# Is the claim “this coin is fair” falsifiable?

Wikipedia says,

The claim "No human lives forever" is not falsifiable since one would have to observe a human living forever to falsify that claim.

Thinking on similar lines, even if the coin is fair, it can produce a arbitrarily long(finite) sequence of continuous heads with finite but small probability.

So if you were to determine whether a coin is not fair, you will have to toss it to eternity to be sure.

The assumption here is that it is not possible to determine the fairness just by looking at the physical properties alone.

Does falsifiability require the process of falsification be finished in finite time?

• How would it help to toss the coin infinitely often? – Phira Jun 16 '11 at 12:47
• @thei for this particular example, P(all heads) approaches zero as number of tosses approaches infinity. For non-mathematicians (and probably for applied mathematicians), P(all heads) equals zero if the number of tosses equals infinity. – Ben Hocking Jun 16 '11 at 14:40
• @Ben Hocking: For infinite trials, zero probability does not mean that it cannot happen. – Phira Jun 16 '11 at 14:42
• @thei: do you mean an uncountably infinite set of trials of infinite length, or merely a countably infinite number of trials of infinite length? If the former, I agree at least in theory, but if the latter, I disagree. – Ben Hocking Jun 16 '11 at 18:44
• @ben:I don't think countability will make a difference. I think the Collatz Conjecture is a case in point. – apoorv020 Jun 16 '11 at 19:33

I am tempted to adopt a Wittgensteinian tack on this one - not saying that it is right but it seems to me an interesting approach for this type of question. For reference, my argument is based on Wittgenstein's observations in the Philosophical Investigations §193-§195.

To ask whether the proposition

(P) This coin is fair

is falsifiable is to confuse two pictures we have of coins. One is the picture we have of the coin as a symbol for all its future applications - that is to say the picture of a physical coin symbolizing an ideal random process that gives one of two results with 50-50 probability. The other picture is that of the coin as an actual physical object - e.g. made of copper, round, thin, liable to friction and tear.

Now the proposition (P) adopts the first picture - the property 'fairness' refers to the symbolic understanding of the coin as encapsulating the whole of its future application (i.e. flips) as a random process giving one of two results with a 50-50 chance.

On the other hand, the following proposition

(Q) Is P falisfiable?

refers to the second picture, i.e. the actual physical realization of the coin. And here the confusion should become apparent. (Q) asks of a symbolic property whether it is empirically falsifiable. And no such answer is forthcoming.

We are often tempted to confuse these pictures in philosophy. It seems as though 'fairness' is somehow in the coin - that it is somehow a property that we ought to be able to uncover. The reason we are "led into temptation" is because we do not take care to separate the two pictures.

The grammar of the word 'fair' is what is at issue here. What makes your question seem meaningful is that what you have in mind is the grammar of fair when we employ it symbolically - e.g. when trying to explain probability to someone. But the only grammar that actually makes the question meaningful is the practical one, i.e. the one that takes into account what compels us to call a loaded dice 'unfair' and a weighted coin 'fair' - namely measurements, production methods etc. And if this grammar makes the question meaningful, it also makes it trivial and the problem, it seems to me, disappears.

You can never falsify statistical claims, tossing the coin infinitely often does not help, either. So the answer is no, these claims are not falsifiable.

But what you can do and what people routinely do is to falsify statistical claim with a certain precise probability, that is where "confidence intervals" come into the picture of statistical tests.

As far as immortality is concerned, you can say with appropriate high confidence that no human can live longer than 150 years which implies mortality with at least equal high confidence.

But I would say that the confidence of mortality is higher than that, since our experience with other complex systems that are instable increases our confidence.

• The claim is fairness implying a statistical equality. So you do not need to show an infinate dataset only a dataset which covers a margin of error of your sample. – Chad Jun 16 '11 at 14:48
• Even with your a claim of no human can live to be 1000 years would be falsifiable even though it is true. A single 1000 year old human would prove it false(even though none is known to exist or likely will). – Chad Jun 16 '11 at 14:53
• Nice answer. However I disagree with your third paragraph. We cannot say with any confidence that no human can live longer than 150 years; what we can say is that, with a high level of certainty, all people so far have died before 150 years of age. Actually, even this statement is a little dodgy. What do we mean by all people? Are we only including the people who lived between 0 CE and the present moment? Are Neanderthals people? If so, how can we be sure Neanderthals didn't commonly live to 150 years? – goblin GONE Aug 19 '13 at 9:46

This is indeed a very good and deep question.

Falsifiability as your secret weapon is indeed foremost a Popperian claim. One of the biggest proponents nowadays are e.g. David Deutsch (see e.g. his new book "The beginning of infinity").

Another author who tries to adapt it to a stochastic environment is Nassim Taleb ("The Black Swan"). Although when I met Nassim a few years ago in London I was especially interested in his current view on this and he grumbled (in his inimitable manner) that Popper won't work in a stochastic environment.

Anyway, trying to reconcile both views I would answer as follows: The fairness is falsifiable up to a certain lower bound. This lower bound could even be deterministic! So we are talking about a fundamental uncertainty principle. One hot candidate is the Cramér–Rao bound.

A very readable (even funny sometimes) article can be found here: http://astro.temple.edu/~powersmr/vol7no3.pdf

Generally the question also tackles the question of what randomness really is, but I won't go into this here...

The question is, interpreted literally, neither answerable nor useful to ask.

Operationally, what people mean when they ask such questions is something like, "Assuming that this coin contains no temporally-varying internal state that will alter its behavior upon flipping (at least not on the time scale that I care about), is there evidence for the hypothesis that this coin will systematically come up on one side more frequently / more times than the other?"

And the answer there is, yes: from statistics, to whatever degree of certainty you want, if you flip it enough times; from physics, to whatever degree of measurement accuracy you can afford and is possible within the limits of quantum mechanics (and to the extent that you believe physics).

One has to be careful about what one really is asking when one tags the real world with simple prepositional phrases. The world is not a good model of prepositional logic. (What is a "coin", anyway?)

I think Wikipedia is off-target here also. "No human lives forever" is a perfectly interpretable phrase and on the basis of evidence we can assign a truth value to it (i.e. "true") with a degree of accuracy that approaches the accuracy with which we can answer any question. That it is not strictly falsifiable should not be terribly worrying, since even things that are supposedly strictly falsifiable ("this wall is solid") are subject to all sorts of problems including whether one can adequately define the meaning of the words, perceptual illusions, incorporation of universal properties in the definition ("X is solid if and only if for all Y such that (whatever), Y cannot pass through X") that require infinite testing etc. etc.. All this means as a practical matter is that you shouldn't utterly reject any claim, regardless of how implausible, if it has really good evidence for it.

The main problem as I see it, is the definition of "this coin".

You can prove, for reasonable definitions of "prove", that "this coin" is fair, by tossing it enough times and showing it has a 50/50 outcome with a reasonable confidence interval. (Or prove that it is unfair by tossing it and showing that with a reasonable confidence interval, the coin does not have a 50/50 outcome). But "this coin" will in that case mean only the coin which you tossed as it was when you tossed it. It doesn't show that it will continue to be fair, since somebody can tamper with the coin, in which case it stops being fair.

But in normal parlance, we would still call it "this coin", although from the scientific/statistical/philosophical standpoint it is no longer the same coin once it has been tampered with.

In other words, you can only prove that the coin was fair, not that it will be fair.

• You do not need to prove it true to be falsifiable. You need to be able to define a state where it is false and could be measured false. – Chad Jun 17 '11 at 14:29
• @Chad: Yes. But will it remain false in the future? The coin could be fixed so it doesn't remain unfair. So the answer works both ways. – Lennart Regebro Jun 18 '11 at 5:22
• the question is not about the future the question is about the present. The statement is not "This coin WILL be fair" it is "This coin IS fair". – Chad Jun 20 '11 at 13:10
• @Chad: But that present became the past as soon as you said it, so that view is pointless. Normally it is assumed that the statement "X is Y" is something that is true not only in the present, but will continue to be true for the foreseeable future. – Lennart Regebro Jun 20 '11 at 15:47

From a scientific point of view you can only rely on statistics. For example, if you were to toss a coin we all agree that it would not be possible to toss the coin an infinite number of times. The purpose of statistics is, precisely, to determine a sufficient number of experiments to infer some kind of information about a phenomenon under observation.
For example, in the case of the coin, you may want to use the Bayes' Theorem to determine whether it is likely or not that a fair coin would produce the sequence of heads and tails you are observing. From statistics you can determine the number of tosses (or experiments) you need to make sure the probability you compute is sufficiently reliable. As your demands, in terms of reliability, get closer and closer to 100%, the number of tosses you need to make increases to infinity.
Your argument about the fact that an all head sequence is always possible with some positive probability is correct. In fact, it does happen with probability one simply because you may model the system as a Markov chain with recurrent states; indeed, any sequence of finite length will eventually occur. However, this is not the point. The point is, precisely, what you said: different events do have some probabilities to occur. This is the key observation statisticians make use of to determine the size of a sufficiently large sample they can possibly use to derive the statistics they need (in your case, the probability that the coin is fair given a sequence of coin tosses).
We know that, in principle, any prediction might be wrong even if the confidence is 99.9% but in practice, statistics is all you can use.

It is certianly falsifiable. If a coin can be shown statistically to favor one side over the other consistantly in a statisically signifigant data set. And the results can be duplicated consistantly then the claim can be shown as false. The size of the dataset is going to depend on the margin of error. The greater the margin of error (or wider of the statistical distribution) the larger the dataset will need to be.

Does falsifiability require the process of falsification be finished in finite time?

No, but the in order to be falsifiable you must have the ability to demonstrate a false condition. In order to determine is someone has lived for ever we need to measure the length of their entire life. In order to measure the entire length of someones life it would have to end. If it the life ends then it is not in a false condition.

If a coin will land on its head 9 out of 10 times And i can demonstrate this repeatedly and independant of biasing conditions, then i can demonstrate a false. If it is biased at a much lower rate say by 1 per 1 million. It would require a much greater dataset but i can still show the bais confirming a false condition.

• How is this statistical argument different from the type of "falsifiability" available to the "no human lives for ever" claim? – Ami Jun 16 '11 at 13:24
• To prove it false it would require a dataset that includes an infinate amount of time. Seeing as we only have a finite amount of time to measure this is not possible. – Chad Jun 16 '11 at 13:49
• Downvoting. As others have pointed out, the statement is only falsiable if you deny that low probability events can take place. So, you can say with large confidence that a coin is fair/not fair, but it's not a logically provable conclusion. – apoorv020 Jun 16 '11 at 19:16
• @apoorv020 It was not asked if it was provable. It was asked if it was falsifable. I can show it false. There for it is falsifiable. I may be able to prove that no human can live forever. That doesnt mean that i can show it to be false. – Chad Jun 16 '11 at 19:46
• @Chad:My problem is basically that you cannot interpret anything conclusively from coin tosses. Even for large number of tosses, a fair coin and an unfair coin can (theoretically) produce the same results. So looking at the results, you can't say that the coin was unfair and unfair. You may guess (or maybe estimate is a better word), whether the coin is fair or not, but it's still an estimate. – apoorv020 Jun 16 '11 at 20:20

A lot of the answers above seem flawed to me in that they assume, usually without stating the assumption, that the coin tosses are stationary in time. Stationary in time means that the underlying random process doesn't change over time. So one would have an incredibly low probability of tossing the coin 10,000 times and getting 90% heads, and then later tossing the coin another 10,000 times and getting 90% tails.

But consider how I might build an unfair coin using modern technology. My unfair coin would be a tiny machine with voice activated controls, keyed to my voice only. I'd have one word that turned off the bias, so that while the coin was in this "fair" state it would indeed behave indistinguishably from a "fair" coin. But when it came to making the big bets, I would have another word which activated the coin's bias. Then make my money, then I'd utter the word to set the coin back to "fair" again.

A strong argument could be made that even an infinite number of tosses might not determine if this coin is fair. If I never utter the word to set the coin to biased, then even an infinite number of tosses will behave indistinguishably from a fair coin.

I suspect some philosophers might consider my fair coin "cheating" for the purposes of this question. I would counter that such a concern is not sensible. The entire point of an unfair coin is to "cheat." By unfair, we essentially mean "behaving differently than naive expectations, but in a way predictable to those with knowledge of the unfairness." If a particular form of unfairness is not predicted by the philosopher, as we say in software this is a feature, not a bug.

My thoughts are close to those of @mwengler Statistics are useful precisely because there's no need to verify every singular manifestation of a phenomena. Statistics work with samples, you only have to determine what is the size of the sample to have a safe result. If you have to collect infinite samples, statistics loses its use. Statistics is not about 100% certainty, that is why it is so useful in areas like biological research, as life is full of irregularities. You may develop a medicine for some sort of disease that is useful for most people but will not work for me. That is how it goes. No guarantees. It is more like some sort of a game.

So is it falsifiable? I propose a linguistic approach. If to say "the coin is fair" you needed a statistic certainty, you could only say "maybe this coin is fair". Semantically speaking, this cannot be denied through negation, as "maybe this coin is not fair" has the same value.

I'd say, through statistics it is not falsifiable.

• +1 for pointing out that a statement may have linguistic truth, that is separate from its absolute logical truth or falsehood. – Dawood ibn Kareem Jun 24 '12 at 0:18

This claim is not falsifiable if you let your hands be tied by an implicit rule (not mentioned in your question): "the only investigative tool is tossing the coin repeatedly". Of course, you can measure where the coin's center of mass is or inspect that random number generator program you have for obvious faults.

I would argue that your last question, though a matter of semantics(essentially you are asking what is the definition of falsifiability), can be answered in the positive (falsifiability requires the process of falsification to be finite).

The reason is that there are practical differences between the two definitions of falsifiability that can be derived by allowing/disallowing the infiniteness of the falsification process. An example is that the statement "All men are mortal" is unfalsifiable when disallowing infinite falsification process. If you think about it, falsifiable is used in the sense that something can be disproved/falsified. Since infinite processes don't really end, you can't actually falsify a statement using them. So if you allow infinite falsification processes, this means that statements that you can't be refuted will be deemed falsifiable.

One can say that falsifiable and refutable are not the same thing. However, in this article by Popper, he says:

One can sum up all this by saying that the criterion of the scientific status of a theory is its falsifiability, or refutability, or testability.

So the accepted sense of falsifiability is the same as that of refutability, and that requires that the falsification process be finite. Note that this sense of the word is similar to the requirement that algorithms be finite in computer science (so that a falsification process is actually analogous to an algorithm, and falsification analogous to solvable).

As to the coin problem, even if infinite falsification processes were allowed, there is not even a terminating condition. That is, even if the coin were not fair and all tosses came up heads, you cannot logically declare that the coin was unfair. This is different from the statement "X is immortal", since you could conclusively say that the statement is false if X died. Both these statements are unfalsifiable, but there is still some order of difference in their unfalsifiability.

• The existance of a single false state makes it falsifiable. So I can show a 2 headed coin will always show heads. I do not need infinate. In addition the generalnes of the claim fair, as opposed to exactly 50% or "Perfectly Fair" which would have different criteria makes this falsifiable as simply being able to demonstrate a bias of the specific coin is possible. – Chad Jun 17 '11 at 13:02
• @Chad:We are not supposed to use physical characteristics of the coin, so double headed coin case is ruled out. – apoorv020 Jun 17 '11 at 14:34
• The rest of my arguement is still valid. – Chad Jun 17 '11 at 14:37

Falsifiability is from a scientific viewpoint worthless anyway. Statistically, what you would put forward is a null hypothesis (the coin is fair) and an alternate one (it isn't). You would then compute a test statistic (in this case, the proportion of heads with respect to the number of flips) and see how likely that is under the assumption of the null hypothesis. The less likely, the more reasonable it is to reject the null hypothesis. This is the scientifically sound decision.

In fact, considering that most scientific research is statistical in nature, your example demonstrates why no scientist makes use of falsifiability in any profound way during his research. It would be totally useless, one would never arrive at scientific conclusions.

• Actually while not being able to probe a theory false does not make it true. Proving it false does make it false. – Chad Jun 17 '11 at 12:58
• How do you know a theory has been disproven? Does the "falsifiability" principle answer that question? I don't think so. That's why you'll never come accross the principle when studying science, except maybe if you take a philosophy of science course. – Raskolnikov Jun 17 '11 at 13:05
• Actually generally you try to prove a theory by showing it works in various controls. If the theory breaks down under certian controls consistantly then it is shown false. For instance if I could show a scenario where Energy does not equal the speed of light squared, then i could prove Einstien's theory wrong. While a theory does not need to be falsifiable to be valid, if it is then showing it to be false is a way to disprove it. – Chad Jun 17 '11 at 13:19
• I strongly disagree. You'll never falsify Einstein's theory. Not anymore than you can falsify Newtonian mechanics. What will happen is that Einstein's theory will become an approximation to another deeper theory. That's all. – Raskolnikov Jun 17 '11 at 13:43
• A false result is not required for something to be falsifiable. You simply need to be able to define a state where the theory is false. In Einstiens formula it is simple a state where E!=Mc^2 . With Newton if I can show an action that has an Enhanced and Same Vector reaction then i can show it false, because the claim is EVERY action has an equal and opposite reaction. – Chad Jun 17 '11 at 14:04

Update : Neither the claim nor it's negation can be either validated or shown to be false. It is example of a statement that can not be verified within the system that is stated.Just consider it's opposite : Is it possible to show a coin not to be fair? It's proof suffers from the same problem as the original claim.

Your question needs to distinguish ( at least ) between two ways of defining a fair coin :

Physical and Statistical fairness. There is no gaurantee that Physically fair coin will be Statistically fair, or vice versa. It seems that question provides it's own circular reasoning for it being fairn and that is in infinite number of tosses giving a 50/50 outcome. With that definition it is easy to see that no coin can be proven fair or unfair in finite time (due to the definition of the fairness used).

• I have mentioned in the question that an underlying assumption is that it's not possible to determine fairness by physical inspection – AIB Jun 17 '11 at 10:33
• Fairness is a general term generally accepted to be a generally equal chance or distribution. In fact the non specifity of this question is part of what makes it falsifiable. Because you do not need to show any specific just a general distribution. A statement of you will always have a 50/50 chance flipping this coin would not be falsifiable. – Chad Jun 17 '11 at 12:56
• @AIB : Taking out what is not possible, all we left with is the statistical definition of fairness, that relies on the properties of infinitely many coin tossing outcomes, and that answers it's own question. It causes circular reasoning to be used in answering. – jimjim Jun 17 '11 at 21:56

Actually, you could simply measure the coin properly and produce a statement such as "This coin is fair to [insert tolerance]."

No coin is completely fair, and the fairness of a coin cannot be determined in an absolute sense by experiment. However, if the distribution over a thousand or so tosses is fair, the coin behaves like a fair coin, and so the question of whether it is a fair coin is sort of irrelevant.

Likewise with the statement "All men are mortal." The evidence supporting this conclusion is extremely high, and the evidence against it is non-existent. So the statement "All observed human lifespans are both finite and shorter than x years" is definitively true, and can be used as a reasonable predictor of future human lifespans. Furthermore, no human has lived forever by definition of forever, and given certain physical constraints (like damage from cosmic rays) it can said definitively that no human, as we currently understand the term, can live forever.

That being said, I could also say that for a highly local definition of the word "forever", all humans live forever.

"This coin is fair" is weak in an important way as a falsifiable proposition, but for the most part is as strong as any scientific conclusion as far as falsification goes.

The one sense in which it is weak is "fair" might generally be interpreted as a value judgement. However, let me assume that here "fair" means that the coin was constructed as are typical coins from the U.S. mint and has not been modified from that constructed form by other than the same wear and tear issues experienced by the vast majority of coins from the US treasury. This is a technical definition of fair, theoretically falsifiable by someone who monitors the coin 24/7 from its creation until now.

Actually, the above paragraph shows how "fair" can be made falsifiable. Interestingly, it does not require an infinite number of tosses of the coin or even a single toss of the coin. It involves knowing a nearly complete history of the coin from its creation until now.

Now can a coin be determined to be "fair" or not by tossing it repeatedly and analyzing the results of that? If it is unfair in a straightforward way (its tosses have time-stationary statistics and a mean rate of heads different from 50%), then this will be detected with arbitrarily high confidence in a finite length of time. But if it is "unfair" in a more subtle way (for example, it is a voice-activated machine which is inert and therefore "fair-like" most of the time, but can be activated to be unfair by some mechanism), then it can not be determined to be unfair if its unfairness is not activated during its testing.

My conclusions:

1. "fairness" as I define it here (which I think is pretty close to how we intuitively use it) is in principle falsifiable.
2. Only by making a probably-unfalsifiable claim of time-stationarity can a "fair" coin be determined to be fair with high confidence, by tossing it. A triggerable unfairness can never be ruled out by tossing it.
3. The test for "fairness" is more properly complete knowledge of the coin's history than anything as limited as tossing it a bunch of times.

Does falsifiability require the process of falsification be finished in finite time?

Awesome thinking. The best answer I know of is no. I'll make my case.

Take the complement of your question:

Does truifiability require the process of truification be finished in finite time?

This is equivalent to your question if one simply negates the claim under test. In other words, is it verifiable that there exists a human who lives forever?

More generally, if a claim is true, is it only verifiable if the search space is effectively finite?

Using a different example so as to avoid the explicit requirement of infinite time, take the claim that

a certain element never seen before by mortals exists.

If the universe is infinite in extent, the search for it would never terminate until either (a) it is found or (b) the searchers give up. If it does not exist, condition (a) will never be met, and the only way the search will end is by giving up the search, having neither proved nor disproved the claim. If it does exist, condition (a) is a possibility, however, depending on its rarity or the difficulty of verification, condition (b) is also possible, which again would neither prove nor disprove the claim. The only possible proof in this case is positive proof: At least by the sole means of direct examination of the search space, disproving the existence of such an element would be impossible, hence the claim of its existence is unfalsifiable.

Now it is clear that there are at least two possible realities for an unfalsifiable claim:

1. It is either verifiable and therefore true, or
2. It is not verifiable (and may be false).

Even though no finite time limit can be given within which a search through an infinite space for a rare element can prove that it exists, at least if it does exist, it may still be proven to exist in finite time, and therefore its existence is verifiable. The same argument applies to any falsification which could take an unbounded amount of time. It may still be falsifiable, although the expense of falsification is unbounded.

How do humans deal with such issues of observational rarity in practice? Through the very human-sounding attributes of desire, belief, and patience. It's pretty practical, and pretty apparent that if you really want to find gold, and if you believe you can or will find it, you're much more likely to keep looking for it than someone who either doesn't want it so much, or who doesn't believe he will or even might find it.

It's partly your choice

So, at a personal level, the observational falsifiability of claims within a potentially infinite search space depends on such subjective, intentional properties such as patience, confidence, and belief. This illustrates a blind spot.

A given claim is either true or it is not true. An unfalsifiable claim could be true, or false. An unverifiable claim could be true, or false. Engaging in the search for a positive or negative proof is betting one's resources and energy on the outcome which may be uncertain from the start. A person who never gives up the search for a difficult truth of great value would be shown to be one of the wisest of the race. A person who never gives up the search for a proof something that is untrue would be among the most foolish. Hence we have such a sharp division of opinions and even persecution in matters both scientific and religious, and it is truly said that a genius is often not appreciated in his own time, and no prophet is accepted in his own country.

Anyway, you've just proved that subjective attributes such as faith, belief, and patience necessarily take an important place in the process of scientific discovery.

We can't number the life-saving breakthroughs that others thought couldn't be done. We also might have difficulty enumerating the vast list of false beliefs people hold that will never be proven; for example, abiogenesis.

You've also hit upon the Decidability Problem of computer science. In essence it says that there may be true statements that exist but which cannot be proven in finite time through methods of exhaustive search, because the search space is infinite. One can decide to embark upon the search for something that may or may not exist. If it does exist, although the tests may appear to be infinite, one's faith is eventually going to be rewarded. If it does not exist, the real test of wisdom is how quickly one abandons the search.

Back to the coin

For the example you gave, the best a person can do in finite time is to take sufficient measurements so as to justify a level of confidence (which is a belief) in a degree of fairness, to some finite level of precision. Of course the possibility is nonzero that any such conclusion is entirely wrong, unless a property can be determined in finite time. The fair coin is actually a problem that to some degree suffers from a problem of being neither falsifiable nor truifiable, that is, through sampling alone one could never determine that it is fair or unfair. Depending on the degree of precision and confidence (these are subjective choices), one could tentatively "prove" or "disprove" the coin's fairness based upon experience, and even assign a probability to that judgment. With predetermined margins for error, the probability of either fairness or unfairness could be vanishingly small, and so this is a matter for confidence: How confident do you want to be?