=1, and at least one deity exists.

If the question is false, it would eliminate a small sliver of the possibilities. Really if any of the six could be eliminated, or certainly determined, I'd like to see where it would lead?

While the question might not be determinable in any sense, obviously, I'd just like to hear takes on this! Hopefully something of value has been added?

***Upon further thought, I believe that epsilon would've equaled zero in Bertrand Russell's brand of agnosticism! He used an example of a teacup floating in space, and the odds that it would be there. Had the man put more time and energy into it, the examples would've gotten even more bizarre! In mathematical terms, the probability, as he would've put more and more time into it, would've APPROACHED zero while whatever was under consideration would've TECHNICALLY remained possible! This in itself is a question of whether the characterization of Bertrand Russell's theological views can be described in this way!?

  • As with most arguments about dieties, its all about definition. I'm a dedicated atheist, but even I would put the probability of a simulation-sysop type deity at considerably higher than 0. – Lee Daniel Crocker May 2 '18 at 17:23
  • I feel at least from a pragmatic perspective, that, at the least, the probability has been getting CLOSER to zero, and is OVERESTIMATED still currently. Perhaps it is where I live? – Paul Burchett May 2 '18 at 17:34
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    Hi, welcome to Philosophy SE. Please visit our Help Center to see what questions we answer and how to ask. "I'd just like to hear takes on this" questions that invite personal opinions are off-topic on this site, we are taking more specific and pointed questions that are more or less objectively answerable. There is no probability that can be assigned to an event when there is no meaningful sample space available, which is the case here. – Conifold May 2 '18 at 18:23
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    "questions that are more or less objectively answerable" Then why is there a theology topic at all on this board, if we are to concern ourselves with what is solely objective? I offered a disection, and 6 possibilities, for at least some. Some questions are extremely difficult to keep under 150 characters, you know! – Paul Burchett May 2 '18 at 20:47
  • This is not a message board but a moderated Q&A site. The focus is on questions addressed in existing literature, and probability of deities is not a topic that theologians or philosophers would typically address, especially in such combinatorial form. You may be able to connect it to something philosophical but as is it looks more like a very loosely phrased math puzzle. – Conifold May 2 '18 at 22:28

As defined in mathematics no. If you work on the numbers, you find that even when you account for the idea of infinity, a probability of 0 still means something simply cannot occur. What you are looking for are infintessimals, which do not exist on the real number line.

That being said, the mere concept of "probability that gods exist" does not actually have real meaning. Typical understanding of the world argues that there is exactly one world and the phrase "Gods exist," if it has a truth value, is either true or false. It isn't a probability. For it to make sense of talking about probabilities, we would need to think of our world as a draw from a cosmic random variable, which is typically not how people think about it. It would immediately beg the question of how you constructed this random generator of universes from which you draw probabilities.

Bayesian inference can do something very similar to what you describe. There can be no evidence for gods, but due to your prior, the probability of gods existing can remain non-zero as you update it for new evidence. However, the instant you say "the probability is zero," the meaning of that phrase is absolute and that hypothesis simply cannot occur, by the definition of probability.

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    This is wrong. A nice paradox stemming from the misconception of "probability 0 means impossible" is to consider the case of a random real number X following a uniform distribution on the interval [0,1]. For every real number, the probability of X=a is 0, so it is impossible for X to be any number in the interval [0,1]. But the probability that X is not in the interval [0,1] is also zero so that's impossible too! – user6559 May 2 '18 at 21:13
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    A maybe more intuitive example is that if you flip a freshly minted coin infinitely often, and determine probabilities by frequency counting (taking the asymptotic proportion), then the event "this is the first time the coin is flipped" has probability zero, but it very clearly happens once. – user6559 May 2 '18 at 21:14
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    @Hurkyl You are right, I was thinking of discrete values over yes/no, which is what the OP is looking at. However, you'll find there's funnyness that arises when you explore that uniform version. It doesn't matter how many draws you do, the expected number of times the draw equals a will be 0! – Cort Ammon May 2 '18 at 21:17
  • As for the second comment, "this is the first time the coin is flipped" is really a pretty extreme misuse of probability phrasing. You woudl need an additional random variable to select which event you are processing. Otherwise you're applying statistics to a non-random thing. "The probability that a randomly chosen coin flip is the first flip" would be a valid probabiltiy phrasing. – Cort Ammon May 2 '18 at 21:18

It may depend on how you define both probability and possibility.

If you define them in a similar framework. For example, a thing has 0.2 probability if it is the case in 20% of all possible world*. A thing is possible if there is at least one possible world where it is the case. Then, to say that an event has a probability of 0 would be to say that there does not exist a possible world such that the event occurs in that world. This would thereby deny the possibility of that event at all.

*These conclusions don't depend on a sort of Modal Realism, but I'm merely using the possible worlds as a useful fiction for the purposes of explanation.

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    If I choose a random real number between 0 and 1, I have a zero probability of picking exactly 0.5, or any other exact number for that matter. But it is still very much in the realm of possibility that I select 0.5. It's counter intuitive, but an event with probability 0 does not necessarily mean the event is impossible! – Nuclear Wang May 2 '18 at 17:38
  • This is why I add that it is crucial how one defines each of these terms. For starters, logic possibility is vastly different for mere metaphysical possibility and physical possibility. – BeingOfNothingness May 2 '18 at 18:05
  • Although you'll have to explain what you mean by your example involving real numbers – BeingOfNothingness May 2 '18 at 18:06
  • Pick a number, any natural number. If it is the one I'm thinking of, you get a prize! Assign a probability other than 0 to you choosing the right one, and we've got contradictions. However, still possible! – Paul Burchett May 2 '18 at 21:53
  • Am I missing something? What are the two relata of the contradiction? – BeingOfNothingness May 3 '18 at 11:30

Probability is relative to evidence. It think it would be going too far to say that there is absolutely no probability of God's existence but that's really beside the point. Which is that there can, could or might be zero - absolutely no - probability of God's existence on given evidence or 'current information'. Yet unless the concept of God involves a logical impossibility, God's existence is still logically possible. Zero probability (on current information) is consistent with logical possibility.

But, of course, it's open to you to include the logical possibility of God's existence in 'current information'. But if you do that, the probability of God's existence on current information is, ex hypothesi, not zero.


Yes, it remains possible.

The existence of a deity is a plausible event. Even if it is mathematically improbable, it remains possible.

Impossibility implies zero probability, but the reverse is false: zero probability does not mean impossibility.

  1. Let's say E is an event, P is the probability measure.

E has zero probability of occurring if P(E) = 0.

E is impossible if E=∅.

  1. Let's say you pick an interval of numbers (0-100 or anything else). There are an infinite number of real numbers between 0 and 100. Due to that, the chance that you would randomly pick one of them correctly (let's say you must randomly pick the same one as I did: 64.195831319481958391894832539510313871240[...millions more decimals...]) is zero. However, one of them will surely be picked, so it was not impossible for you to make the pick.

Therefore, the chance that you select in advance the one that I picked is zero, but it is not impossible.

  • Why would the chance be zero? Is it not the case that the chance would be infinitesimal, and merely rounded to zero? – BeingOfNothingness May 6 '18 at 18:07
  • @BeingOfNothingness This is better explained in terms of ranges. The probability percentage you will hit a subrange from x to y is y-x. A single point is but a special case of this principle where x=y; in that case, the probability is 0 exactly. A comment section's too short for a full explanation, so I'll just refer you to the wiki article on almost surely. – H Walters May 6 '18 at 19:11

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