# What is the Null Hypothesis for a negative claim?

If my claim is: There are no purple swans.

How should this look like in a null/alternative hypothesis form?

Like this?

Null: There are purple swans

Alternative: There are no purple swans

Or like this?

Null: There are no purple swans

Alternative: There are purple swans

Or am i missing something?

• From a Bayesian point of view, of course, there's no privileged null hypothesis; only a distribution (determined by your priors) over the space of hypotheses. In this view, negating the hypothesis merely flips the distribution. May 20, 2018 at 17:20
• Like I said, Likelihood, depend on your prior understanding of the situation; you can add 'Bayesian' to make yourself sound clever, but it still the same idea. May 21, 2018 at 21:52

You don't use the term "null hypothesis" for pure facts. It is used in statistics, when you claim there is correlation between two events. Like "swans living near coal mines tend to be purple more often than swans living elsewhere". Here the null hypothesis is "living near coal mines doesn't affect whether swans are purple or not".

• Are you sure? Because in courts there is the "presumption of innocence" which works just like the Null Hypothesis, but is called differently. What do you think? May 20, 2018 at 15:24
• Since there is a single person in court, using statistics and statistical terms wouldn't be appropriate. May 20, 2018 at 22:55
• @WatchHat: I think you're conflating the statistical concept of a null hypothesis with the logical concept of the burden of proof. The whole point of the null hypothesis is that it is easy to falsify because it makes very specific predictions about the data you should expect to observe. If you observe data which violates those predictions, then the null hypothesis may be rejected. The presumption of innocence, by contrast, is difficult to falsify because you have to disprove every (reasonable) account in which the accused is innocent. May 20, 2018 at 23:38
• @Kevin Presumption of innocence may be hard to falsify, or it could be easy to falsify. Depends on the evidence, just like in statistics. If there's a video recording of a crime, then disproving the innocence becomes relatively easy. May 21, 2018 at 6:27
• @Dmytro: That's irrelevant. My point is that presumption of innocence is not the same as the null hypothesis. May 21, 2018 at 14:57

The "null" in "null hypothesis" refers to there being no relationship between (usually two) variables. In standard statistics terminology, what you claim / predict / hope for has no bearing on which hypothesis is null.

As gnasher729 already points out, your claim is not quite in a suitable form to use this terminology, but we could rework it as "a bird that is a swan has almost no chance of being purple". Notice that, at least in this form, you are actually claiming a relationship: you are saying that knowing the type of a bird (first variable) may allow us to predict its purpleness (second variable) more confidently. The corresponding null hypothesis would claim no relationship: "swans have the same chance of being purple as other birds". (So knowing the bird is a swan doesn't help us reason about its purpleness.)

That said, if your claim was "swans have the same chance of being purple as other birds", it would still be the null hypothesis. The natural alternative hypothesis would be "swans have a different chance of being purple than other birds", but in this case you might test the hypotheses differently than normal because the usual tests favor the null, which is what a healthy skepticism wouldn't want in this situation.

You're missing likelihood and this comes from what we already know and understand about the world. In that world no-one would bother formulating a null hypothesis about the existence or non-existence of purple swans as every school boy or girl knows there are no such animals.

And dyeing a swan purple does not help to get around this. That would be merely cheating.

• Downvote: the OP is clearly using purple swan metaphorically to indicate something that can theoretically exist but hasn't been observed.
– user935
May 20, 2018 at 21:59
• I went so far as to google "every school boy or girl knows there are no such animals" to see if this answerer was sarcastically quoting someone famous on the subject of the impossibility of black swans; but no, I can only conclude that he's being sincere(ly wrong). May 21, 2018 at 2:48
• @Quuxplusone: If I was quoting somebody, I would have mentioned it; there's no need to google 'every schoolboy knows ...'; it's a circumlocution that is generally understood amongst native english speakers. May 21, 2018 at 22:08
• @MoziburUllah It's not common in the US, although we do say "everyone knows...". Of course, there are plenty of things "everyone knows" that aren't true (snopes.com has a long list of these "urban legends"), so even if "every school boy" knows something, that doesn't make it true.
– user935
May 24, 2018 at 22:00

Phrasing this as a null hypothesis question is awkward (but not impossible), because it's an all-or-none proposition.

Null hypothesis: it is not the case that purple swans exist

Experiment: collect swans, observe color

Alternative experiment: collect purple things, see if they are swans

Even more alternative experiment: collect things, see if they are purple and swans

Conclusion: if more than 0% of the observed swans are purple, we can reject the null hypothesis; if 0% of the observed swans are purple, we cannot reject the null hypothesis (ie, there may or may not be purple swans).

Of course, all this assumes we:

• Have a definition of swans that does not preclude them from being purple.

• Have a definition of purple: which colors are close enough to purple to be considered purple.

• Have a definition of "purple swan" consistent with the above two definitions:

• What percentage of the swan has to be purple?
• How do we measure this percentage?
• Is the percentage measured when the swan is in repose? flying?

Of course, I'm guessing you're using "purple swan" metaphorically to mean:

• You have a nonempty set T (eg, "all things")

• You have a predicate q on that set (eg, "thing is a swan"), and at least one item, t1 in T, such that q(t1) holds.

• You have another predicate p on the set (eg, "thing is purple"), and at least one item t2 in T (t2 may or may not be the same as t1), such that p(t2) holds.

• You can't show that `for all t in T, p(t) -> -q(t)` is true or false, nor can you show `for all t in T, q(t) -> -p(t)` is true or false. In other words, you can't show whether one of the two conditions precludes the other.

• More compactly, you can't show `for all t in T, -(p(t) and q(t))` is true or false (its negation is `there exists t in T such that p(t) and q(t)`). In other words, you can't prove that purple swans do or do not exist.

You then select items randomly from T (if T is infinite, this may require the Axiom of Choice), test p(t) and q(t) and reject the null hypotheses (there are no t such that p(t) and q(t)) if you find even one t such that p(t) and q(t).

• I have had null hypothesis testing drilled into my brain throughout graduate school but I've never seen it expressed in terms of sets. Do you have a reference that goes into more detail about this relationship? May 21, 2018 at 0:50
• Actually, the set theory part was more for the posters who thought you meant "purple swan" literally, not about the null hypothesis. I was trying to explain in set notation the concept of "something that can exist in theory, but has never been observed" (ie, something whose existence can't be disproven, but hasn't been proven by observation either)
– user935
May 21, 2018 at 15:24
• -1: For obfuscatory use of set theory. May 21, 2018 at 22:01
• @MoziburUllah I suspect the downvote is retaliatory, but whatever. Set theory is the basis of all mathematics and thus all statistics. i also added a couple of other possible experiments. Somewhere there's a discussion on how the existence of a non-purple non-swam (eg, a black crow) strengthens the hypothesis there are no purple swans, but I haven't looked for it.
– user935
May 22, 2018 at 18:23
• That just a foundational point of view. It's not usually neccesary, and often isn't. For example, I was just looking at a text about the use of tensors in the foundations of electromagnetism - a highly mathematical subject, and yet no sets were visible. May 24, 2018 at 21:37

For any testable hypothesis, I like to sneak in the words "we know that..." before the statement. Then the null is "we don't know that...", as in, "we don't know that there are no purple swans."

To answer this, we need to explore the question of why you want a null hypothesis in the first place. What does that mean? What is its purpose.

A null hypothesis is typically what someone would expect if they were to believe the conventional wisdom of the times. In a scientific experiment, one designs the experiment to have the potential to reject the null hypothesis. Phrasing that differently, the experiment is structured to demonstrate a good reason to doubt the null hypothesis. If you were living in Galleleo's time, it would be conventional wisdom that a heavier ball will fall faster. One sets out to reject this hypothesis by running an experiment which yields data that suggests otherwise.

One can indeed make null hypotheses which are not aligned with the conventional wisdom of the times. One can make a "null hypothesis" that purple swans exist. And, if you gather data to refute this, anyone who believes there were purple swans now has some troublesome data which refutes their beliefs. But with very few people arguing for purple swans, such a hypothesis would not make very many waves in the scientific community.

The trick is to remember that the scientific method's construction of a null hypothesis and an alternate hypothesis is just formal rigor. In the strictest readings of the scientific method, one never actually accepts a theory. There's merely theories that have not been falsified yet. Thus, when one has done some work that makes one believe they have a new hypothesis, they don't get to prove that they are right. Instead, they must prove that the conventional wisdom is wrong, and after succeeding in doing so, they are given the privilege of suggesting what the next theory should be (the alternate hypothesis).

Now with that understanding of what the purpose of the null hypothesis is, we can see that it is reasonable to develop a null hypothesis to use with your claim "There are no purple swans." Now the "conventional wisdom" approach is tricky here, because you're really trying to make an experiment to show something that everyone already expects (which is rather boring). However, there's no reason we can't do it.

In the general pattern, we assume our claim is the alternate hypothesis. The null hypothesis must be something which contradictory to our alternate hypothesis. Typically this is easy, because we're trying to refute the status quo, so everyone will tell you what the null hypothesis should be. In this case, without a clear consensus that purple swans are a thing, it's a bit harder. We have options:

Alternate Hypothesis: There are no purple swans
Example options for the null hypothesis:

• There are purple swans
• One can assign the color purple to swans
• "Swan" is a word which is sufficient to specify an entity (which may have a color)

Practically speaking, the first one is the most natural. However, I wanted to point out others to show that all you need to do for a null hypothesis is to identify a hypothesis which can be refuted by data which is not your claim.

From that point, you could construct a test. Refuting "There are purple swans" is a tricky task, but it could be done with some reasonable level of rigor. An assay of populations in the wild which fails to demonstrate a single purple swan would be considered sufficient evidence in some fields.