# Does every possible event have non-zero probability?

Almost every human being would agree that 2 + 2 != 5. In a sense, this is a logical impossibility.

However, almost every human being would also agree that pigs can't fly. Some, however, are adamant in suggesting that this event still has a non zero probability. But why is this? Secondly, what does it mean for it to be the case? If this is by definition breaking a physical law, shouldn't its physical probability be 0? If it is not 0, what should its probability be?

Let's say that one asserts that the probability of pigs flying is 1 in 10^50. I can then come up with an example that has the same probability: a person predicting a number that is spit out by a random number generator that generates a random number between 1 in 10^50. Of course, the difference would be that predicting a number between 1 in 10^50 would not break any known physical laws. But pigs flying would. No matter how low of a number you come up with to represent the probability of pigs flying, I can come up with a similar example with the same probability except that the example would not break any physical laws.

So what is the correct answer to this? Or is this very question assuming that there is a correct answer? After all, in a way, the concept of probability itself seems manmade and subject to interpretation as compared to say the laws of physics.

• No. If the sample space is infinite possible events can have probability zero. The probability of breaking physical laws can be non-zero when logical possibility is considered in epistemic contexts. How small depends on how well the law is established, our knowledge is not absolute. When physical possibility is considered it will not just be zero, but the breaking is altogether impossible, simply by definition. The "correct answer" only exists after you specify the sample space and the probability measure on it, and that is "subject to interpretation", specifically, modeling what is known. Feb 19 at 6:20
• "Probability" is a problematic concept in philosophy and I don't think an answer can be satisfactory without addressing this head-on. See plato.stanford.edu/entries/probability-interpret
– usul
Feb 19 at 18:28
• There is an analogous problem for probability of one: stats.stackexchange.com/questions/590861/… Feb 20 at 1:13
• Asserting that the probability of a pig flying is 1 in 10^50 implies a physical possibility. If you assert that it is a physical impossibility then that implies the probability is 0 Feb 20 at 10:19

The answer is no. Mathematically, if you have a continuous random variable, the probability of getting any one of its values is zero, but you can still get one, so zero probability does not necessarily imply impossibility.

However, impossibility does imply zero probability. When you roll a conventional dice in the conventional way it can only land face up bearing a number between one and six- there is zero probability of it bearing the number twenty seven, for example. This is because the probability space for the experiment consists of what is called a sigma-algebra (set of subsets) over the set of possible outcomes Omega = {1, 2, 3, 4, 5, 6}, and only subsets of Omega may be assigned non-zero probability.

As for pigs. If you take the saying at face value, and ignore pigs in planes, pigs whipped into the air by hurricanes etc, it is impossible for a pig to fly, so the probability of a pig flying is zero.

• Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed. Feb 22 at 11:51
• I think your first two paragraphs are great. The answer is clearly no: impossible implies zero probability, but the converse is not true. I don't understand how you can state so confidently that pigs can not fly however. For the first 15 years of my life I didn't know that earwigs could fly and would have confidently given your same answer that the probability of an earwig flying is zero. Later I learned they could, and now I would give an answer of "a probability very close to 1" for earwigs being able to fly. How can you be so sure that you can claim probability 0 for pigs flying? May 17 at 11:49
• @DrPhil millions of anatomical experiments are conducted on pigs every day, so far no evidence has been found for any anatomical structure in pigs which could enable them to fly Jul 7 at 13:45
• The idea that physical quantities have exact values is quite contentious, to say the least. All we have is probability distributions for their values, becoming ever narrower as our measurements improve.
– user66959
Jul 24 at 11:07

In Bayesian theory, the principle that every logically possible proposition has a probability that is never exactly zero, nor exactly one, is called the Cromwell Rule.

The idea behind the principle is that if you interpret the probability of a proposition as a degree of credence and use Bayesian conditioning to update it, then a probability of zero or one could never be updated to anything else. A tiny probability might be revised to a large one with sufficient evidence, but a zero probability cannot. So, assigning a zero probability amounts to saying: I hold this to be false, and no amount of evidence could ever make me change my mind.

This is unsatisfactory, since it is always possible that we are mistaken. Even in the case of logical truths, one might argue that it is still appropriate to use probabilities other than zero or one, since we are not logically omniscient and we can make mistakes in the logic. In the case of flying pigs, and other physical laws, it is possible that our current understanding of those laws is inaccurate or incomplete, and we might revise them in future. Maybe pigs could be genetically engineered to fly one day.

• If we both pick a uniformly random real number between 0 and 1. What is the probability that we picked the same number? Isn't it possible, but with exactly 0 probability? Feb 19 at 14:18
• If you wish to talk of the probability of an event such as choosing a real number at random in the interval 0 to 1, then it would be more precise to say that its probability is measure zero. This refers to the way in which measure theory is used to define the space of possible values of a random variable. Though, if you want to be really fancy, you could use Internal Set Theory and say that the probability of choosing a standard real number is infinitessimal but non-zero. Feb 19 at 20:57
• Assuming that probabilities of 0 and 1 can be assigned to contradictions and logical truths respectively is not unreasonable. But if we adopt a thoroughgoing fallibilism, I would say that even then, we might be mistaken, so I would rather not. The history of human thought shows us that it would be unwise to assume that anything is absolutely certain. Feb 19 at 21:40
• I don't want to keep spamming your interesting answer so I've created a chat room.
– user64708
Feb 22 at 9:28
• There are uncountable reals in the interval [0, 1], but only a countable number of computable reals. So yes, in the experiment described each possible outcome has a non-zero probability Mar 9 at 7:37

Given that continuous random variables are perfectly well-defined, it follows trivially that there are logically possible events with probability zero. As an example, consider the probability that a uniform random variable is rational. Since the rationals have Lebesgue measure zero, this probability is equal to zero.

### Your mistake is thinking that probability is relevant

one asserts that the probability of pigs flying is 1 in 10^50

... and one has made one's first mistake which makes everything that follows moot.

Flying pigs is not a concept that lends itself to probabilistic reasoning. Flying pigs is a binary logical state - either they exist, or they don't. Even if you were to invoke random gene mutations, you have the binary problem that unless those gene changes confer some kind of evolutionary advantage along the way, they don't propagate. And in the case of flying pigs, I think I can make a case that they simply wouldn't. (I'll expand on that if you want, but for now I'll just assert it.)

This is the famous "black swan" principle. As far as Europeans knew, all swans were white. Extrapolating the existence of a black swan from an all-white population simply doesn't work. And discovering that an entirely separate population of swans were all-black could not have been guessed through probability either.

The point of a black swan, like a flying pig, is that you can't assign a probability to its existence until you see at least one of them. Until you do, the probability of its existence in any mathematical framework is exactly zero - not low, but literally zero.

• @irecorsan Which proves that a Bayesian framework (or any probabilistic framework) is only valid for contexts in which probabilities can reasonably be estimated. Sure you can arbitrarily assign a very small probability and do the calculations based on that. But pick a different very small probability out of your ass and you get a different result. The ability to do calculations does not mean that the result is representative of reality. Feb 20 at 0:24
• @irecorsan ... There's nothing wrong with arbitrarily picking a probability if you're running an insurance plan, say. In that case the probability of a black swan event (e.g. COVID) relates the risk you're prepared to accept. But in that case you're taking your risk profile and working backwards to decide what black swan probability you'll accept. That's different from the OP trying to work forwards from some arbitrary small-but-non-zero probability. Feb 20 at 0:31
• @NotThatGuy The point is though that assigning any probability to it is wrong. Probability by its very nature requires extrapolation from previous events, and if you don't have a previous event then you can't do that. It is literally impossible to estimate. Like I said in a comment, there are times when you could pick a probability and work backwards if you know the answer you want to get, but there's no context in which you can pick a probability, work forwards, and get a useable result. Feb 20 at 11:43
• @Graham "Probability by its very nature requires extrapolation from previous events" - frequentist statistics does. Bayesian statistics can try to estimate probability indirectly. We could, for example, observe colour variation in other animals or try to understand the mechanism causing this, to understand how plausible it would be to find a black swan. "[The probability] is literally impossible to estimate" seems to contradict "[The probability] is exactly zero" - "zero" sounds a lot like an estimate of probability to me (more than an estimate, in fact, a definitive value). Feb 21 at 10:44
• While I think one could come with probabilities in general, one might however say that pigs flying, for example, is simply too far from any facts of reality and contrary to our understanding of reality, to be able to come up with any sort of reasonable guess at a probability, other than to say the claim is extraordinarily unlikely (although I wouldn't say that's equivalent to a zero probability). However, this doesn't necessarily prevent you from comparing claims based on how far they are from our understanding of reality. Feb 21 at 14:36

For me the problem is not mathematical but linguistic.

First define "pig", second define "fly"

Pig

Some people define a pig as (among other things) a creature that cannot fly. If you accept this definition then the probability of a pig flying is zero. On the other hand, there is a Pacific island where every mammal is described as some form of pig (in fact for them pig is synonymous with what we call mammal). Thus, for them a bat would be a flying pig.

Fly

What do you mean by "fly"? We can all fly vertically in a downward direction. Do we only count creatures as flying if they have flapping wings? This would exclude so called flying squirrels because they don't have wings and can only glide. Does flying include being transported in an aeroplane? - If so pigs can definitely fly and I'm sure they have.

Conclusion

Define "pig" and "fly" and then ask again. A sufficiently tight pair of definitions will dictate the answer unequivocally.

• A pig (Sus domesticus) is an omnivorous, domesticated, even-toed, hoofed mammal. The commonly accepted definitions of "pig" would not include every mammal, and "fly" would not include falling (it may include being transported in an airplane, but this most likely isn't the intended meaning). You intentionally interpreting the words in the statement in obscure ways, doesn't change the intended meaning. You're turning it into a linguistic problem, when it isn't one (at least not to the degree presented - species are blurry). Feb 20 at 9:57
• "A sufficiently tight pair of definitions will dictate the answer unequivocally" - This seems to just be kicking the can down the road. Pick a few pairs of definitions and answer the question for those? Feb 20 at 10:02
• @NotThatGuy - I am claiming that it is indeed a matter of linguistics until suitable definitions are made. Feb 20 at 16:32
• @NotThatGuy - Okay here are my arbitrary definitions: - "Fly:- To move through the air unsupported by the ground, artificial means, or air currents at a constant height above the Earth's surface for at least one mile." "Pig:- A farm animal that is used for meat, doesn't have wings and can't fly" - Under those definitions, the question "Can pigs fly?" can be answered No. Feb 20 at 16:45
• @NotThatGuy - If pigs learned to fly, they would no longer be pigs under my first definition. There would then be a choice. Either give the ones that can fly, a new name, e.g. "Fligs", or update the definition of pig to include flying pigs. Feb 20 at 17:50

Almost every human being would agree that 2 + 2 != 5. In a sense, this is a logical impossibility.

The term "impossibility" only makes sense if you are talking about statistical variables.

For statements like "2+2=5", the correct term is not "possibility", but "truth value". And the truth value for this statement is simply "false". There is no belief involved here. "2+2=5" only makes sense if you define what the symbols mean, and give axioms (which are basically arbitrary definitions as well - the ground rules, so to speak). You either can derive that "2+2=5" is true, or false, but there is no probability involved. Everybody who gets the result that this statement could be true has either misunderstood the definitions, or is using different definitions (or is just being silly for whatever reason).

However, almost every human being would also agree that pigs can't fly. Some, however, are adamant in suggesting that this event still has a non zero probability.

[guessing a 50 digit number correctly on the first try]

So what is the correct answer to this? Or is this very question assuming that there is a correct answer?

The answer to this is that the term "probability" those people use is either not well defined, or they use different definitions of it, or they just ignore any effort at a definition.

If they use the usual mathematical definition of the term, they have to define what they mean by pigs flying, in mathematical terms. What is the stochastical variable? What is the experiment? What exactly do they mean with "pig" and "fly"?

By experience, when people throw these kinds of arguments around, they are primarily trying to entertain - either as a kind of Gedankenexperiment, or simply to receive clicks in their YouTube channel, Podcast, or sell a book. The onus is on them to explain what they actually mean, which in turn would open up their statement to scientific reasoning, which would lead to either a clear mistake on their side; or the revelation that they define the terms in some unsuspected way.

• I love that last paragraph... Feb 21 at 10:12

The reason why some people might suggest that pigs could fly, even though it seems highly unlikely and goes against our understanding of the laws of physics, is because the concept of probability is not always black and white. In some cases, it is possible to assign a numerical probability to an event, even if the event seems unlikely or impossible based on our current knowledge and understanding of the world.

However, assigning a non-zero probability to an event that breaks known physical laws raises questions about the nature of probability and how it relates to our understanding of the world. In general, when we talk about probabilities, we are referring to the likelihood of something happening based on our current knowledge and understanding of the underlying mechanisms involved. If an event goes against our current understanding of the laws of physics, then it is reasonable to assume that the probability of that event happening is extremely low or even zero.

In the case of pigs flying, it is difficult to assign a specific probability because the event goes against our understanding of the laws of physics. Some might argue that the probability is effectively zero, while others might argue that it is infinitesimally small but still greater than zero. However, as you noted, there are alternative events that can be assigned the same low probability without violating known physical laws.

Ultimately, the question of whether pigs can fly or not is not just a matter of probability, but also of the underlying mechanisms involved. If we were to discover new physical laws or mechanisms that made pig flight possible, then the probability of pigs flying would no longer be zero. Until then, it is reasonable to assume that the probability is effectively zero based on our current understanding of the world.

It is also worth noting that the concept of probability is indeed man-made and subject to interpretation. The way we assign probabilities depends on the assumptions and models we use to describe the world. However, probabilities can still be a useful tool for making predictions and understanding the likelihood of different outcomes.

Mathematicians have the concept of 'almost certainly' (conversely 'almost never') to describe an event that has a probability of 1 (or 0) even if there are valid exceptions.

The first example I saw of this is a random walk. If you take a random walk on a table, with equal probability of going one way or the other in equally sized steps, then you will 'almost certainly' fall off the table at some point, but you can write down a sequence (left-right-left-right... etc) that keeps you on the table forever.

I think you are led onto a false path partly by the typical problems with natural language that is woefully inadequate for rigorous reasoning.

"Can pigs fly" is a sentence with only three words, but all of them are ill-defined.

Additionally, you do not specify time and location, both of which are crucial.

Can.

"Can" may mean "able to survive survive" (as in "can humans breathe helium") or "able to perform an act actively" (as in "can I lift a 100 pound weight"). This difference is crucial when we talk about whether pigs "can" fly: The active act is very impossible, but there is a finite chance that they survive flying through the air, depending on the altitude and the manner of landing. How we interpret "can" is connected to our interpretation of the next word, "fly".

Another common problem with "can" questions is that they may ask either about what we could call an "innate capability" — or about a non-zero possibility of a future event. The two are distinct questions, even though they are connected: Because the term "possible" implies that the world is not pre-determined but includes random events of different probabilities, every "possible" event has a finite chance to happen during some finite time interval in the future (and a probability of 1 if we say "ever"!) And if somebody answers "yes" to the question "can you do X?" they may well be asked to prove it, by performing the act.

One problem with the idea that "can" asks about a non-zero probability of something actually happening in the future is that the future may be very different from the present, so that different things become possible.

Fly.

"Fly" generally only means that something is airborne. Among the possible ways to be airborne is flying by actuating wings, like a bird, or staying suspended on a ballistic trajectory, like a missile, or being suspended by the wind, like a kite. There is probably also a measure of time involved because "jumping" is not "flying": We are in the air, but only briefly, and then we fall down. Competitions involving "flying" are very careful to define parameters like altitude and duration.

Flying like a cannonball or being lifted by wind is clearly conceivable with pigs. All kinds of objects have been lifted into the air — one of the interpretations of "fly" — by strong winds, among them cows and Dorothy in her house.

Pig.

Are we talking about the Domestic Pig, sus domestica? Or she suborder suina, a much larger group of mammals of various sizes, including possible extinct species? I think none of them could fly actively, but I'm less sure about that than about the sus domestica with which I am reasonably familiar.

The main reason I'm asking is that a wider definition may include future species which may very well develop the ability to fly. Reptiles spawned a flying branch; why not pigs. If we ask whether there is a chance that any pig ever can fly, the answer is certainly yes. We cannot exclude such a development.

Time.

As mentioned, predictions are difficult, especially if... you know.

Location.

The question probably takes it for granted that the pig is on level ground on Earth around sea level, because in different places like the ISS pigs could easily fly.

Discussion.

If we understand the question "naively", imagining a sus domestica attempting to perform powered flight (excluding jumps) on Earth on a calm day at sea level on a level surface, it is physically impossible. But this is not the only way to understand this sentence. In order to get proper answers you need a clear definition of what you are asking. Part of being specific is whether you are asking about a future possibility (which is implied when we talk about probabilities) and whether you include the past. The past is only imperfectly known; the future is essentially unknown.

• "imagining a sus domestica attempting to perform powered flight on Earth on a calm day at sea level on a level surface, the possibility is clearly zero: It is impossible" - I suppose you mean the probability is zero, or there is no possibility, as possibility is non-numeric. But how do you know there is no possibility? Because we haven't seen it? Because there is no known physical mechanism that would allow for this? Neither of those things would make it strictly impossible (it may be impossible according to our understanding of reality, but our understanding may be wrong or incomplete). Feb 21 at 10:17
• Well, if you think we cannot be sure about anything at all because our perception of reality is incomplete or mistaken to the point of being worthless then yes: Everything is possible. But that is probably more than the OP asked for. -- Thanks for the word correction, fixing right now. Feb 21 at 11:37

Quantum physics has something called the totalitarian principle, summed up as "everything not forbidden is compulsory". In other words, any given transition either violates some immutable law, in which case it's completely impossible, or it doesn't break any rules, in which case it has a non-zero probability, and it will happen given enough time. So when it comes to things like nuclear decay or electron tunneling, what you suggest is accepted fact.

Whether we can logically apply the same principle to macroscopic events — whether the probability of your cup of coffee disappearing and reappearing instantaneously three feet to the left is really zero, or merely so small that it wouldn't be expected to happen even once given a trillion cups of coffee left to sit on desks for a trillion years — depends on some questions about the nature of the universe that I'm not qualified to even discuss, but the simplest interpretation is that the probability really is just unbelievably small, and never zero. I think you would find a lot of people who believe in that interpretation on some level, for instance the late novelist Douglas Adams, who demonstrates a fair grasp of the idea in the explanation of the Infinite Improbability Drive.

Application to flying pigs is no trouble at all — as long as you can somehow define a volume in state-space that means "flying pig" to you (fixing the answers to ontological questions about what it means to be a pig, or to fly), we can take an integral of the PDF over that volume (bypassing all trouble with "the probability of specific outcomes in a continuous space is always zero" in the conventional way, assuming that the universe even is continuous and not somehow discretized), and, in theory, find that integral to be nonzero or zero.

2+2=5 - it is possible in logical (atleast that is why you see this), but 2+2=4 - cuz senses, not by logical. "every human being" - usually not understand in logical/sense definition nothing. Because Human being - it is nonsense. The probability to find someone who can distinct senses and logic is 1 to... me.)

Answer: Probability not based on senses, it is just a beautiful holographic theory based on math logic.

For example this math challenge:

100 prisoners need to play the game:

the room have boxes with numbers till 100 inside, but this numbers are changed in random order. prisoners by one ll enter the room and have to find self number by 50 tries. they can decide the strategic before the start, but after everyone should enter/leave the room without any connections with others.

If all prisoners should find their numbers they ll free, if atleast one doesn't - they all ll executed.

what the chance alive for all?

According to the standart theory of prob it is ~ 0% (1/2^100)

but with sense+logic it is about ~30%

• According to standard theory of probability, if they all choose randomly, it's 2^-100. If they follow different strategies, the probability can be more or less than that. Feb 19 at 22:20
• @ralphmerridew it is ok, if you can't solve to 30% Feb 19 at 22:51
• I'm familiar with this problem and know the answer. But "the standard theory of probability" gives an answer dependent on what strategy they use. Feb 19 at 23:47
• @ralphmerridew but why? Feb 20 at 7:52